NASA Technical Memorandum 4766

Probabilistic Micromechanics and

Macromechanics for Ceramic

Matrix Composites

Pappu L.N. Murthy

Lewis Research Center

Brook Park, Ohio

Subodh K. Mital

The University of Toledo

Toledo, Ohio

Ashwin R. Shah

NYMA, Inc.

Brook Park, Ohio

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

_nformation Program

1997

Probabilistic Micromechanics and Macromechanics for

Ceramic Matrix Composites

Pappu L.N. Murthy

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

Subodh K. Mital

The University of Toledo

Toledo, Ohio 43606

Ashwin R. Shah

NYMA, Inc.

Brook Park, Ohio 44142

Summary

The properties of ceramic matrix composites (CMC's) are

known to display a considerable amount of scatter due to

variations in fiber/matrix properties, interphase properties,

interphase bonding, amount of matrix voids, and many

geometry- or fabrication-related parameters, such as ply

thickness and ply orientation. This paper summarizes preliminary studies in which formal probabilistic descriptions of the

material-behavior- and fabrication-related parameters were

incorporated into micromechanics and macromechanics for

CMC's. In this process two existing methodologies, namely

CMC micromechanics and macromechanics analysis and a fast

probability integration (FPI) technique are synergistically

coupled to obtain the probabilistic composite behavior or

response. Preliminary results in the form of cumulative probability distributions and information on the probability sensitivities of the response to primitive variables for a unidirectional

silicon carbide/reaction-bonded silicon nitride (SiC/RBSN)

CMC are presented. The cumulative distribution functions are

computed for composite moduli, thermal expansion coefficients, thermal conductivities, and longitudinal tensile strength

at room temperature. The variations in the constituent properties that directly affect these composite properties are accounted

for via assumed probabilistic distributions. Collectively, the

results show that the present technique provides valuable

information about the composite properties and sensitivity

factors, which is useful to design or test engineers. Furthermore, the present methodology is computationally more efficient than a standard Monte-Carlo simulation technique; and

the agreement between the two solutions is excellent, as shown

via select examples.

Introduction

Ceramic matrix composites (CMC's) are potential candidate

materials for the High Speed Civil Transport (HSCT) structural

components. As such, they will be required to have an assured

life of several thousand hours. Estimating the reliability of

these components is quite a complex process and requires a

knowledge of uncertainties that occur on various scales. Unlike

conventional materials, CMC's have properties that display

considerable scatter because of the uncertainties encountered at

every level. For example, there are uncertainties or scatter

inherent in the constituent (fiber, matrix, and interphase) properties and uncertainties in the fabrication process that result in

scatter in the fiber-volume ratio, the interphase thickness and

properties, the matrix void-volume ratio, and the geometrical

parameters of the laminate--such as ply thickness and the

orientation of the ply. All of these must be formally accounted

for if CMC behavior is to be predicted with assurance that the

component will have the required reliability during its operating life. Consequently, there is a need for analytical tools that

quantify the uncertainty in the "response" variables while

taking into account the inherent scatter in the basic or "primifive" variables. Primitive variables are the constituent properties/parameters that participate at the lowest level (e.g., the

micromechanics level) in defining a global property. The fibervolume ratio and the individual constituent properties such as

fiber modulus, matrix thermal expansion coefficient, and fiber

tensile strength are some of the primitive variables. These are

assumed to be independent and have their own statistical

distributions. Response variables are those that characterize the

composite behavior, such as the longitudinal modulus, the

transverse thermal expansion coefficient, and the longitudinal

thermal conductivity.

NASA TM-4766 1
In the current practice of deterministic approaches, uncertainties are usually accounted for by using safety factors. This

approach can often yield overly conservative designs, thereby

reducing the potential of many advanced composite materials.

Currently, work is underway at NASA Lewis Research Center

to incorporate the probabilistic distribution of materialbehavior- and fabrication-related parameters into the

micromechanics and the macromechanics of CMC's. The

primary objective of this work is to develop an efficient

computational design tool that could account for all the uncertainties in the constituent properties in a more rigorous manner

and also provide the overall composite properties and their

probabilistic distributions. Such information could then be

used to design structural components to meet the necessary life

requirements. In addition to providing more rigor to the analysis than the so-called safety factor approach does, such procedures would enhance interpretation of the experimentally

measured CMC properties that are scattered over a wide range.

Furthermore, the procedure would help identify the dominant

variables, those that most influence a specific response, thereby

providing guidelines for quality control during the fabrication

of these materials. Thus, the methodology could be applied not

only to the design but also to the development of a better material.

