Matrix Composites

z

NASA Technical Memorandu_ 103669 _

.... _'2o

Probabilistic MicromeChafiics_and

Macromechanics of Polymer:Matrix Composites

....................... _ _=_--, - _._

r (NAC, A-TM-103669) PROBARILIqTIC N91-19236 --_"

_ICROMECHANICS AND MACROMFCHANICS OF POLYMER

MATRIX COMPNSITFS (NASA) 20 p CSCL lID

__L .....

Unclas

G3/Z4 0001701

G.T. Mase

GMI Engineering and Management Institute

Flint, Michigan

and

P.L.N. Murthy and C.C. ChamisLewis Research Center

Cleveland, Ohio

Prepared for the

14th Annual Energy'sources Technology Conference and Exhibition_

sponsored by the American Society of Mechanical EngineersHouston, Texas, January 20-24, I991 ......

https://ntrs.nasa.gov/search.jsp?R=19910009923 2018-04-01T07:40:08+00:00Z

PROBABILISTIC MICROMECHANICS AND MACROMECHANICS OF POLYMER MATRIX COMPOSITES

G.T. Mase*

GMI Engineering and Management InstituteFlint, MI 48504-4898

and

P.L.N. Murthyt and C.C. Chamis_NASA Lewis Research Center

Cleveland, Ohio 44135

ABSTRACT

A probabilistic evaluation of an eight-ply graphite/epoxy quasi-isotropiclaminate was completed using the Integrated Composite Analyzer (ICAN) in con-junction with Monte Carlo simulation and Fast Probability Integration (FPI)techniques. Probabilistic input included fiber and matrix properties, fibermisalignment, fiber volume ratio, void volume ratio, ply thickness and plylayup angle. Cumulative distribution functions (CDF's) for select laminateproperties are given. To reduce number of simulations, a Fast ProbabilityIntegration (FPI) technique was used to generate CDF's for the select proper-ties in the absence of fiber misalignment. These CDF's were compared to asecond Monte Carlo simulation done without fiber misalignment effects. It isfound that FPI requires a substantially less number of simulations to obtainthe cumulative distribution functions as opposed to Monte Carlo Simulationtechniques. Furthermore, FPI provides valuable information regarding the sen-sitivities of composite properties to the constituent properties, fiber volumeratio and void volume ratio.

Ecxx

Efll,Ef22 ,Gf12

Em,Gm

F(x) ,pf

f(x)

fx(X)

Gcxy

g

SYMBOLS

composite elastic modulus (Mpsi) about structural axes

fiber elastic moduli about material axes

matrix elastic moduli

cumulative distribution function

probability density function

joint probability density function

composite shear modulus (Mpsi) about structural axes

FPI limit state function

*Assistant Professor, Department of Mechanical Engineering.

Aerospace Research Engineer, Structures Division.Senior Aerospace Scientist, Structures Division.

gl linear approximation of g

g2 incomplete quadratic approximation of g

kf,km,kv fiber volume fraction, matrix volume fraction, void volumefraction

U standardized normal deviate

uniform deviate

Y normal, Weibull or gamma deviate

FPI response function

Weibull distribution parameters

acxx composite thermal expansion coefficient (ppm/°F) about struc-tural axes

r(x) gamma function

gamma distribution parameters

normal distribution parameters

Vfl2,Vm fiber and matrix Poisson's ratios

INTRODUCTION

The properties of the polymer matrix composites display considerable scat-ter because of the variation inherent in the properties of constituent materi-als. Distinct distributions to describe the effects of scatter on compositeproperties facilitate the composite mechanics calculations. For example com-

posite strength is often examined probabilistically by assuming that the plyfailure strength has a specific distribution (usually Weibull) which is thenused in a laminate failure criterion (1). Analysis of this type has the short-coming that different failure mechanisms occurring at a lower level, that is,at the fiber and matrix level are not directly accounted for when the ply fail-ure stress is the primitive random variable.

A better approach to quantify the uncertainties in the behavior of compos-ites would be to account for the variations in the properties starting fromthe constituent (fiber and matrix) level and integrating progressively toarrive at the global or composite level behavior. Typically, these uncertain-ties may occur at the constituent level (fiber and matrix properties), at theply level (fiber volume ratio, void volume ratio, etc.) and the compositelevel (ply angle and lay-up). In this paper, a computational simulation tech-nique is described which accounts for uncertainties at various levels to pre-

dict the behavior of a quasi-isotropic graphite/epoxy (0/45/90-45)s laminate.

