NASA/TM--2003-106316

CARES/LIFE Ceramics Analysis and

Reliability Evaluation of Structures

Life Prediction Program

Noel N. Nemeth

Glenn Research Center, Cleveland, Ohio

Lynn M. Powers

Case Western Reserve University, Cleveland, Ohio

Lesley A. Janosik and John P. Gyekenyesi

Glenn Research Center, Cleveland, Ohio

February 2003

https://ntrs.nasa.gov/search.jsp?R=20030014949 2018-05-15T15:37:36+00:00Z
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NASA/TMm2003-106316

CARES/LIFE Ceramics Analysis and

Reliability Evaluation of Structures

Life Prediction Program

Noel N. Nemeth

Glenn Research Center, Cleveland, Ohio

Lynn M. Powers

Case Western Reserve University, Cleveland, Ohio

Lesley A. Janosik and John P. Gyekenyesi

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

February 2003
This document was initially released in

1993. The formal numbered publication

was delayed until 2003.

Trade names or manufacturers' names are used in this report for

identification only. This usage does not constitute an official

endorsement, either expressed or implied, by the National

Aeronautics and Space Administration.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs._'c.nasa.gov
TABLE OF CONTENTS

Page

1.0 INTRODUCTION ................................................... 1

2.0 PROGRAM CAPABILITY AND DESCRIPTION ............................. 13

3.0 INPUT INFORMATION ............................................... 25

3.1 MSC/NASTRAN Finite Element Analysis .............................. 25

3.2 Finite Element Interface Modules Input Information ....................... 32

3.3 CARES/LIFE Input File Preparation .................................. 33

3.4 C4PEST Module Input Information ................................... 35

3.5 C4LIFE Module Input Information ................................... 55

4.0 EXECUTION OF THE CARES/LIFE PROGRAM ............................ 69

5.0 OUTPUT INFORMATION ............................................. 71

5.1 Finite Element Interface Modules Output Information ...................... 71

5.2 C4PEST Module Output Information ................................. 74

5.2 C4LIFE Module Output Information .................................. 83

NASA/TM--2003-106316 iii
6.0 THEORY ........................................................ 91

6.1 Fast-Fracture Reliability Analysis ................................... 9 l

6.1.1 Introduction ............................................. 91

6.1.2 Volume Flaw Reliability Analysis ............................. 93

6.1.3 Surface Flaw Reliability Analysis .............................. 105

6.1.4 Material Strength Characterization ............................. 111

6.1.5 Estimation of Statistical Material Strength Parameters ............... 118

6.2 Time-Dependent Reliability Analysis ................................ 126

6.2.1 Introduction ............................................. 126

6.2.2 Volume Flaw Reliability Analysis ..................... • ........ 129

6.2.3 Surface Flaw Reliability Analysis .............................. 132

6.2.4 Static Fatigue ............................................ 135

6.2.5 Dynamic Fatigue .......................................... 136

6.2.6 Cyclic Fatigue ........................................... 138

6.2.7 Application of the Paris and Walker Laws ........................ 142

6.2.8 Material Failure Characterization

for Static, Cyclic, or Dynamic Loading ................. 145

6.2.9 Fatigue Parameter Risk of Rupture Compatibility ................... 152

6.2.10 Evaluation of Fatigue Parameters from Inherently Flawed Specimens .... 161

6.2.11 Proof Testing Effect on Component Service Probability of Failure ...... 178

6.2.12 Proof Testing - Off-Axis Loading ............................. 182

7.0 USERS' GUIDE .................................................... 185

NAS A/TM--2003-106316 iv
8.0 EXAMPLEPROBLEMS ............................................. 211

8.1 Fast-FractureExampleProblems.................................... 211

8.1.1 Example 1 - Statistical Material Parameter Estimation ............... 211

8.1.2 Example 2 - Internally Heated and Pressurized Tube ................ 219

8.2 Time-Dependent Example Problems ................................. 231

8.2.1 Example 3 - Fatigue Parameter Estimation ....................... 231

8.2.2 Example 4 - Cyclic Fatigue and R-ratio Sensitivity ................. 241

8.2.3 Example 5 - Isothermal Rotating Annular Disk .................... 248

8.2.4 Example 6 - Thermomechanically Loaded Rotating Annular Disk ....... 262

8.2.5 Example 7 - Proof Testing ................................... 271

8.3 Example Guide ................................................ 288

9.0 APPENDICES ..................................................... 297

A-Cyclic fatigue (g-factor) examples .................................... 297

B-Neutral File Setup ............................................... 301

C-Symbols ...................................................... 305

10.0 REFERENCES .................................................... 311

NASA/TM--2003-1063 i6 v
Introduction

Advanced ceramic components designed for gasoline, diesel, and turbine heat engines are

leading to lower engine emissions, higher fuel efficiency, and more compact designs due to their

low density and ability to retain strength at high temperatures. Ceramic materials are also used for

wear parts (nozzles, valves, seals, etc.), cutting tools, grinding wheels, bearings, and coatings.

Currently, research is focused on improving ceramic material properties and processing, lowering

associated manufacturing costs, properly characterizing material properties, and developing a mature

and validated brittle material design methodology. The emerging materials, particularly silicon

nitride and silicon carbide, consist of abundant and nonstrategic constituents and have the potential

for competing with traditional metals in many demanding applications. For advanced heat engines,

ceramic components have already demonstrated functional abilities at temperatures reaching 1371 °C

-- well beyond the operational limits of most metallic materials.

Unfortunately, ceramics also have several inherent undesirable properties which must be

considered in the design procedure. The most deleterious of these properties is that ceramics are

brittle materials. This lack of ductility and yielding capability leads to low strain tolerance, low

fracture toughness, and large variations in observed fracture strength. When a load is applied, the

absence of significant plastic deformation or microcracking causes large stress concentrations to

occur at microscopic flaws, which are unavoidably present as a result of materials processing

operations or inservice environmental factors. The observed scatter in component strength is caused

by the variable severity of these flaws and by the behavior of sudden catastrophic crack growth

which occurs when the crack driving force or energy release rate reaches a critical value. In

addition, the ability of a ceramic component to sustain a load degrades over time due to a variety of

effects such as oxidation, creep, stress corrosion, and cyclic fatigue. Stress corrosion and cyclic

fatigue result in a phenomenon called subcritical crack growth (SCG). SCG initiates at a preNASA/TM--2003-106316 1
existing flaw and continues until a critical length is reached, causing catastrophic propagation. The

SCG failure mechanism is a load-induced phenomenon over time. It can also be a function of

chemical reaction, environment, debris wedging near the crack tip, and deterioration of bridging

ligaments.

Once the factors that contribute to material failure have been identified and characterized,

ceramic components can be designed for service applications using an appropriate brittle material

design methodology. For this purpose the integrated design computer program CARES/LIFE

(Ceramics Analysis and Reliability Evaluation of Structures LIFE) has been developed to predict the

fast-fracture and/or lifetime reliability of monolithic structural ceramic components subjected to

thermomechanical and/or proof test loading. This design methodology combines the statistical

nature of strength-controlling flaws with fracture mechanics to allow for multiaxiai stress states,

concurrent flaw populations, and subcritical crack growth. CARES/LIFE is an extension of the

CARES program (Powers, Starlinger, and Gyekenyesi, 1992) (Nemeth, Manderscheid, and

Gyekenyesi, 1990) (Pai, and Gyekenyesi, 1988) (Gyekenyesi, and Nemeth, 1987) (Gyekenyesi,

1986), which predicts the fast-fracture reliability of monolithic ceramic components. The

fundamental subsets of the program include: (a) fast-fracture reliability analysis, (b) inert (fastfracture) statistical material parameter estimation, (e) crack growth laws to account for static and

cyclic fatigue, (d) conversion of cyclic boundary loading to an equivalent static state, (e) static,

dynamic, and cyclic fatigue parameter estimation, (f) fatigue parameter compatibility relationships

for various stress states, fracture criteria, and failure probability models, and (g) the effect of proof

testing on component service probability of failure.

Because ceramics fail due to the presence of microscopic flaws, examination of fracture

surfaces can reveal the nature of failure. Fractography of broken samples has shown that these

flaws can be characterized into two. general categories: (1) defects internal or intrinsic to the

NASA/TM--2003-106316 2
materialvolume(volumeflaws) and (2) defects extrinsic to the material volume (surface flaws).

Intrinsic defects are a result of materials processing. Extrinsic flaws can result from grinding or

other finishing operations, from chemical reaction with the environment, or from the internal defects

intersecting the external surface. The different physical nature of these flaws results in dissimilar

failure response to identical loading situations. Consequently, separate criteria must be employed to

describe the effects of the applied loads on the component surface and volume.

Because of the statistical nature of these flaw populations, the size of stressed material surface

area and volume (known as the size effect) affects the strength. By increasing component size, the

average strength is reduced because of the increased probability of having a weaker flaw.

Generally, for metals, the variation of strength is small, and thus the scaling effect is negligible;

however, for materials that display large variations of strength, this effect is not trivial. Hence, if a

ceramic design is based on material parameters obtained from smaller size test pieces, then the

effects of scaling must be taken into account, otherwise a nonconservative design will result.

Another consequence of the random distribution of flaws is that failure of a complex

component may not be initiated at the point of highest nominal stress. A particularly severe flaw

may be located at a region of relatively low stress, yet still be the cause of component failure. For

this reason, the entire field solution of the stresses should be considered. Clearly, it is not adequate

to predict reliability based only on the most highly stressed point.

Traditional analysis of the failure of materials uses a deterministic approach, where failure is

assumed to occur when some allowable stress level or equivalent stress is exceeded. The most

widely used of these theories are the maximum normal stress, maximum normal strain, maximum

shear stress, and maximum distortional energy criteria of failure. These phenomenological failure

theories have been reasonably successful when applied to ductile materials such as metals.

However, these methods do not account for observed variations in ceramic component fracture

NASA/TM--2003-106316 3
stress.Therefore,to assurehighreliabilityinbrittle material design, large factors of safety are

required. This does not allow for optimization of design since the physical phenomena that

determine fracture response are not properly modeled.

Because of its lack of a proper physical basis, the traditional approach to design is not

adequate to predict failure of brittle materials. Consequently, Griffith (Griffith, 1921) (Griffith,

1924) proposed a fracture theory where failure was due to the presence of cracks of specified size

and shape distributed randomly throughout the material. He assumed that no interaction takes place

between adjacent cracks and that failure occurs at the flaw with the least favorable orientation

relative to the macroscopic loading. The Griffith energy balance criterion for fracture states that

crack growth will occur if the energy release rate reaches a critical value. Griffith's theory provides

a sound physical basis to describe the rupture process in an isotropic brittle continuum. However, it

omits the effect of component size on strength because the crack length is not treated as a

probabilistic quantity.

Reliability analysis is essential for accurate failure prediction and efficient structural

utilization of brittle materials subjected to arbitrary stress states. When coupled with the weakestlink model (Weibuli (151), 1939), this approach takes into account not only the size effect and

loading system, but also the variability in strength due to defect distributions. A statistical theory of

failure can be readily incorporated into the finite element method of structural analysis since each

element can be made arbitrarily small such that the element stress gradient is negligible.

Component integrity is computed by calculating element-by-element reliability and then determining

the component survivability as the product of the individual element reliabilities.

For fast-fracture reliability analysis, the first probabilistic approach used to account for the

scatter in fracture strength and the size effect of brittle materials was introduced by Weibull

(Weibull (151), 1939) (Weibull (153), 1939) (Weibuil, 1951). This approach is based on the

NASA/TM--2003-106316 4
previouslydevelopedweakest-linktheory(WLT)(Pierce,1926)(Kontorova,1940) (Frenkel, and

Knotorova, 1943), which is primarily attributed to Pierce, who proposed it while modeling yarn

failure. The weakest-link theory is analogous to pulling a chain, where catastrophic failure occurs

when the weakest link in the chain is broken. Unlike Pierce, who assumed a Gaussian distribution

of strength, Weibull assumed a unique probability density function known as the Weibull

distribution. It has been shown (Shih; 1980) that the three-parameter Weibull distribution is a more

accurate approximation of ceramic material behavior than the Gaussian or other distributions. Since

three-parameter behavior is rarely observed in as-processed monolithic ceramics, the CARES/LIFE

program uses the two-parameter Weibull model in which the threshold stress (the value of applied

stress below which the failure probability is zero) is taken as zero. The reliability predictions

obtained using the two-parameter model are more conservative than those obtained with the threeparameter model.

To predict the fast-fracture material response under multiaxial stress states by using statistical

parameters obtained from flexural or uniaxial test specimens, Weibull proposed calculating the risk

of rupture by averaging the tensile normal stress raised to an exponent in all directions over the area

of a unit radius sphere (volume flaws) (Weibull(151), 1939) or over the contour of a unit radius

circle (surface flaws) (Gross, 1989). Although this approach is intuitively plausible, it is somewhat

arbitrary. In addition, it lacks a closed-form solution, and therefore, requires computationally

intensive numerical modeling. Subsequently, Barnett and Freudenthal (Barnett, et al., 1967)

(Freudenthal, 1968) proposed an alternative approach usually referred to as the principle of

independent action (PIA) model for finding the failure probability in multidimensional stress fields.

This principle states that the Weibull survival probability of a uniformly stressed material element

experiencing multiaxial loading is equal to the product of the survival probabilities for each of the

tensile principal stresses applied individually. The PIA fracture theory is the weakest-link statistical

NAS A/TM--2003-106316 5
equivalentof the maximum stress failure theory. The Weibull method of averaging the tensile

normal stress and the principle of independent action model have been the most popular methods for

polyaxial stress state analysis, and they have been widely applied in brittle material design

(Margetson, 1976) (Paluszny, and Wu, 1977) (DeSalvo, 1970) (Wertz, and Heitman, 1980) (Dukes,

1971).

However, the Weibull and PIA hypotheses do not specify the nature of the defect causing

failure, and consequently, there is no foundation for extrapolating to conditions different from the

original test specimen configuration. Consequently, the accuracy of these theories has been

questioned, and other statistical models have been introduced (Batdorf, and Crose, 1974) (Evans,

and Jones, 1978) (Batdorf, 1976). The ideas developed by Batdorf and Crose (Batdorf, and Crose,

1974) are important because they provide a physical basis for incorporating the effect of multiaxial

stresses into the weakest-link theory. They describe material volume and surface imperfections as

randomly oriented, noninteracting discontinuities (cracks) with an assumed regular geometry. This

enables the contributions of shear and normal forces to the fracture process to be explicitly treated.

Failure is assumed to occur when the effective stress on the weakest flaw reaches a critical level.

The effective stress is a combination of normal and shear stresses acting on the flaw. It is a

function of the assumed crack configuration, the existing stress state, and the fracture criterion

employed. Accounting for the presence of shear on the crack plane reduces the normal stress

needed for fracture, yielding a more accurate reliability analysis than that of the shear-insensitive

crack model (Weibull's method). Unlike in the deterministic Griffith failure criterion, the size of

the crack in the probabilistic approach need not be considered because it is associated with the

strength of the material.

The search for an accurate fracture criterion to predict fast-fracture response to monotonically

increasing loads leads to the field of fracture mechanics. Many authors have discussed the stress

NAS A/TM--2003-106316 6
distributionaround cavities of various types under different loading conditions, and numerous

criteria have been proposed to describe impending failure. Paul and Mirandy (Paul, and Mirandy,

1976) extended Griffith's maximum tensile stress criterion for biaxial loadings to include threedimensional effects due to Poisson's ratio and flaw geometry, which could not be accounted for in

Griffith's previous two-dimensional analysis. Other investigators (Giovan, and Sines, 1979)

(Batdorf, 1982) (Stout, and Petrovic, 1984) (Petrovic, and Stout, 1984) have compared results from

the most widely accepted mixed-mode fracture criteria with each other and with selected

experimental data. No prevailing consensus has emerged regarding a best theory. Also, most of the

criteria predict somewhat similar results, despite the divergence of initial assumptions. Therefore,

the authors of this report concluded that several alternatives would be available for the sake of

comparison but that the semi-empirical equation developed by Palaniswamy and Knauss

(Palaniswamy, and Knauss, 1978) and Shetty (Shetty, 1987) provides the most flexibility to fit the

available experimental data. In addition, Shetty's criterion can account for the out-of-plane flaw

growth that is observed under mixed-mode loadings. Finally, several different flaw geometries are

provided, but the penny-shaped and semicircular crack configurations are recommended as the most

accurate representations of volume and surface defects, respectively.

A wide variety of materials, including ceramics, exhibit the phenomenon of delayed fracture

or fatigue. Under the application of a loading function of magnitude smaller than that which

induces short term failure, there is a regime where subcritical crack growth occurs and this can lead

to eventual component failure in service. SCG is a complex process involving a combination of

simultaneous and synergistic failure mechanisms. These can be grouped into two catagories: (1)

crack growth due to corrosion, and (2) crack growth due to mechanical effects arising from cyclic

loading. Stress corrosion is due to a stress-dependent chemical interaction between the material and

its environment. Water, for example, has a pronounced deleterious effect on the strength of glass

NASA/TM--2003-106316 7
andalumina.Highertemperaturesalsotend to accelerate this process. Mechanically induced cyclic

fatigue is dependent only on the number of load cycles and not on the duration of the cycle. This

phenomenon can be caused by a variety of effects, such as debris wedging or the degradation of

bridging ligaments, but essentially it is based on the accumulation of some type of irreversible

damage that tends to enhance the crack growth. Service environment, material composition, and

material microstructure determine if a brittle material will display one, none, or some combination

of these fatigue mechanisms.

Because of the complex nature of SCG, models that have been developed tend to be

semi-empirical and approximate the behavior of subcritical crack growth phenomenologically.

Theoretical and experimental work in this area has demonstrated that lifetime failure characteristics

can be described by consideration of the crack growth rate versus the stress intensity factor (or the

range in the stress intensity factor). This is graphically depicted as the logarithm of the rate of

crack growth versus the logarithm of the mode I stress intensity factor. Curves of experimental data

show three distinct regimes or regions of growth. The first region indicates threshold behavior of

the crack, where below a certain value of stress intensity the crack growth is zero. The second

region shows an approximately linear relationship of stable crack growth. The third region indicates

unstable crack growth as the materials critical stress intensity factor is approached. For the stress

corrosion failure mechanism, these curves are material and environment sensitive. This model,

using conventional fracture mechanics relationships, satisfactorily describes the failure mechanisms

in materials where at high temperatures, plastic deformations and creep behave in a linear viscoelastic manner (Evans, and Wiederhorn, 1974). In general, at high temperatures and low levels of

stress, failure is best described by creep rupture which generates new cracks (Wiederhom, and

Fuller, 1985). Creep and material healing mechanisms are not addressed in the CARES/LIFE code.

NASA/TM--2003-106316 8
Themostottencited models in the literature regarding SCG are based on power law

formulations. Other theories, most notably Wiederhom's (Wiederhom, Fuller, and Thomson, 1980),

have not achieved such widespread usage, although they may also have a reasonable physical

foundation. Power law formulations are used to model both the stress corrosion phenomenon and

the cyclic fatigue phenomenon. This modeling flexibility, coupled with their widespread acceptance,

make these formulations the most attractive candidates to incorporate into a design methodology. A

power law formulation is obtained by assuming the second crack growth region is linear and that it

dominates over the other regions. Three power law formulations are useful for modeling brittle

materials: the power law, the Paris law, and the Walker law. The power law (Evans, and

Wiederhom, 1974) (Wiederhom, 1974) describes the crack velocity as a function of the stress

intensity factor, and implies that the crack growth is due to stress corrosion. For cyclic fatigue,

either the Paris law (Paris, 1963) or Walker's (Walker, 1970) (Dauskardt, et al., 1992) modified

formulation of the Paris law is used to model the subcritical crack growth. The Paris law describes

the crack growth per load cycle as a function of the range in the stress intensity factor. The Walker

equation relates the crack growth per load cycle to both the range in the crack tip stress intensity

factor and the maximum applied crack tip stress intensity factor. It is useful for predicting the effect

of the R-ratio (the ratio of the minimum cyclic stress to the maximum cyclic stress) on the material

strength degradation.

Because SCG operates on the pre-existing flaws in the material, the fast-fracture statistical

theories discussed previously are required to predict the time-dependent reliability for brittle

materials. The SCG model is combined with the two-parameter Weibull cumulative distribution

function to characterize the component failure probability as a function of service lifetime. The

effects of multiaxial stresses are considered by using the principle of independent action (PIA)

model, the Weibull normal stress averaging method (NSA), or the Batdorf theory.

NAS A/TM--2003-106316 9
Lifetimereliabilityanalysisaccountingfor subcriticalcrackgrowthundercyclic and/or

sustainedloadsisessentialfor thesafeandefficientutilizationof brittlematerialsin structural

design.Currentlife designmethodologyassumesSCGof mixed-modeloadingis basedona power

functionrelationshipexistingbetweenthecrackpropagationrateandtheequivalentmodeI stress

intensityfactor(Boehm,1989)(Hamanaka,et al., 1987)(Homada,andTeramae,1990)(Thiemeier,

1989)(Sturmer,et al., 1990)(Stunner,et al., 1991).Theliteratureis sparseregardingcrack

velocitymeasurementsfor mixed-modeloadingsof brittlematerials.Whena crackis subjectedto a

combinedmodeloading,it extendsin acurvedor kinkedpaththatreorientsthecrackto a pure

modeI coplanarextensionat thecracktip (Shetty,andRosenfield,1991).Themodelsof Boehm,

Hamanaka,Homada,Thiemeier,andStunnerdonotconsiderthis morecomplexbehaviordueto

paucityof availabledata. Theapproachtakenbytheseresearchersi reasonableif theduration

wheremixed-modeloadingexistsat thecracktip is smallcomparedto thedurationwherethecrack

tip is extendingin puremodeI. In anyevent,theformulationstheyadoptedwill tendto be

conservative.

For corrosion-assistedSCG,time-dependentreliabilityanalysisfor a componentsubjectedto

variouscyclic boundaryloadconditionscanbesimplifiedbytransformingthattypeof loadingto an

equivalentstaticstate.Theconversion,throughtheuseof a constantcalledtheg-factor(Evans,

1980)(Mencik,1984),satisfiestherequirementthatbothsystemswill causethesamecrackgrowth.

Implicit in thisconversionisthevalidityof thecrackgrowthpowerlaw. Theprobabilityof failure

is thenobtainedwith respecto thetransformedequivalentstaticstate.

Priorto placinga componentin service,confidencethatit will performreliablyis usually

demonstratedthroughprooftesting. To a greatextent,this istheacceptedway to assurethe

reliabilityof a component(Evans,andWiederhom,1974)(Wiederhorn,1974)(Ritter,Oates,Fuller,

andWiederhom,1980)(Fuller,Wiederhom,Ritter,andOates,1980)(Srinivasan,andSeshadri,

NASA/TM--2003-106316 10
1982). Ideally,theboundaryloadconditionsappliedto a componentunderprooftestingsimulate

thoseconditionsa componentwouldbesubjectedto inserviceandtheprooftestloadsare

appropriatelygreaterin magnitudeoversomefixedtimeinterval. Thesignificanceof prooftesting

isthatit enablesspecimenswith acertainminimumflawsizeor largerto beeliminatedfromthe

strengthdistribution.Thus,anattenuatedprobabilityof failureisobtainedandthesurvivedcomponentscanbeplacedin servicewith greaterconfidenceintheirintegrity. In practice,however,it is

oftendifficult, expensive,or impossiblefortheprooftestloadconditionsto exactlysimulatethe

serviceloadconditions;therecanexistmisalignmentbetweentheloads,or theprooftestandthe

serviceloadaredifferentmultiaxialstressstates.Thissituationcanbeaccountedfor whenproof

testingdesignmethodologyis incorporatedintothestatisticalfracturetheoriesfor polyaxialstress

states(Service,andRitter,1985)(Hamanaka,Suzuki,andSakai,1990)(Brukner-Foit,Heger,and

Munz,to bepublished).An attenuatedprobabilityof failureiscomputedforthesurvived

components,howevera minimumlife of assuredreliabilitymaynolongerberelavent.

Theprimaryobjectiveof thisreportis to describea publicdomaincomputerprogramthatis

coupledwiththegeneralpurposefiniteelementcodesMSC/NASTRANI, ANSYS2,and

ABAQUS3for predictingthefast-fractureand/orservicelifetimefailureprobabilityof monolithic

ceramiccomponents.Potentialenhancementsto thecodeincludethecapabilityfor transient

analysis,three-parameter Weibull statistics,creepandoxidationmodeling,flaw anisotropy,and

parameterregressionfor multiplespecimensizes.

Thisreportfunctionsasbotha users'andaprogrammers'manual.Theusercanfocus

primarilyonthefirstfive sectionsof thismanualfor backgroundinformationandfor detailed

_MacNeal-Schwendler Corporation,LosAngeles,CA.

2SwansonAnalysisSystems,Inc.,Houston,PA.

3Hibbitt,Karlsson,andSorenson,Inc.,Providence,RI.

NASA/TM--2003-106316 11
descriptions of the program capabilities, setup and execution, input, and output. A users' guide has

been prepared to facilitate execution of the CARES/LIFE program. In addition, seven example

problems are included in the section Example Problems to further illustrate program input, output,

and interpretation of results. These examples also provide program validation and verification. It is

strongly recommended that the user read the theoretical sections as well for a better understanding

of the probabilistic fast-fracture and time-dependent theories employed. All of these sections are

essential to a programmer who may wish to modify the source code. In addition, a list of symbols

has been included to aid the reader.

NASA/TM--2003-106316 12
Program Capability and Description

The CARES/LIFE integrated design computer program predicts the reliability and the failure

probability of a monolithic ceramic component as a function of its service life. CARES/LIFE

couples commercially available finite element programs, such as MSC/NASTRAN, ANSYS, or

ABAQUS with probabilistic design methodologies to account for material failure from subcritical

crack growth (SCG) of preexisting flaws. The code is written in FORTRAN 77 and is divided into

three separately executable modules which perform: (1) statistical analysis and characterization of

experimental data obtained from the fracture of standard laboratory specimens; (2) neutral data base

generation from the MSC/NASTRAN, ABAQUS, and AHSYS finite element analysis programs; and

(3) time-dependent reliability evaluation of thermomechanically loaded ceramic components. Proof

test effects on a survived component are included along with the consequences of off-axis and

multiaxiai loading. Figure 1 illustrates the operational flow of the program. Finite element heat

transfer and linear-elastic stress analysis are used to determine the temperature and stress

distributions in the component. Component reliability for volume (intrinsic) flaws is determined

from finite element stress, temperature, and volume output from two-dimensional, three-dimensional

or axisymmetric elements. Reliability for surface (extrinsic) flaws is calculated from shell element

stress, temperature, and area data. CARES/LIFE produces an optional PATRAN file containing

risk-of-rupture intensities (a local measure of reliability) for graphical rendering of the structure's

critical regions.

The phenomenon of subcritical crack growth (SCG) is modeled with the power law, the Paris

law, and the Walker law. The power law (Evans, and Wiederhorn, 1974) (Wiederhorn, 1974)

describes the crack velocity as a function of the stress intensity factor. For cyclic fatigue, either the

Paris law (Paris, 1963) or Walker's (Walker, 1970) (Dauskardt, et al., 1992) modified formulation of

the Paris law is used to model the subcritical crack growth. The Paris law relates the crack growth

NAS A/TM--2003-106316 13
perloadcycleto therangein thestressintensityfactor. TheWalkerequationrelatesthecrack

growthperloadcycleto boththerangein thecracktip stressintensityfactorandthemaximum

appliedcracktip stressintensityfactor.Thisformulationaccountsfor theeffectof theR-ratio

(minimumcyclestressto maximumcyclestress)onlifetime. Thepowerlaw andtheParislaw

requiretwo experimentallyderivedfatigueparametersN andB whicharematerial/environmental

constants.TheWalkerequationrequiresthreematerial/environmental constantsN, B, andQ.

Steady-statecyclic loadingisaccountedfor by usingtheWalkerlaw,theParislaw,or by employing

g-factors(Mencik,1984)in conjunctionwiththepowerlaw. Theg-factorapproachequatesvariable

cyclic loadingsto equivalentstaticloadings.CARES/LIFEincludesthesinusoidal,square,and

sawtoothloadingwaveforms.Typically,theuseof g-factorsis appropriatefor fiat T-curvematerials

(T-toughnessresistance).

Theprobabilisticnatureof materialstrengthandtheeffectsof multiaxialstressesaremodeled

by usingeithertheprincipleof independentaction(PIA),theWeibullnormalstressaveraging

method,or the Batdorf theory. The Batdorf theory combines linear elastic fracture mechanics with

the weakest-link mechanism. It requires a user-selected flaw geometry and a mixed-mode fracture

criterion to describe volume or surface strength limiting defects. The combination of a particular

flaw shape and fracture criterion results in an effective stress, which is a function of the far-field

stresses, and acts on the crack plane. Figure 2 shows the fracture criteria and flaw geometries

available to the user for both surface and volume flaw analysis. The simple PIA fracture theory

does not require a crack geometry. When the PIA model is used, only tensile principal stresses can

contribute to failure. The Weibull normal stress averaging (NSA) method is independent of crack

geometry. The mode I (opening mode) crack growth is considered, and mode II (sliding mode) and

mode III (tearing mode) effects are neglected. The combination of a particular flaw shape and

fracture criterion results in an effective stress involving far-field principal stresses in terms of

NASA/TM--2003-106316 14
normalandshearstressesactingonthecrackplane. CARES/LIFEincludesthetotalstrainenergy

releaseratetheory(coplanarcrackextension)(Batdorf,andHeinisch,1978).Out-of-planecrack

extensioncriteriaareapproximatedby asimplesemi-empiricalequation(Palaniswamy,andKnauss,

1978)(Shetty,1987).Thisequationinvolvesa parameterthatcanbevariedto modelthemaximum

tangentialstresstheory(Erdogan,andSih,1963),theminimumstrainenergydensitycriterion(Sih,

1974),themaximumstrainenergyreleaseratetheory(Hellen,andBlackburn,1975)

(Ichikawa,1991),or experimentalresults.Forcomparison,Griffith'smaximumtensilestress

analysisfor volumeflawsisalsoincluded.Thehighlightedboxesin figure2 showthe

recommendedfracturecriteriaandflaw shapes.If thenormalstressactingontheflaw planeis

compressive,thennocrackgrowthis assumedto occur.

In CARES/LIFE,if themaximumprincipalstressiscompressive,thenthecorresponding

elementreliabilityis setequalto unity. Typically,brittlematerialsaremuchstrongerin

compressionthanin tension.It is assumedthatthelowertensilestrengthlimit will predominate

overthehighercompressivelimit for a typicalcomponentdesign.If thecompressivestressesare

significant,theyshouldbecheckedagainstlimitingvaluesby othermethods.

For fastfracture,theprobabilisticnatureof materialstrengthisdescribedby thetwoparameterWeibullcumulativedistribution function, which incorporates weakest-link theory. This

relation postulates that inherent material flaws in the component body (volume flaws) and on its

surface (surface flaws) govern the strength response. The component reliability is determined by

integrating the stress over the body. The Weibull stress-volume integral is a function of the scale

parameter, c o, and the shape parameter, m. The scale parameter corresponds to the stress level at

which 63.21 percent of specimens with unit volume or area would fracture. The characteristic

strength, c_o, is similar to the Weibull scale parameter except that it includes the effect of the

NAS A/TM--2003-106316 15
specimenvolumeor area. The shape parameter (or Weibull modulus), denoted by m, is a

dimensionless quantity and measures the degree of strength dispersion of the flaw distribution.

Weibull material parameters, the Batdorf crack density coefficient, and fatigue parameters are

estimated from rupture strength data of naturally flawed specimens. The parameters are obtained

from the fracture stresses of specimens whose geometry and loading configurations are held constant

(30 or more specimens are recommended). A similar number is recommended for fatigue

experiments. The CARES/LIFE program includes closed form solutions for the three- and

four-point modulus-of-rupture (MOR) bending bar (Baratta, Matthews, and Quinn, 1987) and the

pure tensile specimen (Liu, and Brinkman, 1986) under isothermal conditions. For other

conventional specimen geometries, material parameters can be estimated via effective volume and

area calculations (a finite element model of the specimen geometry and loading is required).

Since the material parameters are a function of temperature, various constant-temperature data

sets can be simultaneously input and the corresponding parameter estimates will be calculated and

made available for component reliability analysis. Linear interpolation is performed to obtain values

at intermediate temperatures. More sophisticated interpolation techniques are not used due to the

potential of obtaining erroneous results. Each constant-temperature data set can consist of up to 999

specimens. In addition, each specimen can be identified by its mode of failure-either volume flaw,

surface flaw, or some other mode -- so that parameter estimates for competing failure modes can be

obtained.

CARES/LIFE estimates fatigue parameters from naturally flawed specimens ruptured under

static, cyclic, or dynamic loading (CARES/LIFE does not estimate the Walker R-ratio sensitivity

exponent, Q). Cyclic fatigue parameter evaluation assumes steady state loading and a constant Rratio throughout the specimen. Fatigue parameters can be calculated using either the median value

technique (Jakus, Coyne, and Ritter, 1978), a least squares regression technique, or a median

NASA/TM--2003-106316 16
deviation regression method which is somewhat similar to trivariant regression (Jakus, Coyne, and

Ritter, 1978). The median value technique is a well known estimation procedure based on

regression of the median values of the fatigue data at the various stressing levels or rates. The least

squares regression technique involves a regression on all the fatigue data to establish the parameters.

The median deviation procedure involves minimizing the median deviation of the scatter in the data

versus the crack growth exponent N. In the CARES/LIFE code this minimization is accomplished

by maximizing the time-dependent Weibull modulus versus the crack growth exponent N. The fastfracture strength distribution Weibull modulus, m, and characteristic strength, a 0, are optionally

estimated from the fatigue data for a failure time of one second with constant stress rate loading (or

a lifetime of 1/(N+I) cycles). The fatigue data is transformed to an equivalent fast-fracture strength

distribution. This enables goodness-of-fit testing and using an outlier test. The resulting goodness-offit statistics are applied to the original fatigue data. If inert strength fracture data is simultaneously

input, then the Weibull parameters for this data override those calculated from time-dependent data.

For inert strength fracture (fast-fracture) data, parameter estimation of the biased Weibull

modulus and characteristic strength a 0 can be performed for unimodal or concurrent surface and

volume flaw populations by using least-squares analysis (Johnson, 1964) or the maximum likelihood

method (Nelson, 1982). Because estimates of Weibull parameters are obtained from a finite amount

of data, they contain an inherent uncertainty that can be characterized by bounds in which the true

parameters are likely to lie. Methods have been developed to evaluate confidence limits that

quantify this range with a level of probability as a function of sample size. For the maximum

likelihood method with a complete sample, unbiasing factors for the shape parameter m, and 5- and

95-percent confidence limits for m and the characteristic strength ao, are provided (Thoman, Bain,

and Antic, 1969). For a censored sample, an asymptotic approximation of the 90-percent confidence

NASA/TM--2003-106316 17
limits is calculated.No unbiasingof parametersor estimationof confidencelimits is givenwhen

theleast-squaresoptionis requested.

CARES/LIFEincludesa testthatwill identifypotentialbaddata(outliers)from thetimedependentor inertstrengthfractureexperiments.Thistest,knownastheStefanskyoutliertest

(Stefansky,1972)(Neal,Vangel,andTodt,1987),is basedonthenormaldistributionand,therefore,

its applicationto theWeibulldistributionisnotrigorous.However,it servesasa usefulguidelineto

theuser. Datadetectedasoutliersareflaggedwith a warningmessage,andanyfurtheractionis

left to thediscretionof theuser.

Theability of thehypothesizedistributionto reasonablyfit theempiricaldatais measured

withtheKolmogorov-Smirnov(K-S) andAnderson-Darling(A-D) goodness-of-fittests. Thesetests

areextensivelydiscussedby D'AgostinoandStephens(D'Agostino,andStephens,1986).Thetests

quantifydiscrepanciesbetweentheexperimentaldataandtheestimatedWeibulldistributionby a

significancelevelassociatedwiththehypothesisthatthedataweregeneratedfrom theproposed

distribution. The A-D test is more sensitive than the K-S test to discrepancies at low and high

probabilities of failure. The calculated significance levels are based on the assumption that the

Weibull parameters are chosen independent of the experimental data. For inert strength data the

Kanofsky-Srinivasan 90-percent confidence band values (Kanofsky, and Srinivasan, 1972) about the

Weibull line are given as an additional test of the goodness-of-fit of the data to the Weibull

distribution.

CARES/LIFE automatically calculates the other material parameters necessary for the

reliability analysis. The biased estimate of the shape parameter and the estimated characteristic

strength are used along with the specimen geometry to calculate the Weibull scale parameter. The

Batdorf normalized crack density coefficient is computed from the selected fracture criterion, crack

NASA/TM--2003-106316 18
geometry,andthebiasedestimateof theshapeparameter.Thecomputationof the

material/environmental fatigueparametersi detailedin theTheory section.

To ensurecompatibilityof failureprobabilitiesfor surfaceandvolumeflaw specimens,the

relationshipsbetweenthefatigueparameters(N andB) andthevariousfailurecriteriahavebeen

established.Fromtestspecimendata(uniaxialtension,3- and4-pointbend)compatibilityis derived

by equatingtherisk of ruptureof theuniaxiaiWeibullmodelto theriskof ruptureof thePIA,NSA

or theBatdorfshear-sensitivemodels.This satisfies the requirement that for a uniaxial stress state,

all models produce the same probability of failure as the uniaxial Weibull model. The value of N is

invariant and the value of B is adjusted to satisfy this condition.

Finite element analysis is an ideal mechanism for obtaining the stress distribution needed to

calculate the survival probability of a structure. Each element can be made arbitrarily small, such

that the stresses can be taken as constant throughout each element (or subelement). In CARES/LIFE

the reliability calculations are performed at the Gaussian integration points of the element or,

optionally, at the element centroid. Using the element integration points enables the element to be

divided into sub-elements, where integration point sub-volumes, sub-areas, and sub-temperatures are

calculated. The location of the Gaussian integration point in the finite element, and the

corresponding weight functions are considered when the subelement volume/area is calculated. The

number of subelements in each element depends on the integration order chosen, and the element

type. If the probability of survival for each element is assumed to be a mutually exclusive event, the

overall component reliability is the product of all the calculated element (or sub-element) survival

probabilities.

The component reliability analysis module of the CARES/LIFE program uses finite element

elastostatic output to calculate time-dependent reliability for each element. Volume-flaw-based

reliability is calculated from the previously determined volume flaw material strength parameters

NASA/TM--2003-106316 19
andtheoutputof thestresses,volumes,andtemperaturesfor eachsolidelement.Volumeflaw

analysiscanbeperformedusingbrick,wedge,andtetrahedronisoparametricsolidelements,

triangular(andoptionallyquadrilateral)axisymmetricisoparametricelements,andtriangularand

quadrilateralisoparametricshellelements.FortheMSC/NASTRANfinite elementanalysiscode,

the supported element types are HEXA, PENTA, TETRA, TRIAX6, QUAD4, QUAD8, TRIA3, and

TRIA6. The capability to use shell elements with volume flaw analysis is a restricted option

requiring a zero stress gradient through the element thickness. Surface-flaw-based reliability is

calculated from the surface flaw material strength parameters and individual shell element output of

the two-dimensional surface stresses, areas, and temperatures. Surface flaw analysis can be

performed using quadrilateral and triangular isoparametric shell elements. Modeling with

axisymmetric elements is not permitted for surface flaw reliability analysis. For the

MSC/NASTRAN finite element analysis code, the supported element types are QUAD4, QUAD8,

TRIA3, and TRIA6. Shell elements are used to identify external surfaces of solid elements that

correspond to the component external surfaces important to the reliability analysis. Shell elements

with exclusively membrane properties and negligible thickness (and hence stiffness) are used.

Provision is made in CARES/LIFE to permit the use of the cyclic symmetry modeling option

in MSC/NASTRAN or similar mesh reduction schemes done via component symmetry.

CARES/LIFE also permits the analysis of simultaneously occurring flaw populations in a given

finite element model (multiple ceramic materials or multiple flaw population capability). Elements

not designated as brittle materials are ignored in the reliability computations.

Temperature-dependent statistical material properties are linearly interpolated at each individual

element temperature. Element and nodal identification numbers can be arbitrary. The

risk-of-rupture intensity is also calculated for each element, and these values are sorted to determine

the maximum values. Element risk-of-rupture intensities are written to a data file for processing

NASA/TM--2003-106316 20
with a graphicalinterpreter such as PATRAN (only available if the finite element mesh has been

generated via PATRAN preprocessing).

Proof test methodology is incorporated into the PIA, the Weibull normal stress averaging, and

the Batdorf theories, accounting for the effect of multiaxial stresses. With the Weibull normal stress

averaging and the Batdorf theory, the proof test load need not closely simulate the actual service

conditions on the component. This is important because it allows reliability analysis to be performed

when proof test stresses have not been applied in the same direction and/or location as the service

load stresses. CARES/LIFE simultaneously processes two finite element analysis neutral files

containing the stress analysis results for the proof test and the service load conditions. The duration

of the proof test and the service load are also considered and each load situation can have different

material-environmental constants (Weibull and fatigue parameters).

For a steady-state cyclic load, component reliability analysis can be performed if the ratio of

the minimum cycle stress to the maximum cycle stress is constant throughout the component. When

this ratio is not constant throughout the component, two finite element result files are required for

the reliability analysis. These two files represent the extremes of the cyclic load range. The

capability of performing transient analysis is also planned for a future update of CARES/LIFE. If

temperature cycling is present, resulting in a variation of material properties as a function of time,

the most severe set of parameters will be used in the analysis with the Paris law or the Walker

equation.

The CARES/LIFE coding architecture has been designed into a modular format to facilitate

ease of use, specialization of function, and access to additional finite element codes with a

minimization of effort (Powers, Starlinger, and Gyekenyesi, 1992). The algorithm is split into three

separately executable module types: (1) the finite element data interface modules; (2) the material

parameter estimation module; and (-3) the component reliability evaluation module. With the first

NAS A/TM--2003-106316 21
setof modules, results from finite element analysis (available in the form of standard output files

and/or plot data files) are interpreted. Modules are available for the ABAQUS, ANSYS, and

MSC/NASTRAN finite element programs, which are named ABACARES, ANSCARES, and

NASCARES, respectively. These interpreter programs write element or subelement stress tensors,

volumes (or surface areas for shell elements), and temperatures into a formatted (ASCII) neutral

data base. This feature enables easy data transfer between different computer systems and avoids

repeated (time consuming) executions of the interpreter program. The structure of the neutral data

base is optimized with respect to memory. The finite element data is arranged within the neutral

data base using the following hierarchy-- element groups, elements and subelements.

The second module, named C4PEST, performs the statistical analysis and estimates inert

strength Weibull parameters and fatigue parameters from fracture data of naturally flawed

specimens. The parameter estimates are placed in a data file for subsequent use by the third

module, named C4LIFE, which reads this file and the neutral data base created from the finite

element interpreter program to calculate the component reliability. C4LIFE will read two neutral

data base files simultaneously if proof testing or cyclic loading is invoked. Although all the

modules in CARES/LIFE are separately executed, they are easily coupled through command files to

sequentially execute without user intervention. Modularization of the program minimizes the

memory overhead, creates a more comprehensible code, and decreases the need to repeatedly

process finite element results files.

The component reliability evaluation module (C4LIFE) and the finite element data interface

module NASCARES use a blank common technique to adjust memory size for a particular problem.

As such, there is no limitation on the total number of elements that can be used in a model.

However, larger memory capacity is required for larger problems. A minimum of 2 megabytes of

memory are necessary to execute the programs. The introduction of one large blank common array

NASA/TM--2003-106316 22
(insteadof severalfixed-dimension arrays) in each routine allows the user to easily adapt the

program to the problem size as well as to the available computational resources. The size of the

blank common array is adjusted by resetting one parameter (MTOT) at the beginning of the module

main program. This capability enables executing the program on PC systems and workstations.

Individual arguments within the blank common array are addressed via pointers, which are integer

variables whose values are internally adjusted within the program to optimize memory. The finite

element interface modules ABACARES and ANSCARES use fixed dimension arrays instead of the

blank common technique. Integer parameters are declared at the beginning of the main module and

the subroutines to fix the array sizes. Initial settings support a maximum of 2000 solid and 2000

shell elements.

The material parameter estimation module (C4PEST) does not use the blank common

technique, due to the large amount of variable arguments (and hence pointers) required and the

reduced need to conserve computer memory. Instead, array sizes are declared with parameters

initialized at the beginning of each subroutine. These parameters correspond to the number of

materials (LM), the number of temperature sets (KL), and the maximum number of specimens used

for each temperature (IJK). Module C4PEST has the initial settings of LM = 100, KL = 15, and

IJK = 999. Adjustment of these array sizing parameters must be identically done for every

subroutine within the C4PEST module.

Because input files may be quite large and require a significant amount of execution time,

CARES/LIFE is designed to run in a batch mode environment. Therefore all program input are

contained in control and data files. Data files refer to finite element analysis results files and neutral

data base files. Control files refer to files containing control indices required to control program

execution, material definitions - such as fracture criterion and specimen data or temperature

dependent material parameters, and.flags for printing and postprocessing (e.g. writing the element

NASA/TM--2003-106316 23
risk of ruptureintensitiesto a PATRAN data file). Description and specification of data within

these files, as well as preparation of the finite element problem and program execution are explained

in the following sections of this manual.

NAS A/TM--2003-106316 24
Input Information

The CARES/LIFE program consists of three separate executable modules: (1) CARES

neutral data base creation module (module name of NASCARES, ANSCARES, and ABACARES

for the MSC/NASTRAN, ANSYS, and ABAQUS finite element analysis programs, respectively);

(2) fatigue parameter and Weibull parameter estimation module C4PEST; and (3) component

reliability analysis module C4LIFE. Figure 3 shows the required input files for the various modules

to perform a complete analysis (the default file names are shown in the figure). The required input

information for each of these modules will be discussed in detail. In addition, the preparation of an

MSC/NASTRAN finite element analysis problem for subsequent reliability analysis with

CARES/LIFE will be discussed. For details of preparing a structural model with ANSYS or

ABAQUS, consult the appropriate manuals/papers which are available.

MSC/NASTRAN Finite Element Analysis

NASTRAN (NASA STRuctural ANalysis) is a large, comprehensive, general purpose finite

element computer code for structural analysis which was developed under NASA sponsorship to fill

the need for a universally available analysis program. It was initially released into the public

domain in 1969 through COSMIC (COmputer Software Management and Information Center). In

addition to the NASA-supported version, which is commonly called COSMIC/NASTRAN, there are

several enhanced proprietary versions of this program. The most widely known of these proprietary

versions is MSC/NASTRAN, which was developed and is maintained by the MacNeal-Schwendler

Corporation. This version of NASTRAN includes a consistent set of isoparametric two- and threedimensional elements, rigid elements, superelements for substructural analysis, improved cyclic

symmetry, and numerical analysis and computer science enhancements. MSC/NASTRAN is widely

NASA/TM--2003-106316 25
usedby manyof theworld'slargestcorporations,governmentlaboratories,andcommercialdata

centers.

MSC/NASTRANcreatesandmanipulatesa databaseto solveproblemsby matrixstructural

analysis.It includesrigid formats:thatis, anorderedexecutionof a setof DMAP(DirectMatrix

AbstractionProgramming)instructions.TheCARES/LIFEprogramutilizesresultsfrom onlya very

smallfractionof availableNASTRANanalysiscapability.Sincefast-fracturemechanicaldesignof

high-performanceceramiccomponentsrequirestemperatureandstresssolutionsonly, staticanalysis

resultsfromSolutionSequence61andRigidFormat47(in caseof cyclic symmetry)aremostoften

used.Beginningwith version66,MSC/NASTRANhasintroducednewstructuredsolution

sequences.ThesesolutionsareusefulbecauseDMAProutinesareorganizedto becompatiblewith

anyotherstructuredsolutionsequence.Solutionsnumbered100andhigherarestructured.

SOL 101- SESTATICandSOL 114- CYCSTATXmaybeusedin placeof SOL61andSOL47,

respectively.

TheMSC/NASTRANprogramis controlledbyuser-specifiedinputdata. Thefunctionsof

theMSC/NASTRANinputcanbelogicallydescribedas:(1) Providingusercontroloverthe

MSC/NASTRANexecutivefunctions- theEXECUTIVECONTROLDECK,(2) Providinguser

controloverprograminputandoutput- theCASECONTROL DECK, and (3) Defining the

physical problem consisting of the finite element model, including constraints and loading conditions

- the BULK DATA DECK. Details of preparing a MSC/NASTRAN structural model with the

selected elements can be found in Schaeffer (Schaeffer, 1979) and in the appropriate MacNeaiSchwendler NASTRAN manuals. It is assumed that analysts using the CARES/LIFE program are

fully familiar with MSC/NASTRAN and its capabilities in solving heat transfer and stress analysis

problems.

NASA/TM--2003-106316 26
A newversionof MSC/NASTRAN is made available to the user community every year.

Currently Version 67 is being used on the NASA Lewis computer; consequently, it was the version

used to solve the example problems summarized in the EXAMPLE PROBLEM section of this

manual. A word of caution is in order at this point. Occasionally, coding errors arise in

NASTRAN which, when reported to MacNeal-Schwendler Corporation, are usually corrected in

later releases. MSC/NASTRAN Version 64 and earlier versions had coding errors in calculating

TRIAX6 element volumes. These errors were supposedly corrected in Version 66. Also, for a

HEXA or PENTA element, significant error can be introduced in the calculation of the element

volume when the element geometry has a large amount of curvature, especially when a coarse mesh

is used. Finally, errors in stress computation occurred with Version 64 shell elements when

nonuniform temperatures were imposed on a given element's nodes. This error should also have

been corrected in Version 65, and it does appear that this problem has been fixed.

The following is an example of the MSC/NASTRAN EXECUTIVE CONTROL DECK which

is recommended to obtain satisfactory input for CARES/LIFE postprocessing for a typical problem

involving static analysis:

NASTRAN DAYLIMIT =-1

ID CERAMIC,FRACTURE

APP DISP

SOL 101

TIME 30

CEND

For problems that are modeled with cyclic symmetry, Solution Sequence 101 is replaced by Solution

Sequence 114.

NASA/TM--2003-106316 27
It is assumed that the user is familiar with MSC/NASTRAN, so each statement will not be

explained herein. Previously, for CARES 1 and CARES 2, Solution Sequence 61-

SUPERELEMENT STATICS (or SOL 101, in version 66) and Rigid Format 47-CYCLIC STATICS

(or SOL 114) were used for the analysis because element volumes and areas can be obtained

through a PARAM card located in the BULK DATA. Since the element (and subelement)

volumes/areas are now calculated by the NASCARES finite element interface module, a limit on

which solution sequences may be used is no longer present.

For the CASE CONTROL DECK, a typical problem without the cyclic symmetry option

would contain

TITLE = USER OPTION

SUBTITLE = USER OPTION

SEALL = ALL

SPC = 10

MPC = 11

LOAD = 12

DISP = ALL

STRESS(PRINT) = ALL

ECHO = SORT

BEGIN BULK

A problem with the cyclic symmetry option would typically contain

TITLE = USER OPTION

SUBTITLE = USER OPTION

SET 2 = 0

HARMONICS = 2

SPC = 1

LOAD = 10

SET 5 = 1

NOUTPUT = 5

DISP = ALL

SET 8 = 1 THRU 1000 EXCEPT 200 THRU 300

STRESS (PRINT) = 8

ECHO = SORT

NASA/TM--2003-106316 28
BEGINBULK

Thefollowingthreeitemsareimportanto note. First,multipleloadsubcasesarenotdefined.

CARES/LIFEdoesnothandlestressoutputfrommultiplesubcasesatthistime;therefore,separate

NASTRANexecutionsarerequiredfor eachloadingcondition.Second,elementstressesarerouted

to theprintoutfile whenSTRESS(PRINT)=ALLis specified.All elementstressesmustbeplaced

in theprintfile. Last,ECHO=SORTwill sendthesortedBULK DATA to theprintfile.

CARES/LIFEreadsthenecessaryBULK DATA, suchasmaterialID, elementconnectivity,and

nodaltemperatures, fromtheprintfile.

SelectedcardsfromtheBULK DATA deckare:

BEGINBULK

PARAM,EST,1

TEMPD,2,70.0

TEMP...

GRID...

CHEXA...

CPENTA...

CTETRA...

CQUADS...

CTRIA6...

NASA/TM--2003-106316 29
PSOLID...

PSHELL...

ENDDATA

ThePARAM,ESTcardwill outputelementvolumesandareasto theNASTRANprintoutfile.

As notedpreviously,thiscardonly functionsfor somesolutionsequencesor rigid formats.The

TEMPDcardis requiredif somegridpointsdonothaveanexplicittemperatureassignment.All

nodesmusthave an associated temperature, either through a TEMP, TEMP*, or TEMPD card.

Since this version of CARES/LIFE does not examine cards for their subcase identity, caution must

be exercised not to allow multiple temperature cases. Nodal temperatures are used by CARES/LIFE

to obtain the average element or subelement temperatures, which are not routinely available in

MSC/NASTRAN.

General guidelines for modeling will now be discussed. Ceramics are extremely sensitive to

geometric discontinuities and resulting stress concentrations. Solid elements have the best capability

for modeling regions of high stress gradients, such as fillets and comers, and for outputting detailed

stress maps. A study on the accuracy of NASTRAN solid elements (Case, and Vandegrift, 1984)

concluded that quadratic elements are probably the most accurate and efficient in analyzing threedimensional structures, especially with potential extremes in element aspect ratios. CARES/LIFE

utilizes results from isoparametric two- and three-dimensional, as well as axisymmetric, finite

elements. The HEXA, PENTA, TETRA, and TRIAX6 elements are used for volumerbased

reliability analysis, and the QUAD8, QUAD4, TRIA6, and TRIA3 elements are used for surfacebased reliability analysis. The TRIAX6 element is a special case because it is axisymmetric, and

NASA/TM--2003-106316 30
therefore, it cannot be used for surface reliability analysis since the QUAD8 and TRIA6 elements

cannot be attached to the modeled component surface. For typical designs, the three-dimensional

HEXA, PENTA, and TETRA elements should be used to model the whole ceramic structure,

including thin cross sections such as blades.

The use of the two-dimensional shell elements for surface reliability analysis is primarily to

identify the corresponding external surfaces of the component. Note that shell elements used for

structural modeling cannot be processed correctly by the NASCARES interface module unless

NASCARES is modified to read both upper and lower surface stress states. These elements can be

excluded from the reliability analysis by using a different material property identification number.