The approach taken in the present effort was to combine the

CMC analysis embedded in the computer code CEMCAN

(CEramic Matrix Composite ANalyzer, refs. 1 and 2) with the

fast probability integration (FPI) techniques available in

NESSUS (Numerical Evaluation of Stochastic Structures

Under Stress, ref. 3). The role of CEMCAN was to provide the

functional relationships (micromechanics and macromechanics)

that tie the constituent properties to the equivalent composite

behavior. The role of FPI was to perform probabilistic analyses

by utilizing the properties generated by CEMCAN. The results

are cumulative probability distribution functions (CDF's) for

ply and laminate properties of the CMC's. A CDF is the

relationship defined by the value of a property (response

variable) with respect to its cumulative probability of occurrence. The probabilistic sensitivities of the response variables

to inherent scatter in primitive variables are obtained as

byproducts of the FPI technique. Select examples of CDF's of

composite properties (mechanical, thermal, and strengthrelated properties) and the related sensitivity factors for a

unidirectional SiC/RBSN [08] composite laminate are presented here for illustration.

Probabilistic Ceramic Matrix

Composite Analysis

As previously mentioned, the present work adopts an integrated approach, which is a synergistic combination of two

inhouse-developed methodologies. The first methodology is

concerned with CMC micromechanics and macromechanics.

The second one consists of an FPI technique that takes into

account the uncertainties occurring on various scales in a

composite and computes the cumulative probabilistic distribution of composite global behavior. A schematic of the integrated approach is shown in figure 1. A brief description of the

two methodologies in CEMCAN and FPI are given in the

following sections.

Ceramic Matrix Composite Micromechanics

and Macromechanics

Over the past two decades several micromechanics-based

computer codes have been developed inhouse for polymer

matrix, metal matrix, and CMC analyses (refs. 1,4, and 5). The

computer code that is currently being used inhouse to analyze

CMC's is CEMCAN. CEMCAN utilizes a novel and unique

fiber substructuring technique in conjunction with conventional micromechanics that is based on a mechanics-ofmaterials approach. Reference 1 describes the usage of the

code, and reference 2 describes the various equations embedded in it, along with the theoretical aspects. Because CEMCAN

has a unique fiber substructuring technique, it offers several

advantages over conventional unit-cell-based micromechanics

theories. It more accurately represents micromechanical interfacial conditions and provides much greater detail about local

stresses. As shown in parts of figure l, CEMCAN's methodology consists of an incremental synthesis of the properties,

starting from the constituents--namely, the fiber, the matrix,

and the interphase--to form a slice. The slice-level properties

are obtained by using composite micromechanics equations,

which are represented by simplified closed-form equations.

The slice-level properties are, in general, equivalent elastic

properties such as moduli, Poisson's ratios, thermal expansion

coefficients, and heat conductivities. From slice level to single

lamina level and, subsequently, to laminate level, the code

repeatedly applies classical laminate theory to obtain

composite-level properties and responses. Given a specific set

of loads, the code can progressively decompose, as indicated in

figure 1, retracing the steps followed during synthesis so as to

yield laminate-, ply-, and slice-level responses and constituent

microstresses. The code can predict ply- and laminate-level

thermal and mechanical properties as well as a detailed description of the resulting microstresses caused by an applied load. It

also accounts for nonlinear effects due to material nonlinearities

as well as those due to local stress redistribution resulting from

progressive fracture. By analyzing the response of CMC's, we

can account for fabrication-related parameters. A more detailed

description of this methodology can be found in reference 6.

Probabilistic Simulation

There are a number of approaches available for obtaining a

probabilistic response from a set of independent variables and

the expressions describing the response behavior. Monte-Carlo

simulation is one such fairly common approach to obtaining the

CDF's of composite properties, given the probability

2 NASA TM--4766
Nonlinear

material

properties

Constituents:

fiber, matrix,

and interphase

Integration/synthesis

Fiber

substructuring

Unit

cell

A

_J

l 1

Ill Ill

Slice Lamina/ply

Progressive decomposition

Laminate

Figure 1.mlntegrated probabilistic ceramic matrix composite mechanics approach.

distributions of constituent properties, which are considered as

independent variables. In this technique, randomly selected

values of the input variables, which are based on their known

probabilistic distributions, are used to deterministically compute the value of the response variable. This is repeated, usually

several hundreds of times, to build the response probabilistic

characteristics. In essence, this technique requires a large

number of simulations to generate CDF's of output variables.

Although inherently simple, the large number of output sets

that must be generated to obtain a reasonably accurate CDF of

output variables becomes its obvious disadvantage. Furthermore, if the deterministic computation of the response is

complicated and time-consuming, the computational costs

could become prohibitive. Obviously, to circumvent this

computationally very expensive and time-consuming procedure, more efficient approaches and algorithms are needed.

