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Composites Part B 28B (1997) 167-175 © 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

Computational aspects of modeling complex microstructure composites using GMC Carlos E. Orozco Department of Civil Engineering and Applied Mechanics, University of Virginia, Charlottesville. VA 22903, USA (Received 15 June 1996; revised 12 July 1996) A computationally efficient implementation of the generalized method of cells (GMC) is presented and tested in connection with problems that require large numbers of subcells. The new implementation is found to be substantially faster in predicting the elastic properties of unidirectional composites. Furthermore, the new implementation makes it possible to solve problems with complex microstructures and a degree of refinementthat was not possible before. Cell discretizationswith up to 10,000 subceUswere employed in the numerical experiments. The modeling capabilities of GMC and the new implementation are tested in the context of predicting the effectiveelastic behavior of composites with randomly distributed fiber inclusions. It is shown that GMC accurately predicts the asymptotic behavior of the engineering properties of the composites studied. This is demonstrated by examining the limiting values of the effective engineering properties of the composites studied when the number of subcells is large. © 1997 Elsevier Science Ltd. All rights reserved (Keywords: generalized method of cells; complex microstructure composites)

1 INTRODUCTION Micromechanical models have been used extensively since the 1960s to predict the macroscopic behavior and the effective properties of advanced composites. An extensive account of the characteristics and capabilities of these models can be found in ref. 1. Among the many micromechanical models, the generalized method of cells (GMC) 2, has emerged as an attractive tool to predict the elastic, inelastic, and thermoelastic behavior of a wide variety of composites. One of the most attractive features of GMC is its capacity to produce accurate macroscopic stress-strain responses using a relatively small number of subcells; therefore requiring very little computational effort. As a matter of fact, it has been shown by Wilt 3 that the stress-strain response of composite microstructures with circular fibers can be accurately modeled with a 7 x 7 subcell array. To produce equivalent results using the finite element method, 1088 constant strain finite elements are required 3. The fact that GMC is capable of modeling relatively complex microstructures using a small number of subcells has lead exclusively to dense implementations of the method. With current dense implementations of GMC, it is practical to solve problems with only about 500 subcells. There is, however, increased interest in modeling composites with substantially more complex microstructures which usually require large

numbers of subcells. A class of problems that would generally require fine discretizations of the cell domain is related to the study of damage. Problems such as modeling of progressive damage due to interfacial debonding, or modeling of existing damage due to diametral cracks and their evolution4 may require very fine discretizations. Also, modeling of disperse damage requires consideration of multiple circular fibers within a repeating cell. A second class of problems requiring finer discretizations is related to tailoring the microstructure of the composite to achieve a certain behavior. Among these problems, one can cite the modeling of multiphase composites with fibers of different sizes, different properties, and possibly different shapes. These problems also include the modeling of microstructures with different levels of refinement as well as the modeling of composites with nonuniform fiber spacing, such as composites with macroscopically periodic but locally functionally graded microstructures. Many of these problems require accurate modeling of fiber shape, which in turn requires on the order of thousands of subcells. It is well known that fiber shape and fiber architecture play an important role in predicting the macroscopic behavior of composites, especially with regard to plastic and viscoplastic behavior4'5. This observation opens the possibility of using optimal design techniques to tailor

167

Modeling complex microstructure using GMC." C. E. Orozco X

2

~=N/~

hBI

h

=

B7

/~=1 t_ l-

7=1 7 = 2

I. Figure 1

_i -l

17

1

7=N7

x 3

1. Within each subcell, the gradient of the displacement vector is constant and equal to its value at the centroid of the subcell, i.e. Vu(a'Y) = vu(a'/)[c