MICROMECHANICALANDMACROMECHANICALUNCERTAINTIES

Uncertainties at the Micromechanics Level

To account for uncertainties at all levels of a composite, one has tostart with uncertainties at the fiber and matrix level and use compositemechanics to obtain laminate level response. In the present effort the com-posite mechanics available in ICAN (2) is utilized to obtain the ply/laminatelevel response. At the micromechanics level 29 Parameters (the constituentproperties) are required by ICAN as input (2) (schematic Fig. 1). In additionthree fabrication process variables (Table II) are needed to compute ply prop-erties. For the most part, these properties were considered to be normallydistributed about some mean value. However, the fiber and matrix strengthswere taken to be distributed as a Weibull distribution which is widely accepted

for strength distributions because of its dispersed left tail and sharp righttail which represents experimental data well.

The distribution types and parameters for the fiber and matrix constitu-ent material properties are given in Table I. Using the Monte Carlo simula-tion, these distribution types reproduce histograms (frequency of occurrence)plots as shown in Fig. 2 for fiber longitudinal modulus and in Fig. 3 for

fiber longitudinal strength. It would require testing of 1000 specimens togenerate them experimentally (a rather expensive and time consuming task). Itis worth noting that for the Weibull distribution the mean is not parameter 1nor is the variance parameter 2. In this case the probability density func-tion, mean and variances are given by (3).

where

xB. !

Mean = a-Gr<l + _),

2

Variance-¢_-G[r(1 + _)-r2(1 + _)] (2)

Uncertainties at the Ply Level

The next level of uncertainties enters at the ply level. A typical graph-ite fiber has a nominal diameter of 0.0003 in. which means that a single ply

contains many fibers through the thickness. If an eight-ply graphite/epoxycomposite is considered with a nominal thickness of 0.04 in. each ply will beapproximately 0.005 in. thick. Taking into account an interfiber spacing of0.00005 (for a ply with fiber volume ratio 0.6) in. there are about 15 fibersthrough the thickness of each ply. All of these fibers will have a certainamount of misalignment (random orientation}. To account for this randomnessin probabilistic micromechanics, linear laminate theory is used where each plyis broken down (substructured) into 15 subplies (2). Each of these subplies

J

z

was assumed to be normally distributed about the fiber direction with fiberorientations lying within ±5 ° of the O°-ply direction. The properties of theconstituents are assumed to be the same in the subplies within each ply. Thefiber volume and ply thickness were represented as normally distributed whilethe void volume was represented as a gamma distribution. A gamma distributionwas the proper choice for the void volume ratio because there is no probability

for zero void volume and a bias towards higher void volumes.

As was the case for the Weibull distribution, the parameters given forthe gamma distribution do not directly represent the mean and variance of thedistribution. The probability density function, mean and variance for thegamma function are given by (3).

7,_: -),y k-1f(y) = r--_--_e Y (3)

k __k___Mean = _-, VarianceA k 2

(4)

The ply level distribution parameters are given in Table II.

Uncertainties at the Laminate Level

The uncertainty considered at the laminate level was that of ply orien-

tation and thickness. Each ply in the (0/45/90/-45)s was given a normaldistribution with a 3.33 ° standard deviation about the deterministic angle(Table III).

MONTE CARLO SIMULATION

Given the distributions of Tables I, II and III for fiber and matrix

properties, ply and laminate inputs, the uncertainties in the compositeproperties need to be quantified with appropriate cumulative distributionfunctions (CDF's). One approach to achieve this is to use a Monte Carlo Simu-lation technique. The first step in this process involves running ICAN withrandomly selected input variables from the predetermined probability distribu-tion functions many times. The output comprising of the composite propertiesis saved. The second step consists of processing the various property outputdata to compute the desired CDF. An obvious disadvantage of such an approachis the enormous number of output sets that must be obtained to get reasonableaccuracy in the output CDF's.

Generation of Normal Distributions

In order to generate the input distributions, a uniform deviate (randomnumber between 0 and 1) must first be generated. Rather than use a machineroutine to generate the random number, a portable (machine independent) uni-form deviate routine from Press et al. (4) was used which was based on a three

linear congruential generator method. This routine also had the advantage ofbeing able to reinitialize the random sequence.