For more than a decade NASA Lewis has been engaged in

developing efficient probabilistic methods. As a result of this

intensive program, an FPI (ref. 3) was developed to solve a

large class of engineering problems. In the following paragraphs we describe the problem to be solved and how the FPI

was used to investigate the probabilistic behavior of

ceramic composites.

Let us say that there are n random variables in a problem and

that we want to use probabilistic analysis to compute the

probability of occurrence of a certain response function

Z(X)=Z(X1, X 2..... Xn) O)

where Z represents the response variable and X represents the

random variable. Our aim is to compute the probability that Z

will have a value less than or equal to a given magnitude Z o. To

achieve this goal, the performance function, which describes

how the mechanics of the system behave, can be cast as a limit

state function g(X), which can be described as

g(X) = z(x)- z o (2)

Here the objective would be to compute P[g(X) < 0]. In the case

when Z o describes a limit indicating failure, g(X) is called a

failure function, or in a classical sense, the probability of the

stress exceeding the strength.

Given the probability density functionfx(x ) of the limit state

function g(x), we can formulate the limit state probability

P[g < 0] as

(3)

where f_ describes the domain of integration. This multiple

integration is, in general, very difficult to evaluate analytically.

NASA TM-4766 3
However, FPIhasbeenfoundtobeanexcellent tooltoevaluate

equation(3)efficientlyandaccurately.

FPIisaprobabilisticanalysistoolthatimplementsavariety

ofmethodsforprobabilisticengineeringanalysisanddesign

(seeschematicinfig.2).Ingeneral, FPIrequiresthefollowing:

(1)Theindependent anduncorrelatedinput(design)variablesandtheirprobabilitydistributionsmustbedefined.

Constituent properties, FVR,void-volumeratio(VVR),ply

thickness, plyalignment, andthelikemustbeindependent

variablesthatdeterminethecompositeproperties. Inequation

(1),X represents these variables.

(2) There must be a function (called the performance function) that defines the relationship between the response variable

and the independent variables. Ply or laminate properties are

the response variables in this paper. In equation (l), variable Z

is a dependent variable whose uncertainties must be computed.

The variables that are uncertain in nature are identified as

independent in step (1). Their probability distributions can be

obtained from the available measured data or can be assumed

on the basis of experience and judgment. For most problems, it

is difficult to determine analytical expressions representing a

relationship between independent and dependent variables. In

the case of CMC's, it is very complicated to build relationships

for the ply or laminate properties as a function of constituent

properties, fabrication parameters, and such. CEMCAN code

has a built-in capability to form such relationships, using

micromechanics and macromechanics theory. A performance

function is developed by using a numerical approach. In this

approach an explicit response function is developed by perturbing the independent random variables about their magnitude

and using CEMCAN to compute the response. Discrete evaluations of the response variables for the perturbed values of

independent variables are then fit into a function by using

regression analysis. The uncertainties of a response variable are

quantified in the form of a CDF by the following procedure:

(1) The primitive variables and the corresponding probabilistic distributions are selected. (For example, to generate the CDF

of the composite longitudinal modulus, the primitive variables

could be the fiber modulus, matrix modulus, FVR, and so

forth.) For a given set of values of primitive variables, the

micromechanics and macromechanics in the CEMCAN computer code are used to compute the desired response variable.

(2) The whole process is repeated to generate a table of

response variable values that correspond to the perturbed

values of the primitive variables.

(3) The FPI analysis then uses the previously generated table

to compute the CDF and the corresponding sensitivities of the

response.

FPI computes the CDF of the performance and the probabilistic sensitivity factor. The CDF can be computed at userdefined performance levels or probability levels. Several

methods can be used to compute a CDF and/or failure probability.

Diatdbution type

Random variable

statistics

CEMCAN performance

function z = f(xI , x2, x 3) Output options

//

Y

Xl x2 x3

Fast probability

integration (FPI)

analysis engine

-41

//'

•///

Sensitivity factors Response cumulative

distribution function

(CDF)

Figure 2.--Fast probability integration input-output schematic.