The shell elements are needed by NASCARES to identify the model surface and to obtain

corresponding two-dimensional stress states, areas, and surface temperatures. The shell elements are

attached to the appropriate faces of the solid elements, consistent with the component external

surfaces. Shell elements and solid elements should use the same nodes. An external quadrilateral

face of a solid element should have a QUADS element attached to its nodes. The triangular face of

a PENTA element should share nodes with a TRIA6 element. The shell elements should not

contribute significantly to the structural stiffness of the model. This is achieved by specifying shell

elements with membrane properties only and with a small thickness. Therefore, a typical PSHELL

card would be

PSHELL, 101,300,.000001

The entry 101 in field 2 represents the PSHELL identification number as it appears in the

element connectivity card (CQUAD8 or CTRIA6). The entry 300 in field 3 is the material

identification number for the membrane properties material card. The entry 0.000001 in field 4 is

the thickness of the element; a small nonzero value is used so that the element stiffness contribution

is negligible relative to the solid element. Material identification numbers for bending and

NASA/TM--2003-106316 31
transversesheararelett blank,sothatthoseeffects are excluded. The PSOLID property card is

used for the solid elements and is defined in the usual manner.

After execution of the MSC/NASTRAN problem, the analyst should, have a print file

containing the sorted BULK DATA, element stresses and general problem information. At this

point, all the information required from MSC/NASTRAN to determine the component reliability is

present.

Finite Element Interface Modules Input Information

The results from a finite element analysis (available in the form of standard output files

and/or plot data files) have to be interpreted by an interface program for subsequent reliability

analysis with CARES/LIFE. Due to the different formats used for output of the results in each

finite element program, this interpreter program has to be adapted to the specific finite element

software used; therefore, for each finite element software package, a separate interface program is

prepared. The interface program writes the results of the finite element analysis into a file known as

the neutral data base. Three interface program modules have been developed to process

MSC/NASTRAN, ANSYS, and ABAQUS finite element analysis output files for subsequent

reliability analysis with CARES/LIFE: NASCARES, ANSCARES, and ABACARES, respectively.

Certain information must be available from the finite element output file for processing with

the interface modules. To ensure that this data is available, the user must include the appropriate

flags in the finite element input file so that the data necessary for a reliability analysis is included in

the finite element output file. Since this is dependent on the particular finite element software

package being used, the details of the pertinent information have been addressed in a series of

papers which discuss the NASCARES, ANSCARES, and ABACARES interface modules

individually (Powers, Starlinger, and Gyekenyesi, 1992) (Pintz, 1989) (Edwards, Powers .... , 1992).

NASA/TM--2003-106316 32
Theinterface modules process finite element temperatures, stresses, and volumes/areas from the

finite element analysis output files. If this information is not available in the finite element output

file, the execution of the interface program will be terminated. If the interface program is

successfully executed, a neutral data base file with the default name CARES.NEU (or PROOF.NEU,

if the options of proof testing or cyclic loading are requested) will be generated. The CARES.NEU

file represents the static service loading on the component or the maximum cyclic load condition.

The file PROOF.NEU represents the static proof test load on the component or the minimum cyclic

load condition. The contents, organization, and format of the neutral data base file are discussed in

detail in the Output Information section.

CARES/LIFE Input File Preparation

To control the execution of the CARES/LIFE modules C4PEST and C4LIFE, the user must

prepare an input file (designated as file C4PEST.INP in fig. 3) consisting of the Master Control

Input and the Material Control Input. The Master Control Input is a set of control indices which

directs the overall program execution. It specifies whether material parameter analysis and/or

reliability analysis of a finite element model will be performed. It also specifies the number of

brittle material statistical characterizations. The Material Control Input consists of control indices

and either the rupture data required to estimate the fatigue and statistical material parameters or

direct input of the parameter values themselves, for various temperatures. This input category

includes the choices of fracture criteria and flaw shapes shown in figure 2. On the tape or disks

provided with the program is a file called TEMPLET.INP that can be used to construct an input file

for a particular problem. It is assumed that the CARES/LIFE user has access to a full-screen editor

where block manipulations and character editing can be easily done (the resulting file should be in a

plain ASCII or similar character based format).

NASA/TM--2003-106316 33
Theinput to modules C4PEST and C4LIFE is keyword driven. The keywords can be present

in any order within each input section (except for temperature dependent input, which must be

organized into ascending order of temperature). Keywords must start in the first column of the file.

The beginning of this file is reproduced in figure 4. Note that underneath each keyword a location

is given that specifies where the data value or values are input. An explanation of each keyword is

provided to the right, and a list of available choices is given, if applicable. If integer input is

required, then the input field is defined between asterisks (*), and entries must be right justified.

Real number input is read in an F10.4 or El8.10 format, and asterisks are not present to define the

field width. A maximum of 30 lines between keywords is allowed before an error message is

generated, and therefore, the user can insert short notes as desired. The Master Control Input is

always located at the beginning of the input file.

The CARES/LIFE modules C4PEST and C4LIFE both begin execution by searching for the

keywords associated with the Master Control Input. The end of the Master Control Input occurs

when the SENDX keyword is encountered. Following the Master Control Input, the modules

C4PEST and C4LIFE search for keywords specific to the Material Control Input. The $ENDM and

$ENDT keywords signal the end of temperature independent and temperature dependent data,

respectively. Keywords not found between these SEND_ intervals may assume default values

(specific default values are indicated in the TEMPLET.INP file). Because CARES/LIFE has a

multiple material capability, each section of input for a particular material is separated by a $ENDT

card. The TEMPLET.INP file has only one material characterized. Modifying the file for more

materials involves block copying the Material Control Input sections of the original file, appending

them to the end of the file, and modifying the copied input values accordingly (as well as

appropriately modifying the Master Control Input selections regarding the number of materials in

the reliability analysis). The user is advised to keep an unaltered copy of the TEMPLET.INP file as

NASA/TM--2003-106316 34
a backup.Detailson specific input preparation are described in the Master Control Input and

Material Control Input sections of this manual. Each keyword is discussed in these sections, and

the format field for the input is denoted in parentheses next to the keyword as well as the

appropriate module to which the keyword pertains. It is useful to note that input files previously

used with the CARES fast-fracture code can also be used with CARES/LIFE without modification.

C4PEST Module Input Information

The module C4PEST performs analysis on specimen rupture data to estimate Weibull and

fatigue parameters as well as calculate the other statistical measures as described in section Program

Capability and Description. The default input file name for the C4PEST module is C4PEST.INP.

This file contains two categories of input: (1) Master Control Input, and (2) Material Control Input

(which includes temperature-dependent material fracture data). See the sections C4PEST module

Master Control Input and C4PEST module Material Control Input for a detailed explanation of

the input options of the C4PEST.INP file.

The output from the C4PEST module is placed in the two files (with the default names of)

C4PEST.OUT and CARES.INP. C4PEST.OUT is the results file listing the estimated fatigue and

statistical material parameters as well as other relevant information (goodness-of-fit results, outlier

test, confidence intervals, etc...). The CARES.INP file is the corresponding input file to the C4LIFE

module. This file echoes the Master Control Input and the Material Control Input from the

C4PEST.INP file, except that the temperature dependent specimen fracture data is replaced with the

temperature dependent Weibull and fatigue parameters (direct parameter input option). The C4LIFE

module requires the temperature dependent material parameters as input. It is important to

understand that the CARES.INP input file for module C4LIFE is automatically prepared by module

NAS A/TM--2003-106316 35
C4PESTfromtheC4PEST.INPfile (theuseris notrequiredto preparethisfile, althoughtheuser

maydesireto editthis file andmakechangespriorto executingtheC4LIFEmodule).

C4PEST module Master Control Input. - The Master Control Input section from the

TEMPLET.INP file is reproduced in figure 4. If parameter keywords are omitted, they assume

default values as defined in figure 4. Available keywords may be specific to only one of the two

modules C4PEST or C4LIFE, however, keywords not recognized by a particular module are

ignored. Keywords that will be used with the C4LIFE module should be specified in the input file

C4PEST.INP (for the C4PEST module) because they will automatically be reproduced in file

CARES.INP when module C4PEST is executed.

INTERP (4X,I1) [C4PEST] The control index INTERP is to give the user the option to print

material parameters at interpolated levels of temperature. If interpolated values are requested, then

Lagrange polynomials are used to calculate these values at five intermediate levels between each

user input temperature. These values are output in the CARES.INP file. Since the C4LIFE module

linearly interpolates between the temperatures specified in the CARES.INP file, these additional

values are provided so as to improve the accuracy of the interpolation. The potential penalty for

requesting Lagrangian interpolated values is that poor results may be obtained, therefore interpolated

values should always be verified before proceeding with reliability analysis.

IPRINT (4X,I1) [C4PEST and C4LIFE] The keyword IPRINT controls the printing of element

stresses and experimental fracture stresses. It gives the user the option to control the length of the

program output. If IPRINT = 0, the element stresses and/or specimen fracture data are not printed.

If ][PRINT = 1, all element stresses and/or fracture data are echoed in the program output.

NASA/TM--2003-106316 36
NE (4X,I1) [C4PEST and C4LIFE] NE is a keyword to control program execution. If NE is zero,

then component reliability analysis will not be performed. If the NE control index is 3 (or non

zero) then subsequent component reliability analysis using the C4LIFE module is assumed.

NGP (4X,I2) [C4PEST and C4LIFE] NGP controls the number of Gaussian integration points that

are used in the Batdorf reliability calculations in module C4LIFE. It is also used in the integration

to obtain the Batdorf normalized crack density coefficient kB in module C4LIFE and to

numerically calculate g-factors when closed form solutions are not possible (such as for the sine

wave). The input values can be either 15, 30 or 50. Entries of 30 or 50 will give improved

accuracy but with the penalty of larger CPU requirements.

NMATS, NMATV (4X,I2) [C4PEST and C4LIFE] The keyword NMATS represents the number

of materials for which surface flaw analysis is performed. NMATV represents the number of

materials for which volume flaw analysis is performed. NMATS is associated with shell elements,

and NMATV is associated with either solid, shell, or axisymmetric elements. This association is

achieved via the finite element material ID numbers (see keyword definition MATID in the

Material Control Input section). A component consisting of one material may have one set of

statistical material parameters to characterize the surface and another set for the volume, for which

NMATS = 1 and NMATV = 1. Statistical material parameters are a function of processing,

microstructure, and environment. The CARES/LIFE program modules are capable of analyzing a

single material with multiple statistical material characterizations or many materials with multiple

statistical material characterizations. For example, if a single material component has two different

surface finishes, then NMATS = 2 is used because two different sets of statistical material

NASA/TM--2003-106316 37
parametersarerequired.Theonlyconstraintis that the total of NMATS + NMATV must be less

than 101.

TITLE (72A1) [C4PEST and C4LIFE] The input associated with the TITLE keyword is

reproduced in the program output files C4PEST.OUT and CARES.OUT for problem identification.

TIME (El0.4) [C4PEST and C4LIFE] The keyword TIME allows the user to specify the duration

of the service loading on the component. This input should be in units of seconds (for power law)

or cycles (for the Paris law and Walker law). If TIME = 0.0 then fast-fracture reliability is

calculated.

$ENDX [C4PEST and C4LIFE] The keyword $ENDX signifies the end of the MASTER

CONTROL INPUT.

C4PEST module Material Control Input. - A sample of the Material Control Input from

the TEMPLET.INP file is reproduced in figures 5 and 6. Note that the Material Control Input

actually consists of two different data partitions. The data contained in both figures 5 and 6 are for

a single material. In figure 5, temperature independent control indices, material constants, and

geometric variables are shown. In figure 6, the temperature-dependent fracture data are listed. The

temperature-dependent fracture data or temperature-dependent values of the Weibull and fatigue

parameters are always placed immediately following the temperature independent control indices for

that material. The total number of Material Control Input sections must be equal to the sum of

NMATS + NMATV from the Master Control Input. Note that keywords that are not found assume

default values. Available keywords may be specific to only one of the two modules C4PEST or

NASA/TM--2003-106316 38
C4LIFE. However,all keywordsthatwill beusedwithmoduleC4LIFEshouldbespecifiedin file

C4PEST.INPbecausetheywill automaticallybeprintedin file CARES.1NPwhenmoduleC4PEST

is executed.

It shouldalsobenotedthatthematerialPoisson'sratiois arequiredinput(for shearsensitive

Batdorfmodelsonly). Othertemperature-dependent physicalandmechanicalproperties- suchas

Young'smodulus,thermalconductivity,thermalcoefficientof expansion,andspecificheat- are

requiredfor finite elementanalysisbutarenotusedforreliabilityevaluation.It is assumedthat

Poisson'sratiois constantandtemperatureindependentinthereliabilityevaluations.

C4PEST module Material- and specimen-dependent (temperature independent) data: The

following keywords are the control indices, material indices, and geometric variables necessary for

calculation of volume and surface flaw statistical parameters as shown in figure 5.

DL1, DL2, DH, DW (FI0.4) [C4PEST] If ID1 = 2 or 5 (that is, if statistical material parameters

are to be determined from three- or four-point MOR flexure specimens), then the specimen

dimensions must be input. All dimensions must be input in units consistent with the finite element

analysis. DL1 represents the length between the two outer symmetrical loads. DL2 is the length

between the two inner central loads, DH is the total height of the test specimen cross section, and

DW is the total width of the test specimen cross section. Note that when DL2 is set to zero, then

the solution for three-point bending is obtained, and when DL1 and DL2 are of equal value then the

pure bending beam solution is obtained. This input selection is specific to the C4PEST module.

ID1 (4X,I1) [C4PEST and C4LIFE] ID1 is a control index for specifying the form of the data to

be input for obtaining the statistical, material parameters. Either the Weibull shape and scale

NASA/TM--2003-106316 39
parametersaredirectlyspecified(ID1 = 3) or experimental fracture data are input (ID1 = 1, 2, 4, or

5). The fracture data can be from three-point modulus-of-rupture bend bars (IDI= 2 or 5),

four-point modulus-of-rupture bend bars (ID1 = 2 or 5) or tensile test specimens (ID1 = 1 or 4).

Optionally, the fracture data can be from any specimen geometry (ID1 = 1 or 4), however this

would require that the specimen effective volume or area (see VAGAGE keyword) be input. If the

fracture data are assumed to be all from one failure mode (all volume flaws or all surface flaws),

then ID1 -- 1 or 2 can be chosen. If ID1 = 1 or 2, then CARES assumes that the fracture origins

are consistent with the II)4 input index. If ID1 = 4 or 5, then fracture origins must be supplied with

the fracture data. The ID1 keyword is primarily for the C4PEST module. This input value must be

set to a positive integer, when used with the C4LIFE module. Note that if module C4PEST is run

prior to module C4LIFE, then ID1 is automatically set to the value of 3 when module C4PEST

creates the CARES.INP input file for module C4LIFE.

ID2 (4X,I1) [C4PEST and C4LIFE] The control index ID2 is for selection of a fracture criterion.

Note that some caution must be exercised here because the maximum tensile stress criterion (ID2 =

2) is only used with volume flaw analysis (ID4 must be set to 1). All choices available, except the

PIA model, are used with the Batdorf integration routines. Shetty's mixed-mode fracture criterion

(11)2 = 5) is recommended for both surface and volume flaw analysis (see also keyword C).

1])3 (4X,I1) [C4PEST and C4LIFE] The 1])3 control index is for selection of a crack geometry.

Note that caution must be exercised here since some of the available choices are applicable either

for surface flaw analysis or for volume flaw analysis only. For instance, the penny-shaped crack is

only applicable for volume flaw analysis. The shear sensitive notch crack and the semi-circular

edge crack are only applicable for surface flaw analysis. The Griffith crack is available for both

NASA/TM--2003-106316 40
surfaceandvolumeflaw analysis. The penny-shaped crack is recommended for volume flaw

analysis and the semicircular crack is recommended for surface flaw analysis. See figure 2 for a

matching of crack geometry with fracture criteria.

t_

ID4 (4X,I1) [C4PEST and C4LIFE] 111)4specifies if parameter estimation and subsequent

component reliability analysis is volume or surface flaw based. The value input must be consistent

with the ID2 and ID3 selections. From the fracture data supplied by the user, the Weibuli shape

and scale parameters, along with the normalized Batdorf crack density coefficient consistent with the

]81)4 input value, are estimated and are subsequently made available for processing with finite

element data. The entry for I])4 must be consistent with the finite element analysis material card ID

specified in MATID.

IKBAT (4X, I1) [C4PEST and C4LIFE] IKBAT selects the method of calculating the normalized

Batdorf crack density coefficient, k--sa. If IKBAT = 0, then the crack density coefficient is set to the

value that is the solution for the normal stress fracture criterion, regardless of the fracture criterion

and crack geometry selected by the user for subsequent component analysis. If IKBAT = 1, then

the crack density coefficient is calculated based on the fracture criterion and crack geometry

selected by the ID2 and 11)3 keywords. IKBAT = 0 gives more conservative reliability predictions

than IKBAT = 1 does. Setting IKBAT = 1 is recommended.

OUTLIE (F10.4) [C4PEST] The input associated with the OUTLIE keyword corresponds to the

significance level for which outliers of a sample distribution are to be detected. This keyword is

specific to the C4PEST module. The value input can range between 0.1 (corresponding to a

significance level of 0.1 percent)and 10.0 (corresponding to a significance level of 10.0 percent).

NASA/TM--2003-106316 41
An outlierdetectedat the 1.0 percent level of significance indicates that there is a 1.0 percent

chance that the data point is actually a member of the flaw population, and a 99.0 percent chance

that the data point is an outlier. The outlier test is based on a normal distribution (symmetrical

distriution about the mean) and does not rigorously apply to the Weibull distribution (skewed

distribution about the mean). Results from the outlier test serve only to warn the user. Detected

outliers should not be discarded or censored without proper justification (results from the outlier test

alone does not constitute sufficient justification to remove or censor data).

MATID (1X,I7) [C4PEST and C4LIFE] MATID is the material identification number associated

with the statistical material parameter data. The value input should correspond to the material

identification number listed in the CARES/LIFE neutral data base file for a given element.

Therefore, the value for MATID must match the material ID designated in the finite element

analysis. If only parameter estimation is being performed, then MATID should be an arbitrary

integer value.

MLORLE (4X,I1) [C4PEST] The MLORLE input is only used with the C4PEST module and is

active for fast-fracture and time-dependent data (see keyword SOL in the section

Temperature-dependent fracture or statistical material parameters data; the Data Table form

of input). It is the control index for the method of estimation of Weibull parameters. For fastfracture data these parameters correspond to shape parameter m and characteristic strength Go. For

static and cyclic fatigue data this corresponds to a time-dependent Weibull modulus na and

characteristic time t0. For dynamic fatigue data this corresponds to a time-dependent Weibull

NAS A/TM--2003-106316 42
modulus m (N + 1) andcharacteristictime_. SeesectionTheory for adefinitionof these

terms.

PR (F10.4) [C4PEST and C4LIFE] PR is Poisson's ratio. It is assumed to be temperature

independent.

TITLE (72A1) [C4PEST and C4LIFE] The input associated with the TITLE keyword is

reproduced in the program output files C4PEST.OUT and CgLIFE.OUT for material identification.

VAGAGE (F10.4) [C4PEST] VAGAGE is the gage volume or gage area of a tensile test

specimen. This quantity can also be input in the Data Table (see section Temperature-dependent

fracture or statistical material parameters data; the Data Table form of input). If ID4 = 2 and

ID1 = 1 or 4 (that is, if surface flaw analysis is specified and the statistical material parameters are

to be determined from specimen fracture data), then the gage surface area or effective area of the

specimen must be specified. If ID4 = 1 and ID1 = 1 or 4 (that is, if volume flaw analysis is

specified and the statistical material parameters are to be determined from specimen fracture data),

then the gage volume or effective volume of the specimen must be specified. It is recommended

that the Data Table be used to input this quantity since the effective volume or area can be

temperature dependent quantities. See keyword PFCONV used with module C4LIFE for additional

information.

$ENDM [C4PEST and C4LIFE] The keyword $ENDM signifies the end of a section of the

Material Control Input. The temperature-dependent specimen fracture data or the Weibull shape and

scale parameters are assumed to immediately follow.

NAS A/TM--2003-106316 43
C4PEST module temperature-dependent fracture or statistical material parameters

data: Immediately following the SENDM keyword, which signals the end of the material's

temperature-independent data, the temperature-dependent experimental fracture data or material

parameters are input. Data must be arranged so that they correspond to ascending order of

temperature. These data enable interpolation of the statistical material parameters to other

temperatures. Figure 6 shows an example of the input for experimental fracture stress data for

module C4PEST. The essential feature of figure 6 is that the data is organized in a table, called the

Data Table (DTABLE) for a discrete temperature. The tabular input format for the Data Table is

an upgrade feature from the previous CARES codes. It is therefore recommended that the Data

Table format is always used, however for completeness the input style associated with the CARES

codes is maintained with CARES/LIFE (input files prepared for CARES1, CARES2, or CARES3

can be used with CARES/LIFE). The Data Table is fully described in the following section of this

manual (see section titled Temperature-dependent fracture or statistical material parameters

data; the Data Table form of input).

The keywords used with experimental fracture stresses (i.e. DTABLE, MOR, and NUT) are

only recognized by the C4PEST module. The material parameters calculated from this module are

output in the CARES.INP data file for subsequent use by the C4LIFE module. A sample of the

temperature dependent data section of the CARES.INP file is shown in figure 7. An explanation of

all the keywords, including a separate section describing the Data Table, is provided in the

following text.

DTABLE [C4PEST] This keyword denotes the beginning of the Data Table for a given

temperature. The $ENDD marks the end of the Data Table. See the section titled

NASA/TM--2003-106316 44
Temperature-dependent fracture or statistical material parameters data; the Data Table form

of input for a complete description of this input format.

MOR (3A1,3E18.10) or (3E18.10) [C4PEST] The input format associated with this keyword is not

recommended with the CARES/LIFE code, but is included for completeness. It was previously used

with the CARES 1, CARES2, and CARES3 codes. MOR indicates that experimental inert fracture

stresses (fast-fracture data) will be input. Fracture stresses can be input in random order for each

temperature; however, the number of entries must correspond with the NUT keyword. There are

two styles of input. If ID1 = 1 or 2 (that is, if the fracture data are assumed to be a complete

sample), then fracture stresses only are input. The input format is 3E18.10. IflD1 = 4 or 5 (that is,

if the fracture origins and the fracture stresses are to be input), then the input format is

3A1,3E18.10. The 3A1 represents three fields of single alphanumeric characters. This field is for

fracture origin input. An "S" indicates a surface flaw origin. A "V" represents a volume flaw

origin. A "U" indicates an unknown flaw origin. Each fracture stress has a corresponding

fracture origin. Each line of fracture data consists of three fracture origins followed by their

respective failure stresses. Fracture data values should be unique, and multiple identical values

should not be input (change one value slightly). This keyword is specific to the C4PEST module.

NUT (3X,I3) [C4PEST] The input format associated with this keyword is not recommended with

the CARES/LIFE code but is included for completeness. NUT is the sample size of the

experimental fracture data for the temperature indicated by TEMP (or TDEG) and always precedes

the MOR keyword. NUT is specified if ID1 does not equal 3 (statistical material parameters are

not being directly input). Different numbers of specimens are permitted at different temperatures.

This keyword is specific to the C4P.EST module.

NASA/TM--2003-106316 45
PARAM (2E18.10) [C4PEST and C4LIFE] PARAM signals that the Weibull shape parameter, m,

and Weibull scale parameter, a o, will be input for the temperature indicated by the TEMP (or

TDEG) keyword. Referring to figure 7, the Weibull shape and then the Weibull scale parameters

are entered at the indicated space with a format of 2E18.10 (denoted by M and SP, respectively in

the figure). The Weibull shape parameter is dimensionless. The Weibull scale parameter has units

of stress x (volume) vmv for volume flaw analysis and units of stress x (area) vms for surface flaw

analysis.

TEIVIP or TDEG (F10.4) [C4PEST and C4LIFE] TEMP (or TDEG) is the input keyword for the

temperature of the fracture data or of the statistical material parameters that immediately follow.

Temperature can be specified in any units but must be consistent with the finite element analysis.

$ENDT [C4PEST and C4LIFE] The keyword $ENDT signals the end of the

temperature-dependent data. Another section of the Material Control Input follows, if required.

Temperature-dependent fracture or statistical material parameters data; the Data Table

form of input: The Data Table describes an input format for the time-dependent and fast-fracture

rupture data, an example of which is shown in figure 6. This is the recommended input style for the

C4PEST module. One Data Table exists for each discrete temperature level. Data Tables are

arranged in ascending order of temperature. Input within the Data Table is keyword driven. A brief

description of these keywords is shown in figure 6. A line of input in the Data Table is broken up

into five input areas, denoted by TEST, FLAW, FIELD1, FIELD2, and FIELD3. TEST is the input

area for the keyword. FLAW is the input area for integer data associated with the SOL and

CYCLIC keywords and fracture origin data for the FAST, STAT, CYCL, and DYNA keywords.

NASA/TM--2003-106316 46
FIELD1,FIELD2,andFIELD3defineinputareasforrealnumberinputssuchastemperature,

fracturetimes,stressingrates,etc. BlankentriesfortheTEST,FLAWandFIELD1inputareas

assumetheidentityof thepreviouslineor someotherpreviouslydefined(default)value. The

specific rules are explained for the individual keywords.

CYCL The CYCL keyword indicates that steady state cyclic fatigue data will be input. The

fracture origin is assigned in the FLAW field. Only a single entry is allowed in this field

(placement is arbitrary). An entry of S, U, or V indicates either a surface, unknown or volume flaw,

respectively. A blank entry will default the flaw origin to the type consistent with the 11)4 keyword.

If 1:1)4 = 1, then a volume flaw is assigned. If ID4 = 2, then a surface flaw is assigned. An entry

of U could indicate an unknown flaw origin, an unspecified competing failure mode, or an unfailed

(truncated or run out) specimen. It is recommended that data censoring be used only when SOL =

2 and even then results should be verified, if possible, by other means. The magnitude of the

extreme fiber stress (or maximum stress in the component) is input in FIELD1. For cyclic loading

this corresponds to the peak (maximum) cyclic stress at the highest stressed point in the specimen.

If this field is blank or the entry is less than 1.0E-30, then the value from the last user specified

FIELD1 entry for a CYCL card is used. The specimen times to failure (in seconds) for the power

law or cycles to failure for the Paris and Walker laws are input in FIELD2 and/or FIELD3 (see the

CYCLIC keyword). A blank entry or an entry whose value is less than 1.0E-30 is ignored. Both

FIELD2 and FIELD3 entries are assigned the fracture origin input in the FLAW field.

CYCLIC The presence of this keyword signals that fatigue parameter Bw is to be calculated from

steady state cyclically loaded specimen data. The data for the individual rupture specimen is input

via the CYCL keyword. The crack growth law and cycle waveform are input in the FLAW field

NASA/TM--2003-106316 47
(theParis law and Walker law are independent of the cycle waveform). The R-ratio (the minimum

cycle stress divided by the maximum cycle stress) is input in FIELD1. It is assumed that the Rratio is invariant (constant) at every location in the specimen. Alternatively the user may also

directly input a g-factor in FIELD1 provided the FLAW field contains an entry of 1. Negative

values for the R-ratio are allowed to be input; however for the Paris and Walker laws a warning

message is issued in this situation. A negative value of the R-ratio will predict additional strength

degradation for the Paris and Walker laws. For the power law no crack extension occurs when the

normal stress acting on the flaw is compressive. The traditional fracture mechanics approach

assumes that no additional strength degradation occurs due to the presence of compressive stresses

for any of the crack growth laws. For component reliability analysis CARES/LIFE may or may not

predict additional strength degradation for negative R-ratios. The user is advised to examine the

options listed with the CYCLIC keyword described in the Material- and specimen-dependent

(temperature independent) data section of the Master Control Input. If the power law is selected

then the period of the cycle in units of seconds must be input in FIELD2 and the loading exponent

imposed on the waveform is optionally input in FIELD3. The FIELD3 input for the power law is

specifically for calculation of the g-factor for rotating components where the angular speed varies

with time and has a waveform indicated with the FLAW field input (an input value of 2.0 is for a

rotational load, an input value of 1.0 calculates the g-factor in the usual way). If a value of 0 is

entered in the FLAW field, then the Paris and Walker laws are selected. The value of the Walker

R-ratio sensivity exponent, Q, is input in FIELD3. For this case if FIELD3 is leti blank or a value

is zero is entered then the Paris law is implimented. If a positive nonzero real number is entered in

FIELD3 then the Walker law is implimented.

NAS A/TM--2003-106316 48
DTABLE This keyword denotes the beginning of the Data Table for a given temperature. The

$ENDD marks the end of the Data Table.

DYNA The DYNA keyword indicates that dynamic fatigue (constant stress rate) data will be input.

The fracture origin is assigned in the FLAW field. Only a single entry is allowed in this field

(placement is arbitrary). An entry of S, U, or V indicates either a surface, unknown or volume flaw,

respectively. A blank entry will default the flaw origin to the type consistent with the 1])4 keyword.

If 1])4 = 1 then a volume flaw is assigned. If ID4 = 2 then a surface flaw is assigned. An entry of

U could indicate an unknown flaw origin, an unspecified competing failure mode, or an unfailed

(truncated or run out) specimen. It is recommended that data censoring be used only when SOL =

2 and even then results should be verified, if possible, by other means. The stressing rate

(stress/second) at the highest stressed point in the component is input in FIELD1. If this field is

blank or the entry is less than 1.0E-30, then the value from the last user specified FIELDI entry for

a DYNA card is used. The specimen ultimate failure stresses (at the highest stressed point in the

component) are input in FIELD2 and/or FIELD3. A blank entry or an entry whose value is less

than 1.0E-30 is ignored. Both FIELD2 and FIELD3 entries are assigned the fracture origin input in

the FLAW field.

FAST The FAST keyword indicates that inert strength or fast-fracture data will be input. The

fracture origin is assigned in the FLAW field. Only a single entry is allowed in this field

(placement is arbitrary). An entry of S, U, or V indicates either a surface, unknown or volume flaw,

respectively. A blank entry will default the flaw origin to the type consistent with the ID4 keyword.

If 1])4 = 1 then a volume flaw is assigned. If ID4 = 2 then a surface flaw is assigned. An entry of

U could indicate an unknown flaw origin or an unspecified competing failure mode. The specimen

NASA/TM--2003-106316 49
ultimatefailurestresses(atthe highest stressed point in the specimen) are input in FIELD1 and/or

FIELD2 and/or FIELD3. A blank entry or an entry whose value is less than 1.0E-30 is ignored.

The FIELD1, FIELD2 and FIELD3 entries are assigned the fracture origin input in the FLAW field.

Fast-fracture and fatigue data may both be simultaneously present in the Data Table. Weibuli

parameters determined from fast-fracture data will override Weibull parameters estimated soley from

the fatigue data.

PARAM The keyword PARAM signals that the Weibull shape and scale parameter will be input

for the temperature indicated by the TEMP keyword. The Weibull shape parameter m is input in

FIELD1. The Weibull scale parameter is entered in FIELD2. The Weibull shape parameter is

dimensionless. The Weibull scale parameter has units of stress x (volume) I/_"vfor volume flaw

analysis and units of stress x (area) l/msfor surface flaw analysis. The characteristic strength can be

optionally input in FIELD3. Note FIELD2 or FIELD3 can be left blank. Values input via the

PARAM keyword override Weibull parameters generated from fatigue data.

RANGE The RANGE keyword is used to specify a range of values for the crack growth exponent

N in which to search for a solution when the median deviation estimation technique (ISOL = 2) is

selected. If RANGE is not invoked, then the default range of 2.1 to 150.0 is assumed. The bounds

of the range are input in FIELD1 and FIELD2, in any order. The limitation of the numerical

solution method is that convergence on a value has a tolerance of 0.001 of the span of the range.

Consequently, if further accuracy (in absolute terms) is required then the range bounds must be

decreased. Numerical problems may be encountered for values of N approaching 2.0 and for large

values of N. If a solution is retumed that has a value identical to one of the range bounds, then the

NASA/TM--2003-106316 50
searchrangemustbeexpanded(or shifted).Thereis noguaranteethatanobtainedsolutionis

unique.

RSTRES and +STRES These keywords are not used with parameter estimation from specimen

rupture data (not used with module C4PEST), but rather are transfered to the CARES.INP file when

the module C4PEST is executed (see the keyword RSTRES in section Temperature-dependent

fracture or statistical material parameters data). If material residual stresses (not loading

induced) are present, the keyword RSTRES allows the user to input a stress tensor that is added to

the mechanically induced stresses in the component (it is not added to the fracture stresses of the

experimental fracture specimens). This option requires a consistent material reference frame

(neutral file stresses should be organized in a material coordinate system reference frame). The

stresses are input in two lines. The RSTRES keyword uses FIELD1, FIELD2, and FIELD3, for the

X, Y, and Z components, respectively. The continuation line uses the +STRES keyword with

FIELD1, FIELD2, FIELD3 for the XY, YZ, and ZX components, respectively. For shell elements

only the X, Y, and XY components are used. The RSTRES option is recommended only when

IPRI_C = 2, otherwise the user is limited to input of hydrostatic stresses ( X = Y = Z; 0.0 = XY =

YZ = ZX ) in order to obtain valid results.

STAT The STAT keyword indicates that static fatigue data will be input. The fracture origin is

assigned in the FLAW field. Only a single entry is allowed in this field (placement is arbitrary).

An entry of S, U, or V indicates either a surface, unknown or volume flaw, respectively. A blank

entry will default the flaw origin to the type consistent with the 11)4 keyword. If !1)4 = 1, then a

volume flaw is assigned. If 1I)4 = 2, then a surface flaw is assigned. An entry of U could indicate

an unknown flaw origin, an unspecified competing failure mode, or an unfailed (truncated or run

NASA/TM--2003-1063 i 6 51
out)specimen.

resultsshouldbeverified,if possible,byother means.

maximum stress in the component) is input in FIELD1.

It is recommended that data censoring be used only when SOL = 2 and even then

The magnitude of the extreme fiber stress (or

If this field is blank or the entry is less

than 1.0E-30, then the value from the last user specified FIELD1 entry for a STAT card is used.

The specimen times to failure (in seconds) are input in FIELD2 and/or FIELD3. A blank entry or

an entry whose value is less than 1.0E-30 is ignored. Both FIELD2 and FIELD3 entries are

assigned the fracture origin input in the FLAW field.

SOL This keyword is used to select the method of solution for obtaining the time-dependent power

law parameters (see figure 6) from fatigue specimen data. The choice is input in the FLAW field of

the Data Table and can be placed anywhere within the field. If the SOL keyword is not specified,

then the default is the median value technique (SOL = 0) using least squares linear regression.

Note if SOL = 0 that data assigned fracture origins not consistent with the ID4 index are ignored

(don't censor data with this option). The median value technique is the least accurate estimation

procedure available in CARES/LIFE.

If SOL = 1, then the least squares regression technique is invoked. This technique performs

linear regression on all the fatigue data (as opposed to only the median values) to establish the crack

growth exponent N.

When SOL = 2, the median deviation method is chosen. The median deviation method uses

Weibull analysis to minimize the scatter in the data to establish the crack growth exponent N. As

such, two different solution routes are possible, maximum likelihood or least squares, depending on

the setting of the M1,ORLE keyword in the temperature independent section of the Material

Control Input. The RANGE keyword should be used with this solution option. Data censoring is

NASA/TM--2003-106316 52
allowedalthoughcautionshouldbeexercisedin interpreting results (especially if competing failure

modes have dissimilar crack growth exponents).

For the least squares, median value, and median deviation technique the inert strength

distribution Weibull modulus m and a characteristic strength c_0, is optionally estimated from the

fatigue data. If inert strength fracture data is simultaneously input, then the Weibull parameters for

this data override those calculated from time dependent data.

TENIP This keyword is used to denote temperature. The temperature is input in FIELD1. Only

one assignment of temperature is permitted in the Data Table. Data Tables must be arranged in

ascending order of temperature.

TPARAM The keyword TPARAM signals that the power law parameters N and Bw (and

optionally Q for the Walker law) will be input for the temperature indicated by TEMP. The crack

growth exponent N and the constant BWare entered in FIELD1 and FIELD2, respectively. The

Walker R-ratio sensitivity exponent Q is optionally entered in FIELD3. The crack growth exponent

is dimensionless. The crack growth constant has units of (second x (stress) 2) for the power law and

(cycle x (stress) 2) for the Paris and Walker laws. The Walker R-ratio sensivity exponent is

dimensionless.

VAGAGE

Equivalently, it also represents the fast-fracture effective volume, Ve, or area, Ae, of some

arbitrary specimen geometry and loading (hence it becomes a temperature dependent quantity).

the section titled Material Strength Characterization of the section Theory for a definition.

effective volume and area are used in the calculation of the Weibull scale parameter, go.

The keyword VAGAGE represents the gage volume or area of a tensile test specimen.

See

The

If ID4 = 2

NAS A/TM--2003-106316 53
andID1 = 1or 4 (thatis, if surfaceflaw analysisis specifiedandthestatisticalmaterialparameters

areto bedeterminedfrom specimenfracturedata),thenthegagesurfaceareaor fast-fracture

effectiveareaof thespecimenmustbespecified.If ID4 = 1andID1 = 1or 4 (thatis, if volume

flawanalysisis specifiedandthestatisticalmaterialparametersareto bedeterminedfrom specimen

fracturedata),thenthegagevolumeor fast-fractureeffectivevolumeof thespecimenmustbe

specified.Theeffectivevolumeor surfaceareaareinputin FIELD1. ConsultthesectionMaterial

StrengthCharacterization of this manual for the definition of the effective volume and area.

VAGTIM The keyword VAGTIM represents the gage volume or area of a tensile test specimen.

Equivalently it also represents the stress-normalized effective volume, Vco or area, Aco of some

arbitrary specimen geometry and loading (hence it becomes a temperature dependent quantity). See

the section titled Material Failure Characterization For Static, Cyclic Or Dynamic Loading of

the section Theory for a definition. The stress-normalized effective volume and area are used in the

calculation of the fatigue parameter, Bw. If 1])4 = 2 and ID1 = 1 or 4 (that is, if surface flaw

analysis is specified and the fatigue parameters are to be determined from specimen fracture data),

then the gage surface area or stress-normalized effective area of the specimen must be specified. If

ID4 = 1 and ID1 = 1 or 4 (that is, if volume flaw analysis is specified and the fatigue parameters

are to be determined from specimen fracture data), then the gage volume or stress-normalized

effective volume of the specimen must be specified. The stress normalized effective volume or

surface area are input in FIELD1.

$ENDD This keyword signals the end of the Data Table for a given temperature.

NASA/TM--2003-106316 54
C4LIFE Module Input Information

The output from the C4PEST module is placed in the two files (with the default names of) !

C4PEST.OUT and CARES.INP. The CARES.INP file is the corresponding input file to the C4LIFE

module. This file echoes the Master Control Input and the Material Control Input from the

C4PEST.INP file, except that the temperature dependent specimen fracture data is replaced with the

temperature dependent Weibull and fatigue parameters (direct parameter input option). The C4LIFE

module requires the temperature dependent material parameters as input. It is important to

understand that the CARES.INP input file for module C4LIFE is automatically prepared by module

C4PEST from the C4PEST.INP file (the user is not required to prepare this file, although the user

may desire to edit this file and make changes prior to executing the C4LIFE module).

The module C4LIFE performs the reliability analysis of the component from the

CARES.NEU and optionally PROOF.NEU neutral data base files along with the CARES.INP input

file (default file names are shown). The overall structure and input options of the CARES.INP file

are similar to those for the C4PEST.INP file. Refer to the C4LIFE module Master Control Input

and C4LIFE module Material Control Input sections of this manual for further information

regarding the individual input items. As previously stated, the CARES.INP file is typically

generated from the C4PEST module (as opposed to the user manually preparing this file). The

CARES.NEU and PROOF.NEU neutral data base files are generated from the module NASCARES

(for MSC/NASTRAN), ANSCARES (for ANSYS), or ABACARES (for ABAQUS). The

NASCARES, ANSCARES, or ABACARES programs only create one neutral data base file per

execution. The file PROOF.NEU represents the static proof test load on the component or a cyclic

load on the component. The C4LIFE module can be executed without the PROOF.NEU file. The

CARES.NEU file represents the static service loading on the component or a cyclic load condition

and is always a required input file for the C4LIFE module. For cyclic loads the file CARES.NEU

NAS A/TM--2003-106316 55
would contain the maximum stresses and the file PROOF.NEU would contain the minimum stresses

due to the cyclic load. If the R-ratio is constant throughout the component then only the

CARES.NEU neutral file containing the maximum cyclic stresses is needed for the reliability

analysis (the R-ratio is then input by the user).

C4LIFE module Master Control Input. - The Master Control Input section from the

TEMPLET.INP file is reproduced in figure 4. If parameter keywords are omitted, they assume

default values as defined in figure 4. Available keywords may be specific to only one of the two

modules C4PEST or C4LIFE, however, keywords not recognized by a particular module are

ignored. Keywords that will be used with the C4LIFE module should be specified in the input file

C4PEST.INP (for the C4PEST module) because they will automatically be reproduced in file

CARES.INP when module C4PEST is executed.

FACTOR (El0.4) [C4LIFE] The keyword FACTOR allows the user to input a loading factor to

multiply all element (or subelement) stresses. These adjusted values are used to calculate the

subsequent reliability. Note, adjusted stresses are not echoed in the program output. This keyword

is specific to the C4LIFE module.

[POST (4X,I1) [C4LIFE] The control index [POST is for PATRAN postprocessing of the module

C4LIFE component reliability analysis. If [POST = 1 then a data file of averaged (average of the

subelements of an element) element risk-of-rupture intensities is created ( default file name is

CARES.PAT). Note that use of this file requires that the finite element mesh was created with the

PATRAN program (i.e., the phase II model data is still available). If [POST = 0 then this file is

not created. The [POST keyword is specific to the C4LIFE module.

NASA/TM--2003-106316 56
IPRINC (4X,I1) [C4LIFE] IPRINC isthecontrolindexfor passingstressesto thereliability

algorithms.If IPRINC = 1thentheprincipalstressescalculatedin subroutineEIGENareused.

Thisis a requirementfor thePIA model. If IPRINC = 2 thenthestresstensoris passedto the

Batdorfmodels.Thisoptionallowsconsiderationof theeffectof prooftestingor cyclic loading

whentheboundaryloaddirectionand/orlocationchanges(IPROOF= 1shouldalsobespecified).

Thiskeywordis specificto theC4LIFEmodule.

IPRINT (4X,I1) [C4PESTandC4LIFE] ThekeywordIPRINT controlstheprintingof element

stressesandexperimentalfracturestresses.It givestheusertheoptionto controlthelengthof the

programoutput. If IPRINT = 0, theelementstressesand/orspecimenfracturedataarenotprinted.

If IPRINT = 1,all elementstressesand/orfracturedataareechoedintheprogramoutput.

IPROOF (4X,I1) [C4LIFE] ThecontrolindexIPROOFis usedasa flagsignalingwhetheroneor

twoneutraldatabasefilesareto bereadbytheC4LIFEmodule.If IPROOF = 0, thenonly one

file isto beread. Thisfile (CARES.NEU)containseitherthestressanalysisof thestaticloador the

peakstressesof thecyclic load. In thiscase,for cyclicloadingtheR-ratio(min/maxcyclestress)is

assumedconstanthroughoutthecomponentandtheappropriateloadinginformationmustbeinput

usingtheCYCLIC keyword. If IPROOF= 1thentwoneutralfilesareto beread(CARES.NEU

andPROOF.NEU).Theseneutralfileswill eitherrepresentprooftestingor cyclic loading. For

reliabilityanalysiswithprooftesting,thentheprooftestneutraldatabasefile is PROOF.NEUand

theserviceloadneutralfile is CARES.NEU.ThePROOF.NEUandCARES.NEUfiles containthe

stressanalysesof theprooftestloadandserviceload,respectively.Forreliabilityanalysiswith

cyclic loadingwheretheR-ratioisnotconstantthroughoutthecomponent,thentwo neutraldata

basefilesarerequiredto computetheR-ratioonanelement-by-element basis. In thiscase,the

NASA/TM--2003-106316 57
PROOF.NEU neutral data base file contains the minimum cyclic stresses and the CARES.NEU

neutral data base file contains the maximum cyclic stresses. The IPRINC and T1MEPT keywords

are also required if IPROOF = 1. In this case IPRINC = 2 should be used for either proof test or

cyclic loading. For proof testing, the TIMEPT keyword is used to assign the duration of the proof

test and the TIME keyword is used to assign the duration of the of service load. For cyclic loading,

TIMEPT and TIME are assigned the same value.

NE (4X,I1) [C4PEST and C4LIFE] NE is a keyword to control program execution. If NE is zero,

then component reliability analysis will not be performed. If the NE control index is 3 (or non

zero) then subsequent component reliability analysis using the C4LIFE module is assumed.

NGP (4X,I2) [C4PEST and C4LIFE] NGP controls the number of Gaussian integration points that

are used in the Batdorf reliability calculations in module C4LIFE. It is also used in the integration

to obtain the Batdorf normalized crack density coefficient kB in module C4LIFE and to

numerically calculate g-factors when closed form solutions are not possible (such as for the sine

wave). The input values can be either 15, 30 or 50. Entries of 30 or 50 will give improved

accuracy but with the penalty of larger CPU requirements.

NMATS, NMATV (4X,I2) [C4PEST and C4LIFE] The keyword NMATS represents the number

of materials for which surface flaw analysis is performed. NMATV represents the number of

materials for which volume flaw analysis is performed. NMATS is associated with shell elements,

and NMATV is associated with either solid, shell, or axisymmetric elements. This association is

achieved via the finite element material ID numbers (see keyword definition MATID in the

Material Control Input section). A component consisting of one material may have one set of

NAS A/TM--2003-106316 58
statisticalmaterialparameters to characterize the surface and another set for the volume, for which

NMATS = 1 and NMATV = 1. Statistical material parameters are a function of processing,

microstructure, and environment. The CARES/LIFE program modules are capable of analyzing a

single material with multiple statistical material characterizations or many materials with multiple

statistical material characterizations. For example, if a single material component has two different

surface finishes, then NMATS = 2 is used because two different sets of statistical material

parameters are required. The only constraint is that the total of NMATS + NMATV must be less

than 101.

NS (3X,I3) [C4LIFE] Because it's possible that a finite element model consists of only a fraction

of the total ceramic component (model reduction via symmetry), the NS keyword allows for

multiplication of the model geometry by the appropriate number of times when the reliability of the

entire component is desired. For cyclic symmetry, NS corresponds to the number of segments

required to reproduce the whole component. Specifically NS is used to multiply element or

subelement volumes and areas in the reliability calculations. This keyword is only used with the

C4LIFE module.

TITLE (72A1) [C4PEST and C4LIFE] The input associated with the TITLE keyword is

reproduced in the program output files C4PEST.OUT and CARES.OUT for problem identification.

TIME (El0.4) [C4PEST and C4LIFE] The keyword TIME allows the user to specify the duration

of the service loading on the component. This input should be in units of seconds (for power law)

or cycles (for the Paris law and Walker law). If TIME = 0.0 then fast-fracture reliability is

calculated.

NASA/TM--2003-106316 59
TIMEPT (El0.4) [C4LIFE] The keyword T1MEPT is used to specify the duration of the loading

for the optional neutral data base file PROOF.NEU (defaulted name). This option is activated when

the keyword IPROOF is set equal to 1. If cyclic loading is assumed then the value input should

equal the value input for the TIME keyword. For proof test reliability analysis the keyword

TIMEPT allows the user to specify the duration of the proof test loading on the component. In this

particular case the value input must be in units of seconds (since Paris law, Walker law, or g-factors

have not been made as input options with proof testing). Only static loadings are assumed for the

proof test and service condition. If TIMEPT = 0.0 then only the fast-fracture proof test loading is

considered. The TIMEPT keyword is used only with the C4LIFE module.

$ENDX [C4PEST and C4LIFE] The keyword $ENDX signifies the end of the MASTER

CONTROL INPUT.

C4LIFE module Material Control Input. - A sample of the Material Control Input from

the TEMPLET.INP file is reproduced in figures 5 and 7. Note that the Material Control Input

actually consists of two different data partitions. The data contained in both figures 5 and 7 are for

a single material. In figure 5, temperature independent control indices, material constants, and

geometric variables are shown. In figure 7, the temperature-dependent Weibull and fatigue

parameters are listed. The temperature-dependent Weibull and fatigue parameters are always placed

immediately following the temperature independent control indices for that material. The total

number of Material Control Input sections must be equal to the sum of NMATS + NMATV from

the Master Control Input. Note that keywords that are not found assume default values. Available

keywords may be specific to only one of the two modules C4PEST or C4LIFE. However, all

NASA/TM--2003-106316 60
keywordsthat will be used with module C4LIFE should be specified in file C4PEST.INP because

they will automatically be printed in file CARES.INP when module C4PEST is executed.

It should also be noted that the material Poisson's ratio is a required input (for shear sensitive

Batdorf models only). Other temperature-dependent physical and mechanical properties - such as

Young's modulus, thermal conductivity, thermal coefficient of expansion, and specific heat - are

required for finite element analysis but are not used for reliability evaluation. It is assumed that

Poisson's ratio is constant and temperature independent in the reliability evaluations.

C4LIFE module Material- and specimen-dependent (temperature independent) data: The

following keywords are the control indices, material indices, and geometric variables necessary for

calculation of volume and surface flaw statistical parameters as shown in figure 5.

C (F10.4) [C4PEST and C4LIFE] The value of the empirical constant _ that is required for

Shetty's mixed-mode fracture criterion (noncoplanar strain energy release rate criterion) is assigned

using the C keyword. This fracture criterion is specified with the 1])2 keyword in the Master

Control Input (ID2 =5).

CYCLIC (4X,I1,/,E10.4,/,E10.4,/,E10.4) [C4LIFE] Invoking the CYCLIC keyword in the

temperature independent section of the Material Control Input indicates that the component

reliability analysis performed with module C4LIFE will also include the effects of cyclic fatigue

degradation (see also the CYCLIC keyword described in the section titled Temperature-dependent

fracture or statistical material parameters data; the Data Table form of input, which is used to

estimate fatigue parameters from cyclic fatigue rupture data using module C4PEST). The integer

input (IWAVE) designates the cracl_ growth law (Paris and Walker laws, or the power law) and the

NAS A/TM--2003-106316 61
loadingcyclewaveform(square,sawtooth,andsine). Therealnumberinput(RCYC)assignstheRratio (the minimum cycle stress divided by the maximum cycle stress) for the material. This input

assumes the value of the R-ratio is constant at every point in the body and that only one neutral data

base file is being processed (IPROOF = 0). If two neutral data base files are to be examined

(IPROOF = 1) then the input for RCYC is ignored. This is because the R-ratio is now computed

for each element in the model (by comparing the two neutral files). Negative values for the R-ratio

are allowed to be input (via RCYC) when IPROOF = 0; however for the Paris and Walker laws

this means that additional material degradation is predicted due to the compressive stress component

(an appropriate warning message is issued). When IPROOF= 1 and a negative R-ratio is computed

at an element; no additional degradation due to the compressive stresses is assumed (equivalent to

the R-ratio having a value of zero) for all the crack growth laws (this is the additional fracture

mechanics approach). The real number input for the period of the cycle (PERIOD) is associated

with the power law and must be in units of seconds. Using the CYCLIC keyword with the power

law means that g-factors, which equate the cyclic loading to an equivalent static loading situation

are to be computed. It is useful to know that when the power law is invoked with the sawtooth

waveform and the value of RCYC is set to zero, then the g-factor is 1/N+I, which is equivalent to

dynamic (constant stress rate) loading. The user is warned that calculating g-factors with a sine

wave is numerically intensive when processing two neutral files (IPROOF = 1) with the Batdorf

model (when ID2 is not equal to 4). If the power law is selected then the loading exponent QCYC

is optionally input. The QCYC input for the power law is specifically for calculation of the g-factor

for rotating components where the angular speed varies with time and has a waveform indicated

with the QCYC value input (an input value of 2.0 is for a rotational load, an input value of 1.0

calculates the g-factor in the usual way).

NASA/TM--2003-106316 62
ID1 (4X,I1) [C4PESTandC4LIFE] ID1 is a control index for specifying the form of the data to

be input for obtaining the statistical material parameters. Either the Weibull shape and scale

parameters are directly specified (ID1 = 3) or experimental fracture data are input (ID1 = 1, 2, 4, or

5). The fracture data can be from three-point modulus-of-rupture bend bars (IDI= 2 or 5),

four-point modulus-of-rupture bend bars (ID1 = 2 or 5) or tensile test specimens (ID1 = 1 or 4).

Optionally, the fracture data can be from any specimen geometry (ID1 = 1 or 4), however this

would require that the specimen effective volume or area (see VAGAGE keyword) be input. If the

fracture data are assumed to be all from one failure mode (all volume flaws or all surface flaws),

then ID1 = 1 or 2 can be chosen. If ID1 = 1 or 2, then CARES assumes that the fracture origins

are consistent with the 1])4 input index. If ID1 = 4 or 5, then fracture origins must be supplied with

the fracture data. The 1D1 keyword is primarily for the C4PEST module. This input value must be

set to a positive integer, when used with the C4LIFE module. Note that if module C4PEST is run

prior to module C4LIFE, then ID1 is automatically set to the value of 3 when module C4PEST

creates the CARES.INP input file for module C4LIFE.

11)2 (4X,I1) [C4PEST and C4LIFE] The control index 1])2 is for selection of a fracture criterion.

Note that some caution must be exercised here because the maximum tensile stress criterion (ID2 =

2) is only used with volume flaw analysis (1O4 must be set to 1). All choices available, except the

PIA model, are used with the Batdorf integration routines. Shetty's mixed-mode fracture criterion

(1])2 = 5) is recommended for both surface and volume flaw analysis (see also keyword C).

1])3 (4X,I1) [C4PEST and C4LIFE] The ID3 control index is for selection of a crack geometry.

Note that caution must be exercised here since some of the available choices are applicable either

for surface flaw analysis or for volume flaw analysis only. For instance, the penny-shaped crack is

NASA/TM--2003-106316 63
onlyapplicablefor volumeflaw analysis. The shear sensitive notch crack and the semi-circular

edge crack are only applicable for surface flaw analysis. The Griffith crack is available for both

surface and volume flaw analysis. The penny-shaped crack is recommended for volume flaw

analysis and the semicircular crack is recommended for surface flaw analysis. See figure 2 for a

matching of crack geometry with fracture criteria.

ID4 (4X,I1) [C4PEST and C4LIFE] 1])4 specifies if parameter estimation and subsequent

component reliability analysis is volume or surface flaw based. The value input must be consistent

with the 111)2 and ID3 selections. From the fracture data supplied by the user, the Weibull shape

and scale parameters, along with the normalized Batdorf crack density coefficient consistent with the

111)4input value, are estimated and are subsequently made available for processing with finite

element data. The entry for 1])4 must be consistent with the finite element analysis material card ID

specified in MATID.

