NASA / TM--1998-208492

Probabilistic Modeling of Ceramic

Matrix Composite Strength

Ashwin R. Shah

Sest, Inc., North Royalton, Ohio

Pappu L.N. Murthy

Lewis Research Center, Cleveland, Ohio

Subodh K. Mital

University of Toledo, Toledo, Ohio

Ramakrishna T. Bhatt

U.S. Army Research Laboratory, Lewis Research Center, Cleveland, Ohio

Prepared for the

39th Structures, Structural Dynamics, and Materials Conference and Exhibit

cosponsored by AIAA, ASME, ASCE, AHS, and ASC

Long Beach, California, April 20-23, 1998

National Aeronautics and

Space Administration

Lewis Research Center

September 1998

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PROBABILISTIC MODELING OF CERAMIC MATRIX COMPOSITE STRENGTH

Ashwin R. Shah

Sest, Inc.

North Royalton, Ohio 44133

Pappu L.N. Murthy

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

Subodh K. Mital 1

The University of Toledo

Toledo, Ohio 43606

Ramakrishna T. Bhatt

U.S. Army Research Laboratory

Lewis Research Center

Cleveland, Ohio 44135

ABSTRACT

Uncertainties associated with the primitive random variables such as manufacturing process (processing

temperature, fiber volume ratio, void volume ratio), constituent properties (fiber, matrix and interface), and

geometric parameters (ply thickness, interphase thickness) have been simulated to quantify the scatter in the first

matrix cracking strength (FMCS) and the ultimate tensile strength of SCS-6/RBSN (SiC fiber (SCS-6) reinforced

reaction-bonded silicon nitride composite) ceramic matrix composite laminate at room temperature. Cumulative

probability distribution function for the FMCS and ultimate tensile strength at room temperature (RT) of [0] 8, 102/

902] s, and [+452] s laminates have been simulated and the sensitivity of primitive variables to the respective strengths

have been quantified. Computationally predicted scatter of the strengths for a uniaxial laminate have been compared

with those from limited experimental data. Also the experimental procedure used in the tests has been described

briefly. Results show a very good agreement between the computational simulation and the experimental data.

Dominating failure modes in [0] 8, [0/90] s and [+45]s laminates have been identified. Results indicate that the first

matrix cracking strength for the [0] 8, and [0/90] s laminates is sensitive to the thermal properties, modulus and

strengths of both the fiber and matrix whereas the ultimate tensile strength is sensitive to the fiber strength and the

fiber volume ratio. In the case of a [+45] s laminate, both the FMCS and the ultimate tensile strengths have a small

scatter range and are sensitive to the fiber tensile strength as well as the fiber volume ratio.

Key words: Probability theory, simulation, probability density functions, ceramic matrix composites, sensitivity

CDF

CMC

COV

CTE

SYMBOLS

cumulative probability distribution function

ceramic matrix composite

coefficient of variation

coefficient of thermal expansion

_ResidenlResearch Associate, NASA Lewis Research Center, Cleveland, Ohio44135.

NASA/TM-- 1998-208492 !
Efl 1

E m

FVR

FMCS

SfllT

Sis

Sins

SmT

t i

VVR

O_fl 1

Ot12 2

O(m

fiber modulus in the longitudinal direction

matrix modulus

fiber volume ratio

first matrix cracking strength

longitudinal fiber strength in tension

interphasc shear strength

matrix shear strength

matrix strength in tension

ply thickness

void volume ratio

fiber thermal expansion coefficient in 1 ! direction

fiber thermal expansion coefficient in 22 direction

matrix thermal expansion coefficient

INTRODUCTION

Aircraft engine propulsion system components are subject to a complex state of thermomechanical loading

cycles in an aggressive combustion environment. Owing to their oxidation resistance, strength and microstructural

stability, silicon carbide fiber-reinforced silicon-based ceramic matrix composites are prime candidate materials for

high temperature components such as combustor liners, nozzle vanes, and blades.

Ceramic matrix composites (CMC) are heterogeneous brittle materials consisting of fibers, matrix, and

coating/interphase, and are fabricated by novel fabrication approache_,. Although these materials exhibit a

considerable scatter in their mechanical and thermal properties due to inherent variations in the constituent (fiber,

matrix and interphase) properties and geometric parameters at ply an_ laminate levels, these materials are of interest

to designers because of their near net shape fabricability and high toughness. Designing structural components made

of CMCs with specified reliability is a complex task. Conventional deterministic design methodology accounts for

uncertainties using safety factors or load factors which are at times o,erly conservative. Therefore, there is a need to

develop probabilistic models to predict scatter in the thermomechanic _l properties and the composite strengths

required to design CMC components.