-I

Repeating unit cell with subcells and nomenclature

the microstructure of fiber reinforced composites to achieve desired macroscopic properties and/or behavior. One of the main difficulties of applying optimization techniques to design problems in engineering is the computational cost involved. At the core of the computations is usually the calculation of objective functions and constraints that require solutions of the physical problem multiple times. It is therefore necessary to develop computationally efficient algorithms for the analysis portion of the problem to render the design portion tractable. In modeling advanced composites at the micromechanical level the process of finding the effective properties of the composite and/or its macroscopic behavior constitutes the analysis problem. When using GMC, the analysis problem becomes computationally challenging when a large number of subcells is required. As a first step towards the general goal of creating a complex microstructure modeling and tailoring capability for advanced composites, this paper discusses some of the computational aspects of GMC and explores the capabilities of the method to model microstructures with a large number of subcells. Specifically, it presents a sparse implementation of GMC that enables the calculation of effective elastic properties of complex microstructure composites in a fraction of the time required by previous implementations. The new implementation makes it possible to use GMC in connection with problems with a degree of refinement that was not possible before. In this paper, the modeling capabilities of the new GMC implementation are illustrated by studying the asymptotic behavior of composites with randomly distributed fiber inclusions.

2 THE GENERALIZED METHOD OF CELLS The following is a brief summary of the main concepts associated with the generalized method of cells. A more detailed presentation of the method can be found in ref. 2. Figure I shows a representative volume element (RVE)

168

of a unidirectional composite with its fibers directed in the x I direction. This element is assumed to repeat itself in the x2 and x3 directions. In the context of the generalized method of cells this RVE is also identified with a cell. Each cell is divided into N a x N~ subcells, each of which can have different material properties. The dimensions of the cell in the x3 and x2 directions are I and h, and the corresponding dimensions of the subcells are ha and lv (see Figure 1). Mathematically, the generalized method of cells can be conceived to be based on the following assumptions:

(1)

where u - {ul, u2, u3} T is the displacement vector. 2. The entire cell can be mapped into a single point belonging to a homogeneous deformation field6 with displacement w and displacement gradient Vw. 3. The displacement gradient Vw is chosen such that (see Figure 1): OU 1 (/3'7)__ OWl

OXl

....

V/3,7

OX 1

(2)

Ow, h Uilx2=h = Uilx2=O+ Oxz

(3)

Owi l Uilx3=t = Ui[x3=O+ Ox3

(4)

Now, for assumption 1, Uilx~=nand uilx3=t can also be expressed in terms of the subcell gradients as: N ~ OUi (aT)

U, lx2=h = u, lx2=0 +

ha

(5)

N't Oui (a,7)

Uilx3=l = Uilx3=O+ "-7-~-~X3

17

(6)

From equations (3) and (5), it can be inferred that:

u,

Oui (a'Y)

~ ha-~x2x2

Owi = Oxzh, .. .'7 = 1,N 7

(7)

and from equations (4) and (6):

EN~ I ~ Oui (a'y) OWl Ox3 = Ox3 l,.../3 = I, N a "7

(8)

Equations (7) and (8) implicitly satisfy the displacement continuity conditions at all subcell interfaces. These equations can also be obtained by directly applying these displacement continuity conditions 2.

2.1 Effective average strains The effective average strains in the composite are defined in the usual way 1'2, i.e. eij = ~ L...a Z

-r

a

I, , (13"/) ,,atTe#

(9)

Modeling complex microstructure using GMC: C. E. Orozco or,

where the subcells strains are:

l (Oui (~7) Ouj(/37)~ ~7) --'2 \Oxj +~xi )

Owl

gii

Similarly, from equation (8), N~ (/37)-

g33l =

( O U i (f17)

2

(24)

N~

(11)

0U2(/3,.,/)\ q-~"~X1 )hfll, y

~_. E33 17, . . . fl = 1, N/3 7

(23)

and, V ' =(/3~)t'7, gl3l =/__.~13

-- OX 1

Also, from equations (9) and (10), one obtains for

1 Nv Nfl 1 El2 = ~ [ ~ 7 ~ O X

gi2h

(10)