IKBAT (4X, I1) [C4PEST and C4LIFE] IKBAT selects the method of calculating the normalized

Batdorf crack density coefficient, kaa" If IKBAT = 0, then the crack density coefficient is set to the

value that is the solution for the normal stress fracture criterion, regardless of the fracture criterion

and crack geometry selected by the user for subsequent component analysis. If IKBAT = I, then

the crack density coefficient is calculated based on the fracture criterion and crack geometry

selected by the 1])2 and 11)3 keywords. IKBAT = 0 gives more conservative reliability predictions

than IKBAT = 1 does. Setting IKBAT = 1 is recommended.

MATID (1X,I7) [C4PEST and C4LIFE] MATID is the material identification number associated

with the statistical material parameter data. The value input should correspond to the material

NASA/TM--2003-106316 64
identification number listed in the CARES/LIFE neutral data base file for a given element.

Therefore, the value for MATID must match the material ID designated in the finite element

analysis. If only parameter estimation is being performed, then MATH) should be an arbitrary

integer value.

PR (F10.4) [C4PEST and C4LIFE] PR is Poisson's ratio. It is assumed to be temperature

independent.

PFCONV (F10.4) [C4LIFE] When a non-zero value is input ( 0.000001 < PFCONV < 0.999999

), then the Weibull scale parameter, cro, as well as the effective volume, V_, or area, Ae, of the finite

element model of the component is computed (a neutral data base file is required and IPROOF =

0). The real number assigned to PFCONV corresponds to some level of probability of failure of

the component. The value input must correspond with the specified loading on the finite element

model. The user is warned that the calculated Weibull scale parameter requires that the effective

volume and area are invariant quantities versus the magnitude of the loading (this fact should be

verified for non standard specimen geometries). Appropriately modifying the input Weibull

modulus (via the PARAM keyword) to a value of Nm/(N-2) will yield the stress normalized

effective volume, V a, and stress normalized effective area, A_f. The computed effective volume

and area can be used as input for VAGAGE (see section Temperature-dependent fracture or

statistical material parameters data; the Data Table form of input) with module C4PEST. The

computed stress normalized effective volume and area can be used as input for VAGTIM (see

section Temperature-dependent fracture or statistical material parameters data; the Data

Table form of input) with module C4PEST. The PFCONV keyword requires that the specimen

model is isothermal or that the material properties are invariant over the range of temperature

NASA/TM--2003-106316 65
loading. Fortheuserthis meansthatmaterialpropertiesfor a rangeof differenttemperatures

shouldnotbeinput(only onesetof parametersfor onetemperatureshouldbeinput). Thiskeyword

is onlyactivewith theC4LIFEmodule.

TITLE (72A1) [C4PEST and C4LIFE] The input associated with the TITLE keyword is

reproduced in the program output files C4PEST.OUT and C4LIFE.OUT for material identification.

$ENDM [C4PEST and C4LIFE] The keyword $ENDM signifies the end of a section of the

Material Control Input. The temperature-dependent specimen fracture data or the Weibull shape and

scale parameters are assumed to immediately follow.

C4LIFE module temperature-dependent fracture or statistical material parameters data:

Immediately following the $ENDM keyword, which signals the end of the material's temperatureindependent data, the temperature-dependent Weibull and fatigue parameters are input. Data must

be arranged so that they correspond to ascending order of temperature. These data enable

interpolation of the statistical material parameters to other temperatures. Figure 7 shows an example

of the input for these parameters module C4LIFE. An explanation of these keywords is provided in

the following text.

PARAM (2E18.10) [C4PEST and C4LIFE] PARAM signals that the Weibull shape parameter, m,

and Weibull scale parameter, Go, will be input for the temperature indicated by the TEMP (or

TDEG) keyword. Referring to figure 7, the Weibull shape and then the Weibull scale parameters

are entered at the indicated space with a format of2E18.10 (denoted by M and SP, respectively in

the figure). The Weibull shape parameter is dimensionless. The Weibull scale parameter has units

NAS A/TM--2003-106316 66
of stressx (volume)_/mvfor volumeflaw analysis and units of stress x (area) I/msfor surface flaw

analysis.

RSTRES (3E18.10,/,318.10) [C4LIFE] If material residual stresses (not loading induced) are

present, the keyword RSTRES allows the user to input a stress tensor that is added to the

mechanically induced stresses in the component (it is not added to the fracture stresses of the

experimental fracture specimens). This option requires a consistent material reference frame

(neutral file stresses should be organized in a material coordinate system reference frame). The

stresses are input in two lines having a 3E18.10 format with the X, Y, Z, components on the first

line and the XY, YZ, and ZX components on the second line, respectively. For shell elements only

the X, Y, and XY components are used. The RSTRES option should be used only when IPRINC

= 2, otherwise the user is limited to input of hydrostatic stresses ( X = Y = Z; 0.0 = XY = YZ =

ZX ) in order to obtain valid results.

TEMP or TDEG (F10.4) [C4PEST and C4LIFE] TEMP (or TDEG) is the input keyword for the

temperature of the fracture data or of the statistical material parameters that immediately follow.

Temperature can be specified in any units but must be consistent with the finite element analysis.

TPARAM (3E18.10) [C4LIFE] TPARAM signals that the power law or Paris law fatigue

parameters N and Bw will be input for the temperature indicated by TEMP (or TDEG). The

Walker law fatigue parameters N, Bw, and Q (where Q is the dimensionless Walker R-ratio

sensitivity exponent) are also input with this keyword. Referring to figure 7, the crack growth

exponent N and then the constant Bw are entered at the indicated space with a format of 2E18.10.

The crack growth exponent N is dimensionless. The crack growth constant Bw has units of

NAS A/TM--2003-106316 67
(secondx (stress)2)forthe power law or (cycle x (stress) 2) for the Paris and Walker laws. The

IWAVE control parameter of the CYCLIC keyword (see the section titled C4LIFE module

Material- and specimen-dependent (temperature independent) data of the Master Control Input)

controls the choice of crack growth law. If IWAVE = I or the CYCLIC keyword is not present in

the input file then the power law is automatically assumed with static fatigue. If IWAVE > 1, then

the power law is used with cyclic loading. If IWAVE = 0 and the input field for Q (36X,E18.10) is

left blank or has a zero entry then the Paris law is implemented. If I'WAVE = 0 and the input field

for Q (36X,E18.10) is a positive nonzero real number then the Walker is law is implemented.

$ENDT [C4PEST and C4LIFE] The keyword $ENDT signals the end of the

temperature-dependent data. Another section of the Material Control Input follows, if required.

NASA/TM--2003-106316 68
Execution of the CARES/LIFE Program

The CARES/LIFE integrated design computer program has two primary objectives: (1)

postprocessing MSC/NASTRAN, ANSYS, or ABAQUS finite element static analysis output to

predict component reliability and (2) analysis of specimen fracture data to determine estimates of

the statistical material and/or fatigue parameters. To facilitate user interaction with the program,

CARES/LIFE is divided into three separately executable modules which perform: (1) statistical

analysis and characterization of experimental data obtained from the fracture of laboratory

specimens; (2) neutral data base generation from the MSC/NASTRAN, ANSYS, or ABAQUS finite

element analysis programs; and (3) time-dependent reliability evaluation of thermomechanically

loaded ceramic components. Although all three modules in the CARES/LIFE program are

separately executed, they may easily be coupled through command files to sequentially execute

without user intervention. The disks which are provided with this manual include the source code

for each of the three modules comprising the CARES/LIFE program. The CARES/LIFE program

modules are programmed in standard FORTRAN 77 and must be compiled by the user before they

can be executed. The CARES/LIFE code requires that input files for all three modules be prepared

prior to exeCution. Specific examples of the files associated with each of the three modules are

presented in the Users' Guide section, along with an explanation of how the variables in the files

are designated and/or manipulated. In addition, the Users' Guide section contains several examples

of templet file keyword settings which illustrate the keywords necessary for performing parameter

estimation and component reliability analysis with the C4PEST and C4LIFE modules.

NASA/TM--2003-106316 69
Output Information

The output from the CARES/LIFE program consists of separate input/output files created

from the various independent program modules--C4LIFE, C4PEST, and the finite element interface

programs ABACARES, ANSCARES, and NASCARES. The ABACARES, ANSCARES, and

NASCARES programs couple to the finite element analysis codes ABACUS, ANSYS, and

MSC/NASTRAN, respectively. These interface programs and the parameter estimation program

C4PEST each produce a data file that input into the C4LIFE program to predict component

reliability. The C4LIFE program outputs two files, one of which is a results file and the other is a

PATRAN compatible data file of the element risk-of-rupture intensities, which enables graphical

rendering of the structures critical regions. The contents of the input/output files from each

CARES/LIFE module are described in the following three sections of this manual.

Finite Element Interface Modules Output Information

The results from the finite element analysis (available in the form of standard output files

and/or plot data files) have to be interpreted by an interface program. Due to the different formats

used for output of the results in each finite element program this interpreter program has to be

adapted to the specific finite element software used; therefore, for each finite element software

package a separate interface program is prepared. The interface program writes the results of the

finite element analysis into a file known as the neutral data base (the default name of this file is

CARES.NEU or also PROOF.NEU).

The concept of a neutral data base allows for the coupling of CARES/LIFE to several finite

element packages. A further advantage of this approach is the ease of transfer of this database

(ASCII file) to different computer systems. Its salient feature is that it has a common format (hence

the word neutral) for all finite element programs. The contents of this file include the stress tensors

NASA/TM--2003-106316 71
andtemperatures at the element Gaussian integration points as well as volumes or areas associated

with these integration points. Only one finite element load case is permitted in the file (it is

expected that future versions of CARES/LIFE will allow multiple load cases); therefore for proof

testing or cyclic fatigue then two separate neutral files are required (CARES.NEU and

PROOF.NEU). For proof testing, one neutral file contains the proof test loading case and the other

neutral file contains the service loading case. For cyclic fatigue each neutral file would represent

one extreme in the range of imposed loading on the component over a cycle. The finite element

interface program creates only one neutral data base file per execution.

Appendix B shows a sample of the organization of the neutral data base. The structure of the

file is optimized with respect to memory. The finite element data is arranged using the following

hierarchy: element groups, elements, and subelements. A subelement refers to the volume or area

associated with a particular element Gaussian integration point. The stress state is assumed constant

within the domain of the subelement. The element group data contain information regarding the

number of elements within the group. Interpreter control parameters, such as the number of element

groups to be considered in the reliability analysis, flags for accounting for volume and surface flaw

analysis, etc., precede the finite element data.

The introduction of comment lines (defined by 'COM' as the first 3 characters of an input

line) allows for a more friendly substructuring of the neutral data base. Based on the global

hierarchy outlined above, the neutral file data is arranged into records with standard FORTRAN

formats: the first record (format A80) specifies the title for the particular reliability evaluation. The

next entry (format 215) contains the number of element groups NUMEL (both volume as well as

surface (shell) elements) to be considered in the reliability analysis and the flag NUMELB

indicating combined volume as well as surface flaw reliability analysis for shell elements.

NASA/TM--2003-106316 72
DependingontheparameterNUMEL,a loopoverall elementgroupsis startedwiththenext

statement(format415):theidentificationnumberof theelementgroupconsidered(INUEG),the

elementype(IEGTYP)correspondingtoPATRANelementtypecodes,thenumberof elements

withinthis elementgroup(IGREL)anda shellelementypeindicator(ISHELL)haveto beinput.

If shellelementsareusedintheanalysis,theindicatorISHELLhasto besetto 1. Information

pertainingto theelementlevelis readin a loopoverthenumberof elements(INVEQ)within the

elementgroupconsidered(format415,3F15.0).

Theelementleveldatacontainstheidentificationnumberof theelementconsidered

(IELNUM),thenumberof subelementswithinthiselement(ISUBEL),thematerialidentification

numberfor volumeflaw analysis(MATINP),thematerialidentificationnumberfor surfaceflaw

analysis(MAT2), theelementvolume(ELVOL),theelementhickness(ELTHIC)only incaseof

shellelements,andtheaveragedtemperatureof thiselement(ELTEMP).Theidentificationnumber

of theelementshouldbethesameasthecorrespondingelementnumberinthefinite elementmesh,

if furtherpostprocessingof thereliabilityresultsareplanned.Thenumberof subelementswithin

thiselementdependsontheGaussianintegrationorderchosenin thefinite elementanalysisandon

theelementype. If stressdataisonly availableatthecenterof theelement(standardcasein

NASTRANanalysespreparedfor previousCARES versions), one subelement is specified

(ISUBEL=I). The material identification numbers must correspond to the material numbers used in

the finite element analysis as well as the material numbers in the control file (containing the Weibull

parameters). Element volume and element temperature are necessary only for element group

information summaries in the printout file. For the reliability evaluation the subelement data are

directly used.

A third loop is started for reading at the subelement level. In case of volume type elements

(ISHELL=0 on Element Group Entry) the identification number of the subelement (ISUNUM), the

NASA/TM--2003-106316 73
subelementvolume(SUBVOL),thesubelementtemperature(SUBTEM)andthefull stresstensorat

theintegrationpoint(definedby axx,6yy,6=, cxy, Cyz and Crxzin the local coordinate system) have to

be specified (format I5,5F 15.0,/,35X,3F 15.0).

In the case of shell type elements (ISHELL=I on Element Group Entry) plane stress

conditions are assumed. For that reason only axx, 6yy and cxy are necessary for the definition of the

subelement stress tensor. Instead of the subelement volume, the subelement area has to be specified

(I5,5F15.0): ISUNUM, SUBAREA, SUBTEM, crxx, cy r and a_r.

Typically, subelement volumes and areas are not included with standard finite element output.

Thus, the volume or area of each subelement (corresponding to a Gauss integration point) is

calculated in the interpreter program using the shape functions inherent to the element type. In the

usual context of finite element methods, the volume of a three-dimensional element, (i. e., brick,

wedge, pyramid, or tetrahedron), the volume of an axisymmetric element (i. e., quadrilateral or

triangular), and the area of a shell element (i. e., quadrilateral or triangle) are calculated at_er

transformation into the natural coordinate space (Bathe, 1982). See section Theory (and also

Powers, Starlinger, and Gyekenyesi, 1992) for information regarding calculation of subelement

volumes and areas.

C4PEST Module Output Information

The C4PEST module reads specimen rupture data from the C4PEST.INP input file

(C4PEST.INP is the default file name) and subsequently estimates the inert strength (fast-fracture)

Weibull parameters as well as the fatigue parameters. This information is output in two separate

files. One output file contains the results of the analysis (default file name C4PEST.OUT), which

includes parameter estimates, outlier testing, goodness-of-fit statistics, etc... The other file is a data

file (default file name CARES.INP), which is input into the C4LIFE module. It contains the

NASA/TM--2003-106316 74
parameterestimatesalongwithprogramcontrolindices.TheCARES.INPfile is anexactcopyof

theC4PEST.INPinputfile, excepthatthespecimenrupturedatais replacedwith theparameter

estimates(viakeywordsPARAM, TPARAM, andTEMP asdescribedin sectionTemperaturedependentfracture or statisticalmaterialparametersdata). ThesectionInput Information

providesa descriptionof thecontentsof thisdatafile.

Theresultsfile C4PEST.OUTcontainsasummaryof theparameterestimationanalysis

performedby theC4PESTmodule.Thebeginningof thisfile containsanechoof thechoices

selected(or defaultvalues)fromtheMasterControlInput.The PRINTA subroutine echoes these

data. The results of the analysis of each flaw population for a corresponding material follow,

starting with the echo of the Material Control Input (from the PRINTB subroutine). The timedependent data and analysis are output first (PRINTQ subroutine) and the inert strength (fastfracture) data and analysis are output second (PRINTP subroutine).

The time-dependent specimen data is output in ascending order of temperature (which is also

the way it must be input in file C4PEST.INP). The specimen dimensions are given at each

temperature for either four-point bending, three-point bending, tensile specimen gage volume, tensile

specimen gage area, specimen effective volume or specimen effective area (effective volume and

area are calculated for some arbitrary specimen geometry as prescribed with the keyword PFCONV

in the Master Control Input). The specimen effective volume or area can change with temperature

if the Weibull modulus changes with temperature (see section Material Strength

Characterization).

A summary of the cyclic fatigue control data (see keyword CYCLIC in section

Temperature-dependent fracture or statistical material parameters data; the Data Table form

of input) is given. This output echoes the user's selections for the crack growth law (power law,

Paris law, or Walker law), the cycle waveform (power law only), the period of the cycle, and the RNASA/TM--2003-106316 75
ratio(theratioof minimumcyclestressto maximumcyclestress).TheR-ratiois assumedinvariant

at all locationsin thespecimen(for theC4LIFEmoduletheR-ratiomayvarythroughouthe

component).If thepowerlaw is selectedthentheg-factoris calculatedfor thewaveformand

printed(seesectionCyclic Fatigue). The g-factor can be viewed as a multiplication factor of time

(for cyclic loading) to obtain an equivalent failure time for a static (invariant) load. CARES/LIFE

calculates the g-factor based on this static stress having a magnitude equal to the maximum cycle

stress. The C4PEST module determines the value of the fatigue parameter Bw using the g-factor

(see section Evaluation of Fatigue Parameters from Naturally Flawed Specimens) and the

maximum cyclic failure stresses in the specimens.

Prior to the printing of the specimen rupture data, if the median deviation solution is used the

bounds of the range to search for a solution for the crack growth exponent N is listed (see

keyword RANGE and SOL in section Temperature-dependent fracture or statistical material

parameters data; the Data Table form of input). The total number of specimens and the testing

temperature is printed, followed by a table echoing the fatigue specimen fracture data. For static

and cyclic fatigue data the table includes the maximum stress and the duration of loading in units of

seconds for the power law and cycles for the Paris and Walker laws. The data are ranked according

to the highest rupture stress followed by the smallest duration for which the load is applied. For

dynamic fatigue the data are ranked according to the lowest stressing rate followed by to the lowest

fracture stress. The ranking according to this hierarchy is indicated by the integer in parenthesis.

The failure origin is indicated with a "V" for volume flaw, an "S" for surface flaw, and a "U" for an

unknown flaw origin.

The time-dependent specimen fracture data are transformed to equivalent inert (fast-fracture)

strengths (see section Evaluation of Fatigue Parameters from Inherently Flawed Specimens),

which are then printed in a table. The transformation depends on the estimated fatigue parameters

NASA/TM--2003- !06316 76
and the Weibull modulus m' and characteristic strength o0 estimated solely from the fatigue

data. This approach depends on the extrapolation of the time dependent rupture data to an

equivalent fast-fracture time and should be used with caution. The characteristic strength o0 is

arbitrarily defined as the value of the specimen failure stress where the probability of failure is

0.6321 and the time to failure is 1.0 second for dynamic loading (constant stress rate loading). For

static loading this is equivalent to a time to failure of I/(N+I) seconds and for cyclic loading its

1/(N+I) cycles. The transformed inert strengths are arranged in ascending order of magnitude. The

integer number in parenthesis cross references to the original specimen data of the previous table.

Using the Stefansky outlier test any transformed inert strength data value that deviates

substantially from the rest of the data is detected as an outlier, and its corresponding significance

level is printed. The lower the significance level, the more extreme is the deviation of the data

point from the rest of the distribution. A 1-percent significance level indicates that there is a

1-in-100 chance that the data point is actually a member of the same population as the other data,

assuming a normal distribution (the skewness of the Weibull distribution can falsely influence

results, causing a data point to have a lower significance level than it should). Detected outliers can

be cross referenced to the original specimen data. The results from the outlier test should only be

considered as a warning to the user and any subsequent action should be carefully considered.

Plotting the data on a Weibull scale helps in confirming the identity of outliers. It is recommended

that outliers not be removed from the data set unless the significance level is below 1-percent and a

physical justification can found for such action. Detected outliers may indicate a competing failure

mode is present.

With the conclusion of the specimen data (specimen geometry, specimen fracture data,

transformed fracture data, and outlier test) versus temperature, a table containing the estimated

NAS A/TM--2003-106316 77
fatigueparametersN and Ac (or Ad) along with statistical parameters m' and o'0 obtained

solely from the fatigue data follow. The fatigue parameter Bw is not included in this particular

table since the calculated value may depend on additional fast-fracture data given later. Listed in

order of increasing temperature, the failure mode of the flaw population is identified, either volume

or surface, the type of experiment is identified, either "STAT" for static fatigue, "CYCL" for cyclic

fatigue, or "DYNA" for dynamic fatigue, and the solution method "SOL" is printed. The fatigue

exponent is under the column marked "N" and the natural logarithm of the fatigue constant A c, or

A d (whichever is appropriate) is under the column marked "LN A". Ac and A d are determined based

on a 50.0 percent probability of failure (they describe the median behavior of the data). The

remaining columns of estimated parameters pertain to the statistical distribution of the fatigue data

(the Weibull parameters estimated directly from the fatigue data). These values depend on the

Weibull parameter estimation method (see keyword MLORLE in section Material- and specimendependent (temperature-independen 0 data) as much as they depend on the particular fatigue

parameter estimation method. The columns marked "M BIASED" and "M UNBIASED" indicate

the transformed inert distribution biased and unbiased Weibull modulus m', respectively.

Unbiasing of the Weibull modulus is not provided if least-squares estimation is used (MLORLE ---

1). For maximum likelihood estimation the unbiasing is based on factors from Thoman, Bain, and

Antle (Thoman, Bain, and Antle, 1969). The column marked "CHAR. STR." indicates the

characteristic strength o0 of the transformed inert data. As previously explained this characteristic

strength is arbitrarily defined as the value of the failure stress where the probability of failure is

0.6321 and the time to failure is 1.0 second for dynamic loading (constant stress rate loading). For

static loading this is equivalent to a time to failure of 1/(N+I) seconds and for cyclic loading its

NAS A/TM--2003-106316 78
1/(N+I) cycles. Thecolumnswiththeheadings"MEAN"and"STD.DEV." aretheWeibullmean

andstandarddeviationof theinertdistributionparametersm' and o0.

Following the table of the parameter estimates is the results from the Kolmogorov-Smirnov

and Anderson-Darling goodness-of-fit testing on the transformed inert strength data and the Weibull

#

parameters m' and o 0. These values of the goodness-of-fit statistics are applied to the original

fatigue data. The Kolmogorov-Smirnov (K-S) goodness-of-fit test is done for each data point, and

the corresponding K-S statistics (D ÷ and D) and significance level are listed. Similarly, the K-S

statistic D for the overall population is printed along with the significance level. This overall statistic

is the absolute maximum of individual specimen data D ÷ and D factors. For the Anderson-Darling

(A-D) goodness-of-fit test, the A-D statistic A 2 is determined for the overall population and its

associated significance level is printed. The lower the significance level, the worse is the fit of the

experimental data to the proposed distribution. For these tests, a 1-percent level of significance is

interpreted as a 1-in-100 chance that the specimen fracture data were generated from the estimated

distribution. A word of caution must be inserted regarding the interpretation of the printed

significance levels. These values are based on the assumption that the fatigue and Weibull

parameters are determined independent of the experimental data. This assumption is not true here

and therefore the printed values for the significance level may need revision. The significance level

for these tests should be viewed as a relative measure of goodness-of-fit and not as an absolute

measure. Further research is required in order to resolve this inconsistency.

If fast-fracture (inert strength) experimental data is input (see keyword FAST in section

Temperature-dependent fracture or statistical material parameters data; the Data Table form

of input), then the results of the Weibull analysis versus temperature is printed (via subroutine

PRINTP). The specimen dimensions are given at each temperature for either four-point bending,

NAS A/TM--2003-106316 79
three-pointbending,tensilespecimengagevolume,tensilespecimengagearea,specimeneffective

volumeor specimeneffectivearea(effectivevolumeandareaarecalculatedfor somearbitrary

specimengeometryasprescribedwith thekeywordPFCONVin theMasterControlInput). The

specimeneffectivevolumeor areacanchangewith temperatureif theWeibullmoduluschanges

with temperature(see section Material Strength Characterization). The inert strengths are

arranged in ascending order of magnitude with proper failure mode identification. An "S" indicates

a surface flaw, a "V" indicates a volume flaw, and a "U" indicates an unknown flaw origin.

Following this table the Stefansky outlier test is used to check for outliers in the inert strength data

(the previous comments regarding the interpretation of the results of the outlier test also apply here).

Any value that deviates substantially from the rest of the data is detected as an outlier, and its

corresponding significance level is printed. Detected outliers may indicate a competing failure mode

is present.

After the fast-fracture specimen data (specimen geometry, specimen fracture data, and outlier

test) a table containing the estimated Weibull statistical parameters follow. The estimation method

is identified, either least-squares or maximum likelihood (see keyword MLORLE in section

Material- and specimen-dependent (temperature-independent) data) prior to beginning the table.

The parameters are listed in order of increasing temperature and the failure mode of the flaw

population is identified, either volume or surface. The columns marked "M BIASED" and "M

UNBIASED" indicate the fast-fracture distribution biased and unbiased Weibull modulus m,

respectively. Unbiasing of the Weibull modulus is not provided if least-squares estimation is used.

For maximum likelihood estimation the unbiasing is based on factors from Thoman, Bain, and Antic

(Thoman, Bain, and Antle, 1969). The columns identified with "UP M" and "LOW M" indicate the

ninety percent confidence bounds of the Weibull modulus. These bounds are only applicable for the

maximum likelihood procedure anct are also based on factors developed by Thoman, Bain, and

NASA/TM--2003-106316 80
Antic (Thoman,Bain,andAntic, 1969).Thecolumnmarked"CHAR.STR."indicatesthe

characteristicstrength o0 of the specimen data. The characteristic strength is the value of the

fracture stress where the probability of failure is 0.6321. The columns identified with "UP C.S."

and "LOW C.S." indicate the ninety percent confidence bounds of the characteristic strength. These

bounds are only applicable for the maximum likelihood procedure and again are based on factors

developed by Thoman, Bain, and Antic (Thoman, Bain, and Antic, 1969). As a rule of thumb the

confidence bounds are significantly larger (as a fraction of the estimated parameter) for the Weibull

modulus than for the characteristic strength. The columns with the headings "MEAN" and "STD.

DEV." are the Weibull mean and standard deviation of the fast-fracture distribution parameters m

and a e.

The Kolmogorov-Smirnov (K-S) goodness-of-fit test is done for each fast-fracture data point,

and the corresponding K-S statistics (D ÷ and D) and significance level are listed. Similarly, the K-S

statistic D for the overall population is printed along with the significance level. For the

Anderson-Darling (A-D) goodness-of-fit test, the A-D statistic A 2 is determined for the overall

population and its associated significance level is printed. The previous comments regarding the

interpretation of results of these tests also apply here.

The next table that is generated by C4PEST from the PRINTP subroutine contains data to

construct Kanofsky-Srinivasan 90-percent confidence bands about the Weibull distribution. The table

includes fast-fracture rupture stress data, the corresponding Weibull probability of failure values, the

90-percent upper and lower confidence band values about the Weibull line, and the median rank

value for each data point. These statistical quantities are calculated with either tabular values or

approximating polynomial functions. Experimental fracture data lying outside of these bands are an

indication of poor fit to the Weibuil distribution.

NASA/TM--2003- !06316 81
Thelasttablegeneratedforthematerial(from thePRINTPsubroutine)liststhematerial

parametersthatareusedin thecomponentreliabilitycalculationsasafunctionof temperature.

Thesevaluesarepassedto moduleC4LIFEviatheCARES.INPfile. Thetableliststhebiased

Weibullmodulusm,theWeibull scaleparameterOo,thefatigueexponentN, thefatigueconstant

Bw,andthecrackgrowthlaw. ThefatigueconstantBwisalwaysprintedregardlessof thefracture

criterionchosen(Bwis alsoprintedin theCARES.INPdatafile for moduleC4LIFE). TheC4LIFE

programmodulewill automaticallycalculateBafrom Bwforthechosenfracturecriteriasothatthe

useris notrequiredto repeatedlyruntheC4PESTmoduleeverytimeacomponentreliability

analysisis performedusinga differentfracturecriteria(thisstatementis only truewhenthe

specimenis notmultiaxiallyloaded).Thevaluesgiveninthetablecorrespondto theexperimental

temperatures input (and optionally to five additional interpolated sets of values between each input

temperature if the INTER keyword is set to 1 in the Master Control Input). If fatigue specimen

rupture data is available but not fast-fracture specimen rupture data, then the table lists

e t

m' , Oo , N , and B w (in this ease B, is calculated using m and O'o). If fatigue and fast-fracture

specimen rupture data is available (or the Weibull parameters are directly input using keyword

PARAM in section Temperature-dependent fracture or statistical material parameters data;

the Data Table form of input), then the table lists m , Oo, N , and Bw (in this case B_ is

calculated using m and t_o). The headings For this table are as follows; the Weibull modulus m is

denoted by the column headed by "M", the scale parameter Crois headed by "SP", the fatigue

exponent N is headed by "N" and the fatigue constant B w is headed by "B".

NASA/TM--2003-106316 82
C4LIFE Module Output Information

The C4LIFE module reads control indices, Weibull, and fatigue parameters from the

CARES.INP file; the finite element information is read from the neutral file data base (or from two

neutral files when proof testing or cyclic loading are considered). This information is used to predict

component reliability. The results from this analysis is output in a file with the default name of

CARES.OUT. A separate data file is also generated with the default name of CARES.PAT,

containing the element risk-of-rupture intensities. This data file is PATRAN compatible so that

graphical rendering of the components critical regions is possible.

The reliability analysis results file CARES.OUT begins with an echo of the choices selected

(or default values) from the Master Control Input. The PRINTA subroutine echoes this data. For

each flaw population of a material the Material Control Input is echoed for the temperature

independent information (from the PRINTB subroutine) and then its printed for the temperature

dependent material parameters (from the PRINTP subroutine). The crack growth law is identified

followed by the material parameters which include the Weibull modulus m, the normalized Batdorf

crack density coefficient kaa' the Weibull scale parameter ao, and if the loading duration is nonzero (TIME > 0.0), then the fatigue exponent N and the fatigue constant BB or Bw (whichever is

appropriate). The Weibull modulus m is denoted by the column headed by "M", the normalized

Batdorf crack density coefficient _ column is headed by "K", the scale parameter _o is headed by

"SP", the fatigue exponent N is headed by "N" and the fatigue constant BB or B w is headed by "B".

Information on the selected fracture criterion and crack shape is printed for shear-sensitive fracture

models. Crack shape is not required for the NSA (Batdorf shear-insensitive model) or for the PIA

model, and it need not be identified for those cases.

NASA/TM--2003-106316 83
Informationregardingcyclic loadingis echoedin thetemperatureindependentsectionof the

MasterControlInput(seekeywordCYCLIC in sectionMaterial- and specimendependent

(temperature-independent)data). Thisdataincludesthechoicesfor thecrackgrowthlaw (power

law,Parislaw,or Walkerlaw) andcyclewaveform.Theperiodof thecycle(in seconds)andthe

R-ratio(ratioof minimumcyclestressto maximumcyclestress)is printed. TheR-ratioisassumed

constantthroughouthecomponentif only oneneutralfile is beingprocessed(IPROOF = 0). If

two neutralfilesarebeingprocessed(IPROOF= 1),thentheR-ratioiscomputedat every

subelementandfor all crackorientations.In thiscasethentheprintedvaluefor theR-ratioin the

MasterControlInputis ignored.

Followingtheechoof thecontentsof theCARES.INPinputfile (theMasterControlInput

andMaterialControl Input) is the listing of the finite element analysis data read from the neutral

file. This data comes from the primary neutral file (default file name CARES.NEU). Any

J

information regarding the (optional) second neutral file PROOF.NEU is not printed. Note two

neutral files are used by C4LIFE only for the case of proof testing or cyclic loading (IPROOF = 1).

The output from the neutral file begins with an echo of the problem title that was input by the

user. Next, tables are output showing each element number, averaged element temperature, and the

(undeformed) element volume or area. This information is arranged according to element group.

The total number of elements in the group and the group number is printed at the beginning of the

table and the total element group volume or area is printed at the end of each table. Element groups

are arranged according to element type and not to a particular material designation, therefore

elements may be listed that are not used in the reliability analysis. If the element group pertains to

solid type elements (brick, wedge, or tetrahedron) or two-dimensional axisymmetric elements

(quadrilateral and triangular), then the element volumes are listed. If the element group pertains to

shell elements (quadrilateral, triangular) then the element area is printed. Note that volume flaw

NASA/TM--2003-106316 84
analysismaybeperformedwithshellelements(assumingnothrough the thickness stress gradients

exist), however element thicknesses or volumes are not output for this situation.

After the element group tables of temperatures, volumes and areas, the number of array

elements used to store information in the blank common block A and the total size of this common

block is listed. Blank common block A is the main storage array for material data and finite

element data. This information is efficiently placed in this array through the use of integer

addresses called pointers. The highest pointer number, which is the value listed in CARES.OUT,

indicates the index of the array element where the last item of data is stored. The difference

between this number and the limit (MTOT) or total size of the array indicates the amount of unused

memory. The size of this array is controlled by the parameter MTOT, which is declared at the

beginning of the C4LIFE source code. Increasing or decreasing the value of this parameter

increases or decreases the amount of memory required to run this module. This is a user friendly

feature added to enable tailoring the code to the available machine memory or to the size of the

finite element problem. If the amount of memory allocated is insufficient to run the problem, then

an error message is printed along with this table.

Since in a large finite element mesh the stress output could be excessive, printed element

stress tables in CARES are optional (see keyword IPRI_T in section Master Control Input). The

stress analysis results are printed for each group of elements (from subroutine PRINTC). The

element group number is identified along with the integer identification of the element type. As

stated previously, the element stresses that are printed are from the primary neutral file and elements

may be listed that are not used in the reliability analysis. This information can consist of either

centroidal or subelement level stresses for each element. The subelement stress is the stress at the

Gaussian integration point and therefore the number of subelements per element depends on the

integration order of the element. The centroidal stress is the stress at the center of the element and

NASA/TM--2003-106316 85
is the only data given for the element (considered the equivalent of one subelement per element).

Using centroidal stresses will dramatically reduce execution time of the C4LIFE module, however

the penalty is reduced accuracy in the reliability analysis when element stress gradients are present.

The use of subelements to perform the reliability analysis is recommended. A compromise

approach is to use centroidal stresses for preliminary design and subelement stresses for the final

design. For each line of output the element number, and subelement number are identified along

with the stresses and subelement temperature. The output stresses are either principal stresses or the

tensor stresses read from the neutral file. Principal stresses are calculated when keyword IPRINC =

1 and tensor stresses are used when IPRINC = 2. Typical practice with isotropic monolithic

materials is to use principal stresses, however for certain situations such as proof testing with offaxis loading or cyclic loading where the boundary loading conditions vary with time, then a global

reference frame is required. For solid elements the three principal or the six tensor stress

components are printed. For shell elements only two principal stresses are printed or the three

tensor stress components are printed. For principal stresses these two components appear in the first

and second principal stress columns, the third principal stress column has a zero value listed. For

the tensorial stress components only the X, Y, and XY values are used, the other components YZ,

and ZX have zero values listed.

The results of the reliability analysis follows the output of the element and subelement

stresses. First the information is presented in tabular form for each flaw population of each material

(from subroutine PRINTC) and then given for the whole component (from subroutine PRINTO).

The component level information may include a summary of the reliabilities of all the elements

depending on if more than one volume flaw population or more than one surface flaw population is

specified. Any given finite element may have multiple flaw populations defined, and therefore

several separate tables (representing the different flaw populations) are presented containing

NASA/TM--2003-106316 86
reliabilityinformationregardingthat same element. Thus for a particular material number

(MATID) that references a finite element, there may be several concurrent flaw populations

specified and the total element reliability is the product of the reliabilities of the individual flaw

populations for that element. Reliability results are always presented at the element level and not at

the subelement level. The element reliability is the product of the subelement reliabilities. The

printout of element reliabilities can be suppressed via the IPRINT keyword in the Master Control

Input.

The reliability analysis for each flaw population of each material is presented. The order in

which this is given is identical to the order presented in the input file CARES.INP; therefore

material results are listed without regard to the value of the identification number or whether the

analysis is for volume flaws or surface flaws. Tables are provided for each element group and are

arranged in ascending order of element number. Only elements with the same material ID as

specified with the MATID keyword are listed. Information given for each element is the survival

probability, the failure probability, the element averaged (if subelements are used) risk-of-rupture

intensity, and the average element temperature material parameters. Using subelements means that

the reliability analysis is done with the material parameters interpolated to the subelement

temperature (although this information is never output). The element averaged material parameters

include the Weibull modulus m, the normalized Batdorf crack density coefficient k-Ba'the Weibull

scale parameter Oo, and if the loading duration is non-zero, then the fatigue parameters N and B B.

The Weibull modulus is denoted by the column headed by "M", the normalized Batdorf crack

density coefficient column is headed by "K", the scale parameter is headed by "SP", the fatigue

parameters are headed by "N" and by "B". Following each element group table is a summary of

elements with the fifteen maximum risk-of-rupture intensity values and element ID's (the number of

NASA/TM_2003-106316 87
valuesprintedcanbechangedby adjustingparameterNPLVdefinedatthebeginningof themain

program).

Finallythesurvivalprobabilityof theflawpopulationof thematerial,whichis theproductof

theelementreliabilities,andthefailureprobabilityisprintedalongwith the maximum effective

stress found (the effective stress, _c, and not the effective stress transformed to a time value of zero,

a_cq.o). If the PFCONV keyword is activated (PFCONV = 1 and material parameters are given for

only one temperature in the temperature dependent Material Control Input section of file

CARES.INP), then the effective volume, effective area and scale parameter is determined for the

material (for the flaw population). These calculations are based on the maximum principal stress

found in the component and not on the maximum effective stress. For the non-coplanar crack

extension fracture criteria the maximum effective stress can be larger than the highest principal

stress. In order for these numbers to be valid the keywords TIME and TIMEPT must be set to

zero. It is recommended that PFCONV option only be used for typical specimen geometries such

as beams under flexure, C-rings, O-rings, tensile specimens, biaxially loaded specimens like ring-onring or pressure loaded, etc. For these geometries the effective volume and area are invariant versus

applied loading. For other geometries this may not be true and thus separate analyses with at least

two different magnitude of loading must be attempted with the C4LIFE module to verify that the

effective volume and area are indeed invariant. For the calculation of the scale parameter of the

material to be correct the value input for PFCONV must correspond to the failure probability

associated with the imposed level of loading. The effective volume and area require that the

Weibull modulus m be input. To obtain the stress normalized effective volume Vcf and area Aef

then a modified Weibull modulus of magnitude Nm/(N-2) is substituted for the Weibull modulus

in the temperature dependent Master Control Input (in this case the calculated scale parameter has

no meaning and the value input for.the keyword PFCONV can be some arbitrary value).

NAS A/TM--2003-106316 88
If morethan one volume flaw population or more than one surface flaw population is defined

in the CARES.INP file (that is, if either keyword NMATS or NMATV is larger than one) then

additional tables are generated at the element group level listing the element reliabilities, failure

probabilities and risk-of-rupture intensities. All elements are listed for which reliability analysis was

performed. If an element has more than one statistical flaw population defined for it, then the listed

numbers represent the product of the survival probabilities for each of the flaw populations. The

values shown for the risk-of-rupture intensities are the same values that are output to the PATRAN

compatible data file CARES.PAT. Following the element group tables is a summary of elements

with the fifteen maximum risk-of-rupture intensity values and element ID's (the number of values

printed can be changed by adjusting parameter NPLV defined at the beginning of the main

program). Also included is the probability of failure and survival over the component surface

and/or volume, whichever is appropriate. Finally, the overall component probability of failure, as

well as the component probability of survival, are printed.

NASA/TM--2003-106316 89
Theory

Fast-Fracture Reliability Analysis

Introduction

The use of advanced ceramic materials in structural applications requiring high component

integrity has led to the development of a time-dependent probabilistic design methodology. This

method combines three major elements: (1) linear elastic fracture mechanics theory that relates the

strength of ceramics to the size, shape, orientation, and growth of critical flaws; (2) extreme value

statistics to obtain the characteristic flaw size distribution function, which is a material property; and

(3) material microstructure. Inherent to this design procedure is that the requirement of total safety

must be relaxed and that an acceptable failure probability must be specified.

The statistical nature of fracture in engineering materials can be viewed from two distinct

models (Tracy, 1982). The first was presented by Weibull and used the weakest-link theory as

originally proposed by Pierce (Pierce, 1926). The second model was also analyzed by Pierce

(Pierce, 1926) and, in addition, by Daniels (Danieis, 1945). This second model is referred to as the

"'bundle" or "parallel" model. In the bundle model, a structure is viewed as a bundle of parallel

fibers. Each fiber can support a load less than its breaking strength indefinitely but will break

immediately under any load equal to or greater than its breaking strength. When a fiber fractures, a

redistribution of load occurs and the structure may survive. Failure occurs when the remaining

fibers can no longer support the increased load. The weakest-link model assumes that the structure

is analogous to a chain with n links. Each link may have a different limiting strength. When a

load is applied to the structure such that the weakest link fails, then the structure fails. Observations

show that advanced monolithic ceramics closely follow the weakest-link theory (WLT). A

component fails when an equivalent stress at a flaw reaches a critical value which depends on the

NASA/TM--2003-106316 91
fracturemechanicscriterion,crackconfiguration,crackorientation,andthecrackdensityfunctionof

thematerial.In comparisonwiththebundlemodel,WLT is, in mostcases,moreconservative.

Weibull'sWLT modeldoesnotconsiderfailurecausedby purelycompressivestressstates.

Phenomenological observationsindicatethatcompressivestressesdonot playa majorrole inthe

failureof ceramicstructuressincethecompressivestrengthof brittlematerialsis significantly

greaterthantheirtensilestrength.Theeffectof a predominantcompressiononfailureisassumed

negligiblein theCARES/LIFEprogram.Thisis doneby comparingcompressiveandtensile

principalstressesin eachfinite element.Whena principalcompressivestressexceedsthreetimes

the maximum principal tensile stress in a given element, the compressive stress state predominates,

and the corresponding element reliability is set equal to unity.

One of the important features of WLT is that it predicts a size effect. The number and

severity of flaws present in a structure depends on the material volume and surface area. The

largest flaw in a big specimen is expected to be more severe than the largest flaw in a smaller

specimen. Another consequence of WLT is that component failure may not be initiated at the point

of highest nominal stress (Davies, 1973), as would be true for ductile materials. A large flaw may

be located in a region far removed from the most highly stressed zone. Therefore, the complete

stress solution of the component must be considered.

Classical WLT does not predict behavior in a multiaxial stress state. A number of concepts

such as the PIA, Weibull's normal stress averaging method, and Batdorf's model have been applied

to account for polyaxial stress state response. Batdorf's model (Batdorf, and Crose, 1974) assumes

the following: (1) microcracks in the material are the cause of fracture, (2) cracks do not interact,

(3) each crack has a critical stress _rcr which is defined as the stress normal to the crack plane

which will cause fracture, and (4) fracture occurs under combined stresses when an effective stress

tr_ acting on the crack is equal to _,. For an assumed crack shape, _ can be obtained through

NAS A/TM--2003-106316 92
theapplicationof a fracturecriterion. Theseconceptsareusedinconjunctionwith techniquesto

obtainvariousstatisticalmaterialparametersnecessaryforfast-fracturereliabilityanalysis.

Volume Flaw Reliability Analysis

Consider a stressed component containing many flaws, and assume that failure is due to any

number of independent and mutually exclusive mechanisms (links). Each link involves an

infinitesimal probability of failure APfv. Discretize the component into n incremental links. The

probability of survival Psv of the ithlink is

(P,v)_ = [1 - (AVrO]_ (1)

where the subscript V denotes volume-dependent terms. The resultant probability of survival of

the whole structure is the product of the individual probabilities of survival

n

P,v = rl(P,v)_ = l_I[I - (aPrv) ]_

i=l i=l

= exp [- (APrv)i ] = exp -

i=l t=l

(APfv),

(2)

Assume the existence of a function Nv(a), referred to as the crack density function,

representing the number of flaws per unit volume having a strength equal to or less than a. Under a

uniform tensile stress, _, the probability of failure of the ith link, representing the incremental

volume AV i, is

(APfv)_ = [Nv(t_ ) AV L (3)

and substituting into equation (2), the resultant probability of survival is

NASA/TM--2003-106316 93
Psv= exp[-Nv(a) V ]

(4)

and the probability of failure is

Pfv = 1 - exp[-Nv(O ) V ]

(5)

where V is the total volume. If the stress is a function of location then

Pfv = 1-exp I-Iv Nv(a)dV ] = 1- exp(-Bv)

A term called the risk of rupture by Weibull and denoted by the symbol Bv is commonly

used in reliability analysis. Equations similar to (5) and (6) are applicable to surface-distributed

flaws where surface area replaces volume and the flaw density function is surface area dependent.

Weibull introduced a three-parameter power function for the crack density function Nv(o),

Nv(a ) = - aov

(6)

(7)

where Ouv is the threshold stress parameter, which is usually taken as zero for ceramics. This

parameter is the value of applied stress below which the failure probability is zero. When this

parameter is zero, the two-parameter Weibull model is obtained. The scale parameter OoV then

corresponds to the stress level where 63.2 percent of specimens with unit volumes would fracture.

The scale parameter has dimensions of stress x (volume) _/mv,where m v is the shape parameter

(Weibull modulus), a dimensionless parameter that measures the degree of strength variability. As

m v increases, the dispersion is reduced. For large values of m v (m v > 40), such as those obtained

for ductile metals, the magnitude of the scale parameter corresponds to the material ultimate

NAS A/TM--2003-106316 94
strength.These three statistical parameters are material properties, and they are temperature and

processing dependent.

Three-parameter behavior is rarely observed in as-processed monolithic ceramics, and

statistical estimation of the three material parameters is very involved. The CARES/LIFE program

uses the two-parameter model. The subsequent reliability predictions are more conservative than for

the three-parameter model since we have taken the minimum strength of the material as zero.

The two-parameter crack density function is expressed as

Nv(c ) = c _- kwvomV (8)

and when equation (8) is substituted into equation (6), the failure probability becomes

Pfv = 1- exp(-k.vIvomvdV )

(9)

where k,,v = (OoV) "mV is the uniaxial Weibull crack density coefficient. Various methods have been

developed to calculate OoV and m v for a given material by using fracture strength data from

simple uniaxial specimen tests (Pai, and Gyekenyesi, 1988).

The two most common techniques for using uniaxial data to calculate Pfv in polyaxial stress

states are the PIA method (Barnett, et al, 1967) (Freudenthal, 1968) and the Weibull normal tensile

stress averaging method (Weibull, 1939, two references). In the PIA model, the principal stresses

o, > o 2 > o3 are assumed to act independently. If all principal stresses are tensile, the probability of

failure according to this approach is

Pfv = 1 - exp [-kwv Iv(O_*" + 02

°"

+ o_n"'dVl (10)

NASA/TM--2003-106316 95
Compressiveprincipalstressesareassumednotto contributeto thefailureprobability.

shownthatthisequationyieldsnonconservativeestimatesof Pfv in comparisonwith theWeibull

normalstressmethod(Batdorf,1977).

ThefailureprobabilityusingtheWeibullnormaltensilestressaveragingmethod,whichhas

beendescribedthroughanintegralformulation(Gross,andGyekenyesi,1989),canbecalculated

from

exp(Ik )

It has been

(11)

where

_mv - fA°7 vdA

On fAdA

The area integration is performed in principal stress space over the surface A of a sphere of

unit radius for regions where o,, the projected normal stress on the surface, is tensile. The

polyaxial Weibull crack density coefficient is k,,_v. The relationship between k_v and !%,v is

found by equating the failure probability for uniaxial loading to that obtained for the polyaxial stress

state when the latter is reduced to a uniaxial condition. The result is

k a,v = (2m v + 1) kwv (12)

Batdorf and Crose (Batdorf, and Crose, 1974) proposed a statistical theory in which attention

is focused on cracks and their failure under stress. Flaws are taken to be uniformly distributed and

randomly oriented in the material bulk. Fracture is assumed to depend only on the tensile stress

acting normal to the crack plane; hence, shear insensitivity is inherent to the model. Subsequently,

NASA/TM--2003- !06316 96
Batdorf and Heinisch (Batdorf, and Heinisch, 1978) included the detrimental effects of shear traction

on a flaw plane. Their method applies fracture mechanics concepts by combining a crack geometry

and a mixed-mode fracture criterion to describe the condition for crack growth. Adopting this approach, the CARES/LIFE program contains several fracture criteria and flaw shapes for volume and

surface analyses (fig. 2).

Consider a small, uniformly stressed material element of volume AV. The incremental

probability of failure under the applied state of stress E can be written as the product of two

probabilities,

APrv (_ ,o ,AV) = APivP2v (13)

where APiv is the probability of the existence in AV of a crack having a critical stress between

oct and oct + Aoc,. As previously noted, critical stress is defined as the remote, uniaxial fracture

strength of a given crack in mode I loading. The second probability, P2v, denotes the probability

that a crack of critical stress o,, will be oriented in a direction such that an effective stress crc

(which is a function of fracture criterion, stress state, and crack configuration) satisfies the condition

o r > OCr. The effective stress is defined as the equivalent mode I stress a flaw would experience

when subjected to a multiaxial stress state that results in modes I, II, and III crack surface

displacements.

The strength of a component containing a flaw population is related to the critical flaw size,

which is implicitly used in statistical fracture theories. Batdorf and Crose (Batdorf, and Crose,

1974) describe AP_v as

APw = AV dNv(Cr") do (14)

do

NASA/TM--2003-106316 97
and P2v is expressed as

P2v =

( E,oc)

47_

(15)

where Nv(o,) is the Batdorf crack density function and f_(Z,oCr) is the area of the solid angle

projected onto the unit radius sphere in principal stress space containing all the crack orientations

for which o c > o¢r. The constant 4n is the surface area of a unit radius sphere and corresponds to

a solid angle containing all possible flaw orientations.

The probability of survival in a volume element AV i is

Y2 (_,cy) dNv(a )

(P,v)i = exp -AV 4rt _ do,

i

(16)

where

given stress state. Hence, the component failure probability is

Pfv=l-exp-Iv [°i-f_ (_'°c')4n dNv(°_)d_']do_, dV}

oc,,_ is the maximum effective stress a randomly oriented flaw could experience from the

(17)

The Batdorf crack density function

usually approximated by a power function (Batdorf, and Heinisch, 1978).

crack density function of the form

Nv(o) = ksvO_ V

Nv(O',) is a material property, independent of stress state, and is

This leads to the Batdorf

(18)

where the material Batdorf crack density coefficient kav and the Weibull modulus m v are

evaluated from experimental inert strength fracture data. Batdorf and Crose (Batdorf, and Crose,

1974) initially proposed a Taylor series expansion for Nv(o,), but this method has computational

NASA/TM--2003-106316 98
difficulties. A more convenient integral equation approach was formulated and extended to the use

of data from four-point MOR bar tests (Rufin, Samos, and Bollard, 1984). Note that Nv(ocr ) has

units of inverse volume.

Although the Weibull (eq. (8)) and Batdorf (eq. (18)) crack density functions are similar in

form, they are not the same. The Weibull function simply depends on the applied stress

distribution, o, and is the only term other than the volume necessary to calculate Prv. The Batdorf

function depends on the mode I strength of the crack, or,, which is probabilistic and must be

integrated over a range of values for a given stress state. Furthermore, to obtain Pfv, a crack

orientation function, P2v, must be considered in addition to the density function and the volume.

Finally, the Batdorf coefficient kBv cannot be calculated from inert strength data until a fracture

criterion and crack shape are chosen-in contrast to the Weibull coefficient l%v, which depends only

on the data.

To determine a component probability of failure, P2v (eq. (15)) has to be evaluated for each

elemental volume AV i, within which a uniform stress state _ is assumed. The solid angle D(_,o,)

depends on the selected fracture criterion, the crack configuration, and the applied stress state. For

multiaxial stress states, with few exceptions, f2(Y_,o,) must be determined numerically. For a sphere

of unit radius (fig. 8), an elemental surface area of the sphere is dA = sin a d[3 dot. Project onto

the spherical surface the equivalent (effective) stress oc(Z,ot,[3 ). The solid angle f)(Z,o¢,) is the

area of the sphere containing all of the projected equivalent stresses satisfying oc >_o,. Noting the

symmetry of o_ in principal stress space, and addressing the first octant of the unit sphere, then

(19)

where

NASA/TM--2003-106316 99
H(oo, o=) = I o >_o=

H(%,o=) = 0 a=< a¢r

Substituting into equation (17) and integrating with respect to o., the component failure probability

becomes (Batdorf, Fundamentals of the Statistical Theory of Fracture, 1980)

(20)

where

Nv(oJ =

m V

o_ (x,y,z,a,13)

For a given element, o,(x,y,z, ct,13) is the projected equivalent stress over the unit radius sphere in

principal stress space as shown in figure 8.

Equation (20) circumvents the involved numerical integration of f_(E,ac, ) as developed in the

original CARES program (Nemeth, Manderscheid, and Gyekenyesi, 1990). Equations (17) and (20)

are equivalent formulations; however, equation (20) is more convenient for computational purposes

with few exceptions (Batdorf and Crose, 1974). Therefore, CARES/LIFE applies equation (20) to

obtain the component probability of failure.

Assuming a shear-insensitive condition, fracture occurs when o. = oe > ¢_,, where a. is the

normal tensile stress on the flaw plane. However, it is known from fracture mechanics analysis that

for a flat crack, a shear stress z applied parallel to the crack plane (mode II or III) also contributes

to fracture. Therefore, the effective stress oe is a function of both a. and x.

Selecting an arbitrary plane in principal stress space (fig. 8) and imposing equilibrium

conditions yields the following equations:

1_2 = ((_le)2 4- (o2m) 2 + (o3n): (21)

NAS A/TM--2003-106316 100
o = o,g = + _2m2 + %n 2 (22)

and

1:2 = o 2 - c2 (23)

where o is the total traction vector acting on the crack plane and the direction cosines e, m, and n

are given in figure 8 in terms of trigonometric functions of ct and 13.

From the selected fracture criterion and crack configuration, % is obtained as a function of

_, tz, and 13. Batdorf and Heinisch (Batdorf, and Heinisch, 1978) give effective stress expressions

for two flaw shapes by using both Griffith's maximum tensile stress criterion and Griffith's total

eoplanar strain energy release rate criterion G v Arranged in order of increasing shear sensitivity,

for the maximum tensile stress criterion the effective stress equations are

(Ye = -_

, (o+ (24)

for a Griffith flaw and

(1 - 0.Sv)

(25)

for a penny-shaped flaw, where v is Poisson's ratio.