4 NASA TM-4766
InadditiontotheCDF'softheresponse, theFPItechnique

providesadditional informationregardingthesensitivityofthe

responsewithrespect to theprimitivevariables.Sensitivity

informationisveryusefulforstudyingtheprobabilisticvariationof theresponse. Notethatthesensitivityobtainedhere

shouldnotbeconfusedwiththeconventional deterministic

sensitivitiesoftheresponse. Sensitivityinprobabilisticanalysisconsistsofaproduct oftwoparts.Thefirstparthastheusual

meaning, akintodeterministicsensitivity. Thesecondpartis

theresultofthescatterinaprimitivevariable, anditaccentuates

thedeterministicsensitivity. Themagnitudeofthesensitivity

factorprovidesawaytoranktheimportanceoftheindividual

physical variables. Thesignofthesensitivityfactorindicates

howaspecificrandomvariableinfluencesthefailureprobability. Forexample,a positivesensitivityvalueindicatesan

increaseinfailureprobabilitywithanincreaseintherandom

variable.Thus,thesensitivityinformationobtainedfromFPIis

veryusefulfromthedesignpointofview.Thescatterofthe

significant variablescanbecontrolledtoimprovethereliability.Thus,weakphysical variableswithlargeuncertainties may

haveprobabilisticsensitivityfactorsthataremoreimportant

thanstrongphysical variableswithsmallstandarddeviations.

Variableswithnoscatter(deterministic) willobviouslyresult

inzerovaluesforthesensitivity, whichimpliesthatheresponse

scatterisunaffectedbysuchvariables.

Results and Discussion

The material system chosen for this work was SiC/RBSN

(silicon carbide SCS-6 fiber in a reaction-bonded silicon nitride

matrix). This composite system is known to have avery compliant and weak interphase. The mean values of the constituent

properties are shown in table I, which is taken from reference 7.

The uncertainties in the composite system occur at various

levels owing to both the inherent scatter in the material properties and the uncertainties associated with the manufacturing

processes. They may occur at constituent (fiber, matrix, or

interphase) level, at ply level (FVR, VVR, thickness of the

interfacial region, etc.) and at composite level (ply angle, layup,

etc.).

However, in this preliminary work, uncertainties in the

constituent material properties and those at ply level were

considered to quantify uncertainties in some of the compositelevel properties for a unidirectional [0] 8 composite. Specifically, in this work, the FVR, fiber modulus (Ef), matrix modulus

(Era) , interphase modulus (Ei), interphase thickness (ti), coefficients of thermal expansion of the fiber, matrix, and interphase (af, a m, and cti), thermal conductivities of the fiber,

matrix, and interphase (Kf, Km, and Ki), and the room temperature tensile strengths of the fiber, matrix, and interphase (Sf, Sm,

and Si) are considered random; other parameters were assumed

to be deterministic. The interphase is a distinct region with a

finite thickness that arises as a result of fiber coatings or as a

"reaction-zone" caused by a chemical reaction between the

fiber and the matrix material as these composites are processed

at very high temperature. In any case, interphase material is

assumed to have its own properties, which are different from

both fiber and matrix properties. For illustrative purposes the

assumed distribution types and parameters for the selected

random variables are shown in table II. However, the actual

distributions for primitive variables were based on experimentally measured values whenever available. The micromechanics

and macromechanics embedded in the CEMCAN computer

code were used to compute composite properties. The micromechanics equations are not shown here for the sake of brevity,

but are readily available in reference 1. The FPI technique was

used to obtain CDF's of the required properties. FPI offered

very valuable additional information in the form of sensitivity

factors, which represent the sensitivity of output variable

uncertainty to the uncertainty in the selected primitive random

variables.

Three composite properties, the longitudinal modulus Ecl l,

inplane transverse modulus Ec22, and inplane shear modulus

Gc12, were chosen to illustrate the probabilistic mechanical

behavior of the composite. The composite coefficients of

thermal expansion 0_cl I and t_c22 and the composite thermal

conductivities Kcl 1 and gc22 were chosen as the composite's

representative probabilistic thermal behavior. In addition, the

probabilistic behavior of unidirectional composite longitudinal

tensile strength at room temperature was also examined. The

CDF's of these response variables, along with their sensitivity

factors with respect to the appropriate primitive variables, are

shown in figures 3 to 18 and are discussed in the following

paragraphs.

The CDF of composite longitudinal modulus Ecl 1 and its

probabilistic sensitivity to the various primitive variables are

TABLE L----CONSTITUENT PROPERTIES OF SiC/RBSN COMPOSITE a

Property

Modulus, GPa (Mpsi)

Poisson's ratio

Shear modulus, GPa (Mpsi)

Coefficient of thermal expansion, K-l (°F-l)

Thermal conductivity, W/m-K (Btu/ft-hr °F)

SiC fiber

390 (56.6)

0.17

117 (17)

4.1 xl0 4 (2.3x 10 4)

22x 1o 4 (12.7x 10 4)

RBSN matrix

It0 (15.95)

0.22

45 (6.5)

2.9x 104 (1.6x 104)

5x 10 -6 (2.9x 10 4)

Interphase

3.5 (0.5)

0.22

1.4 (0.2)

2.0x10-6 (1.1x 104)

2.0x 10 _ (1.2x 10 4)

aFiber and matrix properties are taken from reference 7 and interphase properties are based on calibration explained in

reference 7.