4

The uniform deviate was used to generate a normal deviate by the Box-Muller method (4). With the aid of the normal distribution f(y) given by

2_Y._

1 2f(y)dy = --e dy

4"2-_

and the transformation between uniform deviates Xl,X 2 and the variablesYl,Y2 given by

Yl = _ In x I cos 2_x 2, Y2 = 4-2-2" In x I sin 2_x 2,

the inverse transformation can be written as

2 2Yl÷Y2

2 1 Y2

xI _ e , x2 = _-_ arctan Yl

The Jacobian of this transformation is

r Fwhich shows that each y is distributed normally. This shows that Eq. (6)leads to an explicit formula for calculating a normal deviate.

(s)

(6)

(7)

(8)

Generation of Weibull Distribution

To generate a Weibull distribution from a uniform deviate one can inte-

grate the probability density function and then solve for the Weibull deviate.This gives

1

y = B[-In(l - x)] a (9)

as a point from the Weibull distribution where x is a uniform deviate.

Generation of Gamma Distribution

To generate a deviate from a gamma distribution, a uniform deviate, x,was taken and then the zero of the function

Xkw(y) =Zr-_e-Xyyk-ldx - x

o

(io)

was found. This was numerically inefficient because it involved numerical

integration and root finding by the bisection method, but the program ran withsufficient speed to overlook this fact.

The program ICAN was modified so that the properties shown in Tables I,II and III were given a value from their respective distributions. Outputfor the layer and composite properties were saved for 200 samples. While200 samples is probably not enough to converge to the actual CDF, the resultsdo show a good qualitative trend. The cumulative distribution functions were

constructed from these samples for three typical composite properties. Theselected composite properties are the composite longitudinal modulus Ecx x,the composite compressive strength Scxxc, and the composite thermal expansioncoefficient _cxx. These CDF's are shown in Figs. 4 to 6. By its symmetry,the CDF of Ecx x appears to be normally distributed while Scxxc exhibits aWeibull shape (5).

FPl SIMULATION

An alternative approach to obtain the required cumulative distribution

functions is to use Fast Probability Integration (FPI) program (6). FPI helpsgenerate the required CDF's quicker with reasonable accuracy and a lot lessnumber of sample output data. Also, it generates more information than whatcan be expected from a Monte Carlo simulation. The additional information

that FPI offers is the output variable sensitivity information based on theprobabilistic inputs.

A brief overview of FPI is given below. The reader is advised to referto (6) for a detailed discussion.

Consider a response function

Z(X) = Z(XI,X 2 ..... Xn)

where Xl,...,X n are random variables. Also, define the function

(11)

g=Z(X)- Zo--O (12)

as the limit state with Zo a real value of Z(X). The CDF of Z at Zo isequal to the probability that [g _ 0], If the probability of a desired out-put, pf, is defined by

pf = P[g<O] (13)

an exact solution of pf can be obtained from

pf = ..._f (X)dX_ (14)

6

where fx(_) is the joint probability density function and _ is the regiondefined _y [g £ 0].

The evaluation of the preceding integral is often intractable and this

leads to the need for an approximate method of evaluation pf. In doing this,FPI approximates the function g using a Taylor's series expansion as a linear

n

:ao*E i(oi ui)i=1

(15)

or incomplete quadratic

n n

g2(u) = ao +_ai(u i

i=1

. , . 2

-uil ÷_bi(ui- ui)

i:1

(16)

function where u i is the most probable point (6) of the random variable u i.

Note that the random variables X have been replaced by standardized normalvariables u. The coefficients _f these expansions are obtained numericallyand then the probability Pig(O] is computed.

Because of the approximate form of the g-function, FPI requires at leastn+l or 2n+1 data sets to evaluate the linear or quadratic g-function coeffi-cients a , a i, and b i from which the probability is found. In the presenteffort only the ply level variations in the properties (29), fiber volume ratioand void volume ratio are considered as random variables. This means that at

least 32 (29 constituent properties, fiber volume ratio, void volume ratio +1)ICAN runs are needed for the linear approximation and 63 for the quadratic

approximation. A typical data set to FPI consisted of ICAN run with one per-turbed independent variable while all others remaining at mean value. For thelinear case, the variable was perturbed one standard deviation from its meanvalue. In the quadratic case, the independent variables were perturbed twice,one standard deviation each, on both sides of the mean value.