Using equations (7), (8), (9) and (10), it is possible to show that the effective average strains can be written in terms o f the displacement w, as follows: F r o m definitions (9) and (10), and taking into account (2):

U~ ~. "'" 7 = 1, N 7 = \--" / ' ~ 1~(/3"D 2 "~, /3

" " /3 = 1, N~

(25)

7 gi2:

(12)

Now, direct application o f the definition of average strain to g23, yields: Nfl N7

g23hl =

~

Z

¢23-(f17)t'tn/3t7

(26)

fl 7

or, 1

1

OU i (/3",/)

Equations (20), (21) and (23) to (26), can be written in matrix form as: A G a s = J~; (27)

0U2 (/37)"\

and using equation (7), this becomes:

1,~--~,1(Owl

g12 =

7

7 ~

OW[~l

tOX2 -t- OXl)

(14)

7

or,

l(Owi Ow2"~ ~i2 = ~

(15)

~ OX2 + OXi,]

2.2 Traction continuity conditions

In a similar manner, it can be shown that:

1 /OWl Ow3"k gl3 = ~ tO~-x3+ ~lxl), Ow2 g22 = Ox2 ,

1 ['OW2 ~23 = ~ t ~ X 3

(16) (17)

OW3"~ -1-~2X2),

(18)

and,

OW3 OX3

g33 =

(19)

Equations (7), (8), (9) and (10), can now be used to construct analogous expressions to equations (7) and (8), but in terms o f strains, as follows: F r o m equations (2) and (11): /3,`/ , • .. ~ll ~ Eli

(20)

U~ = / ' ~ ~22 n/3, . . . "y = 1, N 7

(21)

/3

0-(n7) 33 = ,,(/3.7+ v33 1), . . . /3 = I,N~, 7 = 1 , N . ~ - 1

(29)

a(fl7) _(/3+J,'y) :s = t~23 , . . . /3 = 1 , N / 3 - 1, 7 = 1 , N 7

(30)

0-(~7) (/3,7+1), . . . 32 = 0"32

/3 = N/3, 7 = 1 , N 7 - 1

(31)

/3 = 1, Nfl - 1, 7 = 1, N 7

(32)

-(fl7) .(/3,7+1) 31 = a31 , . . . /3 = 1,N/3, 7 = 1 , N 7 - 1

(33)

O.(fl~t) 21

=

,,.(fl+ 1,,,/) v21

,...

AMa s = 0

(34)

where use has been made of the elasticity relationships*:

F r o m equations (7) and (15): 1 [U,~

Assuming that the effective average strains are specified, equation (27) constitutes a system of 2(N/3+ N7) + N/3N.~ + 1 equations containing the 6N/3N7 u n k n o w n subcell strains. The remaining 5NnNT2(N/3 + N7) - 1 equations are provided by the traction continuity conditions at the subcell interfaces 2. These traction continuity conditions are (see Figure 1): 0-(/37) .(/3+1,7), ... /3 = 1, N/3 - 1, 3' = 1, N 7 22 = 0"22 (28)

The foregoing traction continuity conditions can be written in terms of subcell strains as:

F r o m equations (7) and (17):

g22h

where as --- {ail,ai2,...,aN~N'} T contains the subcell strains, and g: = {gll, g22, g33, 2g23, 2g13,2g12, }T contains the effective average strains in the composite. The matrices Ao and J contain information about the subcells and cell geometry, respectively. Their entries are h~s, l~s hfl~ products; hs, Is, hl products; and ones and zeros.

(gUl (/37)

~r (/37) = C(~'y)e(~)

+

&

0U2 (/37)~

)

(35)

where C (/37) represents the elastic stiffness matrix of (22) * Valid in the absence of thermal or plasticity effects

169

Modeling complex microstructure using GMC." C. E. Orozco subcell (fiT), and e(~'r) = {~H, 1122,£33,2~23,21~13,2e12} r. The matrix AM then contains entries of the individual elastic stiffness matrices of all subcells, Equations (27) and (34) can be combined into a single equation as: A e s -- Kg (36) where,

ae+lO le+09

........