NASA TM-4766 5
TABLE n.-- PRIMITIVE INPUT VARIABLES DISTRIBUTION PARAMETERS

Variable Units Distribution Parameter 1 Parameter 2

mean value a coefficient

of variation

Fiber

390 (56.6) Modulus, Ef

Coefficient of thermal

expansion, af

Thermal conductivity, Kf

Tensile strength, $!

GPa (Msi)

10_/K (10_FF)

W/m-K (Btu/ft-hr °F)

GPa (ksi)

Normal

Normal

Normal

Weibull

4.4 (2.4)

22 (13)

2 (285)

0.05

0.05

0.05

0.05

Matrix

Modulus, E m 1 10 (15.95) 0.1

Coefficient of thermal

expansion, a m

Thermal conductivity, K m

Tensile strength, S,_

GPa (Msi)

10*/K (10-6r F)

W/m-K (Btu/ft-hr ° F)

MPa (ksi)

Normal

Normal

Normal

Weibull

2.1 (1.2) 0.1

4.2 (2.4) 0.1

93 (13.5) 0.15

lnterphase

Modulus, E i

Coefficient of thermal

expansion, a i

Thermal conductivity, K i

Tensile strength, S i

Thickness, ti

Fiber volume ratio, FVR

GPa (Msi)

lO_S/K (lOirE)

W/m-K (Btu/ft-hr° F)

MPa (ksi)

Ixm (mils)

Normal

Normal

Normal

Weibull

Normal

Normal

3.5 (0.5)

2.1 (1.2)

2.4 ( 1.4)

80 (l 1.6)

4.2 (0,17)

0.36

aData obtained from reference 2.

0.15

0.1

0.1

0.15

0.2

0.1

shown in figures 3 and 4. The computed mean value ofEcl 1for

this particular composite is 183 GPa with a scatter range of 155

to 224 GPa. In other words, if we were to experimentally

determine the longitudinal modulus of such a composite system, we could expect values anywhere in the range indicated by

the scatter. Such information obviously gives a test engineer

useful insight with which to design and plan the test setup and

the number of tests for a particular material study.

Sensitivity factors provide information for ranking the

importance of various input variables in controlling the output

response and scatter. For example, Ecl I is most sensitive to the

FVR (fig. 4). This is to be expected because the longitudinal

modulus is essentially controlled by fiber-dominated behavior.

Also important are the matrix modulus, fiber modulus, and the

interface thickness. If we wish to control the scatter in the

longitudinal modulus of this particular composite, the biggest

payoffs will result from controlling the scatter in the FVR. On

the other hand, the modulus of the interphase material has no

effect on the scatter of the ply longitudinal modulus. In this

situation, any effort in the processing or elsewhere that will

change the modulus of the interphase will not help control the

scatter in longitudinal modulus. Another point of interest is that

the sensitivities remain constant throughout the probability

range considered. In order to verify these results, a limited

number of Monte-Carlo simulation studies were also undertaken. The CDF for the longitudinal modulus obtained from

10 000 samples is also shown in figure 3. As can be seen from

the figure, the Monte-Carlo results agree very well with those

from FPI. Sensitivity information based on the Monte-Carlo

simulation was not obtained because doing so would have

involved a substantial additional effort with little added value,

and it is beyond the scope of the present study.

Figures 5 to 8 show results pertaining to the inplane transverse modulus Ec22 and the inplane shear modulus Gcl 2. These

figures also show the range of values that these moduli can take.

However, the sensitivities do not appear to remain constant

throughout the range of probabilities. Consequently, importance ranking of the primitive variables can change, depending

on the probability level. For example, in the case of Ec22

(fig. 6), the matrix modulus is the most sensitive primitive

variable at low probability levels, whereas at higher probability

levels, the interphase thickness becomes the most sensitive

parameter--a fact not apparent from the micromechanics equations alone. The trends of the curves for the transverse modulus

and the inplane shear modulus appear to be similar in nature.

Figures 9 through 12 show similar plots for the coefficients

of thermal expansion acl I and ac2 2. For t_cl 1 the predicted

mean is 3.38, and the scatter range is 2.85 to 3.91 ppm/K. The

scatter range is based on three standard deviations from the

mean. An examination of the sensitivity factors indicates that

the most important variable for controlling response scatter is

the thermal expansion coefficient of the fiber. Also, the coefficient of thermal expansion of the matrix, the FVR, and the

matrix modulus appear to have a significant effect on the

6 NASA TM-4766
1.0

0.8

°_

.Q

2 0.6

O.