Three typical composite properties Ecx x, Gcxy, and _cxx were chosen asthe output variables for the study. Since the goal was to have a minimumnumber of ICAN runs, the CBF of Ecx x was calculated using 32 data sets withlinear FPI analysis and 125 data sets with quadratic analysis. With these twocases, the CDF's computed by FPI lay on top of each other indicating that 32data sets will give a good approximation for the CDF of Ecx x.

To identify the computational savings that FPI has over a Monte Carlo sim-

ulation, the CDF's for Ecx x, Gcx v, and _cxx were compared for a 32 sampleFPI case with a 31 and 90 sample _lonte Carlo simulation (Figs. 7 to 9). Itappears that the Monte Carlo simulation is converging to the FPI simulation,but the FPI simulation only needed 32 samples.

7

FPI Sensitivity Output

As was previously mentioned, one advantage of FPI is the sensitivity datathat is produces. Before the actual sensitivity numbers are given for the com-

posite modulus Ecx x, it will be helpful to examine how this quantity is calcu-lated by ICAN.

The modulus Ecx x is the (1,1) entry of the matrix (2)

[Ec] 1-- (Zli+l - Zli)([R 1] T[E1][al])i (17)

where t c is the thickness of the composite, Zli is the distance from thebottom of the composite to the ply, [R 1] is rotation matrix which is a func-tion of the ply angle, [El] is the matrix of the layer elastic constants, dis-tortional energy coefficient. To calculate the ply elastic moduli matrix,[El], the components are calculated from primitive variables Efl 1, Ef2 2, Em,Gfl 2, Gf2 3, Gm, kf, k m, uf12, and _m (2}. So in the case of no ply substruc-turing or ply angle variation, the composite modulus should be a function ofonly these 10 primitive variables. It is noted that k m is calculated by

k m = 1 - kf - k v (18)

and is not listed as a primitive variable. Thus kv will be used instead ofkm in the sensitivity analysis.

For the input random variables given previously, FPI calculated a meanEcx x of p = 5.744 Mpsi with a standard deviation of a = 0.363 Mpsi. Thesensitivities at ±0.3a are given in Table IV. As would be expected, the most

sensitive primitive variables are the fiber modulus Efl I and the fiber volumeratio kf. Primitive variable Gf23 has a zero sensitivity which is consistentwith the definition of Ecx x given in matrix Eq. (17).

CONCLUSIONS

A probabilistic evaluation of an eight-ply quasi-isotropic graphite/epoxy[0/45/90/-45] s laminate was completed using two approaches. The first approachwas to use a Monte Carlo simulation technique. The second approach was to use

fast probability integration technique (FPI). Probabilistic inputs for thisstudy included constituent micromechanical properties, fiber misalignmentwithin a ply, fiber volume fraction, void volume percent and ply angle mis-

alignment for the laminate.

It was demonstrated that the use of the FPI program can greatly reducethe computations needed to generate composite CDF's. FPI was demonstrated by

generating CDF's for Ecx x, Gcx v and acxx for a graphite/epoxy[0/45/90/-45] s composite in theVabsence of fiber misalignment.

8

The results of this investigation indicate that an integrated programcombining IC/kNand FPI is feasible. Such an integrated program offers thepotential for a computational efficient probabilistic composite mechanics

methodology.

REFERENCES

I. Duva, J.M., Lang, E.J., Mirzadeh, F., and Herakovich, C.T., 1990, "A Proba-bilistic Perspective on the Failure of Composite Laminae," presented atthe XI U.S. National Congress of Applied Mechanics, Tucson, AZ, May.

2. Murthy, P.L.N. and Chamis, C.C., 1986, "Integrated Composite Analyzer

(ICAN), Users and Programmers Manual," NASA TP-2515.

3. Mood, A.M., Graybill, F.A., and Boes, D.C., 1974, Introduction to theTheory of Statistics, 3rd Edition, McGraw-Hill, New York, NY.

4. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., 1986,Numerical Recipes, Cambridge University Press, Cambridge, U.K.

5. Stock, T.A., 1987, Probabilistic Fiber Composite Micromechanics, MS The-

sis, Cleveland State University, Cleveland, OH.

6. Probabilistic Structural Analysis Methods (PSAM) for Select Space Propul-

sion Systems Components, NESSUS/FPI Theoretical Manual, 1989, SouthwestResearch Institute, NASA Contract NAS3-24389, Dec.