A formal methodology has been developed by Murthy et al. 1 ] to quantify probabilistic mechanical and

thermal properties, and behavior of CMC. This methodology has beet+ incorporated into a computer code called

Probabilistic CEramic Matrix Composite ANalyzer (PCEMCAN) [2] The code can compute the cumulative

probability distribution function (CDF) of the composite properties, aad quantify their sensitivity to the primitive

random variables. The primitive random variables participate at the tcwest level (e.g. the micromechanics level) in

defining a global composite property. Typical examples of primitive rmdom variables are fiber, matrix and

interphase properties, fiber volume ratio, ply thickness, etc.

The probabilistic simulation approach can also be used as a design tool to reduce material development

costs in optimizing composite properties. Furthermore, the reliability of designed structural elements can be

NASA/TM-- 1998-208492 2
predicted readily. The validity and utility of new approaches, however, need to be properly established through

comparison with the experimental data.

The objective of the current research effort is to use the probabilistic simulation approach to predict the

observed scatter in the first matrix cracking strength (FMCS) and ultimate tensile strength at room temperature for

the three SCS-6/RBSN (SiC fiber rein|orced reaction-bonded silicon nitride composite) laminates, namely [0] 8, [02/

902] s, and [+452] s, and to compare the predicted results with the experimental data. The first matrix cracking

strength is generally used to determine the design allowable whereas the ultimate strength provides an idea of the

margin of safety. The simulation process includes uncertainties in the constituent properties, fabrication process

variables, and the stress free temperature (process) to account for any residual stresses. Initially the methodology has

been calibrated by comparing the simulation results with the experimental data tor a uniaxial laminate. The residual

stresses arising out of the fabrication process temperature affect the strength of a composite laminate. Therefore, the

importance of fabrication process temperature will be highlighted. Also, the failure mode for a given laminate

configuration determines the first matrix cracking strength and ultimate strength. Hence, the significance of failure

mode from a reliability viewpoint will be elucidated in the paper. It is important to realize that the availability of

enough experimental data for different laminate configurations for the purpose of probabilistic verification is

difficult. The probabilistic/deterministic results were compared with experimental data for [0]8 laminate, and for the

remaining laminates only deterministic behavior was compared due to lack of experimental data for a full

probabilistic description of the behavior.

EXPERIMENTAL PROCEDURE

Composite Fabrication

The starting materials for the SiC/RBSN composites were SiC fibers and silicon powder. The 142 lam

diameter SiC fiber used was obtained from Textron Specialty Materials, Lowell Massachusetts. This SiC fiber,

designated as double coated SCS-6 by the manufacturer, contained a mixture of SiC/C on its outer surface. The silicon

powder is procured from Union Carbide, Linde Division, Tonawanda, New York. The SiC/RBSN composites were

produced by a two step process. In the first step, the fiber mats and silicon tapes were hot pressed to prepare preforms.

In the second step the prelbrms were nitrided to convert silicon to silicon nitride matrix. Salient features of the

composite fabrication are given here, but detailed account of fabrication is reported elsewhere [3].

For the fiber mats. the SCS-6 fibers were wound on a cylindrical drum and coated with a solution of a fugitive

polymer binder in an organic solvent. Alter carefully drying, the fiber mats of required dimensions were cut. For the

preparation of silicon powder cloth, silicon powder, an additive (for enhancing nitridation), a fugitive polymer binder,

and an organic solvent were mixed in a high speed blender until a dough was formed. The dough was rolled to the

desired thickness and cut to required dimensions. For the composite preform fabrication, alternate layers of SiC fiber

mat and silicon cloth, were stacked in a molybdenum die and hot pressed in vacuum or in a nitrogen environment at a

suitable combination of temperature and pressure to produce a SiC/Si preforms.