>

"_ 0.4

E

C_

-- FPI

0.2 --

0.0 I - --<_1 I I I 1

125 145 165 185 205 225 245

Composite longitudinal modulus, Ec11, GPa

Figure 3.--Cumulative distribution function of longitudinal modulus Ec11 of [0] 8 SiC/RBSN composite.

0.3

0.2 _

0.1

0.0(

-o.1

:_ .-0.2

® -0.3 O3

-0.4

-0.5;

-0.6

-0.7

--<3"-- FVR

Ef

Em

•--C)--- Ei

ti

m

0.0 0.2 0.4 0.6 0.8 1.0

Probability

Figure 4.--Sensitivity factors for Ecl I for [018 SiC/RBSN

composite.

.O

.O

2

¢Z

Q_

>

O

1.0 D

0.8--

0.6--

0.4--

0.2--

o.o .c - J

30 60 90 120

Inplane transverse modulus, Ec22, GPa

Figure 5._Cumulative distribution function of inplane

transverse modulus Ec22 of [0] 8 SiC/RBSN composite.

0.8

0.6'

0.4 q

0.2

0,0(

-0.2

6)

m _0.4 q

-0.6--

-0.8--

-1.0

0.0

FVR

Em

E i

ti

I I I I I

0.2 0.4 0.6 0.8 1.0

Probability

Figure 6.---Sensitivity factors for inplane transverse

modulus Ec22 of [0]8 SiC/RBSN composite.

JD

e0

.O

2

O.

>=

-5

E

d

1.0 --

0.8 --

0.6--

0.4 --

0.2--

0.o - J

10 20 30 40 50

Inplane shear modulus, Gc12, GPa

Figure 7._Cumulative distribution function of inplane

shear modulus Gcl 2 of [0]8 SiC/RBSN composite.

NASA TM-4766 7
.>

¢n

0.6

0.4

0.2

0.0

---C---

FVR

Ef

Em

Ei

ti

-02 -

-0.4____

-0.6

-0.8

-1.0

0.0 0.2

I I I I

0.4 0.6 0.8 1.0

Probability

Figure 8._Sensitivity factors for inplane shear modulus

Gcl 2 of [0] 8 SiC/RBSN composite.

.O

¢0

¢Z

G)

.>

E

O

'°t

0.8

0.6 --

0.4 --

0.2--

00 [ c_ I t

2.0 2.5 3.0 3.5 4.0 4.5 5.0

Longitudinal thermal expansion coefficient,

_c11, ppm/K

Figure 9.--Cumulative distribution function of coefficient of longitudinal thermal expansion _'cll of [0]8

SiC/RBSN composite.

0.4 m

0.2 _--

0.0 _

•g --0.2

C

-0.4 _

-0"61__ _-

--0.8

0.0

2

E

0

_ FVR

--o-- _

Em

E i

t i

---II--- af

am

0.2 0.4 0.6 0.8 1.0

Probability

Figure 10.inSensitivity factors for coefficient of longitudinal thermal expansion e¢cl I of [0]8 SiC/RBSN

composite.

1.00 -- BB<3"4k3aC_P--O /

--_, -- Monte Carlo

FPI

0.80 -- • FPI mean

value

• FPI scatter /_

0.60 -- range //

0.40 -- _/_

0.20 -- J/I

0.00 A;I .;, _ I I I

1.00 1.42 1.83 2.25 2.67 3.08 3.50

Transverse thermal expansion coefficient,

e¢c22, ppm/K

Figure 11 .mCumulative distribution function of coefficient

of transverse thermal expansion _c22 of [0] 8 SiC/RBSN

composite.

8 NASA TM--4766
FVR

Em

Ei

t i

elf

oLm

Figure 12.---Sensitivity factors for coefficient of transverse thermal expansion OLc2 of [0]8 SiC/RBSN

composite.

1.0 --

5 7 9 11 13

Longitudinal thermal conductivity, Kcl 1, W/m-K

Figure 13.--Cumulative distribution function of longitudinal thermal conductivity Kcl I of [0]8 SiC/FIBSN

composite.