I

........

I

........

I

.?.o.NoNz :, osxN,.,

........

11

/I 1

le+08 le+07

A=

AM ] AG

(37)

and,

,[0]

(38

le+06 le+05 Ie+04

Equation (36) constitutes a system of 6N~N7 equations with 6N/~N~ unknowns (assuming the effective average strains are known). The solution of this system of equations represents the core of the computations required by GMC. Premultiplication of (34) by A -1 yields: ~s = As~

(39)

where As = A-1K is a matrix of so-called concentration factors 7. Entries in As can be partitioned into NaN~ (6 x 6)-submatrices, each relating the effective strains in the composite to the subcell strains. Let each of these (6 x 6) matrices be denoted A (~). Then an analogous equation to equation (39) for each subcell is: ~(~) = A(B~)~;

(40)

Premultiplication of equation (40) by C (~) yields: tr (~n) = C(~n)A~7)~:

le+03 ........

Ie+02

I / 10

100

1000 10000 NUMBEROF SUBCELLS

Figure 2 Sparsity of A for increasingnumber of subcells Table 1 Sparsity of A for increasing number of subcells Subcell arrangement

No. of subcells

Order of A

Entries in A

No. of nonzeros

4× 10 × 20 x 30 x 60 x 100 x

16 100 400 900 3,600 10,000

96 600 2,400 5,400 21,600 60,000

9.22 3.6 5.76 2.92 4.67 3.6

318 2,238 9,278 21,118 85,438 238,398

4 10 20 30 60 100

× × × × x ×

103 105 106 107 108 109

(41)

and application of a definition for average stresses analogous to (9), results in:

a 6N~N~ x 6N~N7 sparse matrix with very few nonzeros. The number of entries in A is given by: Entries in A -- 36(N~NT) 2

6" - ~ z_~ E h~/TC(~n)A~;~n)~ 7

(45)

(42)

and the number of nonzero entries can be approximately calculated as:

which suggests the following definition for the effective average stiffness matrix: 1U~ U~ e = ~ Z E h~ITC(~'Y)A~) (43) n Once the effective stiffness matrix (~ is obtained, it can be inverted to obtain the compliance matrix S. The effective engineering properties of the composite are then calculated as8: E l l = 1/Sll , '/312 = - 3 1 2 / S l l , E22 = 1/S22 ,

Nonzeros = 19.2513 × (No. of subcells) 1026345 (46)

'023 = - S 2 3 / S 2 2 , E 3 3

= 1/$33, G23 =

1/$44, GI3

=

1/~¢55,

and G12 = 1/$66. 3 SPARSITY CHARACTERISTICS Calculation of the effective stiffness matrix of the composite entails the estimation of the concentration matrix As, as: A s = A-1K (44) This equation requires factoring A once, and solving for six right-hand sides (K is of order 6NAN.r × 6). Now, A is

170

Figure 2 shows a plot of these two numbers against the number of subcells. The coefficients in equation (46) were found using linear regression. Data used in Figure 2 are also presented in Table 1. Current implementations of G M C are dense. This means that they do not take advantage of the nonzero structure of A. In a dense implementation, A is stored in its entirety, including all the zeros, and then factored without consideration of its sparsity. In a sparse implementation, the nonzero structure of the coefficient matrix is determined and only the nonzeros are stored and used in the factorization process. Once the coefficient matrix is factored, a series of backward and forward passes with the appropriate right-hand sides is performed to solve the system. In the present study, the fortran factoriz~ttion package ma28.f of the Harwell Subroutine Library w was used to solve the linear systems. t Available from netlib at http://www.netlib.org/index.html