The SiC/Si pretbrms were then transferred to a horizontal nitridation furnace consisting of a recrystallized

A1203 reaction tube with stainless steel end caps. The high purity nitrogen gas was flowed through the furnace before,

during and after nitridation. Nitridation of the composite was performed between 1200 and 1400 °C. The nitrided

composite panel, was ground on emery paper to remove any loose surface Si3N 4 particles. Typical dimensions of the

nitrided panels were 125×50xl.2 mm. Using the above fabrication methodology 10]8, [02/902] s, and [+452] s laminates

were prepared.

Specimen Preparation and Testing

Tensile specimens of dimensions 150x12.7z 1.2 mm were prepared by cutting and grinding the composite

panels with a diamond impregnated abrasive wheel. A pair of glass fiber reinforced epoxy tabs of dimensions

38×12.7xl mm were adhesively bonded at the specimen ends leaving 25 mm as the test gage length. A wire wound

NASA/TM-- 1998-208492 3
straingagewasadhesivelybondedtothespecimengagesectiontomo_litortheaxialandtransverse strains. Thetensile

testswereperformedinanInstronmachineatacross-head speedof I.?,6mm/min.

PROBABILISTICSTRENGTHSIMULATIONMETHODOLOGY

Theresearchdescribedinthispaperfocusesontwomajorissues:(I) thequantificationfscatterinthe

CMClaminatestrengthand,(2)thecomparisonfcomputationally simulatedresultswiththeexperimental data.

(I) Quantification of scatter in the CMC laminate strength methodology.--The approach developed by

Murthy et al. [1] has been used to simulate the probabilistic strength and quantify uncertainties in CMC laminates.

This methodology uses the fast probability integration technique (FPI) developed under a NASA Lewis Research

Center sponsored program [4]. The procedure involves the following steps as depicted in Fig. 1.

( 1) Identify the independent uncorrelated input variables and their probability distributions. Examples of

independent variables are constituent properties, fiber volume ratio, void-volume ratio (VVR), ply thickness, ply

alignment, fiber and matrix strengths, etc.

(2) Develop a function (called the performance/response function) that defines a relationship between a

response variable (first matrix cracking strength, ultimate strength, modulus are typical response variables for

CMCs) and the independent variables. A perturbation analysis approach is used to develop the performance function.

Perturbation analysis involves performing several deterministic analyses for different magnitudes of selective

samples of primitive random variables using the micro/macromechanics theory included in PCEMCAN [2]. The

procedure for computing deterministic strength is described in the paragraphs below. Responses obtained from these

analyses are used to generate a response surface required by FPI for the probabilistic assessment of the response.

IA

Distribution type

Random variable /

statistics rl..e/1__

V

Xl x2 x3

Sensitivity factors

CEMCAN performance

function z =/(Xl, x2, x3) Output options

i

Fast probability

integration (FPI)

analysis engine

i

/

/

J

/jJ _

Response cumulative

distribution function

(CDF)

Figure 1.--Fast probability integration input-output schematic.

I

NASA/TM-- 1998-208492 4
(3)UsetheFPItechniquetocomputethecumulativeprobabilitydistributionfunctionoftheresponseandits

sensitivityotheprimitiverandomvariables.

Deterministic strength computation procedure.--It is worth recording the procedure, assumptions and

underlying concepts used in the perturbation analysis to determine the FMCS and the ultimate strength. A unique

capability available in PCEMCAN computer code [2] to carry out progressive fracture analysis with microstress

redistribution at every incremental load step has been used to determine these strengths (refer to Fig. 2 lot the

schematic description of the integrated approach being discussed here). Every ply in a laminate is divided into

smaller slices of a reasonable size specified by the user. At every load increment, the microstresses in all the

constituents of every slice of every ply are computed and compared against their respective current strengths. A

maximum stress criterion is used for the failure analysis, meaning a constituent has failed if its stress exceeds its

preset strength value. In case of a failure, the modulus of the failed constituent is set to a negligible value for the

subsequent analysis. A slice is assumed to have failed under the longitudinal loading when a fiber has failed. A slice

is assumed to have failed under transverse/shear loading, if matrix/interphase has failed. Thus, in the next step, the

load carried by the failed slice is redistributed to the remaining slices. The load that was carried by the failed slice is

then added appropriately to the laminate load and the laminate analysis is performed again. The process is repeated

until the equilibrium (convergence) between the applied load and a damage state is reached for a given load step.