0.8

0.6

>=

0.4

_ 0.2

0.0

response scatter. The remaining variables have only minimal

influence on the response. This behavior is to be expected since

the longitudinal thermal expansion of a unidirectional composite is essentially controlled by the fibers. The expressions for

the thermal expansion coefficients are, in general, much more

nonlinear because they involve products of moduli, volume

ratios, and constituent thermal expansion coefficients. We,

therefore, decided to check the FPI results with a standard

Monte-Carlo technique as well, and generated 10 000 samples

for the Monte-Carlo simulation. As shown in figure 11 for the

transverse thermal expansion coefficient, the CDF calculated

by using FPI agrees very well with that determined by the

Monte-Carlo method. The predicted mean for txc22 is 2.25 ppm/

K, and the scatter range is 1.58 to 2.92 ppm/K. The sensitivity

plot (fig. 12) shows that the dominant variables for ac22

response are different from those of t_cl 1- Of the sensitivities,

the matrix coefficient of thermal expansion influences the

response probability most, followed by the interphase thickness, the interphase modulus, the matrix modulus, and the fiber

coefficient of the thermal expansion. The fiber modulus, which

was the most important variable for the case of Ctcl1, has little

influence on the ac22 response. Likewise, the interphase thermal expansion coefficient has little influence on the response

scatter. These results are, to some extent, intuitively obvious.

The behavior in the transverse direction is essentially matrix

dominated, and therefore, the matrix properties have a greater

effect on the tXc22response scatter. From these two figures we

can conclude that the scatter in the transverse thermal expansion coefficient can be effectively reduced by reducing the

scatter in the matrix thermal expansion coefficient and the

interphase thickness.

The curves for the longitudinal and the transverse thermal

conductivities are shown in figures 13 through 16. The predicted scatter range for Kcl 1is 7.4 to 11.6 W/m-K and for Kc22

is 4.2 to 7.1 W/m-K. The FVR is the most significant variable

for Kc] 1, whereas for Kc22 it is the matrix thermal conductivity.

All the primitive variables considered appear to significantly

influence the scatter in the longitudinal thermal conductivity

response. However, the thermal conductivities of the fiber and

the interphase appear to have only minimal influence on the

transverse thermal conductivity response.

A limited study has also been undertaken to assess the

probabilistic strength of the composite. The room temperature

ultimate longitudinal tensile strength was investigated for a [08]

SiC/RBSN laminate in which stress is redistributed with progressive fracture. The results are shown in figures 17 and 18.

The CDF of the strength response, as predicted by the FPI,

shows that we could expect a scatter range of 472 to 850 MPa

with a mean of 661 MPa. The variable that most influences the

scatter of the strength is the FVR, which is followed by fiber

strength. Since a higher scatter was assumed in the FVR

(10 percent) than in the fiber strength (5 percent), the ranking

may have been influenced. Matrix strength appears to have the

NASA TM--4766 9
.>

0.3

0.2'

0.1

0.0

-0.1

-0.2

-0.3

-0.4z

-0.5 E

-0.6

-0.7

-0.8

m

H

FVR

----V-- ti

Kf

Km

K i

I I i

0.0 0.2 0.4 0.6 0.8 1.0

Probability

Figure 14.--Sensitivity factors for longitudinal thermal

conductivity Kc11 of [0]8 SiC/RBSN composite.

0.4

0.2

0.0

-0.2

-0.6--

-0.8--

-1.0

0.0 0.2

FVR

Kf

Km

EI------gB

I I I I t

0.4 0.6 0.8 1.0

Probability

Figure 16.--Sensitivity factors for transverse thermal

conductivity Kc22 of [0]8 SiC/RBSN composite.

1.0

0.9

0.8

"_ 0.7

0.6

_ 0.5

_ 0.4

_0.3

0

0.2

0.1

0.0

2 3 4 5 6 7 8

Transverse thermal conductivity, Kc22, W/m-K

Figure 15.--Cumulative distribution function of inplane transverse thermal conductivity Kc22 of [0]8

SiC/RBSN composite.

o_

J_

¢g

J_2

>

-5

E

O

1,0 --

0.8--

0.6=

0.4--

0.2--

oo ss b I

200 400 600 800 1000

Longitudinal tensile strength, MPa

Figure 17.--Cumulative distdbution function of longitudinal tensile strength Scll of [0]8 SiC/RBSN

composite.

10 NASA TM-4766
FVR

t i

Slit_ f

S11t_ m

Figure 18.---Sensitivity factors for longitudinal tensile

strength Scl I of [0]8 SiC/RBSN composite.

least influence on strength. This would be expected in a brittle

system since the matrix starts cracking at low stress levels. With

an increase in the load, the matrix becomes fully saturated with

cracks and sheds all its load to the fibers. The ultimate tensile

strength, therefore, does not depend on matrix strength. Matrix

strength influences the onset of the nonlinearity in the stressstrain behavior though. Thus, in order to control the scatter in

longitudinal strength, we should exercise stricter quality control on the FVR and the fiber strength. Both of these issues are

related to the fabrication process.

Summary of Results

A methodology that provides a formal means to quantify the

scatter in composite behavior has been described in this paper.