Modeling complex microstructure using GMC: C. E. Orozco ma28.f is really a collection of fortran routines to solve sparse linear systems using Gaussian elimination. Sparse linear solvers require the specification of the exact positions of the nonzero entries of the coefficient matrix. In the case of ma28.f, this information is furnished by means of two arrays containing the row and column indices of the nonzero entries. The actual nonzero entries of the coefficient matrix (in this case, A), and the right-hand sides (in this case, the columns of K) are stored in one-dimensional arrays. The information about the location of the nonzero entries in A is dictated by the displacement and traction continuity conditions described by equations (20)-(26) and (28)-(33). For instance, the rows of K that correspond to equation (25) will have an l in the fifth column (corresponding to the fifth strain, i.e. gl3) and the rows of A that correspond to equation (25) will have the length I~ of subcell 37 in column (icell - 1) * 6 + 5, where icell is the index of subcell/3"7, i.e. icell -----(/3 - 1)N.~ + 7 (47) This index corresponds to a subcell numbering scheme that starts from left to right and from bottom to top (see Figure 1). The positions of the nonzeros corresponding to all other equations are determined in a similar manner. From Figure 2 and Table 1, it is apparent that to solve problems with a large number of subcells, it is imperative to take advantage of the sparsity of A. In fact, a dense implementation of GMC makes it impossible to solve large problems on an average workstation. A problem with 3,600 subcells for instance, would take approximately 3.8 Gbytes of storage for A alone (using double precision). To store the same matrix sparsely requires only 0.68 MBytes. ~ 4 NUMERICAL RESULTS Two types of numerical experiments were performed in the present study. First, the performance of the sparse implementation of GMC was compared to that of a dense implementation. This was done for problems with up to 576 subcells (24 x 24). This was the largest problem that could be solved on a SUN Sparc 20 workstation using the dense implementation. A second set of problems was solved using the sparse implementation on an SGI Indigo2 workstation. The objective of this second numerical experiment was to evaluate the modeling capabilities of GMC when the cell domain tends asymptotically to a transversely isotropic material. Problems with up to 10,000 subcells were solved in this second numerical experiment.

4.1 Sparse and dense implementations To get a concrete idea of the computational savings that are obtained by exploiting the sparsity of A, the In practice, additional space m u s t be allocated to allow for during factorization

fill-in

210.0

I

I

I

I

I

I

I

I

I

'~--I ,~ t----4,

I

I

1

o 200.0 "'~ 190.0

180.0 170.0 160,0 150.0 140.0 130,0 120.0 110,0 100,0

90,0 i 80,0 70.0

•

DENSE IMPLEMENTATION

50,0 40,0 30,0

20.0 10,0

o,o !---~-~s~-"q--~ 0

50

100

150

200

250

300

350

400 450 500 550 600 NUMBER OF SUBCELLS

Figure 3 Performance comparison between sparse and dense implementations of G M C

Table 2 Performance comparison between sparse and dense implementations of G M C Subcell arrangement

No. of subcells

C P U (s) Dense

C P U (s) Sparse

Speed up

4 × 5× 6 × 7 x 8x 9 x 10 x 12 x 14 x 16 × 18 x 20 x 22 × 24 × 100 ×

16 25 36 49 64 81 100 144 196 256 324 400 484 576 10,000

0.08 0.17 0.29 0.44 0.75 1.12 2.21 3.69 6.87 12.12 19.31 29.67 60.36 200.0 -

0.09 0.10 0.14 0.18 0.18 0.28 0.37 0.58 0.89 1.44 2.26 3.12 4.21 5.82 1,563

0.89 1.70 2.07 2.44 4.17 4.00 5.97 6.36 7.72 8.42 8.54 9.51 14.34 34.36 -

4 5 6 7 8 9 10 12 14 16 18 20 22 24 100

performance of the sparse implementation of GMC was compared to that of a dense implementation. This was done for problems with increasing number of subcells. Performance of the two implementations was measured in CPU time. As mentioned in Section 3, the sparse implementation makes use of the fortran factorization package ma28.f of the Harwell Subrouting Library. Results are plotted in Figure 3 and presented in Table 2. Problems with 15 different subcell arrangements were solved using the two implementations, and the CPU times recorded. Figure 4 illustrates some of the subcell arrangements utilized in these numerical experiments. CPU times are for a SUN Sparc 20 workstation with 64 MBytes of RAM and a 1-GByte disc. Notice that the CPU time for the dense implementation grows quadratically with the number of subcells (see Figure 3). This observation is consistent with the fact that the number of entries in A grows quadratically with the number of subcells [see equation (45)]. CPU time for the sparse