The total coplanar strain energy release rate criterion is calculated from

G T = G I + Gll + G m (26)

NASA/TM--2003-106316 101
where G is the energy release rate for various crack extension modes. In terms of stress intensity

factors, the effective stress equation can be derived from (plane strain condition assumed) enforcing

the condition G r = Gc, where G c is the critical strain energy release rate. Thus,

K,c = Kt z + K,_ + K,,_ (27)

I -v

For a Griffith crack, assuming that modes I and II dominate the response with

and Kti=l: _ , where 2a is the crack length, we have from equation (27)

K I=onx/t_

(3"e = _/O"zn + x2 (28)

For a penny-shaped crack at the critical point on the crack periphery, we have

K_=(4_/(2-v))

stress from equation (28) is

KI =2on _('_'_ and

(Sih, 1973), where a is now the crack radius. The resulting effective

2

= (3"n +

(29)

The equations given by Batdorf and Heinisch consider only self-similar (coplanar) crack

extension. However, a flaw experiencing a multiaxial stress state usually undergoes crack

propagation initiated at some angle to the flaw plane (noncoplanar crack growth). Shetty (Shetty,

1987) performed experiments on polycrystalline ceramics and glass where he investigated crack

propagation as a function of an applied far-field multiaxiai stress state. He modified an equation

proposed by Palaniswamy and Knauss (Palaniswamy, and Knauss, 1978) so that it would

empirically fit experimental data. This multimodal interaction equation takes the form

NASA/TM--2003-106316 102
Kl K_

Klc

where K 8 is either Ka or Kill, whichever is dominant, and _2 is a constant adjusted to best fit

the data. Shetty (Shetty, 1987) found a range of values of 0.80 < C < 2.0 for the materials he tested

which contained large induced flaws. As F2increases, the response becomes progressively more

shear insensitive.

Using this relationship with assumed modes I and II dominance for the Griffith crack yields

(30)

(31)

and for a penny-shaped crack, we obtain

o =_. o + o. + C(2 -v)

For a Griffith crack when (2 =.80, .85, 1.0 and 1.15 equation (30) models, respectively, the

following criteria: Maximum strain energy release rate (Ichikawa, 1991), maximum tangential

stress (Erdogan, and Sih, 1963), maximum strain energy release rate (Hellen, and Blackburn, 1975)

and co-linear crack extension.

Similarly, for a penny shaped crack with a material having a Poisson's ratio of about .22 and

C =.80, .85, 1.05 and 1.10, equation (30) models, respectively, the following criteria: Maximum

strain energy release rate (Ichikawa, 1991), maximum tangential stress, maximum strain energy

release rate (Hellen, and Blackburn, 1975), and co-linear crack extension.

(32)

NAS A/TM--2003-106316 ! 03
For a stressed component , the probability of failure is calculated from equation (20). The

finite element method enables discretization of the component into incremental volume elements.

CARES/LIFE evaluates the failure probability at the Gaussian integration points of the element or

optionally at the element centroid. Using the element integration points subdivides the element into

subelements, hence each V i corresponds to the it subelement. In the usual context of finite element

methods, the volume of a three-dimensional element is calculated after transformation into the

natural coordinate space (Bathe, 1982)

v=f? f: det J(r,s,t) dr ds dt (33) 1 1 1

where J is the Jacobian operator and r,s,t are the natural coordinates. The subelement volume is

defined as the contribution of the integration point to the element volume in the course of the

numerical integration procedure. This means that the volume of each subelement (corresponding to

a Gauss integration point) is calculated using the shape functions inherent to the element type. The

stress state in each subelement is assumed uniform. Powers et al. (Powers, Starlinger, and

Gyekenyesi, 1992) gives further details of the subelementing procedure as used in CARES/LIFE.

The numerical solution of equation (20) takes the following form:

o_ (e,_) dA

iffil i

(34)

where n is the total number of subelements. If kBv is element dependent, it would appear inside the

brackets.

NASA/TM--2003-106316 104
Surface Flaw Reliability Analysis

For surface flaw analysis (Gyekenyesi, and Nemeth, 1987), many of the equations from the

Volume Flaw Reliability Analysis section remain the same, except that the statistical material

parameters are a function of surface area instead of volume and the equivalent stresses are projected

onto the contour of a circle of unit radius rather than onto the surface of a unit radius sphere. The

cracks are assumed to be randomly oriented in the plane of the external boundary with their planes

normal to the surface (Batdorf, and Heinisch, 1978).

For surface-flaw-induced failure in ceramic structures, the probability of failure for the

two-parameter Weibull distribution, which is analogous in form to equation (9), is

Prs = 1- exp (-kws fA(_ m' dA) (35)

where k,¢s = (1/OoS)ms is the uniaxial Weibull surface crack density coefficient. The subscript S

denotes the terms that are surface area dependent. Here aoS is the surface scale parameter with

units of stress x (area)l/ms, and A is the stressed surface area. For biaxial stress states, the PIA

model yields

Pts = 1-exp [ -k_s IA (O_' + O2') dA]

(36)

where

For the Weibull normal stress averaging method, the failure probability is expressed as

Pfs = 1 - exp (-kwps iA -o. a. dA )

o_ and a 2 are the principal tensile in-plane stresses acting on the surface of the structure.

(37)

where

NASA/TM--2003-106316 !05
/. m S

dc

-- ms Jc On

O" n --

Here k_s is the polyaxial Weibull crack density coefficient for surface flaws. The line integration

is performed over the contour c of a unit radius circle where the projected normal stress o n is

tensile. The relationship of k_s to k,¢s is obtained by carrying out the integration in equation

(37) for a uniaxial stress and equating the resultant failure probability to that of equation (35) (Pai,

and Gyekenyesi, 1988). This results in

ms r(m s ) vff

ks = ks 1

(38)

where F is the gamma function. Equation (37) is the shear-insensitive case of the more general

Batdorf polyaxial model.

For mixed-mode fracture due to surface flaws, the Batdorf polyaxial failure probability

equation (analogous to eq. (17)) is

Pes = 1 - exp IA tiNs(oct,doer ocrdA (39)

where analogous to equations (14) and (15)

APzs = AA dNs(°¢')do¢_

d_¢r

(40)

and

NASA/TM--2003-106316 106
co (F.,a) (41)

2=

For randomly oriented cracks, 0_(Y.,ac,) is the total arc length on a unit radius circle in principal

stress space on which the projection of the equivalent stress satisfies _, > or,,, and 2_ is the total arc

length of the circle. Similarly as with volume flaws, the Batdorf crack density function is

approximated by the power function,

Ns(C_ ) = kBs(y, m* (42)

where kBs is the Batdorf surface crack density coefficient.

A simplification of equation (39) is obtained by noting the following:

¢o(_,o=) = f02" H(oe, o=) da (43)

where

H(oe, o =) = 1 o > o=

H(o,,o=) = 0 o= < o=

Substituting into equation (39) and noting symmetry (for principal stress space), we obtain

Pfs = 1 -exp[-2 L ( f? Ns(°') dot)dA ]

(44)

where

m S

Ns(%) = 1_o, (x,y,a)

For a given element, ae(x,y,ct ) is the projected equivalent stress over the first quadrant of a unit

radius circle in principal stress space, as shown in figure 9. Equation (44) circumvents the

NASA/TM--2003-106316 107
computationof o(E,o_) andisusedto obtainthecomponentprobabilityof failurein

CARES/LIFE.

Thefinite element method enables discretization of the surface of the component into

incremental area elements. CARES/LIFE evaluates the failure probability at the Gaussian

integration points of shell elements or optionally at the element centroid. Using the element

integration points subdivides the element into subelements, where each A i corresponds to the i_

subelement. The element or subelement area of a two-dimensional element is calculated in similar

fashion to the method outlined for equation (33)

A = ff f_l det J(r,s) dr ds

1 1

(45)

except a two dimensional natural coordinate space is used. Powers et al. (Powers, Starlinger, and

Gyekenyesi, 1992) gives further details of the subelementing procedure as used in CARES/LIFE.

The stress state in each subelement is assumed uniform and the numerical formulation of

equation (44) is

": If; (46)

where n is the total number of subelements. If kBs is element dependent it would appear inside the

brackets.

For the plane stress condition, selecting an arbitrary plane and imposing equilibrium

conditions yields the following equations:

C_2 = ( _,g )2 + ( _2 m )2 (47)

NAS A/TM--2003-106316 108
c = _le2 + o2m2 (48)

and

1;2 = (y2 _ (y_ (49)

where o is the total traction vector acting on the crack plane and the direction cosines m and n

are given in figure 9 in terms of trigonometric functions of ct.

Fracture occurs when the equivalent stress o e > oct. For the shear-insensitive case, fracture

depends only on the value of the normal tensile stress such that oe = o,.

and colinear crack extension, assuming a Griffith crack with K_=cr,

obtain as before

For shear-sensitive cracks

and K_= 1; _ ,we

O" c = _/ O" 2n + 1;2

(50)

whereas for a Griffith notch subjected to plane strain conditions with

=1; _ (Sih, 1973), we obtain

%= I %+2 0.7951 (1 -v)

KI=1.1215 o_ and K m

(51)

Note that the equivalent stress for the Griffith crack is dependent on modes I and II, whereas the

equivalent stress for the Griffith notch is dependent on modes I and III (Gyekenyesi, and Nemeth,

1987).

For noncoplanar crack growth, from equation (30) the effective stress equations for the

Griffith crack and Griffith notch, respectively, are

NAS A/TM--2003-106316 109
(52)

and

(53)

For a semicircularsurfacecrack,K.= 1.366o, V_, KII = 1.241 _ x/a, and K,. = 0.133 x/a (Smith,

Emery, and Kobayashi, 1967) (Smith, and Sorenson, 1974). Since the contribution of Km

it is neglected, and thus the effective stress for this case is

1

2

a n +

is small,

(54)

For the same stress state and identical C, the Griffith crack is the most shear sensitive, whereas the

Griffith notch and the semicircular crack give almost identical predictions.

NASA/TM--2003-106316 110
Material Strength Characterization

Ceramic inert strength due to inherent flaws is described by the simple Weibull uniaxial

cumulative distribution function. For brittle materials, tensile strength, compressive strength, shear

strength, flexural strength, and theoretical strength all have unique meanings and different values.

The theoretical strength is defined as the tensile stress required to break atomic bonds, which

typically ranges from 1/10th to 1/5th of the elastic modulus for ceramic materials. Because of

processing flaws, this strength is never obtained. In an inert environment, a much more meaningful

strength measurement is the inert strength or ultimate tensile strength in uniaxial tension or flexural

testing. In flexural strength testing the bend strength at of a ceramic is defined as the maximum

tensile stress in the extreme fiber of a beam specimen (modulus of rupture, MOR). The main

objective of the CARES/LIFE program is to characterize ceramic strength in terms of the MOR or

pure uniaxial strength, and to use this information with appropriate analysis to predict component

response under complex multiaxial stress states. This section deals with the calculation of the

Weibull scale parameter, 60, and the Batdorf crack density coefficient, kB. Closed form solutions

for ao are given for the uniaxial tensile, and three- and four-point bending specimen geometries.

Also, simplified Weibull equations are described that facilitate evaluation of Weibull parameters

from experimental data. Procedures for obtaining Weibull parameters from experimental data are

described in the following section titled Estimation of Statistical Material Strength Parameters.

Typically for brittle materials, the Weibull parameters are determined from simple specimen

geometry and loading conditions, such as beams under flexure and either cylindrical or flat

specimens under uniform uniaxial tension. For fast-fracture, in an inert environment, the flexural

test failure probability can be expressed in terms of the extreme fiber fracture stress, 6f, by using the

two-parameter WeibuU form given in equations (9) and (35):

NASA/TM--2003-106316 1 iI
a t (55)

Pf = 1 - exp - -_0

where m is the volume or area fast-fracture Weibull modulus and a 0 is the volume or area

specimen characteristic strength. The Weibull scale parameter ao (as defined in equations (7) and

(35) for volume and surface cracks, respectively) is determined from a e , m, the specimen

geometry, and the loading configuration. The scale parameter ao is a material property, whereas

t_0 includes the effects of the specimen dimensions and stress distribution. The characteristic

strength tt0 is defined as the uniform stress or extreme fiber stress at which the probability of

failure is 0.632. The component failure behavior in fast-fracture, equation (55), is only a function

of a t and the empirically determined parameters m and a 0. Procedures such as least squares or

maximum likelihood analysis are used to estimate m and cr0 from experimental fracture data as

described in the section titled Estimation of Statistical Material Strength Parameters.

The uniaxiai inert strength distribution for volume flaws (equation (9)) is expressed in terms

of the extreme fiber fracture stress, af, of the specimen by

Pfv = 1 -exp Ve_ooV }

where an effective volume, V e, is defined by equating the risk of rupture, equation (9), with

equation (55).

V_= L (a(x'Y'Z)/'v. -- dV = (a°----Zv/'V of j !,Oov)

(57)

The effective volume is the equivalent volume under a uniform uniaxial tensile stress of magnitude

c r that is needed to give an identical failure probability as the specimen. Comparing equations

(55) and (56) the volume flaw scale parameter is solved as

NASA/TM--2003-106316 112
aoV= %v V_/my (58)

For the four-point bend specimen geometry shown in figure 10, the tensile stress distribution

in the specimen is

4xyfff L I - L2

_x = 0<x<

(L I - L2)h 2

2yc_f L_ - L 2 LI ÷ L 2

= <x<

c_x h 2 2

4(L I - x)yfff L l + L2

tr = <x<L I

(L I - L2)h 2

(59)

Substituting equation (59) into equation (57) and solving for the effective volume we obtain

wh (L, + mvL2)

We =

2 (m v + 1)2

(60)

The effective volume for the three-point bend specimen geometry is obtained when L 2 = 0 in

equation (60). For uniaxial tensile loading, the effective volume is equal to the gage volume Vg,

which is the uniformly stressed region where fracture is expected to occur.

When a specimen is subjected to multiaxial stresses, the PIA and Batdorf theories are used to

equate the specimen strength to the uniaxial stress state. For the PIA theory the effective volume

used with equation (56) is

V e = o_ v + 0 2 + o 3 dV (61)

where a_ > a 2 > ¢rs are functions of (x,y,z) and negative values are taken as zero. In this case crf

represents the maximum principal stress found in the component. For the Batdorf theory the

effective volume used with equation (56) is defined as

NASA/TM--2003-106316 1 ! 3
V _ rv f =gv dodpdv ¢ _ Of )

(62)

where

KBV

KBV

Lv

(63)

The term kav is the normalized Batdorf crack density coefficient for volume flaws. It is obtained

by equating the risks of rupture of equations (9) and (20), the polyaxial Batdorf theory to the

uniaxial Weibuli model, for an imposed uniaxial stress state or, equivalently, by equating the

effective volumes in equations (57) and (62). Under a uniform uniaxial stress of magnitude o,, this

yields

I_BV =

7_

2IIo"f 0_ _o(c,,a,13)

mY (64)

sin ct dot d[3

This equation is evaluated numerically except for two special cases where a closed form solution is

known to exist. For the shear-insensitive fracture criterion

'O'e(Ol'_t'[_)) = COS 2 IX °l )

(65)

substituting into equation (64) then

kBv = 2mv + 1 (66)

For the coplanar strain energy release rate criterion and the Griffith crack geometry

NASA/TM--2003-106316 114
°_(°l'a'13) / -- cos c_ °i }

(67)

again substituting into equation (64) gives

kav = mv + 1 (68)

For surface flaws the uniaxial inert strength distribution (equation (35)) is expressed in terms

of the extreme fiber fracture stress, of, of the specimen by

Pfs = 1 - exp A_ !,_] I

(69)

where an effective area, Ac, is defined

dA (70)

Comparing equations (55) and (69) the surface flaw scale parameter is solved as

. l/m s

Oos = Oes Ae

(71)

Referring to equation (59) for the four-point bend specimen geometry (fig. 10), the tensile

stress on the beam surface y = h/2, is

2XCYf L l - L2

C_x= O_<x<

(L_ - L2) 2

L I - Lz Ll + Lz

(_x = O'f --< X --< 2 2

2(L I _ x) L t + L2

(3"x = Gf < X < t I

(L I - L 2) 2

(72)

NASA/TM--2003-106316 i 15
Substituting equation (72) and equation (59) for the side surface stress distributions into equation

(70) and performing the integration, the effective area is obtained as

[iL 1 II }

ms+l

m s w

= _ + 1 (w + h)L I

Ac ( m s + 1)2 w + h

(73)

The effective area for the three-point bending specimen geometry is obtained when 1.2 = 0 in

equation (73). For uniaxial tensile loading, the effective area is equal to the specimen gage area Ag

, which is the total specimen surface area of interest.

When a specimen is subjected to multiaxial stresses, the PIA and Batdorf model risks of

rupture are equated to the uniaxial Weibull risk of rupture given by equation (69). For the PIA

model, the effective area used with equation (69) is

A e = ol(x,y) ms + o2(x,y) ms dA (74)

and of represents the maximum principal stress found on the component. For the Batdorf theory the

effective area used with equation (69) is

Ae- _JA f? of ) J

(75)

where

kBs- kBs (76)

NASA/TM--2003-106316 116
Theterm _s is the normalized Batdorf crack density coefficient for surface flaws. In

CARES/LIFE the normalized Batdorf crack density coefficient for surface flaws is found for a

uniaxially loaded specimen by equating the risk-of-ruptures of equations (35) and (44), or

equivalently by equating the effective areas in equations (70) and (75). For a uniform uniaxial

stress of magnitude o_, this yields

I_BS =

7I

2 [ ! 1dot oo(o ,ot)

(11

(77)

This equation is evaluated numerically. A closed form solution is known to exist for the shearinsensitive fracture criterion (Gross, and Gyekenyesi, 1989). Since

(°°(°-'''>/:ol J

(78)

substituting into equation (77) then

_BS

m s v_" F (ms)

=

1

(79)

For the shear-sensitive fracture criterion, Griffith crack geometry, and co-linear crack extension

(80)

again substituting into equation (77) gives

NASA/TM--2003-106316 ! 17
(81) kBs = ¢

2r tms'a) 2

Estimation of Statistical Material Strength Parameters

Selected statistical theories and equations for Weibull parameter estimation are explained in

detail in the reference by Pal (Pai, and Gyekenyesi, 1988). The following is a brief description of

these methods and how they are used in the CARES/LIFE code. For brittle materials the Weibull

parameters are determined from repeated fracture experiments on nominally identical specinaens.

Typically this involves a simple geometry and loading condition, such as beams under flexure or

specimens with either a round or rectangular cross section under uniform uniaxial tension. As a rule

of thumb, a minimum of 30 specimens are required to obtain parameter estimates with a reasonably

narrow standard deviation within which the true values of the parameters are likely to reside. Since

each specimen carries a fixed cost, the experimentalist desires to use analytical methods that

maximize the information that can be gained from a data sample while using the fewest possible

number of specimens. CARES/LIFE accomplishes this by including efficient parameter estimation

schemes as well as statistical measures to quantify the quality of the data.

For fast-fracture, the flexural test failure probability can be expressed in terms of the extreme

fiber fracture stress o t by using a simplified two-parameter Weibull form as described by equation

(55), where m is the volume or area Weibull modulus and er0 is the volume or area specimen

characteristic strength. Although the statistical theories and parameter estimation methods outlined

in the following discussion are expressed in terms of the fast-fracture strength distribution, these

techniques are equally applicable to the time dependent distribution as well.

Before computing the estimates of the statistical parameters, it is essential to carefully

examine the available specimen data to screen them for outliers. Very often, a data set may contain

NASA/TM--2003-106316 118
one or more values which may not belong to the overall population. The statistical procedure used

to detect the outliers at different significance levels is explained in Pai (Pai, and Gyekenyesi, 1988)

and Stefansky (Stefansky, 1972). This outlier test assumes that the data are normally distributed and

from a complete sample. Therefore, the application of this test to the Weibull distribution and

censored statistics is only approximate. CARES/LIFE improves the original technique (Pal and

Gyekenyesi, 1988) by numerically integrating the t-distribution to calculate the critical values (CV)

for significance levels in the range of 0.0% to 10.0% with a resolution of 0.1% (polynomial

approximating functions are no longer used).

Various methods are available to estimate the statistical material parameters from

experimental data for the two-parameter Weibull distribution. The success of the statistical

approach depends on how well the probability density function fits the data. Two popular

techniques used to evaluate the characteristic strength and shape parameter (o 0 and m) from inert

strength data are the least-squares analysis and the maximum likelihood method. Least-squares

analysis is a special case of the maximum likelihood method, where the error is normally distributed

and has a zero mean and constant variance. The least-squares method is not suitable for calculating

confidence intervals and unbiasing factors, which quantify the statistical uncertainties in the

available data.

Equation (55) can be linearized by taking the natural logarithm twice yielding

en en 1 = gn en 1 = fn 1 + m gn or (82)

For the least-squares analysis, it is necessary to obtain the line of best fit with slope m and an

intercept which, as seen in equation (82), is equal to Qn . The failure probability, P_ is

NASA/TM--2003-106316 119
determined by conducting fracture tests on n specimens. The fracture stresses are ranked such that

Ofi < Of 2 < "'" < Ofi < "'" < Ofn. For rank regression analysis, the probability of failure of a

specimen with rank i is

lYDf/'(_fi I, _ ---- i -- 0.3 (83)

n +0.4

By taking the partial derivative of the sum of the squared residuals with respect to m and a 0, and by

equating the derivatives to zero, values of m and a 0 are calculated.

With censored data, one cannot directly use the rank regression analysis as given in equation

(83) because of the competing failure modes. To take into account the influence of the suspended

items, Johnson (Johnson, 1964) developed the rank increment technique. For this technique, all

observed fracture stresses are arranged in ascending order, and rank increment values are calculated

for each failure stress from the following equation:

Rank increment = (n + 1) - (previous adjusted rank) (84)

1 + (number of items beyond present suspended item)

In the CARES/LIFE program for volume flaw analysis, all fracture stresses designated as V's are

considered as failure data; for surface flaw analysis, the S's are considered as failure data. The

new adjusted rank values are obtained by adding the rank increment value to the previously adjusted

rank. These adjusted rank values are then used to calculate the failure probability by using the

median rank regression equation (83). Finally, the estimated Weibull parameters for m and o 0 are

obtained.

Since the distribution of errors from the data is not normal, the maximum likelihood method

is often preferred in Weibull analysis. This method has certain inherent properties. The likelihood

equation from which the maximum likelihood estimates (MLE's) are obtained will have a unique

NASA/TM--2003-106316 120
solution. In addition, as the sample size increases, the solution converges to the true values of the

parameters. Another feature of the maximum likelihood method is that there are no ranking

functions or linear regression analysis when complete or censored samples are analyzed. The

likelihood equation for a complete sample is given by

n llfl m °'fi exp - °'fi

[OoJ [o0J

(85)

The values of m and o 0 which maximize the likelihood function, L, are determined by taking the

partial derivative of the logarithm of the likelihood function with respect to m and a 0. The

estimated values, 61 and o0, are obtained by equating the resulting expressions to zero and

solving the simultaneous equations with the Newton-Raphson iterative technique. The MLE of m

and o o are designated by mv and O0v and by lhs and 6-os for volume flaw analysis and surface

flaw analysis, respectively. For censored statistics we have

ri) en(o

i=l

_ _1 _ en(afi ) _ _1 = 0 (86)

r ,ol m

and

n h

E O.fi

i-I

r

I/m

(87)

where r is the number of remaining specimens failed by the flaw mode for which parameters are

being calculated. For a complete (uncensored) sample, r is replaced by n, which is the total size

of the sample.

NASA/TM--2003-106316 121
TheMLE of theshapeparameterisalwaysa biased estimate that depends on the number of

specimens in the sample. Unbiasing of the shape parameter estimate is desired to minimize the

deviation between the sample and the true population. The unbiased estimate of m is obtained by

multiplying the biased estimate with an unbiasing factor (Thoman, Bain, and Antle, 1969). The

confidence intervals for complete samples can also be obtained (Thoman, Bain, and Antle, 1969).

For censored samples, a rigorous method for obtaining confidence intervals has not yet been

developed because of the complexity of competing failure modes. Confidence bounds for censored

statistics are instead estimated in the CARES/LIFE code from the factors obtained from complete

samples (Pai, and Gyekenyesi, 1988). Confidence bounds enable the user to estimate the

uncertainty in the parameters as a function of the number of specimens. Bounds at a 90-percent

confidence level, and therefore at 5 and 95 percentage points of distribution of the MLE's of the

parameters, have been incorporated into the CARES/LIFE program, with data taken from Thoman,

et al (Thoman, Bain, and Antle, 1969).

Subjective judgement is needed to test the goodness of fit of the data to the assumed

distribution. When graphical techniques are used, it can be very difficult to decide if the

hypothesized distribution is valid, especially for small sample sizes. Therefore, many statistical tests

have been developed to quantify the degree of correlation of the experimental data to the proposed

distribution.

In general, a statistic is a numerical value computed from a random sample of the total

population. The difference between an empirical distribution function (EDF) and a hypothesized

distribution function is called an EDF statistic. There are two major classes of EDF statistics, and

they differ in the manner in which the functional (vertical) difference between the EDF and the

proposed distribution function is considered. The Koimogorov-Smirnov (K-S) goodness-of-fit

statistic D belongs to the supremum class and is very effective for small samples. It uses the

NASA/TM--2003-106316 122
largestverticaldifferencebetweenthetwo distributionfunctionsto determinethegoodnessof fit.

FortheK-Stest,thesampleis arrangedinascendingorder,andtheempiricaldistributionfunction

F,(x) is a stepfunctionobtainedfromthefollowingexpressions:

F(x) = 0 x < X1

F(x) = _i Xi_<x <Xi+1

n

F(x) = 1 X < x

and i = 1,2,3 ....... n-I

(88)

where X l < X 2 <... _ ... < X_ are the ordered fracture stresses from a sample of size n. The

statistic D is obtained by initially evaluating two other statistics, D + and D (the largest vertical

differences when F,(x) is greater than the distribution function F(x) (equation (55)) and the largest

vertical differences when F.(x) is smaller than F(x), respectively). All three statistics are

calculated by using the following expressions:

D" =1_ - F(x),l I

D m_ax iill

-= ]

I

D = (D ",Di = 1,2,...,n

(89)

For ceramics design, the F(x)_'s are equal to Pf's and are calculated from equation (55).

On the other hand, the Anderson-Darling statistic A 2 belongs to the quadratic class and is a

more powerful goodness-of-fit statistic. It evaluates the discrepancy between the two distributions

through squared differences and the use of an appropriate weighting function. The statistic A 2 is

given by

(90)

NASA/TM--2003-106316 123
In thiscase,the Zi's are the predicted failure probabilities obtained from equation (55).

Corresponding significance levels ot are calculated from the D and A 2 statistics. From

previous surveys (Pai, and Gyekenyesi, 1988) there is no specific mention of an absolute accepted

significance level. Therefore, the users must be subjective, using their own judgement in either

accepting or rejecting the hypothesis that the data fit a Weibull distribution. However, a higher

value of et indicates that the data fit the distribution more closely. In addition, Johnson and

Tucker (Johnson, and Tucker, 1992) point out that the basis of the calculation of the significance

level is relevant. In CARES/LIFE the significance level is calculated with the assumption that the

Weibull parameters are calculated independent of the observed strength data. However, the Weibull

parameters are actually estimates based on the experimental data and hence the assumption of

independence is violated. Until appropriate corrections are available for the K-S and A-D tests, the

user must be cautious in interpreting the calculated significance levels in CARES/LIFE. For now, it

is recommended that the significance level be viewed as a relative measure of goodness-of-fit and

not as an absolute measure.

For complete samples, the 90-percent Kanofsky-Srinivasan confidence band values about the

proposed distribution are also calculated to ascertain the fit of the data. These values are similar to

the K-S statistic D centered around the EDF. The bands are generated by

Confidence bands = [F(x) - K(n), F(x) + K(n)] (91)

where

(55).

F(x) is the failure probability obtained by substituting the Weibull parameters in equation

The Kanofsky functions, denoted by K(n), are described in Abernethy (Abernethy, 1983).

Some limitations are intrinsic to a purely statistical approach to design. One problem occurs

when the design stress is well below the range of experimental data as shown in figure 11.

Extrapolation of the Weibull distribution into this regime may yield erroneous results if other

NASA/TM--2003-106316 124
phenomenarepresent.Whentwo flawpopulationsexistconcurrently,butonly one(populationA)

is activein thestrengthregimetested,thepredictedfailureprobabilitymaybeincorrect.

Furthermore,if thethresholdstrengthisnotzero,thestrengthmaybeunderestimated.Finally,an

approachbasedonlyonstatisticscanallowfor stressstateeffectsonly in anempiricalfashion.

NASAfI'M--2003-106316 125
Theory

Time-Dependent Reliability Analysis

Introduction

For ceramics and glasses the ability to sustain a load degrades over time due to a variety of

possible effects, such as oxidation, creep, stress corrosion, and cyclic fatigue. Stress corrosion and

cyclic fatigue are representatives of a phenomenon called subcritical crack growth (SCG). SCG

initiates on a pre-existing flaw and continues until a critical length is reached, causing catastrophic

propagation. This occurs when the equivalent mode I stress intensity factor, K_eq, equals the fracture

toughness, Kic. The SCG failure mechanism is load induced over time. It can also be a function of

chemical reaction with the environment, debris wedging near the crack tip, the progressive deterioration of bridging ligaments, etc. Because of this complexity, the models that have been developed

tend to be semi-empirical and approximate the phenomenological behavior of subcritical crack

growth.

The previous sections assumed no subcritical crack growth occurred prior to failure, and all

failures were assumed independent of time and history of previous thermal-mechanical Ioadings.

The effects of time-dependent subcritical crack growth on component reliability will now be

addressed. Creep and material healing mechanisms are not addressed in CARES/LIFE. Proof

testing (Evans and Weiderhorn, 1974) will improve the reliability of a survived component. This

form of testing results in an attenuated probability of failure and a predicted minimum life

expectancy of the survived components under the service load. This subject is discussed in the

section, Proof Testing Effect on Component Service Probability of Failure.

For the analysis of time-dependent reliability, in addition to the Weibull shape and scale

parameters, the material/environmental fatigue parameters (N and B) are required. The derivations

that follow will develop the time-dependent probability of failure based on the mode I equivalent

NASA/TM--2003-106316 126
stressdistributiondueto thermal-mechanical loading at time tf, transformed to its equivalent stress

distribution at time t = 0. Determination of the fatigue parameters is discussed in the section,

Evaluation of Fatigue Parameters from Inherently Flawed Specimens.

Investigations in the area of mode I crack extension (Paris, and Sih, 1965) have resulted in

the following relationship

Kx_(_,t) = oi_(_,t) y _ (92)

where Y is a function of crack geometry and can vary with subcritical crack growth; however,

CARES/LIFE assumes Y is a fixed geometric constant. 6_eq is the equivalent mode I far field stress

normal to a crack, a(t) is the appropriate crack length, and q-' represents a location (x,y,z) within the

body and the orientation (_t,13) of the crack. In some models, such as the Weibull and PIA,

represents a location only. The equations presented in this section are based on the Batdorf theory

and the PIA model. For the Batdorf theory, W=(x,y,z, ct,13) for volume flaw analysis and W=(x,y,a)

for surface flaw analysis. For the PIA model, W=(x,y,z) for volume flaw analysis and W=(x,y) for

surface flaw analysis.

The crack growth as a function of the equivalent mode I stress intensity factor can be

expressed

da(_,t) _ A K_q(_,t) N (93)

dt

where A and N are material/environmental constants. From equations (92) and (93) we obtain

d a(_,t) N N

- A ox_(_,t) Y a(_,t) s/2 dt

(94)

The relationship at time t between a(t) and a mode I critical effective stress, (Yleq,t is

NASA/TM--2003- !06316 i 27
a(_f,t) -z = Oleq,t(_,t)

Differentiating equation (95) and substituting into (94) results in

N-2

f °leq't"Oleq,tN-3 dOleq,t = -A y2 KIC fof o_(_,t)dt

ox..oe 2

(95)

(96)

where o_,0(W ) is the transformed critical equivalent stress distribution at t=0 and O'teq,f=Oleq(_IJ,tf) is

the equivalent stress distribution in the component at time t=tf. The transformation of the equivalent

stress distribution at time tf to its effective stress distribution O_,q.0(q') at time t=0 (Thiemeier,

1989)(Sturmer, Schulz, and Wittig, 1991) is

°I_q,o(_f)= IfofOleq(q',t) dt °_z(_'te ) ]l0_--2,

(97)

where

2

B = (98)

A y2 KI_-2 (N-2)

N and B are the material/environmental fatigue parameters. The exponent N is dimensionless and B

has units stress 2 x time. The determination of these parameters is addressed in the section

Evaluation of Fatigue Parameters from Inherently Flawed Specimens.

NASA/TM--2003-106316 128
Volume Flaw Reliability Analysis

CARES/LIFE computes the time-dependent reliability of a ceramic component assuming a

crack density distribution which is a function of the critical effective stress distribution. The crack

density coefficient is now time-dependent. For volume flaw analysis the crack density function is

expressed

m V

Nv(O(T)) = kBv o,,,q,O(T) (99)

where kay and m v are constants and the critical effective stress, crjeq.0, is expressed as

[ }

Oi_q(T,t) dt Nv-2 (Nv-2)

OIeq, 0(II/) = fo+ Nv

BBV + Olcq (V,tf)

(loo)

where N v and BBV are material fatigue parameters. Based on a probability of failure model, a

subscripted fatigue parameter, such as BBv for the Batdorf model, is computed. BBv is directly

proportional to B. The various model-dependent subscripted fatigue parameters are all directly

proprotional to B. They are evaluated by satisfying the requirement that for a uniaxial stress state,

all models produce the same probability of failure. For large values of N, all model fatigue

parameters tend to B. The relationship between the B subscripted parameters is discussed in the

section Fatigue Parameter Risk of Rupture Compatibility.

If the boundary load direction and/or location changes with time, the principal stress vectors

change direction with respect to a fixed global coordinate system. If this occurs, the permanent

reference axis becomes the fixed global coordinate system. The normal and shear stresses are

computed with respect to the global coordinate system. The normal stress is

0a = 12 0x + m 20y + n 2 o z + 2(IrnZxy +mn _yz + nl Zzx) (I01)

NASA/TM--2003-106316 129
andthe shear stress is

'r2 = (1% + m 'ry_ + n 'r=) 2 +(1 'ry + m O'y+ n "Uzy)2

+0 _,_ + m _,_ + n oz)2 - o.2

(102)

where 1, m and n are the direction cosines, defined in figure 12. Symmetry conditions permit the

integration of the equivalent mode I stress projection over the upper half of the spherical surface as

shown in figure 12.

The probability of failure for the Batdorf model is

2,_ _ m,, (103)

Pfv(tf) = 1 - exp - [a_,o(tf)] sinct dot d[_ dV

where Oleq,0(tI/) is the critical effective stress distribution as given in equation (100). Oleq.0 (_) is

dependent on the appropriate fracture criterion, crack shape, and time tr. The criteria and crack

shapes available for time-dependent analysis are identical to those used for fast fracture. The

fracture criteria are: Weibull's normal stress averaging (a shear-insensitive case of the Batdorf

theory), maximum tensile stress, total coplanar strain energy release rate, and the noncoplanar

(Shetty) criterion.

For the Principle of Independent Action (PIA) model, the probability of failure is

Pfv(tf) = 1 - exp [-kwv /v(On,o (t]_) + 012,0 (tp) + 013,0 (_1_) )dV ]

where

troiNV(x,y,z,t) dt

Bwv

and i=1,2,3, o_(x,y,z,t), o'2(x,y,z,t ) and o3(x,y,z,t ) are the principal tensile stress distributions.

(104)

NAS A/TM--2003-106316 130
In proof testing of components, off axis testing (proof test loading axis differs from service

loading) frequently occurs requiring a fixed global coordinate system. This occurrence is discussed

and analyzed in the Proof Testing - Off-Axis Loading section.

For a stressed component, the probability of failure is calculated from equation (103). The

finite element method enables discretization of the component into incremental volume elements.

CARES/LIFE evaluates the failure probability at the Gaussian integration points of the element or

optionally at the element centroid. Using the element integration points subdivides the element into

subelements, hence each V i corresponds to the ith subelement. In the usual context of finite element

methods, the volume of a three-dimensional element is calculated after transformation into the

natural coordinate space (Bathe, 1982)

V-- fj f/det J(r,s,t)dr ds dt (105)

! 1 1

where J is the Jacobian operator and r,s,t are the natural coordinates. The subelement volume is

defined as the contribution of the integration point to the element volume in the course of the

numerical integration procedure. This means that the volume of each subelement (corresponding to

a Gauss integration point) is calculated using the shape functions inherent to the element type. The

stress state in each subelement is assumed uniform. Powers et al. (Powers, Starlinger, and

Gyekenyesi, 1992) gives further details of the subelementing procedure as used in CARES/LIFE.

The numerical solution of equation (103) takes the following form:

- )] ksv _

= V i )F st. d d Pfv(tf) 1-exp -_x i.l i

(106)

where n is the total number of subelements. If kav is element dependent, it would appear inside the

brackets.

NASA/TM--2003-106316 131
Surface Flaw Reliability Analysis

CARES/LIFE computes the time-dependent reliability of a ceramic component assuming a

crack density distribution which is a function of the critical effective stress distribution. The crack

density coefficient is now time-dependent.

expressed as

m S

Ns(o(*l')) = kas oi_,o(_')

For surface flaw analysis the crack density function is

(107)

where kas and m, are constants and the critical effective stress, atcq,0, is expressed as

1

t oieq(_,t )dt lqs_2 0_-2)

= + Ole q (rlr,tf)

OIcq'0(l:P) BB s

(108)

where N s and Bas are material fatigue parameters. Based on a probability of failure model, a

subscripted fatigue parameter, such as Bas for the Batdorf model, is computed. BBs is directly

proportional to B. The vai'ious model-dependent subscripted fatigue parameters are all directly

proprotional to B. They are evaluated by satisfying the requirement that for a uniaxial stress state,

all models produce the same probability of failure. For large values of N, all model fatigue

parameters tend to B. The relationship between the B subscripted parameters is discussed in the

section Fatigue Parameter Risk of Rupture Compatibility.

If the boundary load direction and/or location changes with time, the principal stress vectors

change direction with respect to a fixed global coordinate system. If this occurs, the permanent

reference axis becomes the fixed global coordinate system. The normal and shear stresses are

computed with respect to the global coordinate system. By symmetry the projection of the

equivalent stress is over half of the perimeter of a unit radius circle, where

NAS A/TM--2003-106316 132
on = 12 o x + m 20y + 2 lm _xr (109)

and

1:2 = (1 a x + m ._)2 + (1 Txr +m Oy)2_

and I and m are the direction cosines. Symmetry conditions permit the equivalent mode I stress

projection over the half of the unit circle. The probability of failure for the Batdorf model is

(11o)

Prs(tf) = 1- exp [-kBs JfAdOf " °xeq'°(_) ms da dA ]

(111)

where Ol,a.0(W) is the critical effective stress distribution as given in equation (108). O,_q.0(W) is

dependent on the appropriate fracture criterion, crack shape, and time tf. The criteria and crack

shapes available for time-dependent analysis are identical to those used for fast fracture. The

fracture criteria are: Weibull's normal stress averaging (a shear- insensitive case of the Batdorf

theory), total coplanar strain energy release rate, and the noncoplanar (Shetty) criterion.

For the Principle of Independent Action (PIA) model the probability of failure is

011,0(1t m,y) +012,0(7 m.) ) dR ]

where

on,0(_l') =

fotf NS 1

Oi (x,y,t) dt N_-2 ]r_s------5

+ O i (x,y,tf)/

Bws ]

and i=1,2. 6,(x,y,t) and a2(x,y,t ) are the principal tensile stress distributions.

(112)

NASA/TM--2003-106316 133
In proof testing of components, off axis testing (proof test loading axis differs from service

loading) frequently occurs requiring a fixed global coordinate system. This occurrence is discussed

and analyzed in the Proof Testing - Off-Axis Loading section.

The finite element method enables discretization of the surface of the component into

incremental area elements. CARES/LIFE evaluates the failure probability at the Gaussian

integration points of shell elements or optionally at the element centroids. Using the element

integration points subdivides the element into subelements, where each A i corresponds to the i_h

subelement. The area of a two-dimensional element or subelement is calculated in similar fashion

to the method outlined for equation (105)

A = ft f_ det J(r,s) drds (113)

d- 1 ./- 1

except a two dimensional natural coordinate space is used. Powers et al. (Powers, Starlinger, and

Gyekenyesi, 1992) gives further details of the subelementing procedure as used in CARES/LIFE.

The stress state in each subelement is assumed uniform and the numerical formulation of

equation (111) is

Pts (t0 = 1-exp _Ai f0 ms - " ox_,0(tt') d_

i=l i

(114)

where n is the total of number of subelements. If kas is element dependent it would appear inside

the brackets.

NASA/TM--2003-106316 134
StaticFatigue

Static fatigue is defined as the application of a constant load over a period of time. For static

fatigue, the mode I equivalent stress is independent of time, and from equation (97),

OIeq,0(_) = °IeqN-2(_) tf Oleq(_P) + 1 (1 15)

B

For volume flaws (by symmetry, integrating over one octant of the unit radius sphere), the

probability of failure for the Batdorf model is

Pfv(tf) = 1 - exp _ _ [O_q,O(_I')]m" sint_ dot dfi dV }

For the Principle of Independent Action model, the probability of failure is

_,tf,-l-ox,[-_.v:v( ° . mv • mv ]

On,o(_) o12,o(_) o13,o(_) ) dV

where

N V

Oi (x,y,z) tf

Bwv

1

N v-2 Nv-2

4- Oi (x,y,z)

for i=1,2,3.

The probability of failure for surface flaws is analogous to that for volume flaws.

Batdorf model, the probability of failure is expressed as

Pfs(tf) = 1 - exp - So-_ [oleq,0(_P)] ms dot dA }

For the

For the Principle of Independent Action model, the probability of failure is

(116)

(117)

(118)

NASA/TM--2003-1063 !6 135
-- o,,,o(V)+ o, o(V)/ dA1

(119)

where

On,0(tl') =

N S

o i (x,y) tf

Bws

1

N s -2 Ni-2

+ o i (x,y)

for i=1,2.

Dynamic Fatigue

Dynamic fatigue is defined as the application of a constant stress rate _ (q-') over a period of

time t. Thus

oi_i(_g,t ) = _(_) t

(120)

From equation (97)

olm,0(_l') = fo q [6(V) t]N dt B

(121)

For oIm(_,t f) = b(V) tf

[ Oleq(_,tf) N tf °x_q'°(_P) = (N+l) B

(122)

For volume flaws, the Batdorf probability of failure equation is

f$/'$ro (_1 mv sinct da dl3 dV Pfv(tf) = 1-exp Jo .1o [ leq,O "]

(123)

NASA/TM--2003-106316 136
ForthePIA model

Pfv(tf) = X-exP [-kwv fv ( mv + mv + mv ]

On,o(R_) o_,0(_ ) Oi3o(_) ) dV (124)

where

On,oCI') =

N'V

Oi (x,y,z,tf) tf

(Nv+l) Bwv

1

Nv -2 _vv-2

+ o i (x,y,z,tf)

for i=1,2,3.

For surface flaws, the Batdorf model probability of failure is

Pfs(tf) = 1-exp

1t

(125)

For the PIA model

On,o(_) Pfs(tf) = 1-exp {-kws f^( ms +

mS (126)

where

oii,o(_ ) 1

Ns

0 i (x,y,tf) tf Ns-2

_s +1) Bws + °i (x,y,tf)

for i=1,2.

NASA/TM--2003-106316 137
Cyclic Fatigue

Cyclic fatigue is the repeated application of a loading sequence. Analysis of the timedependent probability of failure for a component subjected to various cyclic boundary load

conditions is simplified by transforming that type of loading to an equivalent static state. The

conversion satisfies the requirement that both systems will cause the same crack growth (Mencik,

1984). Implicit in this conversion is the validity of the crack growth equation (93). The probability

of failure is obtained with respect to the transformed static state.

The fatigue parameters can also be determined from cyclic loaded specimens via

transformation to an equivalent static state. Since static and cyclic tests can yield different results

when determining the fatigue parameters, the type of loading used should simulate as closely as

possible the service conditions.

Evans (Evans, 1980) and Mencik (Mencik, 1984) defined g-factors, for various types of cyclic

loading, that are used to convert to an equivalent static state. For periodic loading, T is the time

interval of one cycle, otm(_F) is the equivalent static stress acting over the same time interval, T, as

the applied cyclic load C_qc (_P,t) . The equivalent static stress is defined as

1

Oleq(_ ) = g(tp)N o_h(_g) (127)

where Och(_ ) is a characteristic value of O'leqc(_,t ) . Over one cycle where both load systems cause

the same crack growth

T= fo Oieqc(_,t " ) dt o2s)

where

NASA/TM--2003-106316 138
andtheg-factoris

Oi_q(_) = .f° °I_c(_'t)dt

T

N

= ooh(V) g(V)

(129)

g(V) = .for (°I'q¢(_'t) / N

o_(V) )

T

dt (130)

In CARES/LIFE, the characteristic value O,h(_) of O_eq00F,t) is taken as Oleqc (_), the

maximum stress of the periodic load over the cycle time interval T. For a periodic load over a time

t I

o_(_) tl= f0 h o_,.q¢(_,t)dt =t, I 1

f0 oi_c(_,t) dt

T

= g(W) O_qc (W) tI

(131)

When more than one type of loading is applied to a component, such as a periodic cyclic load and a

static load, the g-factor is based on the effective variation of the combined loading. The g-factor

can vary from element to element and the stress-volume integration is performed over the

hemisphere for volume flaws and on the semi-circle for surface flaws.

For n multiple, but different, cyclic loadings over an interval of time, t.--tf

tn N

al_(_ ) tf = f0tlolN(_,t)dt+ ft?a_(_,t)dt+ .... + fg_ on(V,t)dt (132)

Thus

o_(_) tf = y_ oi,_¢,_(V) N gi(V) 6t i (133)

ill

NAS A/TM--2003-106316 139
where

8t i = t i - ti_ 1

and

gi(_) =

fo Ti

OIeqq(V,t) / N dt 1

T i

Another approach by Mencik is to define an equivalent static time, tos, during which a

characteristic stress (Och (_) chosen as the maximum stress Oieqcmx(l_l/) ) would cause the same

crack growth as the applied cyclic stress aleqc (_P,t) during time fit. Thus

OiNeqcma(_) 6tes(_p) _ 6t fo r N T Oleqc(_,t ) dt = ale, (_p) g(tp) fit (134)

or

6t (tP) = 6t (135)

For multiple, but different, cyclic loading over an interval of time tf=Sh+&2+...+St_

N s _ (I_36)

Oi_qc (_) [6te_1(W)+6t_(_)+ .... +6t_(W)] = Oleqcmu(l_ 7) [ g1(W) /$tl+g2(W ) 6t2+...+gn(_7 ) 6t n J

or

NASA/TM--2003-106316 140
n n

Stesi(V) = _ gi (v) 8t i

i=1 i=1

(137)

Mencik (Mencik,1984) lists g-factors for a variety of waveforms. Table 1 lists the g-factors for

various loadings and waveforms supported by CARES/LIFE. A simple closed form expression for

• the sine wave is not available and consequently a numerical evaluation is required.

CARES/LIFE adopts the approach of equation (131) to compute the time-dependent

reliability. Equation (131), the static equivalent stress distribution, is substituted into equation (97)

where the time t I is replaced by tf. Hence

ol_q,0('_) = Oi_q%.(_t') .g(_) °21_q_(V)B tf +11 _-2

(138)

Formulations for other failure models are analogously developed.

NASA/TM--2003- ! 06316 ! 41
Application of the Paris and Walker Laws

This section will consider the more general case of reliability modeling where the applied

loading varies as a function of time. Subcritical crack growth is a complex phenomenon involving a

combination of simultaneous and synergistic failure mechanisms. These can be grouped into two

classes: static effects and cyclic effects. Static effects refer to the slow propagation of cracks under

cyclic stresses and may be explained by the same environmental and corrosive processes responsible

for suberitical crack growth under static loads. Cyclic effects are functionally dependent on the

number of cycles and on the duration of the cycle. The subcritical crack growth phenomenon can

be caused by a variety of effects, such as debris wedging or the degradation of bridging ligaments,

but essentially it is based on the accumulation of some type of irreversible damage that tends to

enhance the crack growth. Not all materials display cyclic effects. Glasses seem to show only the

static effects, while polycrystalline materials are more susceptible to the cyclic effects. Modeling

for static effects is well established, while the work for cyclic effects is still evolving. Modeling for

cyclic effects in this version of the CARES/LIFE program is based on phenomenological criteria

(Paris law and Walker law) traditionally used for metal fatigue.

Using g-factors to obtain component life is an unconservative practice for materials prone to

cyclic damage. To empirically model cyclic effects Dauskardt et. al. (Dauskardt, James, Porter, and

Ritchie, 1992) (Dauskardt, Marshall, and Ritchie, 1990) suggest the use of the Paris power-law

expression (Paris, 1963), which has traditionally been used in metals design. Dauskardt et al.

(Dauskardt, Porter, and Ritchie, 1992) use the Walker modification (Walker, 1970) of the Paris law

to express the growth increment per cycle (da(_F,n)/dn) as

d a(_F,n) _ Ao Kx_,_(_,n) (AKIe(_,n))Q (139) dn

NAS A/TM--2003-106316 142
where

Kie (_,n) = Oleqc QP,n) Y

and

AKt,(_,n ) = A Oleqc(T,n) Y_ = [oloq¢ (_,n) - ox_==(_,n)] Y_

A o, N, and Q are the cyclic fatigue parameters determined from experiment. The Paris law is

obtained when N and Q are equal in value. The subscripts max and min indicate the maximum and

minimum cycle stress, respectively. From equation (139)

[ f_f0[1-R(_F,n)]Q Ol_qcmlx(1]/,n)dn

(140)

where nf denotes the number of cycles to failure, B is expressed in units of stress 2 x cycle (B is

determined from cyclic data), and

R(V,n) =

For a periodic cyclic stress, R and oim_ u are independent of n, hence

Ol_le'O(tP'nf) = Oleqcmat('tll) [ o_(_, (V) [1B - R(_)] Q nt + 1 ]_-2

(141)

When R = 1 (the static fatigue case) there is no time-dependent degradation of material strength due

to cyclic effects. The probability of failure for cyclic fatigue using the Batdorf model for volume

flaws is

NASA/TM--2003-106316 143
(-2 kBvr ]

Pfv(nf) = 1 -exp_--_jvj ° Jo Oleqe'O(_tlJ_U'f)Sil]EdE d_ dV )

(142)

Analogously, equations may be derived for the PIA model or for surface flaws.

The Paris law should be used with prudence. The effect of R<0 on the crack growth is

assumed negligible. R is assumed to be equal to zero for this case. Assuming that the static effects

and cyclic effects are mutually exclusive events, the component reliability can be described as the

product of the reliabilities calculated for each of these phenomena:

Ps_ = Ps_.o _ P%_ _.,.. (143)

thus, when R=I degradation due to static fatigue is accounted for. In the CARES/LIFE code,

equation (143) may be implemented for a given component reliability analysis by modeling that

component based on two mutually exculusive events. One event is based on strength degradation

due to cyclic effects (Paris law) and the second is based on strength degradation due to static effects

(Power law). The combined component reliability is calculated from equation (143). If the

component is modeled based on one event, then only one degradation mechanism (cyclic or static)

can be considered. Other models that consider the combined cyclic and static effects may be

incorporated into future versions of CARES/LIFE.

NASA/TM--2003-106316 144
Material Failure Characterization For Static, Cyclic, Or Dynamic Loading

For time-dependent fracture under static or cyclic fatigue loading, failure probability can be

expressed in terms of the specimen time to failure t r by using the two-parameter specimen uniaxial

Weibull model (Paluszny, and Nicholls, 1978)

Ecr_(Wo) g(Wo) tf jl=1-exp ft ,l [t0J.1

(144)

where

- in

in -

(N-2)

tO =

N

of (V o) g(V o)

and

o_(V o) g(V o) tf >>1

BII

For static loading g(tF) = 1. At location _Fo, cr_Fo) is the maximum static or cyclic failure stress in

the specimen, g("ISo) is the g-factor, % is the characteristic strength, c_. 0 is the transformed static

inert strength, m is a modified Weibuil modulus, and to is the volume or area specimen

characteristic time. Henceforth a_Wo) is replaced by (if, where location at W o is implied. The

characteristic time is analogous to the characteristic strength. Equation (144) allows the specimen

time-dependent failure response to be described with a simple Weibull equation which is only a

NASA/TM--2003-106316 145
function of tf and the empirically determined parameters m and t0. Procedures such as least squares

or maximum likelihood analysis can be used to estimate these parameters from experimental fracture

data as described in the section titled Estimation of Statistical Material Strength Parameters.

For static or cyclic fatigue, using the uniaxial time-dependent Weibull distribution for volume

flaws with the g-factor approach of equation (130) and for

o_q¢ (x,y,z) g(x,y,z) tf

Bwv

>> 1

then

,exp[-i ]

= - o l_qc,o(x,y,z) dV

OoV mv

(145)

where

o 1_qc,o(X,y,z) =

O1_q¢ (x,y,z) g(x,y,z) tf

Bwv

For the case when the g-factor is constant throughout the specimen, henceforth denoted by g,

equation (145) can be expressed in terms of the maximum static or cyclic failure stress % in the

t_vN v

specimen by multiplyingthe numerator and denominator by of Thus

Pfv(tf) = 1 - exp -V a [G,] J

(146)

where

NASA/TM--2003-106316 146
Nv-2

Bwv Oov

toV - Nv

g Of

and

. \mvNv

o lmc tx,y,z) [

vo,--L o. j, dV (147)

The term Vtf is the modified effective volume when the applied stress distribution is normalized

with respect to o'f. All expressions previously derived for V, are still applicable for V a with the

exception that mvN v should be substituted for mv (see section Fatigue Parameter Risk of

Rupture Compatibility). For constant g-factor, comparing equations (144) and (146) yields the

time-dependent scale parameter relationship

_ l/_v

toy = tov Vef

(148)

The modified effective volume which is used with equation (146) and the PIA model is

avNv

t' / mvN v mvN v mvNv /

Ve f = 1 Jv/O,mc=(x,y,z) + o2mc,Jx,y,z ) + oamc=(x,y,z))dV

(149)

where

o,mc, (x,y,z)> %_qc, (x,y,z) > O3_ie. (x,y,z) > 0

For the Batdorf theory, the modified effective volume analogous with equation (146) is defined as

NASA/TM--2003-106316 147
and

2_v

Vef - mvNv ]

sin ctdec dfl dV

(150)

Nv -2

BBv OoV

t°v - N.

g of

For surface flaws, the resulting equations are similar to those equations previously derived.

Integration is performed over the specimen surface, and the modified effective area Ace is obtained.

Pts(t_) = 1 - exp -A_f_) ]

(151)

where

N s-2

Bws OoS

tos -

Ns

g of

For the uniaxial Weibull distribution

TA{ . x \ I_sN s

Ad= O leqc_---(x'Y) / dA

of )

(152)

The time-dependent scale parameter is

V_s

tos = tos _ef

(153)

The modified effective area for the PIA model is

NASA/TM--2003-106316 148
A, Il l°SN':' ":s ":, '

= /o,_q¢ (x,y) + o_i_ (x,y)) dA L,od _^

(154)

For the Batdorf theory, the modified effective area is

Acf -

dA (155)

and

Ns-2

BBs Oos

tos -

Ns

g Of

Similar to cyclic fatigue, for dynamic fatigue the specimen time to failure can be expressed

using the two parameter Weibull form as

Pf = 1- exp

(156)

where

tOd

(N + 1)B. O0N-2.