It takes into account the scatter in constituent properties and

fabrication-related parameters. This methodology combines

composite micromechanics and macromechanics and a fast

probability integration technique to generate cumulative distribution functions for mechanical, thermal, and strength-related

properties of a unidirectional SiC/RBSN composite. Collectively, the results show that the present technique can provide

quantitative as well as qualitative information that can be used

as a guide in the fabrication and testing of the material. The

present methodology is computationally more efficient than

the standard Monte-Carlo simulation technique, and agreement

between the two is excellent, as is shown in the results. The

probabilistic laminate properties and responses can be used to

design a more rigorous component, thereby avoiding the usual,

overly conservative, conventional deterministic safety factor

approaches. The results provide valuable information on composite properties and sensitivity factors, which is useful to

design, test, and manufacturing engineers.

References

1. Mital, S.K.; and Murthy, P.L.N.: CEMCAN---CEramic Matrix

Composites ANalyzer User's Guide--Version 2.0, NASA

TM-107187, 1996.

2. Mital, S.K.; Murthy, P.L.N.; andChamis, C.C.: Micromechanics

for Ceramic Matrix Composites Via Fiber Substructuring, J. Compos.

Mater., vol. 29, no. 5, 1995, pp. 614-633

3. Wu, Y.-T.: Computational Methods for Efficient Structural

Reliability and Reliability Sensitivity Analysis. AIAA J., vol. 32,

no. 8, Aug. 1994, pp. 1717-1723.

4. Murthy, P.L.N.; Ginty, C.A; and Sanfeliz, J.G.: Second Generation Integrated Composite Analyzer (ICAN) Computer Code. NASA

TP-3290, 1993.

5. Murthy, P.L.N.; and Chamis, C.C.: MEtal Matrix Composite

ANalyzer (METCAN): Theoretical Manual. NASA TM-106025, 1993.

6. Mital, S.K., Murthy, P.L.N.; and Chamis, C.C.: Modeling of

StxessdStrain Behavior of Fiber-Reinforced Ceramic Matrix Composites Including Stress Redistribution. NASA TM-106789, 1994.

7. Bhatt, R.T.; and Phillips, R .E.: Laminate Behavior for SiC FiberReinforced Reaction-Bonded Silicon Nitride Matrix Composites.

NASA TM-101350, 1988.

NASA TM-4766 11
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June 1997 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Probabilistic Micromechanics/Macromechanics for Ceramic Matrix Composites

6. AUTHOR(S)

Pappu L.N. Murthy, Subodh K. Mital, and Ashwin R. Shah

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135- 3191

9. SPONSORINGJMONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

WU-505-63-12

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-10485

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4766

11. SUPPLEMENTARY NOTES

Pappu L.N. Murthy, NASA Lewis Research Center; Subodh K. Mital, The University of Toledo, Toledo, Ohio; Ashwin R.

Shah, NYMA, Inc., 2001 Aerospace Parkway, Brook Park, Ohio 44142 (work funded by NASA Contract NAS3-27186).

Responsible person, Pappu L.N. Murthy, organization code 5210, (216) 433-3332.

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13. ABSTRACT (Maximum 200 words)

The properties of ceramic matrix composites (CMC's) are known to display a considerable amount of scatter due to

variations in fiber/matrix properties, interphase properties, interphase bonding, amount of matrix voids, and many

geometry- or fabrication-related parameters, such as ply thickness and ply orientation. This paper summarizes preliminary studies in which formal probabilistic descriptions of the material-behavior- and fabrication-related parameters were

incorporated into micromechanics and macromechanics for CMC's. In this process two existing methodologies, namely

CMC micromechanics and macromechanics analysis and a fast probability integration (FPI) technique are synergistically coupled to obtain the probabilistic composite behavior or response. Preliminary results in the form of cumulative

probability distributions and information on the probability sensitivities of the response to primitive variables for a

unidirectional silicon carbide/reaction-bonded silicon nitride (SiC/RBSN) CMC are presented. The cumulative distribution functions are computed for composite moduli, thermal expansion coefficients, thermal conductivities, and longitudinal tensile strength at room temperature. The variations in the constituent properties that directly affect these composite

properties are accounted for via assumed probabilistic distributions. Collectively, the results show that the present

technique provides valuable information about the composite properties and sensitivity factors, which is useful to design

or test engineers. Furthermore, the present methodology is computationally more efficient than a standard Monte-Carlo

simulation technique; and the agreement between the two solutions is excellent, as shown via select examples.

14. SUBJECT TERMS

Ceramic matrix composite; Properties; Response probabilistic material behavior;

Response sensitivities; Uncertainties; Scatter; Strength; Mechanical properties;

Thermal properties

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