171

Modeling complex microstructure using GMC: C. E. Orozco

.,

,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

•"L-"~:';;:i=3"".g=~:L:. : "':?i;;.': .:';ii:=..'~.:~V~~.:.:::-.:':_':?.2.~::'~.!;...~i.7i "=*.I.

• = I , . " " " " o . = I * ' , , = * ~ ..,%=. %" -=" -=" % 1 . . °

•~ • ~•.gg ~*•g ==% ="

"

~ "" .,o°

" • •==";=='g • . ° = ,

• "~••g~ -~°•-='° °=~" -=l. • ~ ' o . • ' . • -* ; % . . . . . . ".. •" -" ==~ I°-*=; . ='.".,'=*~" %" ~ I ' . ='=,;,=',~," " . " . .

==~ • I I *

=.I" =% I *.',=" ,

".I" ; ' % * . ' "

"I~" *" ",".%1 .=,g ,'°°

". •

" . ~ " ",-=.

".;%"%'" ;;;'I'..'.~ % % - ; *.'., ~ . ' * ' . l ° ~.

" •°%%,'°."

%'=.;~

,.'Io

. ~ * ° t " ='~ = " . . - ' * ' , I =-=°

" ; ° ~ " " ,="1",.*

.

o.-~°=* =..%1~'."" . ' • °=" =~ ." ".**~Z*%'.•"..

; ' • ~ 1";.=o* " " ; ' . ; '

i• =::".~.:~-':=.:.:~?•r':~:i: .'• !-:.:5"'.!~:.?'..:'::~:!.~!:.:'~:;..:=.'.:'.v.::..-~.~::'.=:•..::: ", I •'°

I * " °'•

='° " % " " • ; °

•;•.~=,I°

I I * * ° ' ; ' . * '~ ~o I % = ' . ,

,.°o~,'==~.~."~.~

"-=%= • • -=. . . . • • * ,= °.===-..~.=gI• ~'.• " * ~ ° " ° . . ' ~ I - = * ; , -~" = * , . . " " ° ~ ' , , • " 1".'• • . " 1 . • ° • - " . . 1 ° , % ° ; * " . • . , . ' ° . : , ° . " " . • * o ; ==.~ " . , " . . . . . ,= .='. • ~. . . . . . ,"•°.°t°. I = = - ~ I . l l , l • ° ° • ===~'.'~• ° ° •

.~ ° " . . ° . ' . . ~ , : '

l - * ~ ' ~ ° ~ • . . ~ l " I o " " "~"- ° ° l .~%~"1°.

=*.;~°Z.~ g~-. ~ . g . • g • % . = ° . . ; - . . ; - - • , . . ; - ; ; . •• "g .=-;;-- . . . •;I;~••;'l, . , I .~ • ~ l ° . I Z , ' I . • ° =% . d • I ° • == • I ' = * • • " . . l ~ I

';.=ii ~::-;=';::~:'; ~="..=i=:=.=:!!:=:~'.=.g=.i'.:=:?'!.:;i:.':L::=?.".;Y.-='=:

:!'::i'.?=.L. ';

Figure 4

Composite cells with varying r a n d o m microstructure. Subcell arrays shown are 10 x 10, 30 x 30, 50 x 50, and 100 x 100. Fibers shown in dark. Fiber volume fraction = 0.5

implementation grows almost linearly with the number of subcells. This is also consistent with the exponent in equation (46) (which is close to 1.0). The SUN Sparc 20 workstation is not particularly fast. The CPU time required by an SGI Indigo2 workstation to solve the 10,000 subcell problem using the sparse implementation is 745 s (compared to 1,563 s for the SUN). It should be emphasized that all data reported here correspond to the computations required to estimate the effective linear elastic properties of the composite only. Inelastic, plastic and/or viscoplastic analysis would require larger storage and larger CPU times. Ways to make GMC computationally efficient for this kind of problem are the subject of current research.