.N

Of

1/(N+I)

tf is the time to failure and ted is the characteristic time (subscript d indicates dynamic fatigue). For

volume flaws substituting (N v + 1) for the g-factor and rearranging equation (146) then

tf mv('Nv+l) p.v t.. :1 I (157)

NASA/TM--2003-106316 149
where

Nv-2 ] l/(Nv+l) 1

.(N v + 1) Bwv OoV _a-_

toav _ = t°ov v ef

of

Note in equation (157) that crf denotes the maximum stress in the specimen and that O is the

stressing rate at that location. The derivation for surface flaws follows a similar line of reasoning.

For the Paris law the analogous relation to equation (144) is

Pf = I-exp

(15s)

where nf is the cycles to failure and no is a characteristic number of cycles. Similar to equation

045)

= - o l_q,o(X,y,z) dV

Oov mv

(159)

where

Oleq,0(x,y,z ) =

Olmc (x,y,z) (1 - R) r% nf l_v------'2

Bwv

Similar to equation (146) then

Pfv(nf) = 1 - exp [-Vef rn'/ vl i_oV ) J

(160)

where

NASA/TM--2003-106316 150
Nv-2 1

Bwv OoV my

nov = = nov Vef

Nv

(1 - R) Nv (If

The derivation for surface flaws follows a similar line of reasoning.

NAS A/TM--2003-106316 151
Fatigue Parameter Risk of Rupture Compatibility

To ensure compatibility of failure probabilities the relationships between the fatigue

parameters (N and B) and the various failure criterions must be established. From uniaxial test

specimen data (simple tension, 3- or 4-point bend) compatibility is derived by equating the risk of

rupture of the specimen uniaxial Weibull equation to the uniaxial Weibull, the PIA model, the

Weibull normal stress averaging, or the Batdorf shear-sensitive formulation.

For volume flaw analysis, the probabilities of failure for the various approaches mentioned

above are:

Specimen Uniaxial Weibull

Uniaxial Weibull

PIA Model

Normal Stress Averaging

Batdorf Model Pf = 1 - exp

Pf = 1 - exp I_ OI,O 

J

Pf = 1 - exp -l_v Ol,o (x,y,z) dV

Pf = 1 - exp -l_v % (x,y,z) dV

-2kBV "2 "2 OleqO(l]/) $ingt dg dis dV

1_

Subscript 0 denotes transformed stress to time t=0.

The basis for compatibility of failure probabilities is the requirement that all expressions

produce the same probability of failure for a uniaxiai stress state as that obtained from the specimen

uniaxial Weibull equation. The value of N remains invariant, whereas the value of B is adjusted to

satisfy this requirement. The approach is similar to that used to obtain the relationships of the crack

density coefficients for fast fracture reliability analysis. The most common experimental test

specimens used in the evaluation of the fatigue parameters are the uniaxial tension, three-point bend,

NASA/TM--2003-106316 152
andfour-pointbendgeometries.ThefatigueparametersN andB areobtainedfromthedatabased

solely on the maximum stress in the specimen at fracture and the time to failure.

For the specimen uniaxial Weibull equation the time dependent transformation equations are

Static Fatigue

Dynamic Fatigue oi,0

Cyclic Fatigue OI, 0

Of v tf Nv-2

O1, 0 =

_v ]___ of tf

B.v (Nv+I)

_v ]_-_ of g tf

Buy

>> 1, where tf is the time to fracture, g is the g-factor used with cyclic loading, err is

the maximum tensile stress in the specimen at fracture, and B,v is the fatigue parameter obtained

from the specimen uniaxial Weibull equation. Equating the risk of rupture of the specimen uniaxial

Weibull model to the uniaxial Weibull model risk of rupture gives

/ "/" Iv(" Of tf o I (x,y,z) tf

°ov uv Bwv °ov }

dV (161)

then

-vclfv(°l xyz') vNvo, dvV,v (162)

where

NASA/TM--2003-106316 153
ve:/oo__) °or o,:iv/.o,_,z>/°- I, of ) _v _,63>

For V=f, the static equivalent stress distribution Ol(x,y,z ) is normalized with respect to the maximum

equivalent static tensile stress af at t_. Thus V a for a given specimen configuration is similar to V=

for the same specimen with the exception that the exponent associated with Vef is mvNv whereas

the exponent associated with V= is my. When there is no stress gradient then V= and V=f are

equivalent. When stress gradients exist throught the specimen then these terms are not equal. For

the three and four-point bend specimen (for three-point bend, the inner span, L2, is set equal to

zero), equating the risk of rupture of the specimen uniaxial Weibuil equation with that of the

Bwv =Buv

uniaxial Weibuli expression

_ ]t/,;,v

vo (-1i ----_ _vUv) j B.v L_-_j

(164)

where

(1 +mv) :z

and

Vef = [ ]

wh (LI +mvNvL2)

2 ( I +mvN v):

For the PIA model the relationship between Bwv and Buv is

{..v). _ _o, +(.(x,y,.,)--]_odV- v. (165)

NASA/TM--2003-106316 154
Equatingtherisk of ruptureof the specimen uniaxial Weibuli to the Batdorf risk of rupture yields

the relationship between Bav and B,v

1 2 ksv

V e 'n:

%qW)

Of

mvNv

sintt da d13 dV (166)

where O_cq(_P) represents the static equivalent stress distribution. For the Weibull normal stress

averaging method, B.wv is substituted for Bav and a_n.o( _t' ) is substituted for _:cq,o(_P).

The relationship between B,_v and BBv is established by equating the risk of ruptures for the

uniaxial Weibull model and the Batdorf model

Swv mv =

L(°I'q(x'Y'Z)of

Of

mvNv

mvNv

dV

sin a d_ d13] dV

(167)

For the uniaxial stress state this expression becomes strictly a function of the fracture criterion. This

can be demonstrated with a shear insensitive fracture criterion (equivalent to the Weibull normal

stress averaging method) where

o_(_) = oi_(x,y,z)cos2 c_ (168)

so that

fvfoi x,Y,z,/mv v Of )dV

2 k'BV f Oleq(X,Y,Z)I mvNv

o, j fo _ fj cos2'vNvccsinccdcc d13 dV

(169)

NASAfFM--2003-106316 155
combined with equation (66) then

2 mv Nv + 1

2mv+l

(170)

For co-linear crack extension with a Griffith crack

OleqQP)= OXeq(X,y,z)COS a

(171)

so that

fv( )Ol_l(X'Y'Z)]mvYv Of dV

2kay r (°"_(x'Y'Z))_vNv [ fo _ fo_COS_aVYvasina dad,] aV Of

dV

(172)

combined with equation (68) then

Bwv ray m v N v + 1 (173)

For an arbitrary fracture criterion expressed as some function of the flaw orientation

Oleq(1P) = OXeq(x,y,z) f(a,13) (174)

then

lz

2 _'Bv _ ;0_ f_vNv(a,_) sin a da d13)

(175)

NASA/TM--2003-106316 156
For surface flaw analysis, the relationship between the specimen uniaxial Weibull fatigue parameter

Bus and the uniaxial Weibull B_, Weibull normal stress averaging B,ws and the Batdorf parameter

Bas is obtained in a similar manner to that used for the volume flaw analysis. Equating the risk of

rupture of the specimen uniaxial Weibull to the uniaxial Weibull equations

then

Ns / ms

of tf =

N s -2

Oos Bus

fA Ns

01 (x,y) tf

N s-2

Bws OoS

msNs

dA (176)

where

-,f,(o,'-x,y,/'"s _Ae, BuS A e of ) A

(177)

= ( ]ms-O0s O°S ) = fdA _(°'i(x'Y) ]ms--of J dA

A,, (178)

For Ac_ the static equivalent stress distribution oz(x,y ) is normalized with respect to the maximum

equivalent static tensile stress of at tf. Thus Act for a given specimen configuration is similar to Ae

for the same specimen with the exception that the exponent associated with A_f is msN s whereas the

exponent associated with A_ is m s. When there is no stress gradient then Ao and A a are equivalent.

When stress gradients exist throught the specimen then these terms are not equal.

For the four-point bend specimen (L2---0 for the three point bend solution)

(L 1 +msL2) (h +w +msW)

Ao = (179)

( 1 +ms) 2

Equating the risk of rupture of the specimen uniaxial Weibull equation to the uniaxial Weibull

formulation

NASA/TM--2003-106316 157
Bws= B.s

L j : B.s A0)

(180)

For the PIA model the relationship between B,,s and B,s is

('B-_)(Bws) '_s - A_I fA [(°I(x'Y) )_sSsof + (°2(x'Y) 1o' ) _sNs dA - AaAe

081)

Equating the risk of rupture of the specimen uniaxial Weibull to the Batdorf risk of rupture yields

the relationship between Bas and B.s

msNs

da dA (182)

where aleq(T) represents the static equivalent stress distribution. For the Weibuli normal stress

averaging method, B.,,v is substituted for BBv and cr_..o( T ) is substituted for O'leq,0( T ).

The relationship between B_ and Bas is established by equating the risk of ruptures of the

uniaxial Weibull model and the Batdorf model

ox_(x,y) ]'asss

~ -- dA

dA

(183)

For the uniaxial stress state this expression becomes strictly a function of the fracture criterion. This

can be demonstrated with a shear insensitive fracture criterion (equivalent to the Weibull normal

stress averaging method) where

NASA/TM--2003-106316 158
oim(_ ) = Oleq(X,y) cos 2 a (184)

so that

fA[ "_f _(Oleq(x'y)lmsNsdA

- ]

(185)

combined with equation (79) then

(Bws/m s = N s r (msNs) r s

I,_TBs] (N s -2)r (msNs + 2) F (ms)

(186)

For co-linear crack extension with a Griffith crack

oi,_(_) = oi_q(x,y) cos a

(187)

so that

IS.s/°,=

t_)

~

;_(°"<x'y'/m_'_'Of J

- /oi <x,y,-[So]

2

kBsf ) msNs cosmsNs _ da dA

"/_ J A Of ]

(188)

combined with equation (81)then

NASA/TM--2003-1063 i 6 159
Bws ms = Nsr( )r(ms 2-1) (189)

For an arbitrary fracture criterion expressed as some function of the flaw orientation

OI_qQP) = o1_q(x,y)f(a) (190)

then

( )ms

2 kss (So_ f ms_s(Ct) da )

(191)

NAS A/TM--2003-106316 160
Evaluation of Fatigue Parameters from Inherently Flawed Specimens

The lifetime reliability of structural ceramic components depends on the history of the

loading, the component geometry, the distribution of pre-existing flaws, and the parameters, N and

B, that characterize subcritical crack growth. These crack growth parameters must be measured

under conditions representative of the service environment. When determining the fatigue

parameters from rupture data of naturally flawed specimens, the statistical effects of the flaw

distribution must be considered along with the strength degradation effects of subcritical crack

growth. A more direct approach is to calculate fatigue parameters from velocity measurements of

an induced crack of known configuration, thereby eliminating the statistical aspects of the flaw

population from the experiment. The weakness of this approach, however, is the difficulty of

getting a notched or indented specimen to behave in an identical manner as the naturally flawed

specimen. CARES/LIFE is developed on the basis that fatigue parameters are most accurately

obtained from naturally flawed specimens. In the following discussion, three methods are described

to estimate these parameters from fatigue data: the median value technique, a least squares

regression technique, and a modification to a method from Jakus (Jakus, Coyne, and Ritter, 1978)

known as trivariant regression analysis. These methods are described in terms of volume flaw

analysis for static (or steady state cyclic) fatigue and for dynamic (constant stressing rate) fatigue

using the power law formulation. The Paris law methodology is given in terms of steady state

cyclic loading. Analogous relations for surface flaws are easily developed by replacing the effective

volume with the effective area.

Static and cyclic fatigue parameter evaluation

Rearranging equation (146) for static fatigue or steady state cyclic loading at a fixed level of

reliability, the specimen time to failure for volume flaws is expressed as a function of the maximum

NASA/TM--2003-106316 161
stress,of, in thespecimen(for cyclic loading this corresponds to oxmc_,_

in the specimen)

tf =

N v -2

BwV Oov -N v

Of

at the highest stressed point

(192)

The g-factor is assumed constant throughout the specimen and for all loading levels.

between the brackets are simplified by replacing them with a constant yielding

-N v

tf = A e of

The terms

(193)

Equation (193) is a convenient expression from which to fit experimental data, thus A c and N v can

be considered as material-environmental parameters. Taking the logarithm of equation (193) yields

en tr = en A - N ven af (194)

Linear regression analysis of the experimental data is used to solve equation (194) for the slope, -

Nv, and the intercept en A c. The fatigue parameter estimation techniques in CARES/LIFE estimate

Ac and N v for a probability of failure fixed at 50% (Ply = 0.50).

For the median value technique, CARES/LIFE uses the median value at each individual

stressing level as data points. Using equation (194) and performing least-squares linear regression

on the set of median values estimates the line corresponding to a failure probability of 50% with

slope -N v and intercept en A_. Details of the least squares solution technique are given in Pai

(Pai, and Gyekenyesi, 1988). The median value estimation method is the least efficient fatigue

NASA/TM--2003-106316 162
parameterestimationtechniquein CARES/LIFE(i.e.,theestimatedparameterhas the largest

confidence interval for a given sample size).

Another fatigue parameter estimation method incorporated in CARES/LIFE is referred to as

the least squares regression technique. This method is similar to the median value technique except

that linear regression using equation (194) is performed with all the fatigue data points (instead of

only the median values). The fatigue parameter N v is obtained from the slope of the regression line.

Assuming that the experimental data is at a sufficient number of discrete levels of applied stress, all

the data failure times tn are transformed to an equivalent failure time tT_at an equivalent single level

of stress, o r. Equating the failure probabilities calculated from equation (146) for data number i

yields

tri = t_ -- (195)

where the subscript T indicates a transformed value. In CARES/LIFE, the value of o r is the

lowest level of applied stress in the data set. With all the data transformed to the various values try,

CARES/LIFE performs Weibull parameter estimation as described in the section titled Estimation

of Statistical Material Strength Parameters, solving equation (144) for m v and tow. Substituting

into equation (193) for a time to failure corresponding to a 50% probability of failure yields

AC

= t°v'r °a" [ _ 1-0.50)J

N v -2

Bwv OoV

mV

1-0.50)/

Vcf J

(196)

where

NAS A/TM--2003-106316 163
Nv -2

Bwv noV

tOrT - 1

Nv _vv

g OT Vef

The third option in CARES/LIFE for estimating fatigue parameters is a modification to a

method used by Jakus (Jakus, Coyne, and Ritter, 1978) called trivariant regression analysis. This

estimation technique is referred to as the median deviation procedure. In this procedure, the fatigue

parameters and Weibull modulus are determined by minimizing the median deviation of the

logarithm of the time. The characteristic strength a0v is assumed known. From equation (195), the

fatigue data is transformed to a single stressing level for an assumed value of N v. Using equation

(144) and the previously mentioned least squares or maximum likelihood estimation methods, the

Weibull parameters m v and toy-r are obtained. With these parameters the median value, tr0.5, is

calculated (i.e., the value for tr when Pfv = 0.50). Using the transformed fatigue data as a discrete

variable the median deviation is defined as (using absolute values)

M.D. = _

1

i=l

Zkl_ntr- _n I(tov.r _n

I -

1

0.50/

./v_v

J = k

1

i_l

I]kl_ntr - _ntro,o I

(197)

for the k data points. The median deviation, M.D., is a measure of dispersion or scatter about the

median. It can also be obtained for the continuous variable defined by the Weibull parameters nav

and taw for ranked probabilities of failure Pn.

The value of N v for which the M.D. is a minimum establishes the solution. The scatter of the

distribution is measured with the Weibull modulus mv since for a fixed value of k the expression

inside the brackets of equation (198) is a constant. CARES/LIFE minimizes equation (198) by

NAS A/TM--2003-106316 164
= 1 Z Qn OVT Qn

M.D. k i=l

I_1 v i_l

maximizing ffav versus Nv.

1 --Pn 1 )l/ray]- _ntTo 5 i

(198)

Qn

This process is iterative, covering an appropriate range of values of Nv.

After a solution for N v is obtained, equation (196) is used to calculate A¢.

The median deviation procedure was investigated with Monte-Carlo simulations of static

fatigue data. For sample sizes of 20 and 30 specimens each, 10,000 simulations were run where the

fast fracture Weibull modulus, m, randomly varied between 2.0 and 30.0, N randomly varied

between 10.0 and 60.0, and the number of stressing levels randomly varied between 2 and the

sample size. The parameters aoV and B u were fixed at 100.0 and 10000.0/N+I , respectively.

The results of the simulations were compared to estimates calculated using the median value

technique for the same fracture data. Examination of the 90% confidence intervals indicated that

the median deviation procedure yielded better results than the median value technique. Using

maximum likelihood estimation (MLE) gave better results than using least squares (LS) estimation

with the median deviation procedure. For the crack growth exponent, N, the 90% confidence

interval from the median deviation procedure (with MLE) was about 70% of the range of the

median value 90% confidence interval, with no bias indicated with either estimator. For the crack

growth constant, Bu, the 90% confidence interval for the median deviation procedure was

dramatically smaller than the median value 90% confidence interval, with negligible bias indicated

with the median deviation method and significant negative bias with the median value method.

NASA/TM--2003-106316 165
ThefatigueparameterBwvisobtainedby comparing equations (192) and (193) for a 50%

probability of failure

Bwv

Nv-2 On 1 oov

(199)

Alternatively, Bwv can be obtained by equating equations (144) and (145)

BwV =

Nv- lltnv

toy g of V d

Nv-2

OoV

(200)

Information on the underlying inert strength distribution can also be obtained from the fatigue

data. Using equation (195) to transform all the fatigue data to a single Weibull distribution and

performing least squares or maximum likelihood analysis establishes the parameters m v and tow.

The fast-fracture Weibull modulus is then solved as

m v = m v (Nv-2) (201)

where the superscript ' denotes a fast-fracture parameter estimated from fatigue data. The fastfracture characteristic strength Oov cannot be estimated from the fatigue data. CARES/LIFE

calculates a characteristic strength, O'0v, based on extrapolation of the fatigue data to a specific

failure time. This time is arbitrarily fixed at 1/(N+I) seconds for static loading (equivalent to 1.0

second for dynamic loading). From equation (195) then

O'ov = oT [tow (Nv + 1)] l_v (202)

NASA/TM--2003-106316 166
Muchmorestatisticaluncertaintyis associated with the determination of the Weibull modulus than

with the characteristic strength for fast-fracture testing.

With the calculated quantities 6'0v and m'v, the fatigue data can be transformed to an

equivalent inert strength distribution by equating the risk of rupture of equations (55) and (144) for

the various transformed fatigue data values

. r_-2 (203)

oni = aov _t0v.r)

where ani represents the i'th transformed inert fracture strength of the specimen. Plotting the

p

ranked values ani ( En en(1/1-Pfv ) versus En _ ) gives useful visual information for the analyst

i

(CARES/LIFE does not include a graphical interface). With O'0v, m'v, and the various ofl' values,

CARES/LIFE performs the outlier test and determines the K-S and A-D goodness-of-fit statistics as

explained in the section titled Estimation of Statistical Material Strength Parameters. The

outlier test and goodness-of-fit statistics in this case are also valid for the transformed fatigue data

(denoted with subscript T) and the original fatigue data as well. Hence the calculated goodness-offit significance levels measure the hypothesis that fatigue data was generated from the parameters

N v, A c, and m' v.

Dynamic fatigue _arameter evaluation

Fatigue parameter estimation methodology for dynamic fatigue is similar to the power law

formulation for static and cyclic fatigue. Rearranging equation (157) for a fixed level of reliability,

the specimen failure stress of for volume flaws is expressed as a function of the of the stressing rate

at the highest stressed point in the Component

NAS A/TM--2003-106316 167
Of =

N v -2

(Nv + 1) Bwv Oov

[ ],,°+

I/(Nv*I)

{_II(Nv+I)

(204)

The terms between the brackets are simplified by replacing them with a constant yielding

of = A d O I/_v÷D (205)

Equation (205) is a convenient expression from which to fit experimental data, thus A d and N v can

be considered as material-environmental parameters. Taking the logarithm of equation (205) yields

1

= + en o (206)

gn of tn A d N v + 1

Median value, least squares and the median deviation technique are used to solve equation (206) as

previously discussed for static and cyclic loadings.

Assuming that the experimental data is at a sufficient number of discrete levels of stressing

rates, all the data failure times tn are transformed to equivalent failure times tr, at a fixed stressing

rate 6 x. Equating the failure probabilities calculated from equation (156) for data number i yields

0 i ] Nv/fNv * 1)

_+. -- t. _+6---_)

(207)

where the subscript T indicates a transformed value. In CARES/LIFE, the value of 6 r is the

lowest stressing rate in the data set. With all the data transformed to the various values tri ,

CARES/LIFE performs Weibull parameter estimation as described in the section titled Estimation

of Statistical Material Strength Parameters, solving equation (156)for mv(Nv+l) and tow.

NASA/TM--2003-106316 168
Substitutingintoequation(205)for a timeto failurecorrespondingto a 50%probabilityof failure

yields

Ad= t0dv'r T 1 t l-0"50JJ I = V + 1) B,_v OoV

l_l( 1 /l'/&v/'#°%+')

t_--_.5o) / ?

vo, j j

(208)

where

t0dVT = t-(Nv

Nv-2] I/(Nv+ 1)

.__)B.vOov

Nv _ l/m v ]

0 T Vef J

The median deviation method for estimating fatigue parameters minimizes the median

deviation of the logarithm of the time to failure. From equation (207), the fatigue data is

transformed to a single stressing rate for an assumed value ofN v. Using equation (156) and the

previously mentioned least squares or maximum likelihood estimation methods, the Weibull

parameters mv(Nv+l) and t0dv-r are obtained. With these parameters the median value, tro_ , is

calculated (i.e., the value for tr when Pry = 0.50). Using the transformed fatigue data as a discrete

variable the median deviation is defined as

M.D. = _1 _'t_tr_ - _n It/ Odv'r l!n 1/] -

k i=l 1 50 I:_l z'1_tr__%_ I <:09, k i=i

for the k data points. The median deviation, M.D., is a measure of dispersion or scatter about the

median. It can also be obtained for the continuous variable defined by the Weibull parameters

mv(Nv+l ) and todvr for ranked probabilities of failure Pn.

NASA/TM--2003-106316 169
M.D. = _i_lk•=

_n f/OdV.r _n 1 -1

Pn /11- Qn tTo5

mv (Nv + 1)

1

1 -Pa

_n

1

1-0.5

(210)

The value of N v for which the M.D. is a minimum establishes the solution. The scatter of the

distribution is measured with the Weibull modulus mv(Nv+l) since for a fixed value of k the

expression inside the brackets of equation (210) is a constant. CARES/LIFE minimizes equation

(210) by maximizing tnv(Nv+l) versus N v. This process is iterative, covering an appropriate range

of values of N v. After a solution for N v is obtained, equation (208) is used to calculate A d.

The median deviation procedure was investigated with Monte-Carlo simulations of dynamic

fatigue data. For sample sizes of 20 and 30 specimens each, 10,000 simulations were run where the

fast fracture Weibull modulus, m, randomly varied between 2.0 and 30.0, N randomly varied

between 10.0 and 60.0, and the number of stressing rates randomly varied between 2 and the sample

size. The parameters Oov and B were fixed at 100.0 and 10000.0/N+I , respectively. The results

of the simulations were compared to estimates calculated using the median value technique for the

same fracture data. The conclusions reached with dynamic fatigue simulations were identical to

those obtained with the static fatigue simulations.

The fatigue parameter Bwv is obtained by comparing equations (204) and (205) for a 50%

probability of failure

NASA/TM--2003-106316 170
BwV =

AdNv+l llr_ v

N v -2

(N v + I) OoV

Vef

Altematively, Bwv can be obtained by equating equations (156) and (157)

Bwv =

Nv+I l/_av

tOdv 6 Nv Ycf

N v -2

(N v + 1) OoV

(211)

(212)

Information on the underlying inert strength distribution can also be obtained from the fatigue

data. Using equation (207) to transform all the fatigue data to a single Weibull distribution and

performing least squares or maximum likelihood analysis establishes the parameters mv(Nv+l) and

tovT. The fast-fracture Weibull modulus is then solved as

m,v = mv (Nv_2) (213)

where the superscript ' denotes a fast-fracture parameter estimated from fatigue data. The fastfracture characteristic strength C%v cannot be estimated from the fatigue data. CARES/LIFE

calculates a 'characteristic strength, C'0v, based on extrapolation of the dynamic fatigue data to a

specific time. This time is arbitrarily fixed at 1.0 second. From equation (207)

. (Nv+ I)/Nv

O'0V = 0 t0dVT

(214)

Much more statistical uncertainty is associated with the determination of the Weibull modulus than

with the characteristic strength for fast-fracture testing.

With the calculated quantities C'0v and m' v, the fatigue data can be transformed to an

equivalent inert strength distribution by equating the risk of rupture of equations (55) and (156) for

the various data values

NASA/TM--2003-106316 171
/ \ Nv+I

Ofli = OOV

(215)

where oaj represents the i'th transformed inert fracture strength of the specimen. Plotting the

w

ranked values Oni ( en en(1/1-Pfv) versus en c ) gives useful visual information for the analyst

w

(CARES/LIFE does not include a graphical interface). With cr'0v, m' v , and the various on, values,

CARES/LIFE performs the outlier test and determines the K-S and A-D goodness-of-fit statistics as

explained in the section titled Estimation of Statistical Material Strength Parameters. The

outlier test and goodness-of-fit statistics in this case are also valid for the transformed fatigue data

(denoted with subscript T) and the original fatigue data as well. Hence the calculated goodness-offit significance levels measure the hypothesis that fatigue data was generated from the parameters

Nv, An, and m'v.

Cyclic fatigue parameter evaluation

Fatigue parameter estimation for the Paris law is identical to the power law formulation for

cyclic fatigue except that cycles replaces time and the g-factor is replaced with (l-R) N. It is

important to point out that the Paris law as formulated in CARES/LIFE assumes that no crack

extension occurs when loading on the crack face is compressive, hence negative R ratio loading is

equivalent to R = 0 loading. For this parameter estimation technique cyclic data is required at two

or more applied loading levels and the R ratio must be held constant for all the data. Rearranging

equation (158) for steady state cyclic loading at a fixed level of reliability, the specimen cycles to

failure for volume flaws is expressed as a function of the maximum static equivalent stress, crf, in

the specimen (crf = Oleqe_ at the highest stressed point in the component)

NASA/TM--2003-106316 172
nf =

Nv-2

Bwv Oov

(I - R) Qv ]"°,

Qn 1

-N v

Of

(216)

The terms between the brackets are simplified by replacing them with a constant yielding

-N v

nf = A c Of

(217)

Equation (217) is a convenient expression from which to fit experimental data, thus A0 and N v can

be considered as material-environmental parameters. Taking the logarithm of equation (217) yields

en nf = en A c - N ven af (218)

Median value, least squares and the median deviation technique are used to solve equation (218) as

previously discussed for the power law methodology.

Assuming that the experimental data is at a sufficient number of discrete levels of applied

stress, all the data cycles to failure n n are transformed to an equivalent number of cycles nx_ at an

equivalent single level of (maximum) stress, c_T. Equating the failure probabilities calculated from

equation (160) for data number i yields

(°j] Nv (219)

nTi = rift _ OT)

where the subscript T indicates a transformed value. In CARES/LIFE, the value of crT is the

lowest level of applied (maximum) stress in the data set. With all the data transformed to the

various values nT_, CARES/LIFE performs Weibull parameter estimation as described in the section

titled Estimation of Statistical Material Strength Parameters, solving equation (158)for m v and

NAS A/TM--2003-106316 173
now. Substitutingintoequation(217)forthenumberof cyclescorrespondingto a 50%probability

of failureyields

l C (-1--/1

(220)

where

nOV T =

N v-2

Bwv OoV

(1 _ R) Qv o Nv T velTv

The median deviation method for estimating fatigue parameters minimizes the median

deviation of the logarithm of the number of cycles. From equation (219), the fatigue data is

transformed to a single stressing level for an assumed value ofN v. Using equation (158) and the

previously mentioned least squares or maximum likelihood estimation methods, the Weibull

parameters nav and now are obtained. With these parameters the median value, _oJ' is calculated

(i.e., the value for n T when Pfv = 0.50). Using the transformed fatigue data as a discrete variable the

median deviation is defined as

M.D. = _

1

i-i E _nn% - Qn ova" _n-

1

0.50 ;'vii =i

1

i=l I]'l_nn.rt - Qn na.o5"

° I

(221)

NASA/TM--2003-106316 174
for the k datapoints. Themediandeviation,M.D.,is a measureof dispersionor scatteraboutthe

median.It canalsobeobtainedfor thecontinuousvariabledefinedbytheWeibullparametersmv

andnowforrankedprobabilitiesof failurePw

1 _kl C_n OVT_n M.D. k i°l 1 )l_v] _

1 --Pn

[QI11_p_._l

(222)

The value of N v for which the M.D. is a minimum establishes the solution. The scatter of the

distribution is measured with the Weibull modulus mv since for a fixed value of k the expression

inside the brackets of equation (222) is a constant. CARES/LIFE minimizes equation (222) by

maximizing m v versus Nv. This process is iterative, covering an appropriate range of values of N v.

Atter a solution for N v is obtained, equation (220) is used to calculate A c.

The fatigue parameter Bwv is obtained by comparing equations (216) and (217) for a 50%

probability of failure

BwV =

A c (1 - R) Qv

N v -2

Oov

Vef

l/my

(223)

Alternatively, Bwv can be obtained by equating equations (158) and (160)

NAS A/TM--2003- !06316 175
Nv I/_v

nov (I - R)Qv of V_e (224)

Bwv = Nv_2

Oov

Informationon theunderlyinginertstrengthdistributioncan alsobe obtainedfrom thefatigue

data. Using equation (219) to transform all the fatigue data to a single Weibull distribution and

performing least squares or maximum likelihood analysis establishes the parameters rn v and now.

The fast-fracture Weibull modulus is then solved as

m'v = v-2) (225)

where the superscript ' denotes a fast-fracture parameter estimated from fatigue data. The fastfracture characteristic strength t_ov cannot be estimated from the fatigue data. CARES/LIFE

calculates a characteristic strength, O'ov, based on extrapolation of the fatigue data to a specific

number of cycles. This number is arbitrarily fixed at 1/(N+I) cycles.

O'0v = or [n0VT(Nv + I)]vNv (226)

Much more statistical uncertainty is associated with the determination of the Weibull modulus than

with the characteristic strength for fast-fi'acture testing.

With the calculated quantities c_'ovand m' v, the fatigue data can be transformed to an

equivalent inert strength distribution by equating the risk of rupture of equations (55) and (158) for

the various data values

' ' 'N_2

oni = Oov t,n0v.r}

(227)

NAS A/TM--2003-106316 176
s

where oni represents the i'th transformed inert fracture strength of the specimen. Plotting the

s

ranked values afl i ( en en(1/1-Pfv) versus gn t_ ) gives useful visual information for the analyst

(CARES/LIFE does not include a graphical interface). With O'0v, m' v , and the various on, values,

CARES/LIFE performs the outlier test and determines the K-S and A-D goodness-of-fit statistics as

explained in the section titled Estimation of Statistical Material Strength Parameters. The

outlier test and goodness-of-fit statistics in this case are also valid for the transformed fatigue data

(denoted with subscript T) and the original fatigue data as well. Hence the calculated goodness-offit significance levels measure the hypothesis that fatigue data was generated from the parameters

N v, Ao, and m' v.

NAS A/TM--2003-106316 177
Proof Testing Effect on Component Service Probability of Failure

Prior to placing a component in service, confidence that it will perform reliably is usually

demonstrated through proof testing. Another method, nondestructive evaluation (NDE), is used to

remove components with discernable but unacceptable flaw sizes (Wiederhorn, and Fuller, 1985).

Applied NDE is still in its early stages of development and at present not widely accepted because

detection of small flaws, which can cause catastrophic failure, is uncertain.

Ideally, the boundary conditions applied to a component under proof testing simulate those

conditions the component would be subjected to in service and the proof test loads are appropriately

greater in magnitude over a fixed time interval tp. After proof testing, the survived component is

placed in service with greater confidence in its integrity and a predictable minimum service life, tmin,

with reliability equal to one.

The objective of the following analysis is to predict the attenuated probability of failure of a

component in service after proof testing and the minimum life expectancy of the proof tested

component. This concept will then be extended to predict the component reliability for off-axis

proof testing (misaligned and dissimilar multiaxial loads). All derivations in this section are for

static fatigue and volume flaws. If the proof test and service stress distributions are cyclic, the gfactor approach is required to transform the stresses to their equivalent static stresses. Analogous

relationships may be developed for the surface flaw solution.

The attenuated probability of failure, Pfav, of a component surviving proof testing for time tp

and subjected to an in service equivalent static stress distribution O_q(tI') over a time interval (tq-tp)

is (Weibull, 1939)

Pfav(tq) = Peiv(tq) - Pfpv(tp)

1-Pfpv(tp)

(228)

NAS A/TM--2003-106316 178
Theterm Pfpv(tp) is the probability of failure of a survived component subjected to a proof test static

equivalent stress distribution, O_eqp(_P),over a time interval denoted by tp. The term P_v(tq) is the

probability of failure of a survived component subjected to a proof test static equivalent stress

distribution al_(_P) over time interval tp and an in service static equivalent stress distribution crXeq(q")

over time interval (tq - tp). The reliability of the survived component increases as the ratio of the

proof test stress to the service stress increases.

For the Batdorf model, the probability of failure of a given component over time interval tp is

[_2 kBv f f,_r_f,_t2ro (_)1 my sin_ da d[3 dV ] (229)

Pfvv(tp) = 1-exp [ n JvJo Jo L lmp,o j

l

where, from equation (97), the transformed proof test stress distribution over time interval t o is

Nv ]l/(Nv-2)

Nv -2 /

ai_,0(_) = ol_(__ ) tp + °I_' (_)1

For the PIA model

where

P_v(tp) = 1-exp I fv mv mv mv ]-k_v (Oipl, 0 + alp2,0 + aip3,O) dV

Nv ] l/(Nv-2)

api (x,y,z) tp Nv-2

Oipi,O(l_) = _ + Opi (x,y,z)

(230)

for i=1,2,3, op_(x,y,z), Op2(X,y,z), and Op3(X,y,z) are the proof test principal tensile stress

distributions.

Alternately, the probability of failure based on the Weibull normal stress averaging method is

NASA/TM--2003-106316 179
where

Pbv(tv) = 1-exp (-k_, v Olpn(X,y,z ) dV )

(231)

--m V

oi_(x,y,z ) =

L Oi_o(te) mv dA

and

Olpn,O(l:l/) --

Nv r_-2 ]r_----5

Olpn(_t) tp +alpn (_P)!

Bnwv )

_p.(x,y,z) is the proof test average normal stress from the projection of _p,(q_), the normal stress,

over the surface area of a unit radius sphere (equation(11)).

The probability of failure of a given component over time interval tq is calculated as follows

for the Batdorf model

-1

- 2 kay n/2 n/'2 mY " /

Peiv(tq)= 1-exp fvfo fo [aI_q,°(_P)] sm a da d_ dV J

(232)

where

%_,o('I')=

Nv Nv ] I/(Nv-2)

Nv-2 ]

oi_(_P)(tq-tp)+Ol_(_P) tp + °I_ (_P)

Bnv

For the PIA model

Pnv(tq) = 1-exp fv mY mY mY 1-kwv (otl,0 + oi2,o + oi3,_)dV (233)

where

NASA/TM--2003-106316 180
Nv

°n,o(te)= oi (x,y,z) N v ] l/(Nv-2)

(tq-tp) + Opi (x,y,z) tp Nv-2

+ 0 i (x,y,z)

Bwv

for i=1,2,3, o_(x,y,z), o2(x,y,z), and a3(x,y,z ) are the principal tensile stress distributions.

For the Weibull normal stress averaging method

Ptw(tq) = 1-exp [-k_v L °_v(x'Y'Z)dV]

where

_m V

o h (x,y,z) =

mv

O_,O(_)dA

and

IN+ N+ Oin (_)(tq-tp) + O1pn(_)tp Nv-2

= + om (tl')

(234)

i,(x,y,z) is the in-service average normal stress from the projection of the normal stress

distribution al.(tI ') over a unit radius sphere. After determining the values of Pnv(tq) and Pfpv(tp), the

service component attenuated probability of failure Pf,v(tq) is computed. Depending on the

magnitude of the service load and time of application, Pnv can be less than Prpv. For this case, the

attenuated probability of failure, Pf.v, is 0.

The minimum life expectancy of a survived component for a static equivalent stress distribution is obtained by satisfying the condition Pnv(tq) = Pfpv(tp). For the Batdorf model, equating

equations (232) and (229) results in

NASA/TM--2003-106316 181
 :35,

where the subscript min denotes the smallest value of the term in equation (235) for all

throughout the component. If at any location the component proof test stress level is less than the

service stress level, then an assured minimum lifetime t_. does not exist and the component can not

be guaranteed to survive for any given time during service loading. When the proof test loading is

identical to the service loading, except for the magnitude of the loads, then the minimum value for

equation (235) occurs at the maximum stressed point in the component. Analogous relations for tmin

may be obtained for the PIA model and the Weibull normal stress averaging method.

Proof Testing - Off-Axis Loading

Often, the proof test loading does not exactly simulate the loading during service. In such

cases the component stress distribution during testing differs from that during service, resulting in

what is known as off-axis loading (misaligned and dissimilar loadings). Equation (228) is used to

calculate the attenuated failure probability for an off-axis tested component. The Batdorf model or

Weibull's normal stress averaging method are applied. Regions where the service stress is

compressive do not enter in the computation of Pepv.

As discussed in the section Time Dependent Reliability Analysis, for off-axis loading, the

load direction and/or load boundary differs from the service load condition. Symmetry of all the

finite elements is now with respect to a global coordinate system. For the Batdorf model, the

probability of failure for a given component over time interval tp is then

[-kBv 2_ J0_[°"_P,O(tF)] 1

f sin {_ da dp dV

Pfvv(tp) = 1 - exp [ 2_ fv fo 2 m_ (236)

and the probability of failure over time interval tq is

NASA/TM--2003-106316 182
(237)

whereal_a_,o(W) andcrlcq,0(W) arethepreviouslydefined transformed stress distributions.

Substituting equations (236) and (237) into equation (228) and simplifying yields the expression for

the attenuated failure probability

Pf, v(tq) = 1-exp --_--= fv fo fo [O,.q.o(W)- O,eqp.o(°x/)] H(xl:t) sinot dot d_ dV (238)

where

H(_) = 1 t=i_(W) < tq - tp

H(tF) = 0 tmi_(W) _ tq - tp

H(W) = 1 Oleq(_ ) > Oleqp(tP)

The Heaviside function is introduced to account for tm_,,which is now evaluated locally at W.

NASA/TM--2003-106316 183
Users' Guide

This section is a guide to the execution of CARES/LIFE and the preparation of all data files

necessary in the reliability analysis. This includes the files associated with the finite element

interface, the user input file for the parameter estimation module, and the user input file for the

reliability analysis module. Examples of each type of file are presented followed by an explanation

of how the variables in the files are designated and/or manipulated. Initially, the finite element

interfaces and how they process information are discussed. This information is stored in a neutral

data base which is subsequently accessed by the reliability analysis module (and the user if the need

arises). As noted later, the user must coordinate the size of the neutral data base and the blank

common array before attempting to execute a reliability analysis for a component. The steps

required to execute the parameter estimation module are presented along with examples of the input

and output files associated with this module. Lastly, the results of a component reliability analysis

are stored in an output file, and the format of this output file along with the details regarding how

the user manipulates the format are discussed.

Each user is given access to the entire CARES/LIFE source code (interface modules, reliability

analysis module, parameter estimation module) so the user can make changes in the code locally.

Since all of the modules contained in the CARES/LIFE algorithm are programmed in standard

FORTRAN77, the user must have access to a FORTRAN compiler. The algorithm has been

programmed in a general fashion, and system specific FORTRAN statements have been avoided.

Finite Element Interface Module

Three interface programs have been developed to process MSC/NASTRAN, ANSYS, and

ABAQUS output files. It is left to the user to ensure that the appropriate flags are included in the

finite element input file (e.g., bulk data deck) so that all data pertinent to a reliability analysis is

included in the finite element output file. The pertinent information is specified in a series of papers

NAS A/TM--2003-106316 185
(^Powers..,Edwards..,andPintz)where the individual interfaces are discussed. Details given in this

section apply to the NASCARES interface program. The preparation of the NASTRAN BULK

DATA file is described in MSC/NASTRAN Finite Element Analysis.

Certain information must appear in the finite element output file for the execution of the

interface. The information includes: the temperatures, stresses, and the volume/area of each finite

element. The user must ensure that this information is included in the finite element output file. If

this information is not provided in the output file, the execution of the interface program terminates.

Before execution of the interface program (given the default name NASCARES.FOR, the user

must initially compile and link the interface program on the host system. In addition, the user must

compile and link the program each time changes are made to the source code. The most common

changes to the source code involve the size of the blank common. The following error

"INSUFFICIENT MEMORY ALLOCATION"

"TOTAL MEMORY ALLOCATED"

"MEMORY NEEDED"

will be generated if the blank common is not large enough to accommodate the data for the analysis.

If this occurs the value of MTOT in the PARAMETER statement must be increased so that it is

greater than the memory needed. Another instance where modification of the source code may be

necessary, is the assignment the finite element output and the neutral data base file to the proper

logical units. These are identified in the source code as LUA = 7 for the finite element output, and

LUB = 8 for the neutral data base file. These files are assigned through the use of an "OPEN"

statement within the source code. Initially, the file names are assigned interactively. The user is

prompted for the filenames when the program is executed. Another option is to delete (or comment

out) those lines of code which query the user for the filename and insert the appropriate file names

into the OPEN statements. This re.quires editing the source code and placing the name of the

NASA/TM--2003-106316 186
MSC/NASTRANoutputfile in the "OPEN" statement assigned to LUA. The "OPEN" statements

reside at the beginning of the interface source code, and appear as follows:

OPEN(UNIT=LUA,FILE='NASTRAN.OUT', STATU S='OLD ')

OPEN(UNIT=LUB,FILE='CARES.NEU',STATUS='NEW')

The user replaces "NASTRAN.OUT" with the name of the MSC/NASTRAN output file as it

appears in the present working directory. Also, the default name given to the neutral data base file

is CARES.NEU. The name of this file can be modified at the discretion of the user.

throughout this manual the default name is used in reference to the neutral data base.

directed to Section CARES.NEU File for a detailed description of the neutral data base file.

However,

The reader is

CARES.NEU File

The neutral data base (CARES.NEU) is the output file created by either of the finite element

interfaces, and it contains several groups of format specific lines (or cards). Note that the user can

insert comments at the beginning of the neutral data base in order to document a specific component

analysis. Comment lines must begin with the letters COM in the first three spaces of the line. This

approach in documenting a specific reliability analysis will not jeopardize the file structure of the

neutral data base, and thus halt the execution of a reliability analysis.

An example of the different fields as they appear in CARES.NEU file is provided below. An

explanation of the terms associated with each field follows the example. The data fields are

identified by the variable names as they appear in the source code of the reliability analysis program

(CARES/LIFE).

The following is a brief description of each parameter and data field in the CARES.NEU file.

NAS A/TM--2003- i06316 187
TITLE CARD - usedto identify the CARES.NEU file by providing a short description of the

model.

TITLE

Ex. 5: Rotating Annular Disk

FORTRAN format: A80

TITLE - The default title card generated by the interface module identifies the finite

element model by obtaining the title from the output. Thus the user must edit the

CARES.NEU file aider the data base has been created and tailor the title card to a

specific component if more detail is required. The default title is also used as the title

for postprocessing.

NASA/TM--2003-106316 188
CONTROLCARD- input the parameters for the analysis.

NUMEL NUMELB

5 0

FORTRAN format: 215

NUMEL - The number of element groups. All element types: volume, surface, and

axisymmetric are included.

NUMELB - Flag for shell element groups with volume processing. If volume

processing is performed, the stress in the shell element is assumed to be constant

through the thickness.

0 - No volume processing of shell element groups.

1 - volume processing of shell element groups.

NASA/TM--2003-106316 189
ELEMENTGROUPCARD- contains three parameters that specify element group information.

INUEG IEGTYP IGREL ISHELL

1 8 .] a0 0

FORTRAN format: 415

INUEG - identification index of the element group.

IEGTYP - an integer that specifies the element type. This integer is intended to facilitate the

visualization of the CARES/LIFE results. The user has the option to assign values which are

appropriate for a given application. For the current versions of NASCARES and

ABACARES, the element type corresponds to PDA/PATRAN's shape parameter. These

values are:

3-

4-

5-

6-

7-

8-

TRIANGULAR

QUADRILATERAL

TETRAHEDRON

PYRAMID

PENTAHEDRON (WEDGE)

HEXAHEDRON (BRICK)

IGREL - an integer that specifies the number of elements within the element group.

ISHELL - Shell element flag.

0 - volume element group. For subsequent reading of the neutral file on the

subelement level, the six components of stress will be anticipated for each

integration point.

1 - surface element group. For subsequent reading of the neutral file on the

subelement level, the three in-plane stress components will be anticipated for

each integration point.

An Element Group Card appears for each element group. Note that within the element group, the

data is categorized at two levels which corresponds to the information contained on the ELEMENT

CARD and the SUBELEMENT CARD.

NASA/TM--2003-106316 190
ELEMENTCARD - containsinformationthatidentifieseachelementandincludes

IELNUM ISUBEL MATINP MAT2 ELVOL ELTEMP ELTHIC

1 27 300 0 0.5000 25.0 0.0

FORTRANformat:415,3E15.6

IELNUM - theelementidentificationinteger

ISUBEL- thenumberof subelementswithineachelement.Thenumberof subelementsin

eachelementdependsontheGaussianintegrationorderandtheelementype.

MATINP- thematerialidentificationnumberfor volumeprocessingof thiselement.This

valuemustmatchthematerialidentificationnumber(MATID) whichis definedin the

materialcontrolinputsectionof thetempletfile.

MAT2 - thematerialidentificationnumberfor surfaceprocessingof this element.Thisvalue

mustmatchthematerialidentificationnumber(MATID) whichis definedin thematerial

controlinputsectionof thetempletfile.

ELVOL - elementvolumefor volumeelementsor elementareafor surfaceelements.

ELTEMP- averageelementemperatureor the temperature at the center of the element.

ELTHIC - element thickness, set equal to zero for volume elements.

NAS A/TM--2003-106316 191
SUBELEMENTCARD- contains information that define the subelements of the finite element

model. The subelement card has two forms: one for volume elements and the other for surface

elements.

VOLUME ELEMENT

ISUNUM SUBVOL

0.1072

SUBTEM

25.0

SSTR(1)

_XX

SSTR(2)

_yy

SSTR(3)

SSTR(4)

(Yxy

SSTR(5) SSTR(6)

_yz _ZX

FORTRAN format: I5,5E 15.6,/,35X,3E 15.6

SSTR(1) - 6xx,

SSTR(2) - 6ry,

SSTR(3) - c=,

SSTR(4) - Crxy,

SSTR(5) - Cy_,

ISUNUM - the subelement identification integer

SUBVOL - the subelement volume

SUBTEM - the subelement temperature

the stress component in the x direction

the stress component in the y direction

the stress component in the z direction

the shear stress component in the xy-plane

the shear stress component in the yz-plane

SSTR(6) - C_x, the shear stress component in the zx-plane

NASA/TM--2003-106316 192
SURFACEELEMENT

ISUNUM SUBVOL

1 0.03858

SUBTEM

25.0

SSTR(1)

(_XX

SSTR(2)

O'yy

SSTR(3)

(_xy

FORTRAN format: I5,5E15.6

ISUNUM - the subelement identification integer

SUBVOL - the subelement area

SUBTEM - the subelement temperature

SSTR(1) - a_,, the stress component in the x direction

SSTR(2) - ayy, the stress component in the y direction

SSTR(3) - axy, the shear stress component in the xy-plane

NASA/TM--2003- i 06316 193
Parameter Estimation Module C4PEST

Prior to execution of the parameter estimation module, the user must assign the input and

output files to the correct logical units which are specified in the FORTRAN source code. Three

files are necessary for parameter estimation: the input, the output, and the file created by C4PEST

for subsequent input to C4LIFE. These are identified in the source code as LUA, LUB, and LUF,

respectively. These files are assigned through the use of an "OPEN" statement within the source

code. Initially, the file names are assigned interactively. The user is prompted for the filenames

when the program is executed. Another option is to delete (or comment out) those lines of code

which query the user for the filename and insert the appropriate file names into the OPEN

statements. This requires editing the source code and placing the name of the file into the

appropriate "OPEN" statement. The "OPEN" statements reside at the beginning of the source code,

and appear as follows:

OPEN(UNIT=LUA,FILE='C4PEST.INP',STATUS='OLD',FORM='FORMATI'ED ')

OPEN(UNIT=LUB,FILE='C4PEST.OUT',STATUS='UNKNOWN')

OPENCtYNIT=LUF,FILE='C4LIFE.INP',STATUS='UNKNOWN ')

Note that the user replaces "C4PEST.INP", "C4PEST.OUT", and "C4LIFE.INP" with the names of

the files as they appear in the present working directory. Throughout this manual the default names

are used to reference the files.

C4PEST.INP File

The parameter estimation module requires an input file that contains the failure data, failure

mode (i.e., failure caused by volume or surface defects), type of specimen, and the requested

estimation method. The data must be entered in the fixed format specified in the INPUT

INFORMATION section. The following are examples of the necessary data for time-dependent

NASA/TM--2003-106316 194
analyses.Examplesfor thefollowingareincluded:Fast-Fracture, StaticFatigue,CyclicFatigue,

Cyclic Fatigue:ConstantR-Ratio,andEffectiveVolumeandAreaComputation.

C4PEST.OUT File

The output file for the parameter estimation module is given the default name C4PEST.OUT.

Information contained in this file includes a summary of the parameter estimation procedure, the

values of the goodness-of-fit statistics for each estimation method, and the Weibull parameters for

the different temperature levels.

NASA/TM--2003-106316 !95
Weibull

Templet File Keywords for

Parameter Estimation With Module

Fast-Fracture

(input file C4PEST.1NP keyword settings)

Key [] user must specify, -- default value or user input

C4PEST

Master Control Input:

HE = 0or3

NMATS = []

NMATV= H

NGP -

TIME = O0

INTERP = --

Temperature Independent Material Control Input:

MATED = {J

IDI : [1

ID4 = []

11)2 = N

ID3 = []

IKBAT = I

PR = --

MLORLE = --

OUTLIE = --

(IDI = 4 or 5 is recommended. If specimen effective volume or area is input then IDI = 4)

If parameters are to be obtained from flexure bar specimens (IDI = 2 or 5)

DH = []

DW = []

DLI = []

DL2 = []

Temperature Dependent Material Control Input; The Data

DTABLE.

TEMP = []

VAGAGE = [1

FAST = []

(Used only if IDI = I or 4)

Table:

Keywords that are specific to the C4LIFE module are not listed here. The C4PEST module will ignore keywords that are only used with the

C4LIFE module. These keywords should also be included in the C4PEST INP file if component reliability analysis will be subsequently

performed with the C4LIFE module.

NASA/TM--2003-106316 196
Fatigue

Templet File Keywords for

Parameter Estimation With Module

Key:

Static Fatigue

(Input file C4PEST INP keyword settings)

[] user must specify; -- default value or user input

C4PEST

Master Control Input:

NE = 0or3

t'n_t_TS = [l

r,_tATV-- []

NGP -

TIME = Positive non-zero number

INTERP = --

(Seconds)

Temperature Independent Material Control Input:

MATID = []

IDI = [1

1134 = [1

ID2 = []

ID3 = [1

IK.BAT = I

PR = --

C _ -=

MLORLE = --

OUTLIE = --

(IDI = 4 or 5 is recommended If specimen effective volume or area is input then IDI = 4)

If parameters are to be obtained from flexure bar specimens (IDI = 2 or 5)

DH = [1

DW = []

DLI = []

DL2 = []

Temperature Dependent Material Control Input; The Data Table:

DTABLE:

TEMP = [1

SOL = --

VAGAGE = [] (Used only iflDI = I or 4)

VAGTIM = [] (Used only iflDI = I or 4)

(The values input for the VAGTIM and VAGAGE keywords are identical if the specimen is a

tensile specimen)

RANGE = Search for solution between [] and [] (Used only if SOL = 2)

STAT = [] (Static fatigue specimen rupture data)

FAST = -- (Optional fast-fracture specimen rupture data; Weibull parameters obtained from this data will

override Weibull parameters estimated strictly from the fatigue data)

Keywords that are specific to the C4LIFE module are not listed here The C4PEST module will ignore keywords that are only used with the

C4LIFE module. These keywords should also be included in the C4PEST.INP file if component reliabdity analysis will be subsequently

performed with the C4LIFE module.

NASA/TM--2003-106316 197
Fatigue

Templet File Keywords for

Parameter Estimation With Module

Dynamic Fatigue

(Input file C4PEST.INP keyword settings)

Key [] user must specify; -- default value or user input

C4PEST

Master Control Input:

NE = 0or3

NMATS = []

NMATV= []

NGP = --

TIME = Positive non-zero number (Seconds)

INTERP -

Temperature Independent Material Control Input:

MATID = []

IDI = [1

ID4 = [1

11:)2 = [l

ID3 = []

IKBAT = I

PR

C

MLORLE = --

OUTLIE -

(IDI = 4 or 5 is recommended If specimen effective volume or area is input then IDI = 4)

If parameters are to be obtained from flexure bar specimens (IDI = 2 or 5).