TABLE I. - CONSTITUENT INPUT DISTRIBUTION PARAMETERS FOR ICAN

Ef11Ef22Gfl2Gf23

Vfl2vf23afllaf22PfNfdfCfKf11Kf22Kf33SfTSfCEmCm_mo_mPmcmKmSmT

SmCSmS[3m

Dm

Distribution

Units

MpsiMpsiMpsiMpsiin./in.in./in.

ppm/°Fppm/°FIb/in. 3

in.BTU/lb

(a)(a)(a)

ksiksi

MpsiMpsiin./in.

ppm/°Flb/in. 3BTU/lb

(a)ksiksiksiin./in. 1%

moisturein.2/sec

Type

Normal

FixedNormal

ii

1WeibullWeibullNormal

(b)Normal

weibull

WeibullWeibullNormal

Normal

Parameter 1

p = 31.0p= 2.0p - 2.0p - 1.0p -- 0.20p = 0.25la =0.2p = 0.2p = 0.063p = 10 000p = 0.003p = 0.20p = 580p = 58p = 5813 = 400

[3 = 400

p = 0.500

p = 0.35p = 36p = 0.0443p = 0.25 "p = 1.25p = 15p - 35p = 13p = 0.004

p = 0.002

Parameter 2

¢= 1.5a = 0.10¢ = 0.10cr = 0.05cr = 0.01

,¢

a = 0.003¢ = 0<r = 0.00015o = 0.01cr = 2.9¢=2.9a=2.9cr = 40

¢ = 40o = 0.025

¢ = 0.035a = 4a = 0.0022

_ 0.0125o = 0.06

a = 5= 20= 7

o = 0.0002

o = 0.0001

aE BTU • in./hr/ft2/°F.

bGm is calculated using Em and _m, and isotropy.

10

TABLEI I. - PLY INPUTDISTRIBUTION PARAMETERS FOR ICAN

Units

kf Percentkv PercentOf Degrees

Distribution

type

NormalGammaNormal

Parameter 1

= 60),=2

]_=0

Parameter 2

<7=3k=6

= 3.33

TABLE III.- LAMINATE INPUT DISTRIBUTION PARAMETERS FOR ICAN

Units

e 1 Degreest 1 Inches

Distribution

type

NormalNormal

Parameter 1

p=Op= t o

Parameter 2

= 3.33

a = O.O5t o

TABLE IV. - NONZERO SENSITIVITY

PARAMETERS FOR Ecx x FROM

FPI AT ±0.3a AWAY FROM

MEAN OF p = 5.744 MPSI

Primitivevariable

kfEfllEf22Gf12GmEmGf23vf12_'f12_m

kv

Sensitivityparameter

O. 778.624.260.130•060•036.0

11

COMPONENT

/ _k__ STRUCTURAL j/_# t;7-_:_'_///f-J "_"_._ ANALYSIS..,...t' /;'77.1//h \

I i LAMI INATE J II i THEORY _ " THEORY T I

\- _ _ ""_T /UPWARD_'_ CONSTITUENTS \ M i /

INTEG?RATED _.. . ,,.,,, _ f TROAP'cDEODWcNR"SYNTHESIS.... DECOMPOSITION"

Figure 1.- Integrated composite micro- and macromechanicsanalysis embedded in the computer code ICAN.

400

¢_ 300e.1E

o 200

E

z 100

0

IO

3_ 342

111111

F/11/I

N_4444

137 7_

25.75 27.25 28.75 30.25 31.75 34.75 36.25

12

138

33.25

28o

Fiber Modulus, Mpsi

Figure 2. - Monte Carlo simulation (1000 samples) offiber longitudinal modulus from a normal distribution.

12

E03

0t,=,

E

Z

400

300

200

100

0

349

280

189

m 75 61

v///l_33 _4 9 0IF'7")-)')_

290 310 320 350 370 390 410 430 450

Fiber Tensile strength, ksi

Figure 3.- Monte Carlo simulation (1000 samples) offiber longitudinal strength from a Weibull distribution.

¢-0

.m

t-

t-O

t-i.m

i...

.on

(3.}._>

E

L)

1.0

.8

.6

.4

.2

0

B

D

1 ! I I2 4 6 8 10 12 14

Young's modulus Ecx x,Mpsi

Figure 4. - Cumulative distribution function for composite

([o/45/90/-45]s) modulus Ecx x from Monte Carlo simu-lation (200 samples) which includes ply substructuringeffects.