4.2 Variability of transverse elastic moduli To illustrate the modeling capabilities of the GMC sparse implementation, several problems with randomly

172

generated microstructures were investigated. Microstructures were generated for three different fiber volume fractions: 0.2, 0.5 and 0.7. A total of 22 different subcell arrangements for each of these volume fractions were generated (see Table 3). Each of these arrangements was generated by randomly distributing different size inclusions with the properties of boron fibers, in an aluminum matrix. Figure 4 shows some of the microstructures generated in this fashion. The engineering properties corresponding to the 66 different microstructures were estimated using the sparse implementation of GMC. The objective was to study the asymptotic behavior of the transverse engineering properties of the composite. This has potential applications in modeling materials such as concrete, short-fiber, and particulate composites. For the particular case of unidirectional composites with transverse random microstructures, the effective engineering properties predicted by GMC should tend to

Modeling complex microstructure using GMC." C. E. Orozco Table 3

E22/E22vs number of subcells

Subcell arrangement

No of subcells

vf = 0.2

% Error

vf = 0.5

% Error

vf = 0.7

% Error

4×4 5x 5 6×6 7x7 8×8 9x9 10 x 10 12 x 12 14 × 14 16 × 16 18 × 18 20 × 20 24 × 24 30 x 30 36 x 36 40 × 40 50 × 50 60 x 60 70 x 70 80 x 80 90 x 90 100 x 100

16 25 36 49 64 81 100 144 196 256 324 400 576 900 1296 1600 2500 3600 4900 6400 8100 10,000

1.0347 1.0141 0.9870 1.0000 1.0041 0.9990 0.9957 0.9963 0.9958 1.0113 1.0041 0.9987 1.0018 0.9973 1.0006 0.9990 1.0002 1.0000 1.0004 1.0007 1.0001 1.0002

3.47 1.41 -!.30 0.00 0.41 -0.1 -0.43 -0.37 -0.42 1.13 0.41 -0.13 0.18 -0.27 0.06 -0.1 0.02 0.00 0.04 0.07 0.01 0.02

0.9414 1.0000 0.9761 0.9784 1.0466 1.0808 1.0323 1.0017 0.9924 0.9939 1.0033 1.0014 1.0087 1.0041 0.9970 0.9966 1.0024 0.9949 0.9968 0.9991 0.9997 1.0000

-5.86 0.00 -2.39 -2.16 4.66 8.08 3.23 0.17 -0.76 -0.61 0.33 0.14 0.87 0.41 -0.3 -0.34 0.24 -0.51 -0.32 -0.09 -0.03 0.00

1.0000 1.1932 0.8752 0.9309 0.9739 0.9472 0.9999 1.0330 1.0001 1.0366 1.0216 0.9767 0.9895 0.9897 0.9940 0.9992 1.0026 0.9984 0.9979 1.0018 0.9986 1.0001

0.00 19.32 -12.48 -6.91 -2.61 -5.28 -0.01 3.30 0.01 3.66 2.16 -2.33 -1.05 - 1.03 -0.60 -0.08 0.26 -0.16 -0.21 0.18 -0.14 0.01

% Error

vf = 0.5

% Error

vf = 0.7

% Error

3.83 3.65 3.88 3.91 2.75 1.94 3.50 2.28 1.47 1.40 0.93 1.42 1.30 0.77 0.69 0.65 0.50 0.36 0.34 0.29 0.25 0.26