OH = []

OW = [1

DLI = []

DL2 = []

Temperature Dependent Material Control Input; The Data Table:

DTABLE'

TEMP = []

SOL

VAGAGE = [] (Used only ifIDl =I or 4)

VAGTIM = [] (Used only ifIDl =I or 4)

(The values input for the VAGTIM and VAGAGE keywords are identical if the specimen is a

tensile specimen)

RANGE = Search for solution between [] and [] (Used only if SOL = 2)

DYNA = [] (Dynamic fatigue specimen rupture data)

FAST = -- (Optional fast-fracture specimen rupture data, Weibull parameters obtained from this data will

override Weibull parameters estimated strictly from the fatigue data)

Keywords that are specific to the C4LIFE module are not listed here The C4PEST module will ignore keywords that are only used with the

C4LIFE module. These keywords should also be included in the C4PEST.INP file if component reliability analysis will be subsequently

performed wlth the C4LIFE module

NASA/TM--2003-106316 198
Templet File Keywords for

Fatigue Parameter Estimation With Module C4PEST

Cyclic Fatigue; R-Ratio is Constant Throughout the Specimen

(Input file C4PEST INP keyword settings)

Key: [] user must specify; -- default value or user input

Master Control Input:

NE = 0or3

NMATS = [1

NMATV = []

NGP -

TIME = Positive non-zero number

INTERP = --

(Seconds or cycles)

Temperature Independent Material Control Input:

MATID = []

IDI = []

ID4 = []

ID2 = []

ID3 = []

IKBAT = l

PR = --

C

MLORLE -

OUTLIE = --

(IDI = 4 or 5 is recommended. If specimen effective volume or area is input then IDI = 4)

If parameters are to be obtained from flexure bar specimens (IDI = 2 or 5)

DH = []

DW = []

DL1 = [1

DL2 = []

Temperature Dependent Material Control Input; The Data Table:

DTABLE'

TEMP = []

SOL

CYCLIC = []

VAGAGE = []

VAGTIM = []

RANGE

CYCL

FAST

(Power law' IWAVE = 1 to 5 ; RCYC = [] ; PERIOD = [] , QCYC = [])

(Paris law: IWAVE = 0 ; RCYC = [] ; PERIOD = [] , QCYC = 0 0 or blank entry)

(Walker law: IWAVE = 0 ; RCYC = [] ; PERIOD = [] ; QCYC > 0 0)

(Used only if IDI = I or 4)

(Used only if IDI = 1 or 4)

(The values input for the VAGTIM and VAGAGE keywords are identical if the specimen ts a

tensile specimen)

= Search for solution between [] and [] (Used only if SOL = 2)

= [] (Cyclic fatigue specimen rupture data)

(Power law: Peak cyclic stress and time to failure in seconds)

(Paris law and Walker law Peak cyclic stress and cycles to failure)

= -- (Optional fast-fracture specimen rupture data, Weibull parameters obtained from this data will

override Weibull parameters estimated strictly from the fatigue data)

Keywords that are specific to the C4LIFE module are not listed here The C4PEST module will ignore keywords that are only used with the

C4LIFE module These keywords should also be included in the C4PEST.INP file if component reliability analysis will be subsequently

performed with the C4LIFE module.

NASA/TM--2003-106316 199
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Effective Volume and Area Computation

of the Finite Element Model

(Input file CARES.INP keyword settings)

Key. [] user must specify; -- default value or user input

One neutral file is required (CARES NELl)

Master Control Input:

NE = 3

IPRINC = I

NMATS = []

_a,4ATV = []

NGP = --

NS = [1

IPROOF = 0

FACTOR = =-

TIME = 0.0

TIMEFT = 0 0

Temperature Independent Material Control Input:

MATID = []

IDI = 3

ID4 = [1

11)2 = []

ID3 = []

IKBAT = I

PR

C

CYCLIC'

IWAVE = I

RCYC = 1.0

PERIOD = 1 0

QCYC = l o

PFCONV = [] (Probability of failure for corresponding load, 0.000001 < PFCONV < 0 999999 )

Temperature Dependent Material Control Input:

TDEG = []

PARAM = []

(Only one temperature level should be input)

(the Weibull modulus can be obtained from analysis of specimen data using the C4PEST

module. The characteristic strength and scale parameter are arbitrary but reasonable

input values)

The calculated effective volume or area ts input into the C4PEST module using the VAGAGE or VAGTIM keywords of the Data Table

NASA/TM--2003-106316 200
Reliability Analysis Module C4LIFE

Two data files are required to conduct a reliability analysis of a structural component. They

are the user input file and the neutral data base. The CARES.NEU contains the information obtained

from the finite element output file and is discussed in the CARES.NEU section. The CARES.INP

file contains user specified data and is partitioned into three groups, master control input, material

control input, and temperature dependent data. The master control input contains a set of control

indices which direct the overall program execution. The material control input specifies the material

identification parameters. The temperature dependent data segment of the CARES.INP file contains

data sets with temperature dependent Weibull parameters. The structure of the CARES.INP file is

discussed in the INPUT INFORMATION section.

As was indicated previously, the user must coordinate the size of the neutral data base and

the blank common array used in the reliability module. The user adapts the amount of memory

allocated to the blank common array to the size of the problem and the restrictions of the host

computer. The default size of the one dimensional blank common array (identified as MTOT in the

source code) is 1,000,000 data entries. The size is specified on the PARAMETER card which is

located at the beginning of the source code for the reliability analysis module. This card appears as

follows:

PARAMETER (MTOT=100000,NPLV= 15)

Note that the amount of the blank common array utilized is displayed in the CARES.OUT file.

Thus the user can optimize the value of MTOT.

In addition, the user must assign input and output files utilized by the reliability analysis

module through the use of the "OPEN" command. Three files (CARES.INP, CARES.NEU and

CARES.OUT) must be assigned to logical units. Initially, the file names are assigned interactively.

NAS A/TM--2003-106316 201
The user is prompted for the filenames when the program is executed. Another option is to delete

(or comment out) those lines of code which query the user for the filename and insert the

appropriate file names into the OPEN statements. This requires editing the source code and placing

the names of the files into the "OPEN" statements. Changing the file name requires editing the

source code and changing the following statements

OPEN(LUA,FILE ='C4LIFE.INP',STATUS='OLD',FORM='FORMATTED')

OPEN(LUB,FILE='C4LIFE.OUT',STATUS='UNKNOWN',FORM='FORMATTED ')

OPEN(LUC,FILE='CARES.NEU',STATUS='OLD',FORM='FORMA'[TED ')

An optional second neutral file may be used to specify a proof test load or as the other extreme load

case in a cyclic fatigue problem. This file has the default name "PROOF.NEU" and is defined in

the "OPEN" statement

OPEN(LUG ,FILE=' PROOF.NEU', STATU S=' OLD' ,FORM=' FORMATTED')

If the user requests a PATRAN output file, an "OPEN" statement is provided in the source

code for this purpose. The "OPEN" statement is executed if IPOST=I is specified in the

CARES.INP file. This statement appears near the end of the main routine in the reliability analysis

module as follows

OPEN(LUF,FILE='CARES.PAT',STATUS='UNKNOWN',RECORDTYPE='FIXED',

FORM='FORMAT/'ED',RECL=80)

Note that the default name is CARES.PAT. In addition, the entry "RECL=80" stipulates that the

record length (i.e., line format) in the CARES.PAT file is 80 characters in length instead of 132

characters (the standard format for the C4LIFE.OUT file).

NASA/TM--2003-106316 202
C4LIFE.INP File

Thereliabilityevaluationmodule requires an input file that contains the failure data, failure

mode (i.e., failure caused by volume or surface defects), type of specimen, and the requested

estimation method. The data must be entered in the fixed format specified in the INPUT

INFORMATION section. This file is also generated by the parameter estimation module. The

following are examples of the necessary data for time-dependent analyses. Examples for the

following are included: Fast-Fracture, Static Fatigue, Proof Test, Cyclic Fatigue: Constant R-Ratio,

and Cyclic Fatigue: R-Ratio varies throughout the component..

C4LIFE.OUT File

The reliability module output file is assigned the default name of CARES.OUT. Information

contained in this file include job statistics, component reliability, element group reliabilities, and

element reliabilities. The information is presented in this sequential order.

NASA/TM--2003-106316 203
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Fast-Fracture

(Input file CARES INP keyword settings)

Key. [] user must specify, -- default value or user input

One neutral file is required (CARES,NEU)

Master Control Input:

NE = 3

1PRINC = 1

NMATS = []

t,_4n'rv = fl

NGP = --

NS = [1

IPROOF = 0

FACTOR = --

TIME = 00

TIMEPT = 00

Temperature Independent Material Control Input:

MATID = []

IDI = 3

ID4 = [1

ID2 = []

ID3 = []

IKBAT = 1

PR = --

CYCLIC.

IWAVE = I

RCYC = 1 0

PERIOD = 1.0

QCYC = 1 0

PFCONV = 0 0

Temperature Dependent Material Control Input:

TDEG = []

PARAM = [1

NASA/TM--2003-106316 204
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Static Fatigue Load

(Input file CARES INP keyword settings)

Key: [] user must specify, -- default value or user input

One neutral file is required (CARES.NEU)

Master Control Input:

NE = 3

IPRINC = 1

NMATS = []

NMATV = []

NGP = --

NS = []

IPROOF = 0

FACTOR = --

TIME = [] (Seconds)

TIMEPT = 00

Temperature Independent Material Control Input:

MATID = []

IDI = 3

IO4 = []

192 = []

ID3 = []

IKBAT = I

PR

C m ._

CYCLIC:

IWAVE = 1

RCYC -- I 0

PERIOD = 1 0

QCYC = I o

PFCONV = 0 0

Temperature Dependent Material Control Input:

TDEG = []

PARAM = []

TPARAM = []

NAS A/TM--2003-106316 205
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Proof Test Load

(Input file CARES.INP keyword settings)

Key [] user must specify; -- default value or user must specify

Two neutral files are required. 1) CARES NEU - service load, 2) PR(X)F.NEU - proof test load

Master Control Input:

NE = 3

IPRINC = 2

}_¢ATS = [1

NMATV= []

NGP = --

NS = [1

IPROOF = I

FACTOR = 1.0

TIME -- [1

TIMEPT = []

(Service load duration in seconds)

(Proof test load duratmn in seconds)

Temperature Independent

MATID = []

IDI = 3

ID4 = []

ID2 = []

ID3 = []

IKBAT = I

PR

CYCLIC

IWAVE = 1

RCYC = 1 0

PERIOD = 1 0

0CYC = l 0

PFCONV --- 0 0

Material Control Input:

Temperature Dependent Material

TDEG = [1

PARAM = [1

TPARAM = [1

Control Input:

NASA/TM--2003-1063 ]6 206
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Cyclic Load; R-Ratio is Constant Throughout the Component

(Input file CARES INP keyword settings)

Key: [] user must specify; -- default value or user input

One neutral file is required (CARES NEU) containing peak cyclic stresses

Master Control Input:

NE = 3

IPRINC = 1

NMATS = []

NMATV = []

NGP = --

NS = []

IPROOF = 0

FACTOR = --

TIME -- []

TIMEPT = 0.0

(Power law' seconds)

(Paris law and Walker law cycles)

Temperature Independent Material Control Input:

MATID = []

IDI = 3

IDa = [1

ID2 = []

ID3 = []

IKBAT -- 1

PR = --

C _ --

CYCLIC

IWAVE

RCYC

PERIOD

QcYC

= (Power law I<IWAVE<6)

(Paris law and Walker law IWAVE = 0)

= []

= [] (Seconds)

-- []

PFCONV = 0.0

Temperature Dependent Material Control Input:

TDEG =

PARAM = []

TPARAM = [] (For the Walker law to be invoked the value of Q must be specified

entry defaults to the power law or Paris law)

A blank or zero

NASA/TM--2003-106316 207
Templet File Keywords for

Component Reliability Analysis With Module C4LIFE

Cyclic Load; R-Ratio Varies Throughout the Component

(Input file CARES INP keyword settings)

Key. [] user must specify; -- default value or user input

Two neutral files are required. I) CARES NEU - containing maximum cyclic stresses, 2) PROOF.NEU - containmg minimum cyclic stresses

Master Control Input:

hie = 3

1PRINC = 2

NMATS = []

NMATV = [1

NGP = --

NS = [1

IPROOF = 1

FACTOR = 1.0

TIME = []

TIMEPT = TIME

(Power law. seconds)

(Paris law and Walker law cycles)

(The values input for the TIMEPT and TIME keywords must be identical)

Temperature Independent Material Control Input:

MATID = []

1DI = 3

ID4 = I]

ID2 = [1

ID3 = l]

IKBAT = I

PR

C _ ==

CYCLIC:

IWAVE

RCYC

FE_OD

QcYc

= (Power law 1 <IWAVE<6)

(Paris law and Walker law: IWAVE = 0)

= 1.0 (This input is ignored when two neutral files are used)

= [] (Seconds)

= H

PFCONV = 0.0

Temperature Dependent Material Control Input:

TDEG = [1

PARAM = [1

TPARAM = [] (For the Walker law to be invoked the value of Q must be specified. A blank or

zero entry defaults to the power law or Paris law)

NASA/TM--2003-106316 208
Summary

To summarize, the following abbreviated checklist is provided so that the user can quickly

identify difficulties when executing the modules contained in the CARES/LIFE algorithm.

Interface Module:

Assure the parameter MTOT is large enough to process the neutral data base.

Assure all proper information is contained in the finite element output file.

Assign the finite element output file to the correct logical unit.

Assure the source code has been compiled and linked free of errors after all changes have

been made to the interface module source code.

Parameter Estimation Module:

Assign C4PEST.INP and C4PEST.OUT to the proper logical units in the parameter estimation

source code.

Assure that the failure data appears in the correct format in the C4PEST.INP file.

Assure the source code has been compiled and linked free of errors after all changes have

been made to the parameter estimation module source code.

Reliability Module:

Assure the parameter MTOT is large enough to process the neutral data base.

Assign CARES.NEU and CARES.INP to the correct logical units.

Check the C4LIFE.INP file such that the data corresponds to values contained in the

CARES.NEU file.

Assure the source code has been compiled and linked free of errors after all changes have

been made to the reliability analysis module source code.

NASA/TM--2003-106316 209
Example Problems

Fast-Fracture Example Problems

Example 1 - Statistical Material Parameter Estimation

To validate the methods used to estimate statistical material parameters from fast-fracture

data, we compared results from the fracture of four-point bend bars broken at NASA Lewis and

analyzed by the CARES/LIFE parameter estimation module C4PEST with results independently

obtained by Bruckner-Foit and Munz (Bruckner-Foit, and Munz, 1988) for the International Energy

Agency (IEA) Annex II, Subtask 4 (Tennery, 1987). The lEA Annex II agreement is focused on

cooperative research and development among the United States, West Germany, and Sweden in the

areas of structural ceramics. Subtask 4 of the agreement addresses mechanical property

measurement methods with initial research concentrating solely on four-point flexure testing. Three

different materials were analyzed, namely a hot isostatic pressed (HIPped) silicon carbide (SIC)

from Elektroschmelzwerke Kempten (ESK), West Germany, a HIPped silicon nitride (Si3N4) from

ASEA CERAMA, Sweden; and a sintered silicon nitride from GTE WESGO, USA, although only

results from the ESK and ASEA materials are discussed herein.

In November 1986, 400 HIPped SiC flexure bars from West Germany were distributed by

Oak Ridge National Laboratory (ORNL) (Oakridge, Tennessee) to the five participating U.S.

laboratories, including NASA Lewis. The bars were fractured at these laboratories and the fracture

stress data sets were returned to ORNL as complete data without censoring for different failure

modes. Shortly thereatier, 400 Si3N 4 bars from Sweden were also received by ORNL and

subsequently distributed to the same U.S. laboratories for fracture testing. Again, the fracture stress

data sets were returned to ORNL as complete samples. The number of specimens of a particular

material given to each U.S. laboratory was 80. The specimens had cross-sectional dimensions of 3.5

mm in width and 4.5 mm in height. The specimens were tested in four-point bending with an outer

NASA/TM--2003-1063 i 6 211
spanof 40mm andaninnerspanof 20mm. Thenominaloadingratewas0.5mm/min,andthe

testingtemperaturewasapproximately20°C.

Detailsof thestatisticalanalysesof thesedatasetsaregivenin referencesby Bruckner-Foit

andMunz (Bruckner-Foit,andMunz,1988),andTennery (Tennery, 1987). The results of the 80

silicon carbide flexure bars tested at NASA Lewis, which are shown in table El-T1, were analyzed

with the C4PEST module to calculate the maximum likelihood estimates (MLE's) of the Weibull

parameters. The Weibull parameter values from CARES/LIFE, summarized in table El-T2, match

the predictions from Bruckner-Foit and Munz (Bruckner-Foit, and Munz, 1988)reasonably well.

The SiC fracture data are plotted in figure El-F1 along with the proposed Weibull line and the

Kanofsky-Srinivasan 90-percent confidence bands. Since all of the data are within the 90-percent

bands and the goodness-of-fit significance levels are high, it is concluded that the fracture data show

good Weibull behavior.

ASEA CERAMA HIPped Si3N 4 bars (Tennery, 1987) from Sweden were also fractured at

NASA Lewis, and subsequently, the statistical material parameters were estimated with

CARES/LIFE by using the maximum likelihood method. A comparison of the Si3N 4 results with

those from Bruckner-Foit and Munz (Bruckner-Foit, and Munz, 1988)is also shown in table El-T2.

Agreement between estimates from the two sources is excellent. When the 80 ASEA silicon nitride

bars were analyzed by the CARES/LIFE code as a complete sample, the significance levels of 54

and 35 percent from the Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests,

respectively, were relatively low, indicating a questionable fit to the proposed Weibuil distribution.

The lower significance level for the Anderson-Darling test indicated greater deviation occurring in

the low strength region of the distribution. From the outlier test included in the CPEST module, the

highest strength fracture stress was detected to be an outlier at the 1-percent significance level.

Several of the lower strengths were-flagged as outliers at various significance levels (I, 5, or 10

NASA/TM--2003-106316 212
percent).FigureEl-F2 showsa Weibullplotof thedata.Fromthefigureit appearsthatthedata

arebimodalwith anoutlierpointatthehigheststrength.

Becauseof theobservedtrends,thedatawerere-analyzedassuminga censoreddistribution

andremovingthehigheststrengthoutlierpoint(_f = 817.2MPa)asbaddata. Althoughit is

possiblethatbothfailuremodesweresurfaceinduced,for thesakeof thisexampleit is assumed

thatthelow-strengthfailureswerepredominantlydueto volumeflawsandthatthehigh-strength

specimensfracturedpredominantlybecauseof surfaceflaws.Sinceresultsfrom fractographyof the

individualspecimensto identifythevariousfailuremodeswerenotavailable,thefractureorigins

hadto bearbitrarilyassignedpriorto parameterestimation.Notethatidentifyingindividual

specimenflaw originsis especiallyimportantfor smallsamplesizeswhereaplot of thedatadoes

notyield cleartrends.However,fortheNASALewisSi3N4data,the sample size was large, and

clear trends could be observed, although extra care would be required to determine if the trends

were surface flaw or volume flaw based. From inspection of figure El-F2, we decided to assign the

lowest nine strengths as due to volume flaws and the remainder as due to surface flaws. An input

(templet) file for the C4PEST module was prepared and is provided on diskette along with the

output file from the module. The cracks were arbitrarily assumed to be Griffith cracks, and the total

strain energy release rate fracture criterion was used. This assumption was used only in the

calculation of kBs and kBv. The K-S significance level increased from 0.54 to 0.68, and the A-D

significance level increased from 0.35 to 0.58. This improvement supports the initial assumption of

bimodal behavior. The value of fia changed from 13.4 for the complete sample to m s = 22.8 and fias

= 4.13. The value offr 0 changed from 686 MPa for the complete sample to a0s = 692 MPa and

O0v = 1128 MPa for the surface and volume flaw distributions, respectively. Further improvements

in the goodness-of-fit scores may be gained by correctly identifying the failure mode of the fracture

origins.

NASA/TM--2003-106316 213
Fromequation(68) the normalized Batdorf crack density coefficient for volume flaws is (m v

+ 1) = 4.13 + 1 = 5.13, and from equations (58) and (60) the scale parameter Oov is 17.9 MPa

(m) 3/4t3. For surface flaws the normalized Batdorf crack density coefficient is 6.05, whereas OoS

calculated by using equations (71) and (73) is 461.3 MPa (m) m2s.

For the SiaN 4 fracture data from NASA Lewis, we have obtained goodness-of-fit significance

levels as high as 0.78 and 0.88 for the K-S and A-D tests, respectively, by assuming a particular

bimodal flaw distribution. For this case, 13 volume flaws were assumed, and the MLE's were ms =

21.0, ms = 6.79, dos = 693 MPa, and 6es = 876 MPa. The I3 volume flaws did not correspond to

the 13 lowest fracture strengths. On the basis of these goodness-of-fit scores, it is concluded that

the data show good bimodal Weibull behavior.

It should be noted from figure El-F2 that the assumed volume flaw distribution dominates the

failure response at low probabilities of failure. Therefore, in component design, it is essential to

properly account for competing failure modes; otherwise nonconservative design predictions may

result.

NAS A/TM--2003-106316 214
TABLEEl-T1. EXTREMEFIBERFRACTURE

STRESSESOFESKHIPpedSILICON

CARBIDE(SIC)BARS

Flexure

bar

I

m

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

23

24

25

26

27

28

29

3O

31

32

33

34

35

36

37

38

9

4O

Strength. Flexure Strength.

MPa bar MPa

281.2 41

29 !.0 42

358.2 43

385.4 44

389.0 45

390.8 46

391.8 47

402.8 48

412.5 49

413.3 50

413.9 51

417.8 52

418.2 53

426.9 54

437.6 55

440.0 56

441.0 57

442.5 58

443.8 59

444.9 60

446.2 61

451.5 62

452.1 63

452.7 64

470.4 65

474.1 66

475.5 67

475.5 68

479.2 69

483.5 70

484.8 7 !

486.2 72

488.6 73

492.5 74

493.2 75

496.0 76

505.7 77

511.9 78

512.5 79

513.8 80

516.2

519.8

527.6

530.7

530.7

545.7

548.8

552.7

559.6

562.4

563.3

566. I

566.5

570. I

572.8

575.0

576.1

580.0

582.6

588.0

588.6

591.0

591.0

593.3

598.7

599.6

610.0

612.7

619.9

619.9

622.2

622.3

640.5

649.0

657.2

660.0

664.3

673.5

673.9

725.3

NAS A/TM--2003-106316 215
O

r,_

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NASA/TM--2003-106316 216
lad

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CONFIDENCE

BAND -,,

\

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CONFIDENCE

BAND

I" I I I I I

250 350 450 550 650 750

FRACTURE STRESS, MPA

Figure E1-Fl.--90-percent confidence bands about the Weibull line for ESK HIPped silicon carbide

(SIC). (Fracture stress data generated at NASA Lewis; not all data points shown; L=0.0829.)

i

i

NASA/TM--2003-106316 217
-.J

'tJI---

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e.

.9999

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COMPLETE

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DISTRIBUTION--x

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550 E50 750 850

FRACTURE STRESS, MPA

Figure E1-F2.--Complete sample, assumed censored sample Weibull distributions, and outlier in

ASEA CERAMA HIPped silicon nitride (Si3N4). (Fracture stress data generated at NASA Lewis;

not all data points shown.)

NASA/TM--2003-106316 218
Example 2 - Internally Heated and Pressurized Tube

Jadaan, et.al. (Jadaan, 1990) (Shelleman, 1991) (Jadaan, Shelleman, Conway, Mecholsky, and

Tressler, 1991) (Shelleman, Jadaan, Conway, and Mecholsky, 1991) conducted experimental and

analytical studies on Reaction Bonded Silicon Carbide (SCRB210) tubular components at Penn

State. Ten long tubes (660 mm long, 17.1 mm inner radius, and 21.9 mm outer radius) were rapidly

pressurized to failure at 1200°C in order to evaluate their fast-fracture strength distribution. O-ring

and C-ring coupon specimens (9.5 mm wide, 17.1 mm inner radius, and 21.9 mm outer radius), cut

from the same lot of tubes, were fractured by diametrally compressing them to failure at

temperatures ranging from room temperature to 1200°C. The resulting O-ring and C-ring strength

distributions were used to calculate the Weibuli parameters, m, t_os, and C_ovfor the SCRB210

material.

In this example, the C4LIFE module of the CARES/LIFE code was used to calculate the

predicted failure probabilities for the long tubes using the Weibull parameters of the diametrically

compressed O-ring and C-ring specimens (see Tables E2-T1 and E2-T2). The Weibull moduli were

calculated from strength data using least-squares analysis. The C4PEST module (one of the three

modules comprising the CARES/LIFE Code) was utilized to calculate the Weibull moduli and

characteristic strengths. Subsequently, a scale parameter at a given temperature was calculated from

m and a o corresponding to that temperature in association with effective area and volume

expressions obtained from references 1-4. The Weibull moduli and scale parameters were then

input into the C4PEST code to calculate the Batdorf coefficients. The C4PEST module was also

used to specify the fracture criterion and flaw shape to be used for failure probability calculation.

The long tube components were tested in a tube burst test facility (Shelleman, Jadaan, Butt,

Tressler, Hellman, and Mecholsky, 1992). This facility consisted mainly of four components: (1)

pressure seals, (2) end plates, (3) water cooled end caps (152.4 mm long), and (4) containment or

NAS A/TM--2003-106316 219
testchamber.Thelongtubecomponentwasmountedverticallyinthetestchamberandsupported

at theendsby thewatercooledendcaps.Thepressuresealswereinsertedintotheopenendsof the

tubes. This methodology of supporting and pressure sealing the tubes allowed them to expand

freely in the axial direction, thus preventing the development of axial stresses due to thermal

expansion. A heating element was suspended from the upper pressure seal inside the tube and

connected to a temperature controller. A thermocouple was also suspended from the upper pressure

seal to measure the temperature at the inner surface of the tube. Thermocouples were also placed at

the outer surface along the axial direction of the tube at six locations. Figure E2-F1 shows a typical

thermal profile for a tested tube. The lower pressure seal contained an opening connected to a

pressure booster. For more details regarding the design and operation of the burst test apparatus,

see Shelleman, et al.

Fracture surface analysis (fractography) was performed on the reconstructed tubes after failure

to determine the type of dominant flaws causing failure. Fractographic analysis is a necessary

component of reliability analysis, in order to determine whether area, volume, or combined (area

and volume) reliability analysis should be performed. The SCRB210 tubes were highly decorated

with silicon nodules at the inner surfaces of the tubes, remnant from fabrication. These silicon

nodules induced a more severe flaw population at the inner surfac6s of the tubes than through the

volumes or outer surfaces of the tubes. Indeed, upon fractographic examination of reconstructed

tube fragments, failure was found to be dominantly induced by flaws located at the inner surfaces of

the tubes. Therefore, surface area reliability analysis was used to predict the failure probabilities for

the tubes.

The MSC/NASTRAN finite element program was utilized to calculate the thermomechanical

stresses for the internally heated and pressurized tubes at 1200°C. Symmetry conditions allow

modeling of one quarter of the tube, and thus, 4 segments were required to make up the entire

NASA/TM--2003-106316 220
tubular component. A total of 540 solid elements (HEXA) and 270 shell elements (QUAD8) were

used (figure E2-F2). The shell elements, used with surface flaw reliability analysis, were placed at

the inner and outer surfaces of the tubes. The shell elements shared common nodes with the solid

elements. These two-dimensional elements are used by the CARES/LIFE code only to identify

external surfaces and obtain corresponding in-plane stresses and areas. The contribution of these

elements to the overall stiffness of the model was negligible. The thickness of the shell elements

was 1.0 x 106 mm and only membrane properties were assigned to these elements via the PSHELL

BULK DATA card. Four layers of equally sized solid elements were used to model the bulk of the

tube. Three elements were used in the circumferential direction. Note that the entire volume and

surface area of the tube are stressed in tension.

The tangential stresses at the inner surface of the tube from MSC/NASTRAN finite element

analysis were found to be within 5% of the closed form solution. Fifteen Gauss integration points

(NGP=15) were used for the reliability analysis with CARES/LIFE. The values for Poisson's ratio

and thermal coefficient of expansion were 0.16 and 4.6 x 10.6 °C1, respectively. The Young's

modulus was temperature dependent with a value of 378, 326, and 304 GPa at 25, 1100, and

1200°C, respectively. A radial temperature gradient of 6°C was assumed constant along the entire

length of the tube.

After completing the finite element analysis and creating the MSC/NASTRAN output file

(containing the element stresses and temperatures) the NASCARES module was used to create a

neutral file. This neutral file contained the subelement stresses, volumes (or surface areas for the

shell elements), and temperatures.

neutral file for reliability analysis.

the output file, CARES.INP, of the parameter estimation module, C4PEST.

Subsequently, the C4LIFE module read the data base from the

Furthermore, the C4LIFE program read the data base stored in

Finally, the main

NASA/TM--2003-106316 221
reliabilityprogram,C4LIFE, performedthe reliability calculations by calculating the survival

probability and risk of rupture intensities for all subelements.

The manner in which the tubes were supported and pressurized is important from an

analytical point of view. Because the tubes are not axially restrained, a predominantly uniaxial

tensile stress state results. The tensile axial stresses are negligible compared to the tangential tensile

stresses. The radial stresses are compressive, and thus do not contribute to the failure probability

analysis. Since the stress state is essentially uniaxial, then all fracture criteria/flaw shapes should

yield similar failure probabilities (All failure probability fracture criteria collapse to the Weibull

criterion for uniaxial stress states). This expected result was used to verify the CARES/LIFE code.

Table E2-T3 shows the predicted failure probability, based on various fracture criteria and flaw

shapes, for a typical tube. As can be seen from table E2-T3, all fracture criteria predicted

essentially equal failure probabilities as expected.

Table E2-T4 shows the experimental and predicted failure probabilities for the ten tested

tubes. Experimental failure probabilities were calculated using the equation Pf=(i-0.5)/N where i is

the tube rank number and N is the total number of specimens. The predicted failure probabilities

were calculated at the inner and outer surfaces of the tubes. Failure probabilities corresponding to

failures associated with flaws located at the inner surfaces of the tubes were calculated using the Oring specimen Weibuli parameters. Failure probabilities corresponding to failures associated with

flaws located at the outer surfaces of tubes were calculated using the Weibuli parameters obtained

from testing C-ring specimens in compression. This is because the O-ring test samples flaws

located at the inner surface of the tube, while the C-ring in compression test samples flaws located

at the outer surface of the tube. As stated previously, the inner surfaces of the tubes are highly

decorated with silicon nodules while the outer surfaces are smooth and homogeneous. This

variability of the inner surface flaw-population resulted in large scatter in the strength data for the

NAS A/TM--2003-106316 222
O-ringspecimens,andthusa low Weibullmodulus(seeTableE2-T2). Thecombinationof low

Weibullmoduliandmaximumstressesoccurringat theinnersurfacesof thetubesresultedin high

calculatedfailureprobabilitiesattheinnersurfacescomparedto thoseat theoutersurfacesof the

tubes(TableE2-T4). Thecombinedfailureprobability(lastcolumnof TableE2-T4)istheoverall

failureprobabilitytaking into account both the inner and outer surface areas of the tubes.

Figure E2-F3 shows the experimental failure probability data for the ten tubes as a function of

internal pressure at failure. Figure E2-F3 also includes the combined predicted failure probability

curve for the tubes calculated using the CARES/LIFE code. As shown in figure E2-F3, the

experimental failure probabilities compare very well with the predicted failure probabilities.

NASA/TM--2003-106316 223
TABLEE2-T1. AVERAGESTRENGTHSAND WEIBULLPARAMETERS

FORSCRB2]0 O-RINGSPECIMENS

Temperature

(°C)

Number of

Specimens

Average

Strength

(MPa)

Characteristic

Strengt h+

(ao)

(MVa)

Weibull*

Modulus

(m)

Area Scale*

Parameter

(MPa.m z/m)

25 26 242.9+50.4 263.6 5.3 46.5

800 20 250.6+54.8 275.2 4.5 37.4

1000 20 283.0+42.4 301.9 7.2 79.0

1200 21 300.1 +44.6 320.9 6.9 79.9

+ Characteristic strength corresponds to the O-ring specimen strength value at 63.2% failure

probability

m, ao, and aoA are calculated from strength data based on least square analysis.

NASA/TM--2003-106316 224
TABLE E2-T2.AVERAGESTRENGTHSAND WEIBULLPARAMETERS

FORSCRB210C-RINGSPECIMENSTESTEDIN COMPRESSION

Temperature

(oc)

Number of

Specimens

Average

Strength

(MPa)

Characteristic

Strength +

(ao)

(MPa)

Weibull*

Modulus

(m)

Area Scale*

Parameter

(ao )

(MPa.m 2/m)

25 22 239.2+39.7 255.8 6.8 73.6

800 20 240.6+29.6 253.7 9.1 98.4

1000 26 275.9+48.9 298.4 5.7 68.6

1200 20 318.7+29.8 331.8 12.1 160.7

+ Characteristic strength corresponds to the C-ring specimen strength value at 63.2% failure

probability

m, a o, and aoA are calculated from strength data based on least square analysis.

NASA/TM--2003-106316 225
I!g4.

fl

1124.

O •

1055. " : _

985.

915.

046.

776.

707.

490.

420.

358.

209,

219.

Figure E2-Fl.--Typical thermal profile for a tested tube. 151'1.

i

NASA/TM--2003-106316 226 ,
FigureE2-F2.--Tubularcomponent with finite element mesh.

NASA/TM--2003-106316 227,
TABLE E2-T3. CARES/LIFE PREDICTED FAILURE PROBABILITIES FOR A TYPICAL

INTERNALLY HEATED AND PRESSURIZED SCRB210 TUBE CALCULATED BASED ON

VARIOUS FRACTURE CRITERIA AND FLAW SHAPES.

(pressure at failure = 19 MPa, experimental failure probability = 0.45)

Fracture Criterion and Flaw Shape Predicted Failure Probability

PfA

Normal Stress Averaging Method 0.465

PIA Criterion 0.470

Energy Release Rate Criterion 0.470

(Griffith Crack)

Noncoplanar Strain Energy Release Rate Criterion 0.476

= 0.80

(Semi-Circular Surface Crack)

NASA/TM--2003-106316 228
TABLEE2-T4. EXPERIMENTALAND PREDICTEDFAILUREPROBABILITIES

FORINTERNALLYHEATEDAND PRESSURIZEDSCRB210TUBES

Energyreleaseratecriterion

(Surfaceflawanalysis)

Rank

(i)

Pressureat

Failure

(MPa)

Max. Stress

at Failure

(MPa)

Experimental

Failure

Probability

(i-0.5)/N

CARES/LIFE

Predicted Failure

Probability, Pfs

Inner

Surface

Outer

Surface

Combined

1 13.8 53.4 0.05 0.110 0.001 0.111

2 14.5 56.1 0.15 0.140 0.001 0.141

3 17.7 70.1 0.25 0.353 0.004 0.355

4 18.6 73.9 0.35 0.429 0.005 0.437

5 19.0 75.7 0.45 0.467 0.006 0.470

6 20.7 83.0 0.55 0.621 0.010 0.624

7 23.8 96.3 0.65 0.864 0.024 0.868

8 24.0 97.0 0.75 0.876 0.026 0.880

9 24.2 98.4 0.85 0.891 0.028 0.894

10 28.9 119.3 0.95 0.996 0.090 0.996

NASA/TM--2003-106316 229
1.0

0.8

0.6

0.4

0.2

0.0

• Experimental failure probability

-- Predicted failureprobability

1 5 20

Pressure (MPa)

10 25 30

Figure E2-F3.--Experimental and predicted failure probabilities for the ten tubes as a function of

internal pressure at failure.

NASA/TM--2003-106316 230
Example Problems

Time-Dependent Example Problems

Example 3 - Fatigue Parameter Estimation

This example demonstrates the determination of Weibull and fatigue parameters from

multiaxially loaded specimen rupture data. In addition, Weibull parameters evaluated using fastfracture data will be contrasted to Weibull parameters estimated from fatigue data. Ring-on-ring

loaded square plate specimens made from soda-lime glass were prepared and fractured under

dynamic fatigue loading in a distilled water environment at the NASA Lewis Research Center.

Ring-on-ring loading induces an equibiaxial stress state within the inner loading ring. The plate

specimens measured 50 mm x 50 mm and had an average thickness, h, of 1.50 mm.

The maximum fracture stress at the specimen center was computed using the fracture load, P,

and the equation given by Shetty et al. (Shetty, Rosenfield, Bansal, and Duckworth, 1980)

3 P I2 [ro] (1 - v)(r2 - r_2) 1 (1) crf= _ (I + v),°n -- +

4 n h 2 ri R 2

where the diagonal half length, Rs, was 35.92 mm, the inner radius, ri, was 5.02 mm, the outer

radius, ro, was 16.09 mm, and Poisson's ratio, v, was 0.25. Figure E3-F1 shows the fracture

strengths of the 121 specimens loaded at the stressing rates of 0.02, 0.20, 2.00, and 20.00 MPa/s.

addition, 30 specimens were tested in fast fracture in an inert environment of silicon oil. All flaw

origins resided on the specimen surface (surface flaws).

Using the CARES/LIFE parameter estimation module, the fast-fracture Weibull parameters

and the fatigue parameters were determined. For the fast-fracture strengths measured in silicon oil

the Weibuli modulus, ms, and characteristic strength, Cos, were estimated to be 2.871 and

394.2 MPa, respectively, using the maximum likelihood technique. The least-squares method

In

NASA/TM--2003-106316 23 !
producedestimatesof 2.675and395.3MPa,respectively.Forthefatiguedata,thecrackgrowth

exponent,Ns,wasdeterminedwiththemedianvalue,least-squares, andthemediandeviation

techniques.Theinert strengthWeibullmodulus,m's, anda characteristicstrength,C'os,were

estimatedfromthefatiguedatausingequations(201)and(202),respectively.Usingequation(203)

the fatigue data was transformed to an equivalent inert strength distribution. The ranked data

(median ranking (Pal, and Gyekenyesi, 1988)) is plotted in figure E3-F2, along with the ranked inert

strengths of the fast-fracture specimens. The transformed inert strengths in the figure were

determined using the parameters obtained from the median deviation analysis utilizing equation

(194) and the maximum likelihood method utilizing equation (144). The solid line represents the

transformed fatigue distribution with m' s = 2.344 and cr'0s = 387.4 MPa. The dashed line

represents the fast fracture strengths with ms = 2.871 and Cr0s= 394.2 MPa. Figure E3-F2 supports

the assumption that the fatigue data and the inert strength data are generated from the same flaw

population.

The specimen effective areas, Act and Ae, were evaluated numerically using equations (155)

and (75), respectively, with the CARES/LIFE component reliability analysis module and a finite

element model of the specimen. Because of symmetry, only one quarter of the specimen was

modeled using brick and wedge elements as shown in figure E3-F3. Quadrilateral and triangular

shell elements were attached to the faces of the solid elements corresponding to the specimen

external surfaces. The shell elements contributed negligible stiffness to the model. For this

example, the Batdorf model with the G v (total strain energy release rate) fracture criterion and a

Griffith crack was used in the numerical evaluation. The scale parameter t_oS and the fatigue

constant BBs were calculated from equations (71) and (200), respectively. Table E3-T1 lists the

NASA/TM--2003-106316 232
resultsof thisanalysisfortheoptionusingWeibullparametersestimatedfrom thefatiguedata.

TableE3-T2liststheresultsfortheoptionusingtheWeibullparametersfromthefast-fracturedata.

Thescaleparametervaluesin tablesE3-T1andE3-T2showa significantvariation.Because

of thelow valueof theWeibullmodulus,thesmalldifferencesin theestimatedvaluestranslateto

largedifferencesin thescaleparameter.Thecharacteristicstrengthsestimatedfromthefatiguedata

andthefast-fracturedatawereall within +7.0 MPa of the value of 394.0 MPa (the fast-fracture

value of ges from maximum likelihood estimation) for all methods of estimation.

Summary of the Preparation and Program Execution

Steps for the Glass Plate Example Problem

(See file contents table in Example Guide section for file descriptions)

• Prepare input file 3_PEST.INP for execution with module C4PEST. This file contains the

dynamic fatigue data consisting of the specimen stressing rates and the failure stresses (at the center

of the specimen). Optionally include the fast-fracture data of the plate specimens in the Data Table.

Because the specimen geometry is not a beam in flexure or a tensile specimen, the effective area

must be numerically evaluated using a finite element model of the specimen. Since the specimen

effective area is unknown the 3_PEST.INP input file is prepared using the tensile specimen option

(ID1 = 4). The effective area, A_, and the stress-normalized effective area, Acf, are assigned a

default value of 1.0 (see keywords VAGAGE and VAGTIM in the Data Table). For this problem

the PFCONV keyword should also be activated in the Master control Input (see the comments

regarding preparation of the finite element model, below). If the specimen geometry was a beam in

flexure or a tensile specimen, then Ae and Act are automatically evaluated and the user could directly

proceed with component reliability evaluation following the execution of the C4PEST module.

• Execute the C4PEST module with input file 3_PEST.INP and obtain the output file

3_PEST.OUT. Examine the 3_PEST.OUT file and obtain the Weibull modulus m and the fatigue

exponent N. Calculate the modified Weibull modulus rn = Nm/('N-2). Editing the 3_PEST.INP

input file, prepare another section of the Material Control Input with a new data table including the

modified Weibull modulus ffa (using the PAR.AM keyword). Do not include any specimen rupture

data in this table. Arbitrary values'are input for the Weibull scale parameter _o and the

NASA/TM--2003-106316 233
characteristicstrengtht_0 (input of the value of the characteristic strength of the specimen data is

recommended for both of these parameters). Again assume a tensile specimen geometry (ID1 = 4)

and input 1.0 for A e and Act (keywords VAGAGE and VAGTIM in the Data Table). The fracture

criterion and crack geometry should be consistent with the other Material Control Input section

previously prepared.

• Again execute the C4PEST module with the now modified input file 3_PEST.INP, obtain the

output file 3_PEST.OUT and the data file 3_LIFE.INP.

• Prepare a finite element model of the specimen and loading configuration (in this case

MSC/NASTRAN input file 3_NAS.INP). The magnitude of the loading is recommended to

correspond to some finite probability of failure of the specimen. A good choice is to specify

loading corresponding to the specimen characteristic strength cr0. In this case the keyword

PFCONV in the Master Control Input of file 3_PEST.INP (or also 3_LIFE.INP) would be assigned

the value Pf = 0.63212.

• Execute the finite element program (in this case MSC/NASTRAN) and obtain the output file

(in this case file 3_NAS.OUT).

• Prepare a neutral file by executing the appropriate interface module NASCARES, ABACARES, or ANSCARES (in this case the file 3.NEU is obtained from execution of the NASCARES

module).

• Execute the C4LIFE module with the input control file 3_LIFE.INP and the neutral file

3.NEU. Examine the output file 3_LIFE.OUT (obtained from executing module C4LIFE) and

obtain the calculated effective area A e associated with Weibuli modulus m, and the stress

normalized effective area A a associated with the modified Weibull modulus m.

• Again, edit the file 3 PEST.INP. For the Data Table containing the specimen rupture data,

input the correct values for Ae and Aef using the VAGAGE and VAGTIM keywords.

• Execute the C4PEST module with input file 3 PEST.INP. The output files 3_PEST.OUT and

3 LIFE.INP now contain the correct fatigue and Weibull parameters.

• Perform component reliability analysis with module C4LIFE. Since the specimen had

induced multiaxial stresses, the reliability analysis must be performed using the same fracture

criterion and flaw geometry as that used to obtain the fatigue and Weibull parameters.

NAS A/TM--2003-106316 234
(I

la.

o_

co

gr3

[-,

gd

600

500

.400

300

20O

lOO

Q

• i ' !

I I "

! " - :

z, ! z

m • • •

l I I •

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Figure E3-Fl.--Dynamic fatigue fracture strengths for the ring-on-ring loaded soda-lime glass square

plate specimens.

NASA/TM--2003-106316 235
-g:l

.,g

o

t-,

.¢

t_

g9

gO

80

_o

2C

_o

• /tO

/

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FRACTURE 5"TRES:S (_e.}

Figure E3-F2.--Soda-lime glass fatigue data transformed to an equivalent inert strength distribution

versus inert strengths measured in silicon oil. The solid line and the dashed are maximum likelihood

estimations of the respective distributions.

i

i

NASA/TM--2003-106316 236
Figure E3-F3.--Finite element mesh for the ring-on-ring loaded square plate specimen. The finite

element model reproduces one quarter of the total specimen geometry.

i

NAS A/TM_2003-106316 237
TABLE E3-T1. PARAMETERS ESTIMATED FROM FATIGUE DATA

Method Fatigue Fatigue Weibull Scale

exponent, constant, modulus, parameter,

Ns Bas m' s a'oS

MPa 2 . s MPa • mm _m,

Median Value 12.60 4 445 2.279 5 904

Least Squares 11.24 6 337 2.208 6 707

Median Deviation 11.88 5 994 2.344 5 443

TABLE E3-T2. PARAMETERS ESTIMATED FROM FATIGUE AND FAST-FRACTURE DATA

Method Fatigue Fatigue Weibull Scale

exponent, constant, modulus, parameter,

N s Bss ms Oos

MPa _ • s MPa • mm 2/_s

Median Value 12.60 3 225 2.675 3 915

Least Squares 11.24 8 527 2.675 3 915

Median Deviation 11.88 6 064 2.871 3 305

NASA/TM--2003-106316 238
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NAS A/TM--2003-106316 239
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NAS A/TM--2003-106316 240
Example 4 - Cyclic Fatigue and R-ratio Sensitivity

This example demonstrates the determination of Weibull and fatigue parameters from

cyclically loaded specimen rupture data. In addition, reliability predictions using the power law will

be contrasted to predictions based on the Walker equation for various levels of R-ratio.

This work is based on experiments performed by Liu and Chen (Liu, and Chen, 1991) using

3-mol%-Yttria-Stabilized Zirconia (3Y-TZP). The data is sparse and is used only to demonstrate the

capability of the CARES/LIFE program.

Cyclic fatigue experiments for various levels of R-ratio were performed on tensile specimens.

Five specimens each were tested at R-ratio values of 0.8, 0.5, and 0.0, while six specimens were

tested at an R-ratio value of-l.0. A triangular cyclic stress wave form was used, at a frequency of

1 Hz, for 105 cycles or until failure occurred. The tensile specimens had a 16.0 mm gauge length

and a 6.0 mm gauge diameter. Fractographie examination showed that failure most frequently

occurred at or near the surface and the failure origins were pore type flaws.

Fast-fracture testing was performed using 4-point bending specimens. A total of 11

specimens are used in this analysis with an inner load span of 4.0 ram, an outer load span of 13.6

mm, an average width of 2.577 ram, and an average height of 1.384 mm. Specimens that

significantly deviated from these dimensions were not included in this analysis. Fractography of the

4-point bending specimens indicated that failure occurred from surface or near surface flaws.

The analysis performed with the CARES/LIFE code assumed that failure was due to surface

flaws. Using the CARES/LIFE parameter estimation module the fast-fracture Weibull parameters

and the fatigue parameters were determined. The maximum likelihood estimates (MLE's) of the

Weibull parameters from the 4-point bend specimens were Weibull modulus m = 14.02,

characteristic strength Cr0s= 901.6 MPa, and scale parameter eros= 1079.0 Mpa. mm ° 1427. These

values were used in the subsequentcalculations for the fatigue parameter Bws. The fatigue

NASA/TM--2003-106316 241
parameterswereestimatedusingthemediandeviationtechniqueandtheyareshownin Table E4-T1

for the various R-ratio levels and crack growth laws. The g-factor for the power law formulation

with the sawtooth waveform and a positive R-ratio is

1 - R N+I

g = (1)

(I - R)(N + I)

while for a negative R-ratio

1

g = (2)

(1 -R)(N + 1)

For the Walker crack growth law, Liu and Chen (Liu, and Chen, 1991) determined that a value of Q

= 2.0 for the R-ratio sensitivity exponent was a best fit to the experimental data. Since the

CARES/LIFE code does not estimate the value of Q from experimental data, the value of 2.0 for Q

was used for this analysis.

Specimens that did not fail (runout data) were not considered in the parameter estimation

analysis, although an exception was made for the one specimen at R = 0.0. This runout data was

included because another specimen did rupture at this corresponding level of applied stress. From

the analysis of Liu and Chen (Liu, and Chen, 1991) the fatigue exponent for all the data was

estimated to be N = 21.0. From Table E4-T1, the average value of the fatigue exponent is N =

20.0.

For this example problem the fatigue parameters estimated for R = 0.8 are used to predict the

median (Pfs = 0.5) cycles to failure versus the range of the applied stresses, Ac, for the tensile

specimen geometry. These predictions are shown in Figs. E4-FI and E4-F2 for the power law and

the Walker formulation, respectively, for the various R-ratios corresponding to the experimental

data. The results were obtained using a finite element model of the gauge area of the tensile

NASA/TM--2003-106316 242
specimen and performing the subsequent reliability analysis for various peak stresses and R-ratios.

The finite element model consisted of a single shell element loaded in uniform tension and with an

area equal to the specimen gauge area. The reliability analysis used the noncoplanar strain energy

release rate criterion with a Griffith crack, although for the uniaxial stress state all the failure criteria

in CARES/LIFE are normalized to yield an identical failure probability.

In Fig. EA-F1 the power law is used with the sawtooth loading waveform to predict the effect

of changing the R-ratio. Since the fatigue parameters for R = 0.8 are used in the reliability analysis,

the median line prediction for R = 0.8 shown in the figure is in good agreement with the rupture

data, as expected. However, as the R-ratio becomes smaller the deviation from the data tends to

increase. This trend is not apparent for R = 0.5, but it is apparent for the 0.0 and -1.0 values of Rratio. In this case the deviation from the data progresses in a non conservative manner (material

strength is over predicted). These results for the 0.0 and -1.0 R-ratios seem to indicate that some

cyclic degradation is present.

In Fig. E4-F2 the Walker formulation with Q = 2.0 is used to predict the effect of changing

the R-ratio. The fatigue parameters for R = 0.8 are used and there is good agree-ment between the

median line prediction for this R-ratio and the fatigue data. Good correlation to the data is also

achieved for the 0.0 and -1.0 values of the R-ratio. Some discrepancy is noted for the R = 0.5

specimen data, however, the deviation is on the conservative side (material strength is under

predicted). The results from Fig. E4-F2 reasonably correlate with those of Liu and Chen (Liu, and

Chen, 1991) and support their conclusion that cyclic fatigue degradation has occurred. Figure E4-F2

also shows the results using the Paris law. In this case the predictions for all the R-ratios occur on

the same line (The R = 0.8 median line).

The experimental data apparently showed increased fatigue degradation due to the presence of

compressive stresses (negative R-ratio). The traditional fracture mechanics approach assumes that

NAS AfFM--2003-106316 243
thepresenceof compressive stress does not enhance crack growth. This is assumed true for the

power law formulation shown in Fig. E4-F1. With the Walker formulation this assumption means

that the median line for R = -1.0 would superimpose on the R = 0.0 median line. However to

achieve correlation to the experimental data, the line shown in Fig. E4-F2 for R = -1.0 is calculated

on the basis that there is further material degradation when compressive stress is present. The

current version of the CARES/LIFE code is constructed so that no additional material strength

degradation is predicted if compressive cyclic stresses are present (negative R-ratios), unless the

direct R-ratio input option is used. In that case the code issues an appropriate warning message. If

cyclic fatigue reliability analysis is being performed using the option of simultaneously processing

two finite element results files, then if a negative R-ratio is encountered at a particular element the

code automatically resets the value to zero.

NASA/TM--2003-106316 244
TABLEE4-T1. PARAMETERSESTIMATEDFROMCYCLICFATIGUEDATA OFZIRCONIA

R-Ratio Fatigue Power law Walker eq. Paris law

exponent, constant, constant, constant,

N s Bws , Bws, B_s,

MPa 2 • S MPa 2 • cycle MPa 2 • cycle

0.8

0.5

0.0

-1.0

19.88 5.995 (10) s 1.011 (10) s 3.225 (10) s

19.20 1.892 (10) 6 4.776 (10) 6 3.174 (10)'

15.13 2.532 (10) 5 4.083 (10) 6 4.083 (10) 6

25.79 6.829 (10) I 1.464 (10)' 2.131 (10) I1

NASA/TM--2003-106316 245
[]

0

nn

0

R=_1.0

r'l

R=0.0

^ R=0.5

_ 102

R

R=0.8

I I I I I I

101

100 101 102 103 104 105 106

Cycles to f'_lure, n t

Figure E4-Ft.--Stress amplitude-failure cycle (S-N) curve of 3Y-TZP. Median lines are extrapolated

from R = 0.8 data using the power law.

!

NAS A/TM--2003-106316 246
R=O.O

R=0.8

Figure E4-F2.--Stress amplitude-failure cycle (S-N) curve of 3Y-TZP. Median lines are extrapolated

from R = 0.8 data using the Walker equation.

NASA/TM--2003-106316 247
Example 5 - Isothermal Rotating Annular Disk

This example illustrates the use of the CARES/LIFE program modules for the time-dependent

reliability analysis of a silicon nitride (NC-132) annular disk rotating at various constant and cyclic

angular speeds. The module C4LIFE is used to compute the probability of failure of the annular

disk rotating at constant speeds for both fast fracture (t=-0) and time-dependent (static fatigue)

analyses. The problem is then repeated to determine the time-dependent probability of failure for

the annular disk rotating at various sinusoidal cyclic angular speeds. Results are presented for

various fracture criteria and crack geometries using the PIA and Batdorf models for both volume

flaw and surface flaw analyses.

The disk inside diameter was 0.0127 m, the outside diameter was 0.08255 m, and the disk

was 0.0038 m thick (fig. E5-F1). Analysis was performed on the annular rotating disk at a uniform

1000°C temperature. The material/environmental fatigue parameters, N=30 and B=320 MPa 2 hr,

necessary for time-dependent reliability analysis were estimated using NC-132 data at 1000°C

("Quinn, and Quinn, 1983). The same fatigue parameters were used for both volume flaw and

surface flaw analyses. Weibull material parameters independently evaluated at room temperature

from four-point bend tests of 85 specimens were assumed to be valid for the 1000°C analysis. For

the given "A-size" bar geometry (w = 0.00635 m, h = 0.003175 m, Lm= 0.01905 m, and L2 =

0.009525 m), the Weibull modulus, m v, was 7.65 and the characteristic strength, Cr0v, was 808 MPa

(^Swank, and Williams, 1981). For volume flaws, equations (56) and (60) are used to calculate the

Weibull scale parameter, 6oV = 74.79 MPa (m) 3/7'65. Assuming that fractures in the four-point bend

specimens were caused by surface flaws, the Weibull modulus, m s, was 7.65, the characteristic

strength, Oos, was 808 MPa, and the surface scale parameter, eros, calculated from equations (69) and

(73), was 232.0 MPa (m) 2/7'65. Other required material properties include a Young's modulus of 289

GPa, a Poisson's ratio of 0.219, and a material mass density of 3.25x103 kg/m 3.

NASA/TM--2003-106316 248
Sevendisksweretestedin fastfractureandthe experimental disk Weibull modulus obtained

was 4.95. This value differed considerably from the 7.65 value ("Swank, and Williams, 1981)

based on four-point bend specimen data. A better agreement between disk and four-point bend

specimen Weibull slopes would lead to improved predictions in failure probabilities for the entire

range. Estimation of parameters from small sample sets greatly increases potential deviation from

the true population parameters. Confidence limits are used to measure the intrinsic uncertainties in

parameter estimates from finite sample sizes. The 90-percent confidence limits on m can be

obtained from Thoman, et al. (^Thoman, Bain, and Antic, 1969) For the 85 four-point bend

specimens, with mv = 7.65, the calculated limits are 6.56 and 8.68. Similarly, for the seven

fractured disks and the estimated m v of 4.95, the 90-percent confidence limits are 2.27 and 6.98.