Convergence is established when the mid plane strains, ply strains and slice strains are within 5 percent of those in

the previous iteration. The nonlinearity observed in the stress-strain behavior arises primarily due to the stress

redistribution.

The FMCS is the stress at which the matrix starts failing. In the current analysis, a first ply matrix failure

criteria is used to determine the FMCS. It means that the FMCS is reached when the matrix in all the slices of any

ply have failed. Basically at this point the stress-strain curve starts changing slope since the load is mainly carried by

the fibers in each ply then alter. Even after the FMCS is reached, the analysis is continued to assess the ultimate

strength since the plies that have not failed completely (including fibers) can still carry load. An incremental load is

applied until all the plies in laminate fail completely and can no longer carry further load. At this point, the laminate

is assumed to have reached the ultimate strength.

andinte. hase

Pr°pe iesI ]

Integration/synthesis

Fiber

substructuring

Unit

cell

! t

III II I

Slice Lamina/ply

I/ 1 I/ 1

Progressive decomposition

I

I

Laminate

Figure 2.--Integrated probabilistic ceramic matrix composite mechanics

approach.

NASA/'TM--1998-208492 5
RESULTS AND DISCUSSION

Tensile Behavior of laminates

The room temperature tensile stress-strain behavior of the uniaxial and cross-ply SCS-6/RBSN laminates

arc shown in Figs. 3 to 5. The tensile stress-strain curve show three distinct regions: an initial linear elastic region, a

very short nonlinear region, and a second linear region. The nonlinear signifies the start of the matrix cracking to

saturation when the entire load is carried only by fibers. The stress corresponding to the deviation from the initial

lincarity in the stress-strain curve is considered to be the first matrix cracking stress (FMCS). Indeed, at this stress level,

previous optical microscopy and in-situ x-ray studies [5,6] confirmed formation of a through-the-thickness matrix

crack. From the tensile stress-strain curves, the primary elastic modulus, first matrix cracking stress and strain, and

ultimate tensile strength and strain were noted. The room temperature tensile property data for the three laminates are

summarized in Table I. The data represent an average of 60 specimens of [0] 8 laminates, and 10 specimens of [02/

902} s. and 5 specimens of [+452] s.

The comparison of the experimental data with the deterministic computational simulation results is

depicted in Figs. 3 to 5 and details of the verification procedure are given in Ref. 7. It is seen from these figures that

the computational results compare fairly well with the experimental data.

101.5

-_ 72.5

e

P 43.5 4=J

03

800 -

600 --

o.

-- _ °

400 --

u_ CEMCAN [08]

-- _ CEMCAN [908]

03 200 Experiment [08]

Experiment [908]

14.5 --

I I I I I I

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Strain, e, I:ercent

Figure 3.--Room-temperature stress-strain curves to

failure of [0]8 and [90]8 SiC/RBSN composite.

=;

C/)

==

Or)

43.5 _ ¢ 300 I

200

145 1

0.0 0.1

CEMCAN [108]

.... CEMCAN [458]

• Experiment [458]

[] Experiment [458]

L I I I I

0.2 0.3 0.4 0.5 0.6

Strain, e, percent

Figure 4.--Room-temperature stress-strain curves to

failure of [10]8 and [45]8 laminates of SiC/RBSN.

NASA/TM-- 1998-208492 6
43.5 -- 300 --

29.0

U_

,=

14.5

-- _ 200 CEMCAN

• Experiment

(/)

_ p100 -- o

0 I° 6 I

0.0 0.1 0.2 0.3 0.4

Strain, c, percent

Figure 5.--Room-temperature stress-strain curves to

failure of [-+452]s SiC/RBSN laminate.

TABLE I.--PRIMITIVE RANDOM VARIABLES

DISTRIBUTION PARAMETERS

Variable I

Modulus. ErH

Coefficient of thermal

expansion, ct m

Longitudinal strength

in tension. Sfll. I

Modulus. E,,

Coefficient of thermal

expansion, otm

Shear strength. Sins

Strength in tension S.,_

Shear strength, Sis

Thickness. t,

Fiber volume ratio

Units I Mean Cofficienl

Ivalue of variation

SCS-6 fiber

GPa 390

Distribution

t),pe

ppm/°C 4.1

GPa 2.86

RBSN matrix

GPa 110

ppm/°C 2.2

MPa 55

MPa 84

Carbon interphase

gm 2.5

Process variable

I.... I °,2 1

0.55 Lognormal

0.07 Nomml

0.04 Weibull

14 Weibull

0.06 Normal

0.34 Weibull

26 Weibull

0.34 Weibull

0.25 Weibull

0.05 Normal

Probabilistic Strength Prediction and Verification

Computational simulation of probabilistic strength has been performed using the PCEMCAN code for [0] 8,