13

¢-O

oN

c-

t-O

°w

,m

._>

E

1.0

.8

.6

.4

.2

m

m

_o/I I I I I40 60 80 110 120 140

Compressive strength Scxxc,ksi

Figure 5. - Cumulative distribution function for composite([o/45/90/-45]s) modulus Scxx from Monte Carlo simulation(200 samples_ which includes ply substructuring effects.

14.

CEO

.i.o..w

¢-

c-O

t'_oD

b,-

(/3.u

°w

-1E-I(.3

1.0

.8

.6

! I 1 I2 4 6 8xl 0-6

Thermal expansion coefficient (Zcxxc , ppm/°F

Figure 6. - Cumulative distribution function for composite([o/45/90/-45]s) thermal expansion coefficient _ fromMonte Carlo simulation (200 samples) which includes plysubstructuring effects.

15

t-O

.m

t-

O.i

(3.>e3

E

1.0

.8

.6

.4

.2 0[]

Number- of.

samples

Monte Carlo 31Monte Carlo 90FPI linear 32

1 I I5.0 5.5 6.0 6.5

Young's modules Ecx x , Mpsi


([o/45/90/-45]s) modulus Ecx x simulation with MonteCarlo (no ply substructuring) and Fast ProbabilityIntegration (FPI).

16

1.0

t-O

(.-

_1.-=

t-O

°--

E

o

.8

.6

.4

0.2 []

Numberof

_amples


o I [ I I2.2 2.4 2.6 2.8 3.0 3.2 3.4

Composite Shear modulas Gcxy, Mpsi


([o/45/90/-45]s) Shear modulus GcxySimulation withMonte Carlo (no ply substructuring) and Fast ProbabilityIntegration (FPI).

17

t-O

t-

t-O

.i

t,_

i

E

1.0

.8

.6

.4

.2

0

0[]

Numberof

samples


l I I0.8 1.2 1.6 2.0 2.4

Thermal expansion coefficient _cxy, ppm/°F

Figure 9. - Cumulative distribution functions for the com-

posite ([o/45/90/-45]s) thermal expansion coefficientc_cxxsimulated with Monte Carlo (no ply substructuring) andFast Probability Integration (FPI).

18

National AeronauticsandSpace Administration

1. Report No.

NASA TM-103669

Report Documentation Page

2. Government Accession No.

4. Title and Subtitle

Probabilistic Micromechanics and Macromechanics

of Polymer Matrix Composites

7, Author(s)

G.T. Vase, P.L.N. Murthy, and C.C. Chamis

3. Recipient's Catalog No.

9. Performing Organization Name and Address

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio 44135-3191

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

5. Report Date

6. Performing Organization Code

8. Performing Organization Report No,

E-5876

10. Work Unit No.

510-01-0A

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

Invited paper prepared for the 14th Annual Energy-sources Technology Conference and Exhibition sponsored by

the American Society of Mechanical Engineers, Houston, Texas, January 20-24, 1991. G.T. Vase, GMI

Engineering and Management Institute, Flint, Michigan 48504; P.L.N. Murthy and C.C. Chamis, NASA LewisResearch Center.

16. Abstract

A probabilistic evaluation of an eight-ply graphite/epoxy quasi-isotropic laminate was completed using the

Integrated Composite Analyzer (ICAN) in conjunction with Monte Carlo simulation and Fast Probability

Integration (FPI) techniques. Probabilistic input included fiber and matrix properties, fiber misalignment, fiber

volume ratio, void volume ratio, ply thickness and ply layup angle. Cumulative distribution functions (CDFs) for

select laminate properties are given. To reduce number of simulations, a Fast Probability Integration (FPI)

technique was used to generate CDFs for the select properties in the absence of fiber misalignment. These CDFswere compared to a second Monte Carlo simulation done without fiber misalignment effects. It is found that FPI

requires a substantially less number of simulations to obtain the cumulative distribution functions as opposed to

Monte Carlo Simulation techniques. Furthermore, FPI provides valuable information regarding the sensitivities of

composite properties to the constituent properties, fiber volume ratio and void volume ratio.

17. Key Words (Suggested by Author(s))

Composites; Probabilistic mechanics; Micromechanics;

Macromechanics; Composite mechanics; Laminate theory;

Fast probability integrator; Monte Carlo simulation;

Sensitivities; Properties

18. Distribution Statement

Unclassified - Unlimited

Subject Category 24

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages

Unclassified Unclassified 19

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