Excluding potential experimental errors, it is possible that the rotating disks and the four-point bend

specimens broke because of the same flaw population since their 90-percent confidence limits

overlap between 6.56 and 6.98. Therefore, it cannot be concluded with absolute certainty that the

disks broke from a different flaw population than the four-point bend bars unless further testing is

performed. The apparent differences between prediction and experiment are not significant enough

to cause rejection of the hypothesis that the bars and disks broke because of the same flaw

population.

MSC/NASTRAN finite element models of the annular disk were created for several constant

rotational speeds between 60 000 rpm and 120 000 rpm. For volume flaw analysis, the

MSC/NASTRAN finite element model of the disk consisted of brick elements and used the

structured solution sequence 114, static cyclic symmetry analysis. Because of symmetry, only eight

HEXA elements were used in one 15° sector of the model (fig. E5-F1). Each element spanned both

the thickness and circumferential directions. For surface flaw analysis, the same MSC/NASTRAN

finite element analysis was performed as for volume flaws with the addition of QUAD8 shell

elements to identify external surfaces of the model. The shell elements shared common nodes with

NASA/TM--2003-106316 249
thesolidelements.Thecontributionof these elements to the overall stiffness of the model was

negligible. The thickness of the shell elements was 1.0 x 10.6 m and only membrane properties

were assigned to these elements via the PSHELL BULK DATA card in MSC/NASTRAN. The

various constant angular rotations considered were applied via RFORCE and LOADCYH BULK

DATA cards. The stresses and element volumes calculated using the finite element model were

both within approximately 1 percent of the available closed-form solution for this problem. A

model with twice as many elements along the radial direction was also analyzed to check mesh

convergence, and the resulting stresses were extremely close to those obtained with only eight

HEXA elements.

After each MSC/NASTRAN finite element analysis was executed, and the MSC/NASTRAN

output files were created, the NASCARES module was used to create a neutral file for each angular

speed considered• The information contained in the neutral files (the element/subelement stresses,

volumes/areas, and temperatures) was read by the C4LIFE module for subsequent volume flaw and

surface flaw reliability analyses. The number of Gauss points used in the numerical integration for

reliability was set to 15 (NGP = 15). The shear-sensitive IKBAT = 1 option, with C = 0.82, was

used when performing the reliability analysis to calculate the normalized Batdorf crack density

coefficient based on the various fracture criteria and crack geometries selected.

For the annular disk rotating at various constant angular speeds, reliability calculations were

made as a function of disk rotational speed and time for several fracture criteria and crack

configurations. The fast-fracture (t=-0) and time-dependent (static fatigue) analyses were performed

at constant rotational speeds between 60 000 and 120 000 rpm to cover the entire range of failure

probabilities predicted by the fracture models within CARES/LIFE.

Fast-fracture and time-dependent probability of failure results for the annular disk rotating at

constant angular speeds are shown in tables E5-TI and E5-T2 for volume flaw analysis and surface

flaw analysis, respectively. As expected, these results show that the probability of failure increases

NAS A/TM--2003-106316 250
with bothincreasingangularspeedandtime. Fora givenconstantangularspeedandtime,failure

probabilitiesobtainedby surfaceflawanalysisareconsiderablylessthanthoseobtainedby volume

flaw analysisfor all fracturemodels,indicatingthatfailurewouldmostlikely occurdueto volume

flaws. Themainreasonforthegreatlydecreasedfailureestimatesfor surfaceflaw analysisis the

muchhigherequivalentsurfaceWeibullscaleparameter,OoS.FigureE5-F2is a Weibullplotof

time-dependent probabilityof failureresultsfor theannulardisk rotatingatvariousconstantangular

speedsusingthenoncoplanarstrainenergyreleaseratefracturecriterionwiththeBatdorfmodel.

Forthis case,volumeflaw analysiswasperformedassumingapenny-shapedcrackgeometry,and

surfaceflaw analysisassumeda semi-circularcrackgeometry.Theimportanceof postmortem

fractographyto identifythenatureof thefracture-causingflawsis evidentfrom thetwo widely

differentsetsof answersshownin figureE5-F2forthevolumeflaw andsurfaceflawanalyses.

Theproblemwasrepeatedfor theannulardiskrotatingat varioussinusoidalcyclic angular

speeds.Reliabilitycalculationsfortherotatingannulardiskweremadeasafunctionof cyclic

rotationalspeedandtimefor variousfracturecriteriaandcrackgeometries.Two approacheswere

usedto determinetheprobabilityof failurefor thecyclic fatigueproblem.Forthefirst approach,

reliabilityanalysiswasperformedfor twosinusoidalcyclic angularspeedvariations:(1) 60000

rpmto 70000rpmand(2) 60000rpmto 80000rpm. In thisapproach,reliabilitycalculations

weremadebasedona userinputR-ratioandastressdistributionthroughoutthediskbasedonthe

maximumcyclic stresses.Thesecondapproachwasusedto verify theresultsof thefirst reliability

analysisforthe60000rpmto 80000rpmcycle. In thisapproach,thesinusoidalcyclic angular

speedvariationwastransformedto anequivalentconstantangularspeed,andreliabilityanalysiswas

performedfortheequivalentstaticstressdistribution.

Forthefirst approach,theprocedureoutlinedabovewasfollowedto performthe

MSC/NASTRANfinite elementanalysesandcreatetheneutralfiles requiredfor subsequent

reliabilityanalysiswithCARES/LIFE.Foreachcyclic angluarspeedconsidered,themodule

NASA/TM--2003-106316 251
C4LIFEwasusedwith oneneutralfile (containing information relative to the maximum rotational

speed for the cycle) to calculate the time-dependent probability of failure for both volume flaw and

surface flaw analyses. The reliability analysis was performed using the power law with the period

of one cycle equal to 36 seconds (0.01 hr). Since the stress distribution in a rotating annular disk is

directly proportional to the square of the angular speed, the QCYC keyword was used to assign the

value of 2.0 to the loading waveform exponent. This value is used by the C4LIFE module to

determine the g-factor associated with the given waveform. The ratio of the minimum to the

maximum cycle stress (in this case, the square of the ratio of the minimum to the maximum

rotational speed) was specified using the CYCLIC keyword in the C4LIFE input file.

For this approach, reliability calculations for the rotating annular disk were made as a

function of cyclic rotational speed and time for two sinusoidal cyclic angular speed variations of 60

000 rpm to 70 000 rpm and 60 000 rpm to 80 000 rpm. Results from these analyses are shown in

tables E5-T3 and E5-T4 for volume flaw analysis and surface flaw analysis, respectively, for various

fracture criteria and crack geometries. Figure E5-F3 is a Weibull plot of time-dependent probability

of failure results for the annular disk rotating at a sinusoidal cyclic angular speed varying from 60

000 rpm to 80 000 rpm over a period of 0.01 hr using the noncoplanar strain energy release rate

fracture criterion with the Batdorf model. For this case, volume flaw analysis was performed

assuming a penny-shaped crack geometry, and surface flaw analysis assumed a semi-circular crack

geometry. As expected, the reliability predictions for the disk rotating at this particular cyclic

angular speed are between the predicted reliabilities for constant angular speeds of 60 000 rpm and

80 000 rpm. Also, as observed for the annular disk rotating at constant angular speeds, failure

probabilities obtained by surface flaw analysis are lower than those obtained by volume flaw

analysis for all fracture models.

To verify the results of the above analysis, the sinusoidal cyclic load condition for the 60 000

rpm to 80 000 rpm cycle was transformed to an equivalent static load condition, and reliability

NASA/TM--2003-106316 252
analysiswas performed for the equivalent static problem. As discussed in the section Cyclic

Fatigue, transforming a periodic cyclic load condition to an equivalent static load condition is based

on satisfying the requirement that both systems produce the same crack growth. An equivalent

constant angular speed, 03eq,which satisfies this requirement is calculated for the given cycle

(1)_1 If.or [ (_2C°min)+ (6°_°mi_)sin2r_____t ]2Ndt 1_

_- T

T

where T is the period of one cycle, N is the fatigue crack growth exponent, t is the service time,

corn,x is the maximum rotational speed in the cycle, and 0_m_. is the minimum rotational speed in the

cycle. The equivalent constant angular speed calculated for the 60 000 rpm to 80 000 rpm cycle

was 77 485.20 rpm. For this equivalent constant angular speed, the procedure outlined previously

was followed to perform an MSC/NASTRAN finite element analysis and create the neutral file

required for subsequent reliability analysis. The module C4LIFE was then used to perform timedependent reliability analysis for the equivalent constant angular speed problem. Overall, the results

of this analysis agreed with the results presented for the cyclic analysis in tables E5-T3 and E5-T4.

For short service times, however, the results deviate slightly from those presented in tables E5-T3

and E5-T4, due to the presence of the fast-fracture term which appears in the calculation of the

transformed equivalent stress distribution (equation (97)). As the service time increases, the

influence of this fast-fracture term diminishes.

To verify the equivalence of the two approaches employed for performing the reliability

analysis for the 60 000 rpm to 80 000 rpm sinusoidal cyclic angular speed variation, the g-factors

associated with each of the two approaches were compared. For the first approach, the g-factor

calculated for the 60 000 rpm to 80 000 rpm cycle using the C4LIFE reliability analysis module was

g = 0.147322. For the second approach, the equivalent constant angular speed was used to calculate

the g-factor

NAS A/TM--2003-106316 253
g=

For this case, the g-factor calculated was g = 0.147138. Since the difference in the values of the gfactors obtained for the two approaches is essentially negligible, the equivalence of the two

approaches is verified.

NASA/TM--2003-106316 254
J

Figure E5-F1.--Rotating annular disk with 150 sector finite element mesh. Inner disk radius, ri, 6.35

mm, outer disk radius, ro, 41.28 ram; disk thickness, t, 3.80 mm.

i

NAS A/TM--2003-106316 255
TABLE E5-TI. FAILURE PROBABILITIES OF A ROTATING ANNULAR

DISK-CONSTANT RPM VALUES-VOLUME FLAW ANALYSIS

[NGP=I5; 1KBAT=I; mv=7.65; O.v=74.79 MPa(m)3a65; N=30; B=320 MPa 2 hr]

Angular

Speed,

rpm

60 000

70 000

80 000

90 000

100 000

II0 000

120 000

Service

Time,

hrs

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

i.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

i.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

PIA

criterion

0.0002101

0.0003598

0.0006488

0.001211

0.002270

0.004254

0.002219

0.004408

0.008071

0.01505

0.02804

0.05196

0.01699

0.03829

0.06971

0.1266

0.2242

0.3789

0.09869

0.2349

0.3922

0.6067

0.8263

0.9625

0.4060

0.7771

0.9392

0.9947

0.9999

1.0000

0.8934

0.9992

!.0000

1.0000

1.0000

1.0000

0.9998

1.0000

1.0000

1.0000

1.0000

1.0000

CARES/LIFE Failure Probability

Batdorf Model

Noncoplanar strain

energy release rate

criterion

(C=o.82)

Griffith

crack

0.0001920

0.0003268

0.0005882

0.001098

0.002058

0.003857

0.002028

0.004001

0.007320

0.01365

0.02545

0.04721

0.01554

0.03479

0.06340

0.1155

0.2056

0.3506

0.09058

0.2155

0.3632

0.5708

0.7954

0.9490

0.3787

0.7436

0.9210

0.9914

0.9999

1.0000

0.8707

0.9985

1.0000

1.0000

1.0000

1.0000

0.9996

1.0000

1.0000

1.0000

1.0000

1.0000

Pennyshaped

crack

0.0001817

0.0003095

0.0005573

0.001040

0.001950

0.003655

0.001920

0.003791

0.006936

0.01294

0.02413

0.04479

0.01471

0.03299

0.06017

0.1097

0.1959

0.3357

0.08595

0.2055

0.3479

0.5513

0.7776

0.9404

0.3627

0.7246

0.9097

0.9890

0.9998

1.0000

0.8558

0.9978

1.0000

1.0000

1.0000

1.0000

0.9993

1.0000

1.0000

1.0000

1.0000

1.0000

Griffith

crack

0.0002307

0.0003911

0.0007033

0.001313

0.002460

0.004610

0.002437

0.004785

0.008746

0.01630

0.03036

0.05619

0.01865

0.04147

0.07533

0.1364

0.2405

0.4032

0.1078

0.2520

0.4171

0.6363

0.8500

0.9715

0.4356

0.8036

0.9519

0.9966

1.0000

1.0000

0.9145

0.9996

1.0000

1.0000

1.0000

1.0000

0.9999

1.0000

i.0000

1.0000

!.0000

1.0000

Batdorf Model

Energy release

rate criterion

GT

Pennyshaped

crack

0.0002209

0.003751

0.0006748

0.001260

0.002360

0.004423

0.002333

0.004590

0.008392

0.01565

0.02914

0.05397

0.01786

0.03982

0.07238

0.1313

0.2320

0.3905

0.1035

0.2431

0.4042

0.6211

0.8380

0.9671

0.4217

0.7901

0.9456

0.9957

1.0000

1.0000

0.9050

0.9994

1.0000

1.0000

1.0000

1.0000

0.9999

1.0000

1.0000

!.0000

1.0000

i.0000

NASA/TM--2003-106316 256
TABLE5-T2.FAILUREPROBABILITIES OFAROTATINGANNULAR

DISK-CONSTANT RPMVALUES-SURFACE FLAWANALYSIS

[NGP=I5; IKBAT=I; ms=7.65; Oos=232.0 MPa(m)Zn6_; N=30; B=320 MPa 2 hr]

Angular

Speed,

rpm

60 000

70 000

80 000

90 000

100 000

110 000

120 000

Service

Time,

hrs

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.0

0.1

1.0

10.0

100.0

1000.0

PIA

criterion

0.00003677

0.00006674

0.0001214

0.0002269

0.0004254

0.0007979

0.0003888

0.0008223

0.001515

0.002835

0.005311

0.009940

0.002995

0.007258

0.01343

0.02502

0.04642

0.08531

0.01802

0.04878

0.08899

0.1603

0.2795

0.4593

0.08713

0.2447

0.4079

0.6257

0.8417

0.9685

0.3242

0.7372

0.9179

0.9908

0.9998

1.0000

0.7731

0.9961

1.0000

1.0000

1.0000

1.0000

Griffith

crack

CARES/LIFE Failure Probability

Batdorf Model

Noncoplanar strain

energy release rate

criterion

( C =0.82)

Semicircular

crack

0.00003389

0.00006183

0.0001125

0.0002103

0.0003943

0 0007396

0.0003584

0.0007620

0.001404

0.002628

0.004924

0.009217

0.002761

0.006728

0.01246

0.02321

0.04310

0.07932

0.01662

0.04529

0.08276

0.1495

0.2620

0.4344

0.08060

0.2290

0.3848

0.5978

0.8189

0.9595

0.3031

0.7103

0.9014

0.9870

0.9997

1.0000

0 7452

0.9942

0 9999

1.0000

1.0000

1.0000

0.00003502

0.00006370

0.0001158

0.0002165

0.0004060

0.0007615

0.0003703

0.0007848

0.001446

0.002706

0.005069

0.009489

0.002853

0.006928

0 01282

0.02389

0.04435

0.08157

0.01717

0.04661

0.08511

0.1536

0.2686

0.4439

0.08317

0.2349

O.3936

0.6086

0.8278

0.9631

0.3115

0.7207

0.9080

0.9886

0.9998

1.0000

0.7566

0.9950

10000

1.0000

1.0000

1.0000

Batdorf Model

Energy release

rate criterion

Gr

Griffith

crack

0.00003843

0.00006923

0.0001257

0.0002349

0.0004405

0.0008262

0.0004063

0.0008523

0.001569

0.002936

0.005499

0.01029

0.003130

0.007520

0.01391

0.02590

0.04803

0.08820

0.01882

0.05049

0.09201

0.1655

0.2878

0.4709

0.09087

0.2522

0.4189

0.6386

0.8518

0.9722

0.3360

0.7495

0.9249

0.9922

0.9999

1.0000

0.7878

0.9968

1.0000

1 0000

1.0000

1 0000

NASA/TM--2003-106316 257
0

99

gO

100 000 r/min

•100 000 r/min

80 000 rL/min

60 000 r/mln

60 000__r/rain

1000

figure ES-F2 Ti

peeds "" me-de

tile Ba_-, Analysis per/;, 'Pendent Vro_,-_..

taorf _zorraed .- v=oll i .

denote ,,_,, _Odel. 8 i;._ ,. using th _ ty of fadure _

• Su_zace l_a_=, _ O-.u lines ,_^ e no/iCOn]an - 1For alln.. L .

-- analysis r_c.aote Volu r __ar Strata e "'_ disk rot

_,_ " -. me _ n a "

ml'carcUlar_ ._aw _ ....ergy rolea_^ tmg at con°,-

" Crack) _":ysls (Pen- ._c rate fra_ -.o_aat an,...,

" qY'Shane._ '_mre c,';,- • su_ar

• _ _ Crack)..__ :-crzoz_wkh

"" _ed lines _"

NASA/TM--'_2003-106316
TABLE E5-T3. FAILURE PROBABILITIES OF A ROTATING ANNULAR

DISK-CYCLIC RPM VALUES-VOLUME FLAW ANALYSIS

[NGP=I5; IK.BAT=I; mv=7.65; Oov=74.79 MPa(m)_n6s; N=30; B=320 MPa 2 hr]

Angular

Speed,

rpm

60 000-

70 000

60 000-

80 00

Service

Time,

hrs

0.01

0.10

1.00

10.00

100.00

1000.00

0.01

0.10

1.00

10.00

100.00

1000.00

0.002352

0.003072

0.005246

0.009706

0.01811

0.03370

0.01824

0.02455

0.04228

0.07715

0.1397

0.2459

CARES/LIFE Failure Probability

PIA Batdorf Model

criterion

Noncoplanar strain

energy release rate

criterion

(C=o.82)

Griffith

crack

0.002147

0.002795

0.004759

0.008802

0.01643

0.03059

0.01667

0.02235

0.03841

0.07019

0.1275

0.2257

Pennyshaped

crack

0.002033

0.002647

0.004510

0.008341

0.01557

0.02901

0.01578

0.02118

0.03643

0.06663

0.1212

0.2152

Batdorf Model

Energy release

rate criterion

GT

Griffith

crack

0.002579

0.003347

0.005690

0.01052

0.0196!

0.03647

0.01998

0.02671

0.04577

0.08334

0.1505

0.2635

Pennyshaped

crack

0.002

0.00:

0.00. _

0.01(

0.018

0.03:

0.019

0.025

0.04:

0.08C

0.144

0.254

NAS A/TM--2003-106316 259
TABLE E5-T4. FAILURE PROBABILITIES OF A ROTATING ANNULAR

DISK-CYCLIC RPM VALUES-SURFACE FLAW ANALYSIS

[NGP=I5; IKBAT=I; ms=7.65; 00s=232.0 MPa(m)_6_; N=30; B=320 MPa 2 hr]

Angular

Speed,

rpm

60 000-

70 000

60 000-

80 00

Service

Time,

hrs

0.01

0.10

1.00

10.00

100.00

1000.00

0.01

0.10

1.00

I0.00

100.00

100(3.00

CARES/LIFE Failure Probability

PIA

criterion

0.0004175

0.0005642

0.0009815

0.001824

0.003415

0.006397

0.003268

0.004564

0.008036

0.01491

0.02778

0.05147

Griffith

crack

0.0003854

0.0005223

0.0009096

0.001691

0.003166

0.005930

0.003018

0.004227

0.007450

0.01383

0.02577

0.04780

Batdorf Model

Noncoplanar strain

energy release rate

criterion

(C--o.82)

Semicircular

crack

0.0003979

0.0005384

0.0009367

0.001741

0.003260

0.006106

0.003116

0.004356

0.007670

0.01424

0.02653

0.04918

Batdorf Model

Energy release

rate criterion

GT

Grimm

crack

0.0004357

0.0005863

0.001017

0.001889

0.003537

0.006623

0.003410

0.004740

0.008324

0.01544

0.02875

0.05325

NASA/TM--2003-106316 260
D

I--'4

<l

O

I--4

I--.4

m

<

m

o

99

9O

8O

50

2O

10

5

.t

I I I

0.i 1 I0 io0 io0

SERVICE TIME (hrs)

Figure E5-F3.--Time-dependent probability of failure for annular disk rotating at sinusoidal cyclic

speed varying from 60 000 rpm to 80 000 rpm (period = 0.01 hr). Analysis performed using the

noncoplanar strain energy release rate fracture criterion with the Batdorf model. Solid lines denote

volume flaw analysis (penny-shaped crack); dashed lines denote surface flaw analysis (semi-circular

crack).

NASA/TM--2003-106316 261
Example 6 - Thermomechanically Loaded Rotating Annular Disk

This example illustrates the use of the CARES/LIFE program modules for the time-dependent

reliability analysis of an annular disk subjected to a thermomechanical load at various constant

angular speeds. The module C4LIFE is used to compute the probability of failure of the annular

disk for both fast fracture (t=0) and time-dependent (static fatigue) analyses. Results are presented

for the noncoplanar strain energy release rate fracture criterion using the Batdorf model for both

volume flaw (penny-shaped crack geometry) and surface flaw (semi-circular crack geometry)

analyses.

The annular disk analyzed in this example is a modified version of that in Example 5-

Isothermal Rotating Annular Disk. The disk inside diameter was 0.0127 m, the outside diameter

was 0.08255 m, and the disk was 0.0038 m thick (fig. E6-F1). The finite clement mesh geometry

was identical to that in example 5. For example 5, the disk was entirely made of silicon nitride and

the analysis was performed at a constant temperature. For this example, the disk was composed of

three materials: silicon nitride, silicon carbide, and steel. Elements were assigned material

properties accordingly, and a radial variation in temperature was imposed.

MSC/NASTRAN finite element models of the disk were created for each constant angular

speed considered. The MSC/NASTRAN finite element model of the disk for volume flaw analysis

consisted of brick elements and used the structured solution sequence 114, static cyclic symmetry

analysis. Because of symmetry, only eight HEXA elements were used in one 15° sector of the

finite element model (fig. E6-F1). Each element spanned both the thickness and circumferential

directions. For surface flaw analysis, the same MSC/NASTRAN analysis was performed as for

volume flaws with the addition of QUAD8 shell elements to identify external surfaces of the model.

The shell elements shared common nodes with the solid elements. The contribution of these

elements to the overall stiffness of the model was negligible. The thickness of the shell elements

NASA/TM--2003-106316 262
was1.0x 10.6m andonly membrane properties were assigned to these elements via the PSHELL

BULK DATA card in MSC/NASTRAN.

For this example, the disk consisted of three materials: two were ceramics and one was

metal. The material properties were assigned via PSHELL, PSOLID, MATI, MATT1, and

TABLEM1 cards in the MSC/NASTRAN BULK DATA. The MAT1 card assumes temperatureindependent material properties. The MATT1 and TABLEM1 cards are employed to define

temperature-dependent material properties. Only the PSHELL and PSOLID cards are required for

postprocessing with CARES/LIFE. For this example, the innermost solid element was sintered

silicon carbide with an assigned MAT1 material ID of 302. The three shell elements (I00, 110, and

130) that shared nodes with this element were also assigned to material ID 302. Temperaturedependent material properties (Sturmer, Schulz, and Wittig, 1991) shown in table E6-TI were

assigned to material ID 302 via MATT1 and TABLEM1 cards in the MSC/NASTRAN BULK

DATA. Temperature-independent material properties assigned to the silicon carbide include a

Poisson's ratio of 0.160 and a material mass density of 3.420x103 kg/m 3. The outermost HEXA

element was assigned to MAT1 material ID 301 and had temperature-independent material

properties consistent with steel: a Young's modulus of 20.69 GPa, a Poisson's ratio of 0.3, a

material mass density of 7.834x103 kg/m 3, and a thermal expansion coefficient of 3.70x10 "6°C"t.

The three shell elements (107, 117, and 120) that shared nodes with this element also had material

ID 301. The remaining 6 solid and 12 shell elements were silicon nitride with an assigned MAT1

material ID of 300. The required temperature-independent material properties assigned to the silicon

nitride include a Young's modulus of 289 GPa, a Poisson's ratio of 0.219, a material mass density

of 3.25x103 kg/m 3, and a thermal expansion coefficient of 3.083x10 -6 °C't.

For this example, a thermal gradient that varied with the disk radius was imposed on the

nodes. The temperature varied from 1093°C at the inner bore to 704°C at the outside periphery.

NASA/TM--2003-106316 263
Thegradientwasapproximatelylinear,andtheloadswerearbitrarilyimposed.Thethermalloads

wereassignedvia TEMP and LOADCYN BULK DATA cards. The various constant angular

rotations considered were applied via RFORCE and LOADCYH BULK DATA cards.

Atter each MSC/NASTRAN finite element analysis was executed, and the MSC/NASTRAN

output files were created, the NASCARES module was used to create a neutral file for each angular

speed considered. The information contained in the neutral files (the element/subelement stresses,

volumes/areas, and temperatures) was read by the C4LIFE module for subsequent volume flaw and

surface flaw reliability analyses.

The Weibull and fatigue parameters needed for the reliability analyses are given in tables E6-

T2 and E6-T3. For the silicon nitride, the temperature-dependent Weibull parameters were assumed

based on values obtained for the Weibull modulus at room temperature (Swank, and Williams,

1981)(Nemeth, Manderscheid, and Gyekenyesi, 1990) The temperature-dependent

material/environmental fatigue parameters necessary for the time-dependent reliability analysis were

estimated using values obtained for silicon nitride (NC-132) data at 1000°C (Quinn, and Quinn,

1983). The parameters were directly input at temperature levels of 538°C, 816°C, and 1093°C. For

the sintered silicon carbide, the Weibull and fatigue parameters were obtained from Stunner, et al.

(Sturmer, Schulz, and Wittig, 1991) based on four-point bend test specimen geometry with w =

0.005 m, h = 0.004 m, L_ = 0.040 m, and L 2 = 0.020 m. Weibull parameters at room temperature

were assumed valid for the temperature-dependent analyses. Temperature-dependent fatigue

parameters were estimated from this data and directly input at the three temperature levels given

above. Material 301 was steel and was subsequently ignored in the reliability analysis by not

specifying it in either the Master Control Input or the Material Control Input.

For the thermomechanically loaded annular disk rotating at various constant angular speeds,

reliability calculations were made as a function of disk rotational speed and time for the fracture

NASA/TM--2003-106316 264
criterionand crack configurations considered. The number of Gauss points used in the numerical

integration for reliability was set to 15 (NGP = 15). The shear-sensitive IKBAT = 1 option, with =

0.82, was used when performing the reliability analysis to calculate the normalized Batdorf crack

density coefficient based on the fracture criterion and crack geometries selected. Fast-fracture (t=0)

and time-dependent (static fatigue) reliability analyses were performed at constant rotational speeds

of 2000 rpm, 3000 rpm, 6000 rpm, and 10 000 rpm to examine the range of failure probabilities

predicted by the C4LIFE program module.

Fast-fracture and time-dependent probability of failure results for the annular disk rotating at

constant angular speeds are given in tables E6-T4 and E6-T5 for volume flaw analysis and surface

flaw analysis, respectively. Figure E6-F2 is a Weibull plot of time-dependent probability of failure

results for the annular disk rotating at various constant angular speeds. These results were obtained

using the noncoplanar strain energy release rate fracture criterion with the Batdorf model. Volume

flaw analysis was performed assuming a penny-shaped crack geometry, and surface flaw analysis

assumed a semi-circular crack geometry. As expected, these results show that the probability of

failure increases with both increasing angular speed and time. For a given constant angular speed

and time, failure probabilities obtained by surface flaw analysis are considerably less than those

obtained by volume flaw analysis, indicating that failure would most likely occur due to volume

flaws.

NASA/TM--2003-106316 265
r I

Figure E6-F1.--Thermomechanically loaded rotating annular disk with 15° sector finite element

mesh. Inner disk radius, ri, 6.35 mm; outer disk radius, ro, 41.28 mm; disk thickness, t, 3.80 mm.

TABLE E6-T1. TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

FOR SILICON CARBIDE

Temperature

*C

599.85

799.85

999.85

1199.85

E

GPa

422.6

419.0

415.4

411.8

oc°l

-_ 1.678x10 _

1.678x10 _

1.689x10 _

1.728x10 _

NAS A/TM--2003-106316 266
t_

¢-.q

O"1

'7

[..-,

Z

oz

©

r/3

[.-,

[...,

<

<z

r,i

[.-,

,,d,

,.--1

<

t"_ ¢'xl C'q

"_ wl r,-; _

¢'.,I .....-, Ox

1=:1_ e,.'_ r,_ rq

> _ OO ',,,O

Z ¢_ ¢'q ¢'q

,_, • .

F: "-' ,-'

[...,

<

Z

©

©

<

<

z

z_

>

P'- t'_ P '-

I_ t¢-_ t--.l

t.,_ r_ r_

, o o .

NAS A/TM_2003-106316 267
TABLE E6-T4. FAILURE PROBABILITIES FOR THERMOMECHANICALLY LOADED

ROTATING ANNULAR DISK (VOLUME FLAW ANALYSIS)

Noncoplanar strain energy release rate criterion

Batdorf model

(Penny-shaped crack)

[NGP=15; IKBAT=I; _2=0.82]

Service CARES/LIFE Failure Probability

time,

hrs Angular speed,

rpm

2 000 3 000 6 000 10 000

0.0

0.1

1.0

10.0

30.0

50.0

100.0

300.0

500.0

700.0

900.0

1000.0

0.4691

0.6909

0.8624

0.9665

0.9877

0.9930

0.9971

0.9995

0.9998

0.9999

0.9999

1.0000

0.4728

0.6951

0.8654

O.9678

0.9883

0.9933

0.9973

0.9995

0.9998

0.9999

0.9999

1.0000

0.4937

0.7178

0.8820

0.9743

0.9912

0.9952

0.9981

0.9997

0.9999

1.0000

1.0000

1.0000

0.5489

0.7740

0.9191

0.9866

0.9962

0.9981

0.9994

0.9999

1.0000

1.0000

1.0000

1.0000

NASA/TM--2003-106316 268
TABLE E6-T5. FAILURE PROBABILITIES FOR THERMOMECHANICALLY LOADED

ROTATING ANNULAR DISK (SURFACE FLAW ANALYSIS)

Noncoplanar strain energy release rate criterion

Batdorf model

(Semi-circular crack)

[NGP=15; IKBAT=I; C=0.82]

Service CARES/LIFE Failure Probability

time,

hrs Angular speed,

rpm

2 000 3 000 6 000 10 000

0.0

0.1

1.0

10.0

30.0

50.0

100.0

300.0

500.0

700.0

900.0

I000.0

0.1405

0.3084

0.4720

0.6719

0.7667

0.8076

0.8580

0.9223

0.9448

0.9571

0.9649

0.9679

0.1436

0.3148

0.4804

0.6809

0.7750

0.8153

0.8647

0.9270

0.1615

0.3509

0.5267

0.7285

0.8176

0.8543

0.8978

0.9493

0.9486

0.9602

0.9676

0.9704

0.9659

0.9745

0.9798

0.9818

0.2108

0.4453

0.6392

0.8303

0.9010

0.9269

0.9547

0.9824

0.9897

0.9930

0.9949

0.9956

NASA/TM--2003-106316 269
_d

I",,

,d

t_

t.

O

tt,n

Izl

o

re

.11.

95

gC

80

50

20

"- I©

_.o o(_rpm

I| I a

i.l 10 10() ,UO0

SERVICI_ TIME (l_rs)

Figure E6-F2.--Time-dependent probability of failure for thermomechanically loaded annular disk

rotating at constant angular speeds. Analysis performed using the noncoplanar strain energy release

rate fracture criterion with the Batdorf model. Solid lines denote volume flaw analysis (pennyshaped crack); dashed lines denote surface flaw analysis (semi-circular crack).

NASA/TM--2003-106316 270
Example 7 - Proof Testing

This problem illustrates the use of the CARES/LIFE program modules for determining the

time-dependent reliability of a silicon nitride (NC-132) beam (Fig. E7-F1) subjected to a pure bend

service load aider being proof tested in pure bend or pure tension for various time periods. As

discussed in the section Proof Testing Effect on Component Service Probability of Failure, proof

testing components allows weaker specimens to be eliminated from the strength distribution. Thus,

a component can be placed in service with greater confidence in its integrity. In this example, the

resulting attenuated probability of failure as a function of time is computed using the noncoplanar

strain energy release rate fracture criterion with the Batdorf model for both volume flaw (pennyshaped crack geometry) and surface flaw (semi-circular crack geometry) analyses. Based on a

service load having a fast fracture probability of failure of approximately 0.03, four combinations of

proof test loading and service loading were considered: (1) The beam was proof tested under a pure

bend load condition and then subjected to an in-service pure bend load condition which produces a

maximum stress 25% lower than the maximum proof test stress. (2) The beam was proof tested

under a pure bend load condition and then subjected to an in-service pure bend load condition which

produces a maximum stress 25% higher than the maximum proof test stress. (3) The beam was

proof tested under a pure tension load condition and then subjected to an in-service pure bend load

condition which produces a maximum stress 25% lower than the maximum proof test stress. (4)

The beam was proof tested under a pure tension load condition and then subjected to an in-service

pure bend load condition which produces a maximum stress 25% higher than the maximum proof

test stress. Cases (1) and (2) were designed to illustrate the ideal situation, where the boundary

conditions applied to a component under proof testing identically simulate those conditions the

component is subjected to in service, and differ only in magnitude. Cases (3) and (4) illustrate offaxis proof test loading, i.e., the proof test and service loads are misaligned (pure tension vs. pure

bending).

NASA/TM--2003-106316 271
Thebeamdimensions are given in figure E7-F1. Analysis was performed on the beam at a

uniform 1000°C temperature. The material/environmental fatigue parameters, N=30 and B=320

MPa 2 hr, necessary for time-dependent reliability analysis were estimated using NC-132 data at

1000°C (Quinn, and Quinn, 1983). The same fatigue parameters were used for both volume and

surface flaw analyses. Weibull material parameters independently evaluated at room temperature

from four-point bend tests of 85 specimens were assumed to be valid for the 1000°C analysis. For

volume flaw analysis, the Weibull modulus, m v, was 7.65 (Swank, and Williams, 1981) (Nemeth,

Manderschied, and Gyekenyesi, 1990). The Weibull scale parameter, calculated using equations

(56) and (60), was OoV= 74.79 MPa (m) 3n6s. For surface flaw analysis, the Weibull modulus, m s,

was 7.65 and the scale parameter, calculated from equations (69) and (73), was OOS=232.0 MPa

(m) _'65. Other required material properties include a Young's modulus of 289 GPa and a Poisson's

ratio of 0.219.

MSC/NASTRAN finite element models of the beam were created for both volume flaw and

surface flaw analyses for each of the service and proof test load conditions considered. The

MSC/NASTRAN finite element analyses used the structured solution sequence 101, static analysis.

Because of symmetry, only one quarter of the beam was modeled for each finite element analysis,

as indicated in figure E7-F1. For volume flaw analysis, the finite element mesh of the quarter beam

consisted of twenty HEXA elements in the length (x) direction, eight HEXA elements in the height

(3') direction, and one HEXA element in the thickness (z) direction. For surface flaw analysis, the

same MSC/NASTRAN finite element model was used as for volume flaws with the addition of

QUADS shell elements to identify external surfaces of the model. The shell elements shared

common nodes with the solid elements. The contribution of these elements to the overall stiffness

of the model was negligible. The thickness of the shell elements was 1.0xl0 "6 m, and only

membrane properties were assigned to these elements via the PSHELL BULK DATA card in

MSC/NASTRAN. The pure bend load conditions were induced in the beam by prescribing

NASA/TM--2003-106316 272
displacementsin thelongitudinaldirectionwhichareproportionalto the y-coordinate of the beam at

x=L. The pure tension load conditions were induced in the beam by applying a uniform tensile

force at x=L. For case (1), the beam was proof tested under a pure bend load condition producing a

maximum stress of 722.5 MPa and then subjected to a pure bend service load condition producing a

maximum stress of 578.0 MPa. For case (2), the beam was proof tested under a pure bend load

condition producing a maximum stress of 433.5 MPa and then subjected to a pure bend service load

condition producing a maximum stress of 578.0 MPa. For case (3), the beam was proof tested

under a pure tension load condition producing a maximum stress of 722.5 MPa and then subjected

to a pure bend service load condition producing a maximum stress of 578.0 MPa. For case (4), the

beam was proof tested under a pure tension load condition producing a maximum stress of 433.5

MPa and then subjected to a pure bend service load condition producing a maximum stress of 578.0

MPa.

After the MSC/NASTRAN finite element analyses were executed, and the MSC/NASTRAN

output files were created, the NASCARES module was used to create a neutral file for each load

condition. The information contained in the neutral files (the element/subelement stresses,

volumes/areas, and temperatures) was read by the C4LIFE module for subsequent volume flaw and

surface flaw reliability analyses. The number of Gauss points used in the numerical integration for

reliability was set to 15 ('NGP=15). The option IPROOF=I was invoked to consider the neutral files

for both the service load and proof test load conditions simultaneously. The shear-sensitive

IKBAT=I option, with C = 0.82, was used when performing the reliability analysis to calculate the

normalized Batdorf crack density coefficient based on the fracture criterion and crack geometries

selected.

Time-dependent reliability calculations were made for each of the four combinations of

service and proof test loading consirdered. Attenuated probability of failure results were obtained for

NAS A/TM--2003-106316 273
prooftesttimesof 0.1, 1.0,10.0,and100.0hours.In addition,thetime-dependent probabilityof

failureresultsforthe beamwithoutprooftestingpriorto serviceloadingwerecalculated.

For cases(1) and(2),theprooftestandserviceloadconditionsdifferedonly in magnitude.

Thesetwo caseswerechosento examinethepredictedattenuatedreliabilityof a componentin

serviceafterbeingprooftestedat stresses25%aboveand25%belowthemaximumexpected

service stress. Time-dependent reliability analysis results for case (1) are given in tables E7-T1 and

E7-T2 for volume flaws and surface flaws, respectively. Figure E7-F2 is a Weibull plot of the timedependent probability of failure results obtained for various proof test times examined in case (1).

This plot shows the attenuation which occurs in the time-dependent probability of failure as a result

of proof testing. Time-dependent reliability analysis results for case (2) are given in tables E7-T3

and E7-T4 for volume flaws and surface flaws, respectively. The results for case (2) indicate that

proof testing at stresses 25% lower than the maximum stress expected in service does improve

component integrity. However, this improvement is essentially insignificant when compared with

the results from case (1) where the components are proof tested at stresses 25% greater than the

maximum stress expected in service. For both of the cases where the proof test load conditions

identically simulate the service load conditions, the probability of failure observed for surface flaw

analysis is greater than that for volume flaw analysis, due to the specimen geometry and the Weibull

material parameters.

For cases (3) and (4), the proof test and service load conditions differed not only in

magnitude, but also in direction of application (i.e., off-axis proof testing). The predicted attenuated

reliability of a component in service after off-axis proof testing at stresses 25% above and 25%

below the maximum expected service stress was examined. Time-dependent reliability analysis

results for case (3) are given in tables E7-T5 and E7-T6 for volume flaws and surface flaws,

respectively. Figure E7-F3 is a Weibull plot of the time-dependent probability of failure results

obtained for various proof test times examined in case (3). As in figure E7-F2, this plot also shows

NASA/TM--2003-106316 274
theattenuationwhichresultsfrom prooftesting.Time-dependent reliabilityanalysisresultsfor case

(4) aregivenin tablesE7-T7andE7-T8for volumeflawsandsurfaceflaws,respectively.Figure

E7-F4is a Weibullplotof theresultsobtainedfor variousprooftesttimesexaminedin case(4).

For clarity of presentation, theresultsshowninthisfigurearefor volumeflaw analysisonly;the

resultsfor surfaceflawanalysisaresimilar,althoughgreaterin magnitude,asobservedincase(3).

Theseresultsindicatethatprooftestingat stresses25%belowthemaximumstressexpectedin

servicealsoimprovescomponentintegrityfor off-axisprooftestloading.However,asobserved

previously,theseimprovementsaresmallcomparedto thoseobtainedwhencomponentsareproof

testedat stresses25%greaterthanthemaximumstressexpectedin service.Also,asobserved

previouslyin cases(1) and(2),theprobabilityof failureobservedfor surfaceflawanalysisis

greaterthanthatfor volumeflawanalysisin cases(3) and(4),dueto thespecimengeometryand

theWeibullmaterialparameters.In addition,for all of theoff-axisprooftestedspecimensin cases

(3) and(4),theprobabilityof failureobservedfor surfaceflawanalysisis muchhigherthanthatfor

volumeflawanalysis.Thisobservationis expected,dueto thedifferentstressdistributionsinduced

inthebeamby theserviceandprooftestloadconditionsforthiscase.

Theresultsobtainedforthefourcasesconsideredin thisexampleillustratehowprooftesting

enablesa componentto beplacedin servicewith greaterconfidencein its integrity. Furthermore,

theseresultsshowthatcomponentintegrityissensitiveto theparticularprooftestandserviceloads

thatareapplied.

NASA/TM--2003-106316 275
FigureE7-F1.--Siliconnitride(NC-132)beam.Length,L, 10.0mm;height, h, 4.0 mm; width, w,

1.0 mm.

i

NASA/TM--2003-106316 276
TABLEE7-TI. FAILUREPROBABILITIESFORPROOF-TESTED

BEAM--CASE1(VOLUMEFLAW ANALYSIS)

Batdorf Model

Noncoplanar strain energy release rate criterion

(Penny-shaped crack)

[NGP=15; IKBAT=I; C=0.82; mv=7.65; aoV=74.79 MPa(m)3176_; N=30; B=320 MPa 2 hr]

Service CARES/LIFE Failure Probability

time,

hrs Proof test time, tv,

hrs

No proof testing 0.1 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.02836

0.09214

0.1654

0.2876

0.4706

0.6976

0.0000

0.0000

0.0007664

0.01798

0.1362

0.4607

0.0000

0.0000

0.0001447

0.003541

0.03558

0.2418

0.0000

0.0000

10.00002717

0.0006680

0.007039

0.06608

0.0000

0.0000

0.000005097

0.0001254

0.001330

0.01324

NAS A/TM--2003-106316 277
TABLE E7-T2. FAILUREPROBABILITIESFORPROOF-TESTED

BEAM--CASE1(SURFACEFLAW ANALYSIS)

BatdorfModel

Noncoplanarstrainenergyreleaseratecriterion

(Semi-circularcrack)

[NGP=15;IKBAT=I; C---0.82;ms=7.65;COS=232.0MPa(m)_65;N=30;B=320MPa2hr]

Service CARES/LIFEFailureProbability

time,

hrs Prooftesttime,tv,

hrs

No proof testing 0.1 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.03105

0.1046

0.1868

0.3215

0.5169

0.7445

0.0000

0.0000

0.001074

0.02077

0.1544

0.5066

0.0000

0.0000

0.0002028

0.004091

0.04063

0.2715

0.0000

0.0000

0.00003807

0.0007721

0.008055

0.07522

0.0000

0.0000

0.000007141

0.0001449

0.001522

0.01513

NASA/TM--2003-106316 278
0

1 10 i00

10(30

SERVICE" TI,.%[E (hrs)

Figure E7-F2.--Probability of failure vs. service time for case (1). Solid _lines denote volume flaw

analysis; dashed lines denote surface flaw analysis; (tp = proof test time).

t

NAS A/TM--2003-106316 279
TABLE E7-T3. FAILURE PROBABILITIES FOR PROOF-TESTED

BEAM--CASE 2 (VOLUME FLAW ANALYSIS)

Batdorf Model

Noncoplanar strain energy release rate criterion

(Penny-shaped crack)

[NGP=15; IKBAT=I; C=0.82; mv=7.65; aoV=74.79 MPa(m)3t765; N=30; B=320 MPa 2 hr]

Service CARES/LIFE Failure Probability

time,

hrs Proof test time, lv,

hrs

No proof testing 0.1 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.02836

0.09214

0.1654

0.2876

0.4706

0.6976

0.01952

0.08378

0.1577

0.2810

0.4657

0.6939

0.01271

0.07652

0.1510

0.2753

0.4615

0.6915

0.004884

0.06298

0.1383

0.2643

0.4533

0.6868

0.001107

0.04006

0.1144

0.2435

0.4378

0.6779

NAS A/TM--2003-1063 !6 280
TABLEE7-T4. FAILUREPROBABILITIESFORPROOF-TESTED

BEAM--CASE2 (SURFACEFLAWANALYSIS)

BatdorfModel

Noncoplanarstrainenergyreleaseratecriterion

(Semi-circularcrack)

[NGP=15;IKBAT=I; C=0.82;ms=7.65;CroS=232.0 MPa(m)2n65; N=30;B=320MPa2hr]

Service CARES/LIFEFailureProbability

time,

hrs Prooftesttime,tp,

hrs

No prooftesting 0.1 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.03105

0.1046

0.1868

0.3215

0.5169

0.7445

0.02099

0.09520

0.1782

0.3143

0.5118

0.7418

0.01331

0.08698

0.1707

0.3080

0.5073

0.7395

0.004876

0.07165

0.1565

0.2961

0.4988

0.7350

0.001076

0.04563

0.1297

0.2732

0.4825

0.7263

NASA/TM--2003-106316 281
TABLE E7-T5. FAILURE PROBABILITIES FOR PROOF-TESTED

BEAM--CASE 3 (VOLUME FLAW ANALYSIS)

Batdorf Model

Noncoplanar strain energy release rate criterion

(Penny-shaped crack)

[NGP=15; IKBAT=I; C=0.82; mv=7.65; 60v=74.79 MPa(m)3n'65; N=30; B=320 MPa 2 hr]

Service CARES/LIFE Failure Probability

time,

hrs Proof test time, tv,

hrs

No proof testing 0.1 1.0 10.0 100.0

0.0

0.1

1.0

1.125

1.25

1.5

2.0

5.0

10.0

100.0

1000.0

0.02836

0.09214

0.1654

0.1703

0.1748

0.1829

0.1962

0.2446

0.2876

0.4706

0.6976

0.0000

0.0000

0.0000

0.00001733

0.00006652

0.0001908

0.0004538

0.002075

0.004739

0.04773

0.2461

0.0000

0.0000

0,0000

0.000003264

0,00001253

0.00003595

0.00008561

0.0003937

0.0009076

0.01046

0.09026

0.0000

0.0000

0.0000

0.0000006125

0.000002351

0.000006747

0.00001607

0.00007394

0.0001707

0.002003

0.02006

0.0000

0.0000

0.0000

0.0000001149

0.0000004411

0.000001266

0.000003015

0.00001387

0.00003202

0.0003767

0.003854

NASA/TM--2003-106316 282
TABLEE7-T6. FAILUREPROBABILITIESFORPROOF-TESTED

BEAM--CASE3 (SURFACEFLAWANALYSIS)

BatdorfModel

Noncoplanarstrainenergyreleaseratecriterion

(Semi-circularcrack)

[NGP=15;IKBAT=I; C=0.82;ms=7.65;t_oS=232.0 MPa(m)2n65; N=30; B=320 MPa 2 hr]

Service CARES/LIFE Failure Probability

time,

hrs Proof test time, tp,

hrs

No proof 0. I 1.0 10.0 100.0

testing

0.0

0.1

1.0

1.125

1.25

1.5

2.0

5.0

10.0

100.0

1000.0

0.03105

0.1046

0.1868

0.1923

0.1973

0.2062

0.2211

0.2745

0.3215

0.5169

0.7445

0.00000006397

0.0000001124

0.0008089

0.001011

0.001229

0.001671

0.002558

0.007800

0.01621

0.1253

0.4459

0.00000006397

0.0000001124

0.0001529

0.0001912

0.0002325

0.0003163

0.0004853

0.001499

0.003180

0.03204

0.2239

0.00000006397

0.0000001124

0.00002886

0.00003606

0.00004382

0.00005958

0.00009134

0.0002822

0.0006001

0.006315

0.05968

0.00000006397

0.0000001124

0.000005581

0.000006935

0.000008397

0.00001136

0.00001734

0.00005321

0.0001130

0.001193

0.01189

NASA/TM--2003-106316 283
1 10 I00 lOGO

EERg-[CE TIME (hr-_)

Figure E7-F3.--Probability of failure vs. service time for case (3). Solid lines denote volume flaw

analysis; dashed lines denote surface flaw analysis; (tp = proof test time).

NASA/TM--2003-106316 284
TABLE E7-T7. FAILURE PROBABILITIES FOR PROOF-TESTED

BEAM--CASE 4 (VOLUME FLAW ANALYSIS)

Batdorf Model

Noncoplanar strain energy release rate criterion

(Penny-shaped crack)

[NGP=15; IKBAT=I; C=0.82; mv=7.65; 6ov=74.79 MPa(m)3/765; N=30; B=320 MPa 2 hr]

Service CARES/LIFE Failure Probability

time,

hrs Proof test time, tv,

hrs

No proof testing 0.1 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

1000.0

0.02836

0.09214

0.1654

0.2876

0.4706

0.6976

0.01050

0.06751

0.1392

0.2622

0.4497

0.6838

0.004926

0.05379

0.1227

0.2452

0.4349

0.6740

0.001367

O.O3543

0.09817

0.2177

0.4101

0.6572

0.0002749

0.01682

0.06511

0.1762

0.3690

0.6285

NAS A/TM_2003-106316 285
TABLEE7-T8. FAILUREPROBABILITIESFORPROOF-TESTED

BEAM--CASE4 (SURFACEFLAW ANALYSIS)

BatdorfModel

Noncoplanarstrainenergyreleaseratecriterion

(Semi-circularcrack)

[NGP=15;IKBAT=I; C=0.82;ms=7.65;OOS=232.0MPa(m)2n65; N=30;B=320MPa2hr]

Service CARES/LIFEFailureProbability

time,

hrs Prooftesttime,tp,

hrs

No prooftesting 0.I 1.0 10.0 100.0

0.0

0.1

1.0

10.0

100.0

lO00.0

0.03105

0.1046

0.1868

0.3215

0.5169

0.7445

0.01787

0.08966

0.1720

0.3082

0.5068

0.7389

0.01062

0.07925

0.1612

0.2982

0.4990

0.7345

0.003659

0.06228

0.1431

0.2809

0.4854

0.7266

0.0007885

0.03769

0.1132

0.2514

0.4613

0.7124

NASA/TM--2003-106316 286
8O

=,5O

2C

a° I

&¢-

.oi._

I0

SER%-ICE T.M_.. T,

IOCO

Figure E7-F4.--Probability of failure vs. service time for volume flaw analysis for case (4); (1⁄2 =

proof test time).

NAS A/TM--2003-106316 287
Example Guide

This section is intened to aid the user in preparing the files associated with the finite element

analysis, the finite element interface, the parameter estimation module, and the reliability analysis

module for execution of the example problems presented in this manual. The disks included with

the CARES/LIFE program contain sample files associated with each example problem. These files

typically represent one complete execution of the CARES/LIFE program. Also included are the

input and output files for a finite element analysis (using MSC/NASTRAN) where required, and

where disk space permits. The following tables provide a list of the files which are included, along

with a brief description of the contents of each file. Additional information may be obtained by

referring to the individual example problems presented in this manual.

NASA/TM--2003-106316 288
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NASA/TM--2003-106316 289
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NASA/TM--2003-106316 290
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NASA/TM--2003-106316 291
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NASA/TM--2003-106316 292
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NASA/TM--2003-106316 293
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NASA/TM--2003-106316 296
Appendix A

Cyclic Fatigue (g-factor) Examples

Example 1: Simple Boundary Periodic Rectangular Loading (Table 1)

g

 0,1 ,11roi.o t,/Ndt ( 01 ) = tll

tl +tll tl +tll

where

Oie q = 01

[ t I tx_

Note: If the cyclic loading is a force instead of a stress, force P, replaces a,, and P,, replaces a, io

The equivalent static boundary force Peq replaces O'le q and

Iq+ (PIll N 11

tll

P_ PI [-P-I-I)

1

= = P1 gN

tI + tl_

From this equivalent static boundary load ateq (or Peq), we compute the stress distribution throughout

the component (specimen), a,_q(W). To obtain the probability of failure of a component at time t=t o,

the transformation of a,cq(W) at some time to to its equivalent value a,_q.o(Ug) at t=O is

Oieq,O(_) = IOINeq(_)

B

t O + Oie q (_/) 1

NAS A/TM--2003-1063 !6 297
Fromthis transformedboundaryload,theprobabilityof failureis obtained for time t-=t0, using

the appropriate equation.

Example 2: Sinusoidai Loading (Table 1)

I 1+°11 + °1-°11 sin 2 T t]

where

g

T

dt

and

As in example 1 , if the loads are forces P_ and PH then

p_c(t).. = [P,+P _ + _Ppsin 2 2

for [P'+P't[-2pz+ P-----_ PI-P" N2 sin (2T-'----_t) ]

gT

dt

and the equivalent static load Pcq is

NASA/TM--2003-106316 298
1

P_ = P1 gN

Since N is a non-integer value, g is evaluated numerically.

To obtain the probability of failure at time t=t o, as in Example 1, we compute the stress

distribution a_eq(W) throughout the component based on the equivalent static boundary stress a_q (or

the equivalent static boundary load, Peq) at location Wo- Next, a_q(q') is transformed to a_q,0(W ).

From this transformed stress distribution, the probability of failure is obtained using the appropriate

equation.

NASA/TM--2003-106316 299
Appendix B

Neutral File Setup

THERMOMECHANICALLY LOADED ANNULAR DISK

COM EXAMPLE #4 FROM THE CARES USERS MANUAL

COM

COM EXAMPLE NEUTRAL FILE

COM

2 0

COM FORMAT (215)

COM NUMEL = 2 : NUMBER OF ELEMENT GROUPS

COM

COM IE_ = 1

COM ISUBEL = 27

COM MATINP = 302

COM MAT2 = 0

COM ELVOL = .00054

COM ELTEMP = 1975.0

COM ELTHIC = 0.0

1 0.107827E-04

COM NUMELB = 0 : FLAG FOR SHELL ELEMENT GROUPS WITH VOLUME PROCESSING

COM NUMELB STILL IN EXPERIMENTAL STAGES!!