[0J902] s, and [+452] s SCS-6/RBSN laminates. The mean values, coefficient of variation and the distribution types of

the primitive random variables related to the fiber, matrix, and interphase properties, fiber volume ratio (FVR) and

the ply thickness used in the analysis for the [0] 8 laminate are shown in Table I. The values used for the [02/902] s

and [_452] s laminates are the same as those in Table I except that the mean FVR is 0.36. The distribution types and

magnitude of scatter in these primitive random variables have been assumed based on the engineering judgment and

past experience. There were many more variables associated with the fiber, matrix and interphase properties which

were considered as random in the analysis. However, the variables that were found to have an insignificant effect on

the probabilistic strengths have been omitted from the list for the sake of brevity.

Additionally, the strength simulation accounts for any residual stresses that arise from the fabrication of

these composites. It has been assumed in the present simulation that very little degradation (maximum 10%) of the

constituent properties such as modulus, etc. with respect to the room temperature properties occur within the

processing and maximum use temperature range. Also, the constituents are assumed to be homogeneous, and

isotropic in this work. The importance of process temperature has been highlighted by the sensitivity computations

NA SA/TM-- 1998-208492 7
for the strengths. Detailed description of the results for each laminate configuration is given in the subsequent

paragraphs.

(i) Uniaxial, 101s iaminate.--The first matrix cracking and ultimate strength of a uniaxial laminate have

been simulated using PCEMCAN code. A stress free temperature of 540 °C was assumed in the analysis. The

thermal processing and the mechanical loads were applied incrementally. A reasonably small size load step was used

in the analysis to determine lhilure modes and a more accurate failure strength of constituents. At every load step,

the stresses and strains in the constituent were observed with respect to the failure criteria to determine thilure

strengths.

The CDF tor the computed FMCS and that obtained from the experimental data are plotted in.Fig. 6. The

experimental test data CDF was obtained using 60 test samples. The computationaily obtained mean first matrix

cracking strength is 225 MPa that matches with 221 MPa from the test data. The coefficient of variation (COV) for

the FMCS is about 11.34 percent. The predicted probabilistic scatter in the FMCS range from 124 to 303 MPa as

compared to 172 to 276 MPa obtained experimentally. The experimentally observed scatter based on 60 samples fall

within the predicted scatter.

Sensitivity of the FMCS to the primitive random variables are plotted in Fig. 7. It is seen that the FMCS is

sensitive (in the descending order) to the thermal expansion coefficient (CTE) of the fiber (ocfl l,), longitudinal fiber

modulus (Efl I )' matrix strength in tension (SmT), fiber volume ratio (FVR), matrix thermal expansion coefficient

(_m)' and matrix modulus (Em). In a composite in which the modulus and CTE of the fiber are greater than those of

the matrix, compressive residual stresses are generated in the matrix after cooling the composite from the fabrication

temperature. In this case the compressive residual stress in the matrix increases the FMCS. In contrast, in a

composite in which the modulus and CTE of the fiber are lower than those of the matrix, tensile residual stresses are

generated. This will decrease the FMCS. Therefore, the thermal residual stresses significantly affect FMCS in

composites. Thus, the processing temperature plays an important role in affecting the design strength. Also, to obtain

a reduced scatter in the design strength, the uncertainty in these variables should be controlled. Furthermore,

processing techniques to reduce the residual stresses should be devel,_ped. Note that the stress free temperature is

considered deterministic in the current analysis.

Figure 8 depicts the comparison of probabilistic and experimental CDF of the ultimate strengths. The

predicted mean of 658 MPa compares closely with the experimentally obtained mean of 690 MPa and COV is

1.0 PCEMCAN f

.__ 0.8 ...... Experimental _ •

•_ 0.6 2

e_

_ o.4

_ 0.2

0.0

100 150 200 250 300

First matrix fracture strength, MPa

Figure 6.--Comparison of first matnx cracking

strength CDF simulation with the experimental

data for a [0]8 SCS-6/RBSN composite laminate.

NASA/TM-- 1998-208492 8
SmT

[] 0.999 Probability

• 0.001 Probability I

O_ m

Em

°_fl 1

Efl 1 I

FVR

i I I I I t

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

Sensitivity

Figure 7.--Sensitivity of the first matrix cracking

strength to the primitive random variables of a

[0]8 SCS-6/RBSN composite laminate.

.D

n

.Q

2

¢).

0)

E

d

1.0 =_.

PCEMCAN f_,,--

0.8 ...... Expedmental,; ,,r °"

0.6 _int_ °"

0.4

0.2

0.0

400 500 600 700 800 900 1000

Ultimate strength, MPa

Figure 8.--Comparison of ultimate strength CDF

simulation with the experimental data for a [0] 8

SCS-6/RBSN composite laminate.

14.8 percent. The predicted scatter is from 345 to 966 MPa as compared to that of 414 to 966 MPa obtained

experimentally. Looking at the overall CDF curves, the ultimate strength CDF and scatter are in good agreement.

The sensitivity of ultimate strength to the primitive random variables is shown in Fig. 9 and is mainly

controlled by the longitudinal tensile strength of the fiber (Sfl iT) and the fiber volume ratio. Thus, the ultimate

strength is mainly controlled by the fibers as expected. This is due to the fact that after the matrix starts cracking and

becomes saturated with cracks, thereafter the load is mainly carried by the fibers. Since the fiber tensile strength is

the only variable that controls the ultimate strength, the COV of the predicted ultimate strength is almost the same as

that of the fiber tensile strength. It is worth noting here that the reliable margin of safety tor the uniaxial laminates is

fairly large and the fiber properties play a major role for the FMCS as well as ultimate strength.

NASA/TM--1998-208492 9
[] 0.999 Probability

• 0.001 Probability

I SmT

Sfl 1T

i I I

-1.0 -0.8 -0.6 -0.4 --0.2 0.0

Sensitivity

FVR

Figure 9.--Sensitivity of the ultimate strength to the

primitive random variables of a [0] 8 SCS-6/RBSN

composite laminate.

1.0

Z' 0.8

..Q

0.6

•_ 0.4

"5

E

0.2

0.0

100 150 200 250 300

First matdx strength, MPa

Figure 10._umulative probability distribution

function of the first matrix cracking strength of a

[02/902]2 SCS-6/RBSN composite laminate.

(ii) Cross-ply, [02_/90__2La laminate configuration.--Probabilistic tensile strength of a cross ply laminate has

been computed using the PCEMCAN computer code and the approach discussed earlier. The deterministic

verification for this laminate has been performed in Ref. 7. Using the same properties and a stress free temperature

given in Ref. 7 and listed here in Table I, the CDFs of the first matrix strength and the ultimate strength were

determined. The CDF of the FMCS and its sensitivity to the primitive random variables are plotted in Figs. 10 and

11, respectively. The mean FMCS is 204 MPa and its coefficient of wtriation (COV) is 12.3 percent. It is worth

noting here that the FMCS and its COV are not much different from t le uniaxial case. The scatter range is from 112

to 282 MPa. Also, along the same line the primitive variables that control the first matrix strength are the same as

that for the uniaxial strength, and in the same order of importance. The important variables are _fl 1' Efl 1, SmT' FVR,

_m' Era' and _f'2_2" Once again the stress free temperature controls the design strength, i.e., FMCS for the reasons

discussed above.

NA SA/TM-- 1998-208492 10
SmT

[] 0.999 Probability

• 0.001 Probability

Ol m

Em

af22

af11

Ef11

FVR

t I I I I I

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

Sensitivity

Figure 11 .--Sensitivity of the first matrix cracking

strength to the primitive random variables of a

[02/902] s SCS-6/RBSN composite laminate.

1.0

0.0

250 350 450 550

Ultimate strength, MPa

Figure 12.--Cumulative probability distribution

function of the ultimate strength of a [02/902]s

SCS-6/RBSN composite laminate.