COM FOR THIS CASE - NO VOLUME PROCESSING OF SHELL ELEMENT GROUPS

1 8 8 0

COM FORMAT (415)

COM INUEG = 1 : GROUP NUMBER

COM IEGTYP = 8 : ELEMENT TYPE --- EQUIVALENT TO PATRAN'S SHAPE PARAMETER

COM 2 = BAR (NOT USED WITH CARES)

COM 3 = TRIANGLE

COM 4 = QUADRILATERAL

COM 5 = TETRAHEDRON

COM 6 = PYRAMID

COM 7 = PENTAHEDRON (WEDGE)

COM 8 = HEXAHEDRON (BRICK)

COM IGREL = 8 : NUMBER OF ELEMENTS IN THIS GROUP

COM ISHELL = 0 : SHELL ELEMENT FLAG

1 27 302 0 0.539957E-03 0.197500E+04 0.000000E+00

FORMAT (415,3E15.6)

: ELEMENT NUMBER

: NUMBER OF SUBELEMENTS

: MATERIAL NUMBER FOR VOLUME FLAW ANALYSIS

: MATERIAL NUMBER FOR SURFACE FLAW ANALYSIS

: ELEMENT VOLUME

: ELEMENT TEMPERATURE

: ELEMENT THICKNESS (SET TO ZERO FOR VOLUME ELEMENTS)

0.199436E+04 -0.242574E+04

-0.361891E+04

COM FORMAT (I5,5EI5.6,/,35X,3EI5.6)

COM ISUNY3M = 1 : SLrBELEMENT _ER

COM SUBVOL = .00001 : SUBELEMENT VOLUME

COM SUBTEM = 1994.4 : SUBELEMENT TEMPERATURE

COM SSTR(L),L=I,6 : STRESSES SIGX SIGY

COM SIGXY SIGYZ

2 0.185592E-04 0.197500E+04 -0.411254E+04 -0.343776E+05

-0.251220E+04 0.240778E+03

3 0.124163E-04 0.195564E+04 -0.579934E+04 -0.249061E+05

-0.140549E+04 0.781060E+03

4 0.171642E-04 0.199437E+04 -0.242574E+04 -0.438490E+05

-0.122070E-03 0.000000E+00

5 0.295430E-04 0.197500E+04 -0.411254E+04 -0.343776E+05

0.000000E+00 0.000000E+00

6 0.197646E-04 0.195564E+04 -0.579934E+04 -0.249061E+05

-0.152588E-04 0.476837E-05

7 0.I07827E-04 0.199436E+04 -0.242574E+04 -0.438490E+05

0.361892E+04 0.299505E+03

8 0.185589E-04 0.197500E+04 -0.411254E+04 -0.343776E+05

0.251220E+04 -0.240778E+03

9 0.124161E-04 0.195564E+04 -0.579934E+04 -0.249061E+05

-0.438490E+05 -0.495269E+04

-0.299505E+03 0.307790E+04

SIGZ

SIGZX

0.946542E+04

-0.217892E+04

0.238835E+05

-0.743573E+04

-0.495269E+04

0.307790E+04

0.946542E+04

-0.217892E+04

0.238835E+05

-0.743573E+04

-0.495269E+04

0.307790E+04

0.946542E+04

-0.217892E+04

0.238835E+05

NASA/TM--2003- i 06316 301
I0 0.172524E-04 0.199437E+04

ii 0.296951E-04 0.197500E+04

12 0.198662E-04 0.195564E+04

13 0.274625E-04 0.199436E+04

14 0.472688E-04 0.197500E+04

15 0.316233E-04 0.195564E+04

16 0.172523E-04 0.199437E+04

17 0.296945E-04 0.197500E+04

18 0.198660E-04 0.195564E+04

19 0.I07826E-04 0.199436E+04

20 0.185594E-04 0.197500E+04

21 0.124163E-04 0.195564E+04

22 0.171642E-04 0.199437E+04

23 0.295430E-04 0.197500E+04

24 0.197646E-04 0.195564E+04

25 0.I07826E-04 0.199436E+04

26 0.185591E-04 0.197500E+04

27 0.124161E-04 0.195564E+04

2 27 300 0 0.137444E-02

1 0.262589E-04 0.194436E+04

0.140549E+04 -0.781060E+03 -0.

-0.242574E+04 -0.438490E+05 -0.

-0.361892E+04 -0.762939E-05 0.

-0.411254E+04 -0.343776E+05 0.

-0.251220E+04 0.953674E-05 -0

-0.579934E+04 -0.249061E+05 0

-0.140549E+04 -0.286102E-05 -0

-0.242574E+04 -0.438490E+05 -0

0.000000E+00 0.000000E+00 0

-0.411254E+04 -0.343776E+05 0

0.000000E+00 0.000000E+00 0

-0.579934E+04 -0.249061E+05 0

0.000000E+00 -0.190735E-05 -0

-0.242574E+04 -0.438490E+05 -0

0.361891E+04 0.000000E+00 0

-0.411254E+04 -0.343776E+05 0.

0.251220E+04 -0.152588E-04 -0.

-0.579934E+04 -0.249061E+05 0.

0.140549E+04 -0.381470E-05 -0.

-0.242574E+04 -0.438490E+05 -0.

-0.361891E+04 0.299505E+03 -0.

-0.411254E+04 -0.343776E+05 0.

-0.251221E+04 -0.240778E+03 0.

-0.579934E+04 -0.249061E+05 0.

-0.140549E+04 -0.781060E+03 0.

-0.242574E+04 -0.438490E+05 -0.

-O.122070E-03 O.O00000E+00 -0.

-0.411254E+04 -0.343776E+05 0.

0.000000E+00 0.000000E+00 0.

-0.579934E+04 -0.249061E+05 0.

-0.152588E-04 -0.381470E-05 0.

-0.242574E+04 -0.438490E+05 -0.

0.361891E+04 -0.299505E+03 -0.

-0.411254E+04 -0.343776E+05 0.

0.251220E+04 0.240778E+03 0.

-0.579934E+04 -0.249061E+05 0.

0.140549E+04 0.781060E+03 0.

0.192500E+04 0.000000E+00

-0.181920E+04 -0.754958E+05

-0.618627E+04 0.290242E+03

743573E+04

495269E+04

915527E-04

946542E+04

I06812E-03

238835E+05

209808E-03

495269E+04

122070E-03

946542E+04

610352E-04

238835E+05

320435E-03

495269E+04

000000E+00

946542E+04

122070E-03

238835E+05

732422E-03

495269E+04

307790E+04

946542E+04

217892E+04

238835E+05

743573E+04

495269E+04

307790E+04

946542E+04

217892E+04

238835E+05

743573E+04

495269E+04

307790E+04

946542E+04

217892E+04

238835E+05

743573E+04

-0.213713E+05

-0.257061E+04

27 -0.565651E-03 0.131127E+04

2 4 18

I00 9 0

COM FORMAT

COM IELNUM =

COM ISUBEL =

COM MATINP =

COM MAT2 =

COM ELVOL =

COM ELTEMP =

COM ELTHIC =

-0.II1869E+04

-0.985361E+02

-0.230136E+03

-0.836775E+01

1

302 0.359971E-02 0.197500E+04 0.100000E-05

(415,3E15.6)

I00 : ELEMENT NUMBER

9 : NUMBER OF SUBELEMENTS

0 : MATERIAL NUMBER FOR VOLUME FLAW ANALYSIS

302 : MATERIAL NUMBER FOR SURFACE FLAW ANALYSIS

.00360 : ELEMENT AREA

1975.0 : ELEMENT TEMPERATURE

1.0E-6 : ELEMENT THICKNESS

-0.410745E+05

SIGY

-0.363226E+05

1 0.258785E-03 0.199436E+04 0.715319E+03

COM FORMAT (I5,5E15.6)

COM ISUNUM = 1 : SUBELEMENT NUMBER

COM SUBVOL = .00026 : SUBELEMENT AREA

COM SUBTEM = 1994.4 : SUBELEMENT TEMPERATURE

COM SSTR(L),L=I,3 : STRESSES SIGX

2 0.445423E-03 0.197500E+04 -0.592114E+04

-0.194717E+04

-0.749014E+02

0.432249E+03

SIGXY

0.398923E+03

NASA/TM--2003-106316 302
3

4

5

6

7

8

9

I01

1

0 297991E-03

0 411939E-03

0 709031E-03

0 474350E-03

0 258784E-03

0 445415E-03

0 297990E-03

9 0 300

0.630212E-03

0.195564E+04

0.199436E+04

0.197500E+04

0.195564E+04

0.199436E+04

0.197500E+04

0.195564E+04

0.916289E-02

0.194436E+04

-0.125576E+05

0.715319E+03

-0.592114E+04

-0.125576E+05

0.715319E+03

-0.592114E+04

-0.125576E+05

0.192500E+04

0.650431E+04

-0.315707E+05

-0.410745E+05

-0.363226E+05

-0.315707E+05

-0.410745E+05

-0.363226E+05

-0.315707E+05

0.100000E-05

-0.673899E+05

0.365598E+03

0.000000E+00

0.000000E+00

-0.381470E-05

-0.432249E+03

-0.398923E+03

-0.365598E+03

0.293382E+03

7 0.450427E-02

8 0.177071E-02

9 0.524055E-02

130 9 0 302

1 0.759243E-03

2 0.120858E-02

3 0.759245E-03

4 0.121479E-02

5 0.193372E-02

6 0.121479E-02

7 0.759241E-03

8 0.120858E-02

9 0.759246E-03

0.130000E+04

0.130000E+04

0.130000E+04

0.981743E-02

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.200000E+04

0.639825E+03

0.639825E+03

0.639825E+03

0.200000E+04

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.437730E+05

-0.I00347E+04

-0.I00347E+04

-0.I00347E+04

0.100000E-05

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

-0.796470E+04

0.509900E+00

0.000000E+00

-0.509900E+00

0.I05466E+02

0.000000E+00

-0.I05466E+02

0.000000E+00

0.000000E+00

-0.I19209E-06

-0.I05466E+02

0.000000E+00

0.I05466E+02

NASA/TM--2003-106316 303
APPENDIX C

SYMBOLS

A

A 2

a

B

b

C

C

D

D+_ D"

E

F(x)

FN(x)

G

g

H

h

i

J

K

K(n)

k

surface area; material-environmental fatigue constant

Anderson-Darling goodness-of-fit test statistic

crack half length; penny-shaped crack radius; radius of semi-circular surface crack

risk of rupture; subcritical crack growth constant

the intersection of a line with the ordinate axis

Shetty's constant in mixed-mode fracture criterion

the contour of a unit radius circle in two-dimensional principal stress space

Kolmogorov-Smimov goodness-of-fit test statistic defined as D ÷ or D, whichever is the

largest

Kolmogorov-Smimov goodness-of-fit test statistic

Young's modulus of elasticity

cumulative distribution function of a random variable

empirical distribution function

strain energy release rate

g-factor

step function

total height of four-point bend bar with rectangular cross section

ranking of ordered fracture data in statistical analysis

Jacobian operator

stress intensity factor

Kanofsky-Srinivasan confidence band factors

crack density coefficient; any counter

NASA/TM--2003-106316 305
L

Li

e,m,n

m

N

P

Pf

PI

R

r

r,s,t

ri

ro

T

t

to

V

v

likelihood function

length between outer loads in four-point bending

length between symmetrically applied inner loads in four-point bending

direction cosines of oblique plane normal in principal stress space for the Cauchy

infinitesimal tetrahedron

Weibull modulus, or shape parameter

material-environmental fatigue constant; Weibull crack density function; number of flaws

per unit volume or area

number of links in a structure; any counter; number of cycles; number of four-point bend

specimens at a given temperature

load applied to four-point bend bar specimen

cumulative failure probability

probability of existence in incremental volume or area of a crack with strength < oct

probability of crack with strength < Ocr being so oriented that o c >__oct

ratio of minimum to maximum effective stress in a loading cycle

number of remaining specimens in censored data analysis

finite element natural coordinates

inside radius

outside radius

period of one cycle

time; thickness

time-dependent scale parameter

volume

velocity

NAS A/TM--2003-106316 306
W

X

X

x,y,z

Y

Zi

P

F

A

V

Z

_0

a 1,0"2_0" 2

T

total width of four-point bend bar with rectangular cross section

any variable

ordered statistics

Cartesian coordinate directions

crack geometry factor

predicted failure probability at the fracture strength of the ith specimen

angle between c. and the stress al; significance level

angle between a. projection and the stress cr2 in plane perpendicular to a_

gamma function which is tabulated in mathematical handbooks

increment

increment

material Poisson's ratio

3.1416

applied multidimensional stress state

applied stress distribution; the traction or stress vector on oblique plane of Cauchy

infinitesimal tetrahedron

Weibull scale parameter

tensor stress components; principal stresses (a_ > a 2 > c3)

shear stress acting on oblique plane whose normal is determined by angles ot and

spatial location (x,y,z) and orientation (ct,13) in a component

solid angle in three-dimensional principal stress space for which a e > a,r

to angle in two-dimensional principal stress space for which a e > _.; angular speed

Subscripts:

a applied or service load

NASA/TM--2003- ! 06316 307
B Batdorf

c cyclic; critical

ch characteristic

cr critical

d dynamic fatigue

e effective

ef effective

eq equivalent

f failure; fracture

g gage

I inert

I crack opening mode

II crack sliding mode

III crack tearing mode

i i'th; combined service and proof test loading

max maximum

min minimum

n normal; normal stress averaging

p polyaxial; proof test

Q total or combined

s survival; service

S surface

T transformed

u uniaxial

NASA/TM--2003-106316 308
V volume

w Weibull

0 characteristic

Superscripts:

, inertdistributionparameterestimatedfromfatiguedata

" estimatedparameter

modifiedparameter

normalizedquantity

NASA/TM--2003-106316 309
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Paluszny, A.; and Nicholis, P.F.: "Predicting Time-Dependent Reliability of Ceramic Rotors".

Ceramics for High Performance Applications-II. Burke, J.J., Lenoe, E.N.; Katz, N.R., eds.

Chesnut Hill, Massachusetts, Brook Hill, 1978, pp. 95-112.

Paris, P.C.; Erdogan, F.: A Critical Analysis of Crack Propagation Laws. J. Basic Eng., Trans.

ASME, vol. 85, 1963, pp.528-534.

Paris, P.C.; and Sih, G.C.: Stress Analysis of Cracks. ASTM STP 381, 1965, pp. 30-83.

Paul, B.; and Mirandy, L.: An improved Fracture Criterion for Three-Dimensional Stress States,

J. Eng. Mater. Technol., vol. 98, no. 2, Apr. 1976, pp. 159-163.

Petrovic, J.J.; and Stout, M.G.: Multiaxial Loading Fracture of A120 s Tubes: II, Weibull Theory

and Analysis. J. Am. Ceram. Soc., vol. 67, no. l, Jan., 1984, pp. 18-23.

Pierce, F.T.: "The Weakest Link" Theorems of the Strength of Long and of Composite

Specimens. Text. Inst. J., vol. "17, 1926, pp. T355-T368.

NASA/TM--2003-106316 315
Q

R

S

Pintz, A.; Abumeri, G.H.; and Manderscheid, J.M.: CARES-ANSYS Structural Ceramics

Reliability Analysis-Users Manual. NASA CR (not officially published).

Powers, L.M.; Starlinger, A.; and Gyekenyesi, J.P.: Ceramic Component Reliability With the

Restructured NASA/CARES Computer Program. NASA TM , 1992

Quinn, G. D., and Quinn, J. B., "Slow Crack Growth in Hot-Pressed Silicon Nitride", Fracture

Mechanics of Ceramics. Bradt, R. C., Evans, A. G., Hasselman, D. P. H, and Lange, F. F.,

eds., Plenum Publishing Corp., Vol. 6, 1983, pp. 603-636.

Rufin, A.C.: A Study of Statistical Failure Prediction Models for Brittle Materials Subjected to

Multiaxial States of Stress, M.S. Thesis Univ. Washington, 1981.

Rufin, A.C.; Samos, D.R.; and Bollard, R.J.H.: Statistical Failure Prediction Models for Brittle

Materials. AIAA J., vol. 22, no. 1, Jan. 1984, pp. 135-140.

Samos, D.R.: Experimental Investigation and Probability of Failure Analysis of Pressure Loaded

Alumina Discs, M.S. Thesis, Univ. of Washington, 1982.

Schaeffer, H.G.: MSC/NASTRAN Primer: Static and Normal Modes Analysis. 2nd ed., Schaeffer

Analysis Inc., Mount Vernon, NH, 1979.

Service, T.H.; and Ritter, J.E.; Proof Testing to Assure Reliability of Structural Ceramics.

Fracture Mechanics of Ceramics, Volume 7- Composties, Impact, Statistics, and HighTemperature Phenomena. Bradt, R.C.; Evans, A.G.; Hasselman, D.P.H.; and Lange, F.F., 1985,

pp.255-264.

Shelleman, D.L., "Test Methodology for Tubular Ceramic Components (Fast Fracture Strength

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Shelleman, D.L., Jadaan, O.M., Conway, J.C., and Mecholsky, J.J., " Prediction of the Strength

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Evaluation, Vol. 19, No. 3, May 1991.

Shelleman, D.L., Jadaan, O.M., Butt, D.P., Tressler, R.E., Hellman, J.R., and Mechoisky, J.J.,

"High Temperature Tube Burst Test Apparatus," Journal of Testing and Evaluation, Vol. 20,

No. 4, July 1992.

Shetty, D.K.; et al.: A Biaxial-Flexure Test for Evaluating Ceramic Strengths, J. Am. Ceram.

Sot., vol. 66, no. 1, pp. 36-42, Jan. 1983.

Shetty, D. K.; Rosenfield, A. R.; Bansal, G. K.; and Duckworth, W. H.: Biaxial Flexure Test for

Ceramics, Journal of the American Ceramic Society, Vol. 59, No. 12, pp. 1193-1197, 1980).

Shetty, D.L.; Rosenfield, A.R.; and Duckworth, W.H.: Statistical Analysis of Size and Stress

State Effects on the Strength of an Alumina Ceramic. Methods for Assessing the Structural

Reliability of Brittle Materials; ASTM-STP-844, S.W. Frieman and C.M. Hudson, eds.,

American Society for Testing and Materials, 1984, pp. 57-80.

NASA/TM--2003-106316 316
T

Shetty,D.K.: Mixed-ModeFracture Criteria for Reliability Analysis and Design with Structural

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13, no. 2, 1980, pp. 257-271.

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pp. 469-479.

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1991.

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253-277.

NASA/TM--2003-106316 317
W Walker, K.: Effects of Environmental and Complex Load History on Fatigue Life. ASTM STP

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Weibull, W.: A Statistical Theory of the Strength of Materials. Ingeniors Vetenskaps Akademien

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13, 1978, pp. 1781-1793.

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Ceramics and Glasses in Corrosive Environments. Metal Science, August-September 1980, pp.

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Wiederhorn, S.M.; and Fuller, E.R., Jr.: Structural Reliability of Ceramic Materials.

Materials Science and Engineering, vol. 71, 1985, pp. 169-186.

NASA/TM--2003-106316 318
Oo

LIJ

I---

Z

W

0! o,

W

m

I--

W

u.

m

w

E

m

co

et

0

o

m o

im

NASA/TM_2003-106316 319
GRIFFITH

CRACK

? -

.--2;'

/;//

," /

#l

I

II TOTAL

MAXIMUM i i STRAIN SHETTY'S

TENSILE _ J ENERGY RIXED

STRESS II RELE_EMODE

I '' RAT_ EOUiT I Ol_

SH_-SENSITI_J

WEIBULL

PRINCIPLE OF NORMAL

. INDEPENDENT STRESS

ACT ION AVERAG ING

SHEAR-INSENSITIVE

i

FRACTURE CRITERIA J

SURFACE FLAWS

GRIFFITH GRIFFITH

CRACK NOTCH

/f

#i

TOTAL STRAIN SHETTY'S

ENERGY MIXED

RELEASE MODE

RATE EQUATION

I

I SHEAR SENSITIVE

SEMI- NO CRACK

CIRCULAR SHAPE

SURFACE REQUIRED

CRACK

I ] WEIBULL

PRINCIPLE OF NORMAL

INDEPENDENT STRESS

ACTION

i AVERAGING

I

FRACTURE CRITERIA

SHEAR INSENSITIVE

Figure 2.--Available failure criteria and crack shapes.

NASA/TM--2003-106316 320
FRACTURE STRENGTH

DATA, FLAW TYPES

AND

FRACTURE CRITERIA

Control or templet

Ifile:

I{C4PEST.INP}

ESTIMATE WEIBULL I

FATIGUE

PARAMETERS

[C4PEST]

COMPONENT GEOMETRY, LOADING

AND MATERIAL PROPERTIES

J

I MODEL [PATRAN GENERATION]

i

I H_T TRANSFERANALYSIS 1

Temperature

at each node

ELEMENT STRESS

ANALYS I S

[MSC/NASTRAN]

[ANSYS]

[ABAQUS ]

FEA output file:

nodal stresses,

temperatures, and

element connectivity

NEUTRAL FILE CREATION

[NASCARES]

[ANSCARES]

[ABACARES]

Neutral data base

files:

C_ES NEU)

PROOF.NEU)

Control or templet

file:

IICARES.INP)

| CERAMICS ANALYSIS AND RELIABILITY 1

t EVALUATION OF STRUCTURES

[C4LIFE]

{C_aES-OUT}I IIc_ES.PAT}

VOLUME, SURFACE AND ELEMENT RISK OF

COMPONENT RELIABILITY RUPTURE INTENSITIES

interactive

pre- and

postprocessing

Figure 3 - Block diagram for the analysis and time-dependent reliability

evaluation of ceramic components. This analysis includes the

CARES/LIFE integrated design program, finite element analysis,

PATRAN PLUS pre- and post-processing.

and

NASA/TM--2003-106316 321
--- CARES/LIFE TEMPLET INPUT FILE ---

VERSION 4.0

****_*_****_**********_*****_ _** ***_*****_************_******

* RELIABILITY PREDICTION FOR BRITTLE MATERIAL STRUCTURES *

* --- FAST-FRAC1URE AND TIME-DEPENDENT MODELING--- *

MASTER CONTROL INPUT

TITLE : PROBLEM TITLE (ECHOED IN CARES OUTPUT)

SAMPLE INPUT FOR CARES/LIFE

NE

3

IPRINC

___

1

*_*

: CONTROL INDEX FOR FINITE ELEMENT POSTPROCESSING

(DEFAULT: NE = O)

0 : EXPERIMENTAL DATA ANALYSIS ONLY

3 : COMPONENT RELIABILITY ANALYSIS WITH NEUTRAL FILE

DATA BASE

: CONTROLINDEX FOR ELEMENTSTRESSOUTPUT

(DEFAULT: IPRINC=I)

i : USE PRINCIPAL STRESSES

2 : USE ELEMENT OR MATERIAL COORDINATESYSTEMSTRESSES.

THIS OPTION IS RECOMMENDEDFOR MATERIALS MODELED

WITH THE BATDORFTHEORY. WHENTWONEUTRALFILES

ARE READ (IPROOF = i) THEN IPRINC SHOULDBE SET

TO THIS VALUE

NMATS

Ol

____

: NUMBER OF MATERIALS FOR SURFACE FLAW ANALYSIS

(NMATS+NMATV < 101)

(DEFAULT: NMATS = O)

NMATV

O0

____

: NUMBEROF MATERIALS FOR VOLUMEFLAWANALYSIS

(NMATS+NMATV< I01)

(DEFAULT: NMATV : O)

IPRINT

I

NGP

_'__*

15

: CONTROL INDEX FOR STRESS OUTPUT

(DEFAULT: IPRINT = O)

O : DO NOT PRINT ELEMENT STRESSES AND/OR FRACTURE DATA

I : PRINT ELEMENT STRESSES AND/OR FRACTURE DATA

: NUMBER OF GAUSSIAN QUADRATURE POINTS (15 OR 30 OR 50)

(DEFAUL[: NGP = 15)

Figure 4. Master Control Input section of the templet file

NAS A/TM--2003-106316 322
NS

001

_'____

: NUMBER OF SEGMENTS IN SYMMETRY PROBLEM

(DEFAULT: NS : i)

IPROOF

*_*

0

: CONTROL INDEX FOR RELIABILITY ANALYSIS WITH NEUTRAL FILES

(DEFAULT: IPROOF = O)

0 : READ ONLY ONLY ONE NEUTRAL FILE;

EITHER STATIC OR CYCLIC LOADING RELIABILITY ANALYSIS

(NO PROOF TESTING IS CONSIDERED)

I : READ TWO NEUTRAL FILES;

INCLUDE THE EFFECTS OF PROOF TESTING OR CYCLIC LOADING

IN THE RELIABILITY ANALYSIS

IPOST

___

0

: CONTROL INDEX FOR PATRAN DATA FILE

(DEFAULT: IPOST = O)

0 : DO NOT WRITE ELEMENT RISK OF RUPTURE INTENSITIES

1 : WRITE ELEMENT RISK OF RUPTURE INTENSITIES

INTERP

___

0

___

: CONTROL INDEX FOR TEMPERATURE INTERPOLATED PARAMETERS

(DEFAULT: INTERP = O)

0 : DO NOT WRITE INTERPOLATED MATERIAL PARAMETERS

1 : WRITE INTERPOLATED MATERIAL PARAMETERS

FACTOR

000001.000

: LOAD FACTOR FOR WHICH TO MULTIPLY ELEMENT STRESSES

(DEFAULT: FACTOR = 1.0)

TIME

000001,000

: SERVICE LIFE FOR RELIABILITY ANALYSIS OF THE FIRST NEUTRAL

FILE IN UNITS OF SECONDS OR CYCLES (DEFAULT: TIME = 0.0)

TIMEPT

000000.00

: DURATION OF THE PROOF TEST LOAD IN SECONDS OR NUMBER

OF SERVICE CYCLES FOR RELIABILITY ANALYSIS USING THE SECOND

NEUTRAL FILE (REQUIRES THAT IPROOF = I). FOR CYCLIC FATIGUE

THIS INPUT SHOULD BE IDENTICAL TO THE VALUE INPUT FOR THE

TIME KEYWORD (DEFAULT: TIMEPT = 0.0)

$ENDX : END OF MASTER CONTROL INPUT

Figure 4. Concluded

NASA/TM--2003-106316 323
MATERIAL CONTROLINPUT

MATID

0000300

IDI

5

: MATERIAL IDENTIFICATION NO. FROM THE FINITE ELEMENT

MATERIAL PROPERTY CARD (IF POSTPROCESSING IS NOT

BEING PERFORMED THIS ENTRY SHOULD BE SOME UNIQUE NO.)

(NO DEFAULT)

: CONTROL INDEX FOR EXPERIMENTAL DATA

(NO DEFAULT)

I : UNIFORM UNIAXIAL TENSILE SPECIMEN TEST DATA

2 : FOUR-POINT BEND TESr DATA

3 : DIRECT INPUT OF THE REQUIRED PARAMETERS

4 : CENSORED DATA FOR SUSPENDED ITEM ANALYSIS OF

UNIFORM UNIAXIAL TENSILE SPECIMEN TEST DATA

5 : CENSORED DATA FOR SUSPENDED ITEM ANALYSIS OF

FOUR-POIN[ BEND TEST DATA

ID4

2

: CONTROL INDEX FOR VOLUME OR SURFACE FLAW ANALYSIS

(NO DEFAULT)

I : VOLUME

2 : SURFACE

ID2

_-__-

5

#(__k

ID3

4

-k _ "A"

IKBAT

1

: CONTROL INDEX FOR FRACTURE CRITERION

(NO DEFAULT)

I : NORMAL STRESS FRACTURE CRITERION

(SHEAR-INSENSITIVE CRACK)

2 : MAXIMUM TENSILE STRESS CRITERION (VOLUME FLAW ONLY)

3 : COPLANAR STRAIN ENERGY RELEASE RATE CRITERION

(G SUB T)

4 : WEIBULL PIA MODEL

5 : SHETTY'S SEMI-EMPIRICAL CRITERION (NONCOPLANAR SERR)

: CONTROL INDEX FOR CRACK GEOMETRY

(NO DEFAULT)

I : GRIFFITH CRACK (VOLUME FLAW OR SURFACE FLAW)

2 : PENNY-SHAPED CRACK (VOLUME FLAW ONLY)

3 : GRIFFITH NOTCH (SURFACE FLAW ONLY)

(ID2 = 3 OR 5 IS REQUIRED)

4 : SEMICIRCULAR CRACK (SURFACE FLAW ONLY)

(ID2 = 5 IS REQUIRED)

: CONTROL INDEX FOR METHOD OF CALCULATING BATDORF CRACK

DENSITY COEFFICIENT (K SUB B) FROM TEST DATA

(DEFAULT: IKBAT = 1)

O : SHEAR-INSENSITIVE METHOD (MODE I FRACTURE ASSUMFD)

1 : SHEAR-SENSITIVE METHOD (FRACTURE ASSUMED TO OCCUR

ACCORDING TO THE FRACTURE CRITERION AND CRACK SHAPE

SELECTED BY THE ID2 AND ID3 INDICES)

Figure 5. Material Control Input section of the templet file

NASA/TM--2003-106316 324
PR

00000.2500

C

00000.8000

CYCLIC

___:___ ....

1

____ _k ....

I. 00000000

1.00000000

1.00000000

MLORLE

___

0

___

: POISSON'S RATIO

(DEFAULT: PR = 0.25)

: CONSTANT FOR SEMI-EMPIRICAL MIXED-MODE FRACTURE CRITERIA

(KI/KIC)_(KII/(C*KIC))**2 = ! REF. D.K.SHET_Y

OBSERVED VALUES RANGE FROM 0.8 TO 2. (REF. D.K. SHETTY)

NOIE: AS C APPROACHES INFINITY, PREDICTED FAILURE

PROBABILITIES APPROACH NORMAL STRESS CRITERION VALUES

(DEFAULT C = 1.0)

: CYCLIC FATIGUE BASED COMPONENT RELIABILITY EVALUATION

: CRACK GROWTH LAW AND WAVE FORM (IWAVE) (DEFAULT: IWAVE = I)

: R-RATIO OF MIN/MAX CYCLE STRESS (RCYC) (DEFAULT: RCYC = 1.0)

: PERIOD (IN SECONDS) OF A CYCLE (PERIOD) (DEFAULT: PERIOD = 1.0)

: EXPONENT OF POWER LAW WAVE FORM (QCYC) (DEFAULT: QCYC : 1.0).

IWAVE : 0 :

=1:

=2:

=3:

:4:

=5:

PARIS LAW OR WALKER LAW

POWER LAW; STATIC LOAD

POWER LAW: SQUARE WAVE

POWER LAW; SAWTOOTH WAVE

POWER LAW; I/2 SINE WAVE

POWER LAW; SINE WAVE

NOIE: I) THE R-RATIO INPUT (RCYC) ASSUMES THAT

THE R-RATIO IS CONSTANT THROUGHOUT THE

COMPONENT. THIS INPUT VALUE IS IGNORED

IF A SECOND NEUTRAL FILE IS SPECIFIED

(IPROOF = I) WHERE RCYC IS THEN CALCULATED

FOR EACH ELEMENT.

2) THE WAVE FORM EXPONENT (QCYC) IS USED WITH

THE POWER LAW (IWAVE > I). THE G-FACTOR IS

DETERMINED FOR IHE SELECTED WAVE FORM RAISED TO

THE QCYC EXPONENT. FOR EXAMPLE: IF A ROTATING

COMPONENT HAS A SINUSIODAL VARIATION IN ANGULAR

SPEED, THE STRESS IS DIRECTLY PROPORTIONAL TO THE

SQUARE OF THE ANGULAR SPEED AND QCYC IS SET

TO A VALUE OF 2.0

: CONTROL INDEX FOR METHOD OF CALCULATING WEIBULL

PARAMETERS FROM THE EXPERIMENTAL FRACTURE DATA

(DEFAULT: MLORLE = O)

0 : MAXIMUM LIKELIHOOD

I : LEAST-SQUARES LINEAR REGRESSION

Figure 5. -Continued

NASA/TM--2003-1063 ! 6 325
OUTLIE

00000010.0

PFCONV

O, 00000000

DH

00002.1000

: SIGNIFICANCE LEVEL FOR WHICH OUTLIERS ARE TO

BE DETECTED. INPUT IS IN PERCENT AND MUST BE BETWEEN

THE RANGE OF 0.I PERCENT TO 10.0 PERCENT

(DEFAULT OUTLIE = I0.0)

: FAILURE PROBABILITY TO CALCULATE THE WEIBULL SCALE

PARAMETER FOR THE FINITE ELEMENT MODEL

---> (ASSUMES MATERIAL PARAMETERS ARE CONSTANT VERSUS

TEMPERATURE; ONLY ONE TEMPERATURE DATA SET IS INPUT)

(DEFAULT PFCONV : 0.0 ,THUS QUANTITY NOT CALCULATED)

NOTE: 0.0 < PFCONV < 1.0

: HEIGHT OF THE FOUR-POINT BEND BAR

(NO DEFAULT)

DL1

00040.0000

: OUTER LOAD SPAN OF THE FOUR-POINT BEND BAR

(NO DEFAULT)

DL2

00020.0000

: INNER LOAD SPAN OF THE FOUR-POINT BEND BAR

(NO DEFAULT)

DW

00002.8000

: WIDTH OF THE FOUR-POINT BEND BAR

(NO DEFAULT)

$ENDM : END OF TEMPERATURE-INDEPENDENT MATERIAL CONTROL INPUT

Figure 5. Concluded

NAS A/TM--2003-106316 326
TEMPERATURE-DEPENDENT MATERIAL CONTROL INPUT DATA

FOR THE ABOVE MATERIAL

!!!!!!!]!!!!!!!]!!!!!!!!!!!!!!!!!!!!!!I!!!!!!!!I!!!!!!!!!!!!!!!!!!!!!!(!

PLEASE NOTE THE FOLLOWING:

I. TEMPERATURE-DEPENDENT DATA SETS MUST BE ARRANGED IN ASCENDING ORDER

ACCORDING TO TEMPERATURE (ONE DATA TABLE PER TEMPERATURE).

2. FRACTURE STRESSES FOR A GIVEN TEMPERATURE CAN BE INPUT IN

ARBITRARY ORDER.

3. REGARDLESS OF THE FRACTURE ORIGIN LOCATION, THE FRACTURE STRESS

INPUT VALUE IS IHE EXTREME FIBER STRESS WITHIN THE INNER LOAD SPAN

OF THE MOR BAR.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

............ > DATA TABLE; EXPLANATION OF INPUT

TEST : TYPE OF DATA TO BE INPUT.

: BLANK CHARACTERS DEFAULT TO THE SETTING OF THE

IMMEDIATLY PRECEDING VALUE (THIS APPLIES TO

TEST AND FIELDI ONLY FOR TH[ FAST, STAT, CYCL, AND

DYNA KEYWORDS)

CYCL : CYCLIC FATIGUE SPECIM[N DATA

FIELDI : MAGNITUDE OF FATIGUE STRESS

(MAXIMUM CYCLE STRESS)

FIELD2 : LEAVE BLANK OR LIFETIME TO FAILURE

(UNITS OF SECONDS FOR POWER LAW)

(UNITS OF CYCLES FOR PARIS AND WALKER LAWS)

FIELD3 : LEAVE BLANK OR LIFETIME TO FAILURE

(UNITS OF SECONDS FOR POWERLAW)

(UNITS OF CYCLES FOR PARIS AND WALKER LAWS)

CYCLIC : CYCLIC FATIGUE DATA (IWAVE)

(INPUT IN ALPHA-NUMERIC "FLAW" FIELD)

0 : PARIS LAW OR WALKER LAW

i : DIRECT INPUT OF G-FACTOR

2 : POWER LAW; SQUARE WAVE

3 : POWER LAW; SAWTOOTHWAVE

4 : POWER LAW; 1/2 SINE WAVE

5 : POWER LAW: SINE WAVE

FIELDI : R-RATIO OF MIN/MAX CYCLE STRESS OR

DIRECT INPUT OF G-FACTOR

FIELD2 : PERIOD OF THE CYCLE (SECONDS)

FIELD3 : WALKER R-RATIO SENSITIVITY EXPONENT

OR EXPONENT OF POWER LAW WAVE FORM (QCYC)

WHEN QCYC IS SET TO ZERO BY THE USER AND THE WALKER OR

PARIS LAW IS SELECTED, CARES/LIFE AUTOMATICALLY

DEFAULTS QCYC TO THE VALUE OF THE FATIGUE EXPONENT N,

WHICH DEFINES THE PARIS LAW.

Figure 6. Temperature Dependent Material Control Input

section of the templet fi|e

NASA/TM--2003-106316 327
WHEN THE WAVE FORM EXPONENT (QCYC) IS USED WITH THE

POWER LAW (IWAVE > i), THEN THE G-FACTOR IS DETERMINED FOR

THE SELECTED WAVE FORM RAISED TO THE QCYC EXPONENT.

FOR EXAMPLE; IF A ROTATING COMPONENT HAS A SINUSIODAL

VARIATION IN ANGULAR SPEED, THE STRESS IS DIRECTLY

PROPORTIONAL TO THE SQUARE OF THE ANGULAR SPEED AND

QCYC IS SET TO A VALUE OF 2.0

DYNA : DYNAMIC FATIGUE (CONSTANT STRESS RATE) SPECIMEN DATA

FIELD1 : MAGNITUDE OF STRESS RATE (STRESS/SECOND)

FIELD2 : LEAVE BLANK OR ULTIMATE FAILURE STRESS

OF SPECIMEN

FIELD3 : LEAVE BLANK OR ULTIMATE FAILURE STRESS

OF SPECIMEN

FAST : FAST FRACTURE

FIELDI : LEAVE BLANK OR SPECIMEN FRACTURE STRESS

FIELD2 : LEAVE BLANK OR SPECIMEN FRACTURE STRESS

FIELD3 : LEAVE BLANK OR SPECIMEN FRACTURE STRESS

PARAM : DIRECT INPUT OF FAST-FRACTURE WEIBULL PARAMETERS

FIELDI : WEIBULL MODULUS OR SHAPE PARAMETER

FIELD2 : WEIBULL SCALE PARAMETER

FIELD3 : WEIBULL CHARACTERISTIC STRENGTH

RSTRES : RESIDUAL STRESSES IN THE MATERIAL COORDINATE SYSTEM

FIELDI : X STRESS COMPONENT (3-D AND 2-D)

FIELD2 : Y STRESS COMPONENT (3-D AND 2-D)

FIELD3 : Z STRESS COMPONENT (3-D ONLY)

+STRES : RESIDUAL STRESSES IN THE MATERIAL COORDINATE SYSTEM

FIELDI : XY STRESS COMPONENT (3-D AND 2-D)

FIELD2 : YZ STRESS COMPONENT (3-D ONLY)

FIELD3 : ZX STRESS COMPONENT (3-D ONLY)

RANGE : SEARCH RANGE TO FIND THE FATIGUE EXPONENT N WHEN

THE MEDIAN DEVIATION TECHNIQUE IS SPECIFIED

(SOL = 2 )

(THE DEFAULT RANGE IS 2.1 TO 150.0)

FIELDI : BOUNDARY OF SEARCH RANGE

FIELD2 : BOUNDARY OF SEARCH RANGE

FIELD3 : LEAVE BLANK

SOL : SOLUTION METHOD FOR THE TIME-DEPENDENT DATA

(INPUT IN ALPHA-NUMERIC "FLAW" FIELD)

0 : MEDIAN VALUE LEAST SQUARES REGRESSION

1 : LEAST SQUARES REGRESSION

2 : MEDIAN DEVIATION ITERATION

STAT : STATIC FATIGUE (CONSTANT STRESS) SPECIHEN DATA

FIELDI : MAGNITUDE OF FATIGUE STRESS

FIELD2 : LEAVE BLANK OR TIME TO FAILURE (SECONDS)

FIELD3 : LEAVE BLANK OR TIME TO FAILURE (SECONDS)

TEMP : TEMPERATURE OF SET (ONE ASSIGNMENT PER DATA TABLE)

FIELDI : TEMPERATURE

FIELD2 : LEAVE BLANK

FIELD3 : LEAVE BLANK

TPARAM : DIRECT INPUT OF TIME-DEPENDENT PARAMETERS

FIELDI : FATIGUE CRACK GROWTH EXPONENT N

FIELD2 : FATIGUE CONSTANT B SUB W,

(STRESS**2*SECOND) OR (STRESS**2*CYCLE)

FIELD3 : LEAVE BLANK OR WALKER R-RATIO

SENSITIVITY EXPONENT

Figure 6. -Continued

NASA/TM--2003-106316 328
VAGTIM : TIME-DEPENDENT EFFECTIVE VOLUME OR AREA

FIELDI : EFFECTIVE VOLUME OR AREA

FIELD2 : LEAVE BLANK

FIELD3 : LEAVE BLANK

VAGAGE : FAST-FRACTURE EFFECTIVE VOLUME OR AREA

FIELDI : EFFECTIVE VOLUME OR AREA

FIELD2 : LEAVE BLANK

FIELD3 : LEAVE BLANK

$ENDD : END OF DATA MARKER FOR THE DATA TABLE

FLAW : FRACTURE ORIGIN (ONLY FOR FAST, STAT, CYCL, AND

DYNA KEYWORDS)

: BLANK CHARACTER DEFAULTS TO FLAW TYPE INDICATED BY

THE ID4 KEYWORD

S : SURFACE FLAW

U : UNKNOWN FLAW (COMPETING FAILURE MODE OR RUN OUT)

V : VOLUME FLAW

DTABLE : DATA TABLE (FAST-FRACTURE, STATIC OR DYNAMIC FATIGUE)

-TESI-I-FLAW-I ..... FIELDI ...... I ..... FIELD2 ...... I ..... FIELD3 ...... !

TEMP O.IO0000E+04

SOL

RANGE

FAST

STAT

3.000000E+O0

0.I02300E+04

0.I01100E+04

0 972000E+03

0 943000E+03

0 941000E+03

0 927000E+03

0 921000E+03

0 917000E+03

0 916000E+03

0 915000E+03

0 889000E+03

0 883000E+03

0 883000E+03

0 876000E+03

0 866000E403

0 /84000E+03

0 781000E+03

0 700000E+03

0 650000E+03

0 650000E+03

0 650000E+03

O. iO0000E+03

0.600000E+03

0.600000E_03

0.550000E+03

0.550000E+03

0.550000E+03

0.500000E+03

0.500000E+03

0.500000E+03

1.000000E+02

0.576000E+02

0 432000E+02

0 504000E+02

0 792000E+02

0 313200E+02

0 900000E+02

0 198000E+03

0 360000E+04

0 II52UOE+05

0 176400E+05

0 972000E+03

0.648000E+04

0.248040E+06

Figure 6. -Continued

NASA/TM--2003-106316 329
STAT

$ENDD

U 0.500000E+03 0.I09620E+08

U 0.450000E+03 0.508680E+07

U 0.450000E+03 0.I15704E+08

S 0.400000E+03 0.296640E+07

U 0.400000E+03 0.120024E+08

U 0.400000E+03 0.124236E+08

S 0.750000E+03 0.608400E+01

S 0.700000E+03 O.900000E+OI

S 0.700000E+03 0.273600E+02

S 0.650000E+03 0.360000E+01

S 0.650000E+03 0.370800E+02

S 0.650000E+03 0.122400E+03

S 0.600000E+03 0.756000E+03

S 0.600000E+03 0.169200E+03

S 0o600000E+03 0.345600E+03

S 0.550000E+03 0.121680E+04

S 0.550000E+03 0.219600E+04

S 0.550000E+03 0.144000E+04

S 0.500000E+03 0.I00800E+05

S 0.500000E+03 0.288000E+04

S 0.500000E+03 0.255600E+04

S 0.450000E+03 0.124344E+07

S 0.450000E+03 0.173124E÷08

S 0.450000E+03 0.181152E+08

S 0.700000E+03 0.504000E+01

-TEST-I-FLAW-I ..... FIELD1 ...... I ..... FIELD2 ...... I ..... FIELD3 ...... I

END OF DATA FOR THE ABOVE TEMPERATURE

DTABI.E : DATA TABLE (FAST-FRACTURE, STATIC OR DYNAMIC FATIGUE)

-TEST-I-FLAW-I ..... FIELDI ...... I ..... FIELD2 ...... I ..... FIELD3 ...... I

TEMP 0.110000E+04

SOL 2

STAT S 0.650000E+03

S 0.650000E+03

S 0.650000E+03

S 0.600000E+03

S 0.600000E+03

S 0.600000E+03

S 0.550000E+03

S 0.550000E+03

S 0.550000E+03

S 0.500000E+03

S 0.500000E+03

S 0.500000E+03

S 0.450000E+03

S 0.450000E+03

S 0.450000E+03

S 0.400000E÷03

S 0.400000E÷03

S 0.400000E+03

S 0.400000E+03

S 0.350000E+03

S 0.350000E'+03

0.I08000E+02

0 183600E+02

0 396000E÷02

0 309600E+02

0 504000E+02

0 648000E+02

0 576000E+02

0.165600E+03

0.194400E+03

0.576000E+03

0.972000E+03

0.374400E+04

0.136800E+05

0.144000E+05

0.208800E+05

0.324000E+05

0,385200E+05

0.208080E+06

0.248400E+06

O.134280E+OZ

0.176760E+07

Figure 6. -Continued

NASA/TM--2003-106316 330
S 0.350000E+03 0.203400E+07

U 0.300000E+03 O.B28000E+07

U 0.300000E+03 0.118944E+08

$ENDD

-TEST-I-FLAW-I ..... FIELDI ...... I ..... FIELD2 ...... I ..... FIELD3 ...... I

END OF DATA FOR THE ABOVE TEMPERATURE

DTABLE : DATA TABLE (FAST-FRACTURE, STATIC OR DYNAMIC FATIGUE)

-TEST-I-FLAW-I ..... FIELDI ...... I ..... FIELD2 ...... I ..... FIELD3 ...... I

TEMP 0.120000E+04

SOL 2

STAT S 0.600000E+03 0.360000E+01

S 0.545000E+03 0.154800E+02

S 0.545000E+03 0.201600E+02

S 0.545000E+03 0.252000E+02

S 0.500000E+03 O.9720UOE+02

S 0.500000E+03 0.972000E+02

S 0.500000E+03 0.147600E+03

S 0.450000E+03 0.205200E+03

S 0.450000E+03 0.237600E+03

S 0.450000E+03 0.601200E+03

S 0.400000E+03 0.154800E+04

S 0.400000E+03 0.158400E+04

S 0.400000E+03 0.277200E+04

S 0.350000E+03 0.709200E+04

S 0.350000E+03 0.108720E+05

S 0.350000E+03 0.277200E+05

S 0.317000E+03 0.205200E+05

S 0.317000E+03 0.219600E+05

S 0.317000E+03 0.302400E+05

S 0.317000E+03 0.392400E+05

S 0.317000E+03 0.410400E+05

S 0.317000E+03 0.691200E+05

S 0.250000E+03 0.442800E+06

S 0.250000E+03 0.558000E+06

S 0.248000E+03 0.540000E+06

s 0.200000E+03 0.218888E+07

S 0.200000E+03 0.236880E+07

S 0.200000E+03 0.213480E+07

S 0.200000E+03 0.191880E+07

S 0.266000E+03 0.279000E+06

STAT S 0.317000E+03 0.385200E+05

S 0.317000E+03 0.554400E+05

S 0.317000E+03 0.525600E+05

S 0.317000E+03 0.345600E+05

S 0.350000E+03 0.684000E+04

S 0.350000E+03 0.756000E+04

S 0.350000E+03 0.122_00E+05

S 0.3500U0[+03 0.13680UE+05

S 0.350000E+03 0.140400E+05

S 0.350000E+03 0.126000E+05

Figure 6. -Continued

NASA/TM--2003-106316 331
S 0.250000E+03 0.489600E+06

S 0.250000E+03 0.511200E+06

S 0.250000E+03 0.540000E+06

$ENDD

-TEST-I-FLAW-I ..... FIELDI ...... I ..... FIELD2 ...... I ..... FIELD3 ...... I

END OF DATA FOR THE ABOVE TEMPERATURE

$ENDT : END OF DATA FOR THE ABOVE MATERIAL

Figure 6. -Concluded

NASA/TM--2003-106316 332
TEMP

10o0.0000

PARAM

* ........ M........ * ....... SP........ *

O. 1611065200E+02 O. 1210912550E+04

TPARAM

* ........ N........ * ........ B........ *

0.3113000000E+02 0.8027108872E+03

TEMP

1100.0000

PARAM

* ........ M........ * ....... SP........ *

0.2770502294E+02 0.I036704873E+04

TPARAM

* ........ N........ * ........ B........ *

O. 1836900000E+02 0.4116402524E+05

W ................. * ................. *

TEMP

1200.0000

PARAM

*........ M........ *....... SP ........ *

0.1969015533E+02 O. 1141176912E+04

TPARAM

* ........ N........ * ........ B........ *

0.1215720002E+02 0.6478040059E+05

................. _ ................. *

$ENDT

Figure 7. Temperature Dependent Material Control Input

style for module C4LIFE

NASA/TM--2003-106316 333
o

I 'r

AREA OF PLANE-A

l --- DIRECTION COSINES:

_- cosa

o2 oI m- sina cosB

n- sinasinB

(a)

o I /-dA : sin c{dJ3da

Q i

I

• _", _,_ - _ 0"3

0"2

(b)

(a) In principal stress space.

(b) Projected onto a plane tangent to the unit radius sphere.

Figure _,.---Stresses on Cauchy infinitesimal tetrahedron.

NASA/TM--2003-106316 334
° 1

r-'[ = (01 - 02) .cos o sin cz

' t

o n = o1 cos 2a +o 2 sin 2cz

\

DIRECTION

COSINES:

d

J, = cos u

m = sin o

0 2

Figure cl .--Normal and shear stress as a function of c_ projected onto a tangent

line to the unit radius circle (.in pranc_p_-{ _tress _po.ce).

NASA/TM--2003-106316 335
p .

h12

x

P

I

h

Y

P p

Figure l O--Four-point bend specimen geometry (beam width is w).

NASA/TM--2003-106316 336
DES IGN

STRESS,

ad

I _DATA

I l J' RANGE

I /

I/

/

/ "-'EXTRAPOLATION RANGE

/

/

BASIC DATA

ASSUMED ZERO J_,

THRESHOLD _'I

STRENGTH7

/!

/ /ASSUMED NONZERO

THRESHOLD STRENGTH

I

I

NONZERO THRESHOLD STRENGTH

A AND B-_x '_'__////

',x_/ -

_/'f LPOPULATION

///// m=5

/x,_ POPULATION A

// m: o

UNDETECTED FLAW POPULATIONS

BIAXIAL_/_J/

/,_V ° \\

/ly,_o '--UNIAXIAL

/_/I

/

/

/

STRESS STATE EFFECTS

Figure l 1.--Limitations of experimental data extrapolation and statistical

approach to design.

NASA/TM--2003-106316 337
I = COS O.

m = sin _, cos 13

n = sin,_ sin 13

PY

Z

(7n = l(Tx+m(Ty+n_rz+2(Im_xy+mP_yz+nlTzx)

T = _/(Icrx+m_yx+rVrzx)2+(l_xy+m=y+nTzy)2+(ITxz+mCyz+n_z)2-Crn 2

(a) Cauchy stress components on infinitesimal Tetraledron.

I ......... '_::n'"° i,----...

_--F .... =z

(b) Globial Coordinate System.

Figure 12.--Projection of equivalent stress onto upper half of a

unit radius sphere in the global-coordinate system.

NASA/TM--2003-106316 338
_v

t,r,,,

•-' jj JJ

4_

'7

0

°_

Q

U

°_

I

i

J

,!

b

Q

U

fl

+

_Z

°_

r

L

,k

b

4-_ m

Z

÷

!

°_

I!

v

b

!

"4"

+

t |

N

+

÷

c_

I1

v

NASA/TM--2003-106316 339
,-->.

O

r..) r,.) r,.)

r._ AI V

o

r,.) /_

.<

II II

-- _9 r..)

t_

<

[-..,

•" I["

U

_"

°o

o ._

#I!°°

i

b

f-- +

I

+ I

t_ +

I-

-_r Z _ ----- _

I

° I

• _ It

o_

NASA/TM--2003-106316 340
/fi

©

Z

_'=

AI ,,-..,

,...., _

g,_,

o_

e_

L O

°_

.-=_

O

°_ _,_

E*"

°_

J

°_

V "_n

•" _:Xl)

Iz._

.=.

t-..I_

&

[....

N

I

II

II

II

[-.,

m

4-

AI

g.

,a

13

J

[-..,

ID

+

_ I['-,

II

Z

+

[..,

_ It-.,

°_

t_.,

O

ca)

°_

[--

II

NASA/TM--2003-106316 341
Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0188

Pubhc reporting burden for this collection of rntormatlon is estimated to average 1 hour per response, including the time for reviewing instructions, searching exlsttng data sources,

gathering and mamtarnmg the data needed, and completing and reviewing the collection of mforrnatlon Send comments regarding thts burden estimate or any other aspect of this

collectton of reformation, including suggesbons for reducmg this burden, to Washmgton Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

February 2003 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

CARES/LIFE Ceramics Analysis and Reliability Evaluation of Structures Life

Prediction Program

6. AUTHOR(S)

Noel N. Nemeth, Lynn M. Powers, Lesley A. Janosik, and John E Gyekenyesi

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WB S-22-708-31-09

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-13748

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--2003-106316

11. SUPPLEMENTARY NOTES

This document was initially released in 1993. The formal numbered publication was delayed until 2003. Noel N. Nemeth,

Lesley A. Janosik, and John P. Gyekenyesi, NASA Glenn Research Center; Lynn M. Powers, Case Western Reserve

University, Cleveland, Ohio 44106. Responsible person, Noel N. Nemeth, organization code 5920, 216--433-3215.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 39 Distribution: Nonstandard

Available electronically at http://gltrs.grc.nasa.gov

This publication is available from the NASA Center for AeroSpace Information. 301-621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This manual describes the Ceramics Analysis and Reliability Evaluation of Structures Life Prediction (CARES/LIFE)

computer program. The program calculates the time-dependent reliability of monolithic ceramic components subjected

to thermomechanical and/or proof test loading. CARES/LIFE is an extension of the CARES (Ceramic Analysis and

Reliability Evaluation of Structures) computer program. The program uses results from MSC/NASTRAN, ABAQUS,

and ANSYS finite element analysis programs to evaluate component reliability due to inherent surface and/or volume

type flaws. CARES/LIFE accounts for the phenomenon of subcritical crack growth (SCG) by utilizing the power law,

Paris law, or Walker law. The two-parameter Weibull cumulative distribution function is used to characterize the variation

in component strength. The effects of multiaxial stresses are modeled by using either the principle of independent action

(PIA), the Weibull normal stress averaging method (NSA), or the Batdorf theory. Inert strength and fatigue parameters

are estimated from rupture strength data of naturally flawed specimens loaded in static, dynamic, or cyclic fatigue. The

probabilistic time-dependent theories used in CARES/LIFE, along with the input and output for CARES/LIFE, are

described. Example problems to demonstrate various features of the program are also included.

14. SUBJECT TERMS

Life (durability); Fatigue life; Reliability analysis; Reliability; Fracture strength; Structural

analysis; Component reliability; Finite element method; Ceramics

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

35O

16. PRICE CODE

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102