0.8

.Q

¢0

._ 0.6

9

_ 0.4

E

0 0.2

i

650

The ultimate strength CDF range from 257 to 626 MPa as shown in Fig. 12 with a mean strength of

446 MPa and COV being 13.56 percent. The scatter range and the mean strength have reduced noticeably due to the

fact that less amount of fiber is available in the load direction and it is mainly governed by the Sfl IT and FVR as

shown in Fig. 13. Also, after the matrix is saturated with cracks the load is mainly carried by the 0 ° plies only. At

low cumulative probability levels the importance of both of these variables is the same. However, at higher

cumulative probability level, the S/1 IT dominates due to the larger magnitude of strength. The low probability levels

are important from the reliability point of view. Therefore, it can be said that the probabilistic behavioral pattern is

similar for uniaxial and cross-ply laminates.

_2_ Laminate configuration.--The constituent properties and the probabi[istic data for this laminate

configuration is the same as that for the [02/902] s configuration. The CDF of the first matrix cracking strength and

the ultimate strength for this configuration are plotted in Fig. 14 and their sensitivity to the primitive random

variables are depicted in Fig. 15. It is important to note in this case that the FMCS and the ultimate strengths turn out

to be the same. This is mainly due to the fact that the failure of laminate occurs due to the normal and shear failure

of the matrix/interphase as evidenced by the sensitivity data in Fig. 15. This is the characteristic of a shear failure

NASA/TM-- 1998-208492 I I
Sf11T

[] 0.999 Probability

• 0.001 Probability

FVR

-1 -0.8 -0.6 -0.4 -0.2 0.0

Sensitivity

Figure 13._ensitivity of the ultimate strength to

the primitive random variables of a [02/902]s

SCS-6/RBSN composite laminate.

EoB

.O

.Q

2

O.

._>

E

1.0

0.8

0.6

0.4

0.2

0.0

80 90 100 110 120

First matrix fracture/ultimate strength, MPa

Figure 14.---Cumulative probability distribution function of the first matrix cracking/ultimate strength

of a [45/--45]s SCS-6/RBSN composite laminate.

mode. These failure modes have been confirmed by the optical micro ;copic observations of the failed specimens [5].

Also, the deterministic strength verification of this laminate configun tion has been performed in Ref. 7 and depicted

in Fig. 5. The mean first matrix cracking and ultimate failure strength is 108 MPa and the COV is 4.7 percent. The

range of the scatter is from 81 to 119 MPa which is a very small rang) as expected tor shear mode failures. Also, the

failure strength is sensitive to the matrix and interphase shear strengths and mostly dominated by the matrix at all

probability levels. It is important to note here that controlling the scatter in matrix and interphase shear strengths

may not control the scatter of the failure strength due to the fact that the shear failures are sudden in nature as

evidenced by a very small scatter range.

SUMMARY AND CONCLL SIONS

A methodology to quantify scatter in the first matrix cracking strength and the ultimate strength of the

SCS-6/RBSN composite laminates has been developed and the predictions are compared with the limited

NASA/TM-- 1998-208492 12
Interphase shear strength

[] 0.999 Probability

• 0.001 Probability

Matrix shear strength

-1.0 -0.5 0.0

Sensitivity

Figure 15.--Sensitivity of the first matrix cracking/

ultimate strength to the primitive random variables

of a [-+452]sSCS-6/RBSN composite laminate

experimental data. The experimental results for unidirectional laminates agreed well with the computationally

predicted probabilistic strengths. Two more laminate configurations, [02[902] s, and [+452] s, have also been studied,

however, due to the lack of sufficient data, the probabilistic simulations could not be verified. The deterministic

results did agree well with the experimental results. Collectively, the results provide a valuable insight to material

developers and design engineers. The material developers can utilize the sensitivity information to produce a better

material with lesser scatter. Also, the design engineer can use the probabilistic strength information to quantify the

conservatism in the design and reliability of the component. Based on the results it has been observed that the scatter

in the strengths and their sensitivity to the random variables is controlled by the dominating failure modes and stress

free temperature. In cases when the failure is governed by the normal tension failure in the matrix, the thermal

properties of the fiber and matrix, fiber and matrix modulus, matrix strength in tension and FVR controls the first

matrix cracking strength. Therefore, the design strength scatter can be improved by controlling the variables that

affect the composite manufacturing process. On the other hand, the scatter in the reliable safety margin can be

reduced by controlling the scatter in the fiber tensile strength and the fiber volume ratio.

REFERENCES

I. Murthy, EL.N, Mital, S.K. and Shah, A.R.: Probabilistic Mieromechanics and Macromecbanics for Ceramic

Matrix Composites, NASA TM-4766, June 1997.

2. Shah, A.R., Mital, S.K. and Murthy, P.L.N.: "PCEMCAN (Probabilistic Ceramic Matrix Composite Analyzer)

User"s Guide Version 1.0, NASA TM-206984, 1998.

3. Bhatt, R.T.: "Method of Preparing Fiber Reinforced Ceramic Materials," U.S. Pat. No. 4689188, 1987.

4. Probabilistic Structural Analysis Methods for Select Propulsion Components, Annual Reports I-6, NASA

Contract NAS3-24389, 1985-1990, Southwest Research Institute, San Antonio, TX.

5. Bhatt, R.T. and Philips, R.E.: "Laminate Behavior tor SiC Fiber-Reinforced Reaction Bonded Nitride Matrix

Composites," NASA TM- ! 01350, 1988.

6. Baaklini, G.Y. and Bhatt, R.T.: "In-Situ X-Ray Monitoring of Damage Accumulation in SiC/RBSN Tensile

Specimens," Ceram. Eng. Sci. Proc. 12, (7-8), 1599, 1991.

7. Mital, S.K. Murthy, EL.N. and Cbamis, C.C.: Modelling of Stress/Strain Behavior of Fiber Reinforced Ceramic

Matrix Composites Including Stress Redistribution, NASA TM-106789, Dec. 1994.

NASA/TM-- 1998-208492 13
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

September 1998

4. TITLE AND SUBTITLE

Probabilistic Modeling of Ceramic Matrix Composite Strength

6. AUTHOR(S)

3. REPORT TYPE AND DATES COVERED

Technical Memorandum

5. FUNDING NUMBERS

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

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

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0(OI

WU-523-21-13-00

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-II281

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--1998-208492

11. SUPPLEMENTARY NOTES

Prepared for the 39th Structures, Structural Dynamics, and Materials Conference and Exhibit cosponsored by AIAA,

ASME, ASCE, AHS, and ASC, Long Beach, California, April 20-23, ] 998. Ashwin R. Shah, Sest, Inc., North Royalton,

Ohio 44133; Pappu L.N. Murthy, Lewis Research Center; Subodh K. Mital, University of Toledo, Toledo, Ohio 43606:

and Ramakrishna T. Bhatt, U.S. Army Research Laboratory, Lewis Research Center, Cleveland, Ohio. Responsible

person, Pappu L.N. M urthy, orsanization code 5910, (216)433-3332.

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Unclassified - Unlimited

Subject Category: 24 Distribution: Nonstandard

This publication is available from the NASA Center for AeroSpace Information, (3")1) 621_t)390.

13. ABSTRACT (Maximum 200 words)

Uncertainties associated with the primitive random variables such as man fracturing process (processing temperature, fiber

volume ratio, void volume ratio), constituent properties (fiber, matrix and interlace), and geometric parameters (ply thickness,

interphase thickness) have been simulated to quantify the scatter in the fif _t matrix cracking strength (FMCS) and the ultimate

tensile strength of SCS-6/RBSN (SiC fiber (SCS-6) reinforced reaction-bonded silicon nitride composite) ceramic matrix

composite laminate at room temperature. Cumulative probability distribution function for the FMCS and ultimate tensile

strength at room temperature (RT) of [0] 8, [02/902] s, and [+452] s laminates have been simulated and the sensitivity of primitive

variables to the respective strengths have been quantified. Computationally predicted scatter of the strengths for a uniaxial

laminate have been compared with those from limited experimental data. Mso the experimental procedure used in the tests has

been described briefly. Results show a very good agreement between the computational simulation and the experimental data.

Dominating failure modes in [0] 8, [0/90]s and [+45] S laminates have been identified. Results indicate that the first matrix

cracking strength for the 1018, and [0/90]s laminates is sensitive to the the real properties, modulus and strengths of both the

fiber and matrix whereas the ultimate tensile strength is sensitive to the fiiler strength and the fiber volume ratio. In the case of

a [+451s laminate, both the FMCS and the ultimate tensile strengths have _small scatter range and are sensitive to the fiber

tensile strength as well as the fiber volume ratio.

14. SUBJECT TERMS

Probability theory: Simulation; Probability density functions; Ceramic matrix composites;

Sensitivity

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19

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