1.2295 1.1704 1.2011 1.0907 1.0776 1.0897 1.0520 1.0241 1.0503 1.0447 1.0349 1.0365 1.0434 1.0233 1.0242 1.0240 1.0221 1.0167 1.0135 1.0123 !.0095 1.0081

22.95 17.04 20.11 9.07 7.76 8.97 5.20 2.41 5.03 4.47 3.49 3.65 4.34 2.33 2.42 2.40 2.21 1.67 1.35 1.23 0.95 0.81

1.4383 1.1886 1.2711 1.5092 1.3663 1.2378 1.2631 1.1085 1.0456 1.0817 1.0605 1.1079 1.0711 1.0443 1.0426 1.0370 1.0335 1.0193 1.0174 1.0192 1.0180 1.0157

43.83 18.86 27.11 50.92 36.63 23.78 26.31 10.85 4.56 8.17 6.05 10.79 7.11 4.43 4.26 3.70 3.35 1.93 1.74 1.92 1.80 1.57

Table 4

E22/2(1 + v23)G23vs number of subcells

Subcell arrangement

No of subcells

vf = 0.2

4×4 5x 5 6x6 7x 7 8x8 9x9 10 x 10 12 x 12 14 x 14 16 x 16 18 x 18 20 x 20 24 x 24 30 x 30 36 x 36 40 x 40 50 x 50 60 x 60 70 x 70 80 x 80 90 x 90 100 x 100

16 25 36 49 64 81 100 144 196 256 324 400 576 900 1296 1600 2500 3600 4900 6400 8100 10,000

1.0383 1.0365 0.0388 1.0391 1.0275 1.0194 1.0350 1.0228 t.0147 1.0140 1.0093 1.0142 1.0130 1.0077 1.0069 1.0065 1.0050 1.0036 1.0034 !.0029 1.0025 1.0026

those of a transversely isotropic material as the number o f s u b c e l l s is i n c r e a s e d . F o r a t r a n s v e r s e l y i s o t r o p i c m a t e r i a l o r i e n t e d as i n Figure 1, t h e f o l l o w i n g t w o r e l a tionships must hold: E33 = E22

(48)

E22 G23 -- 2(1 + v23 )

(49)

and

The foregoing two relationships should then be asymptot i c a l l y s a t i s f i e d as t h e r e f i n e m e n t o f t h e m i c r o s t r u c t u r e is increased. To test

this

hypothesis,

the

r a t i o s E33/E22 a n d using GMC. As

E22/2(1-}-v23)G23 w e r e d e t e r m i n e d

mentioned above, boron inclusions in an aluminum matrix were used. However, since the numbers being considered are ratios of engineering properties, the results are independent of the constituents. Results are s h o w n i n Tables 3 a n d 4. T h e r a t i o s a r e a l s o p l o t t e d a g a i n s t n u m b e r o f s u b c e l l s i n Figures 5 a n d 6. A s e x p e c t e d , b o t h r a t i o s t e n d t o 1.0 as t h e n u m b e r o f s u b c e l l s i n c r e a s e s . F r o m Figure 5, it c a n b e o b s e r v e d t h a t t h e r a t i o o f t h e Y o u n g ' s m o d u l i o s c i l l a t e s a r o u n d 1.0, showing no noticeable preference for values above or b e l o w 1.0. N o t i c e a l s o , t h a t b y c h a n c e , t h e 4 × 4 a r r a n g e m e n t c o r r e s p o n d i n g t o t h e 0.7 f i b e r v o l u m e f r a c t i o n p r o d u c e s a r a t i o o f m o d u l i t h a t is e x a c t l y e q u a l t o 1.0 (see Table 3 a n d Figure 5). O n a v e r a g e , h o w e v e r , it is s e e n t h a t t h e r e is a n e r r o r i n t h i s r a t i o a n d

173

Modeling complex microstructure using GMC: C. E. Orozco The error incurred in calculating the ratio of E22/2(1 + v23) to G23 for say, a discretization with 900 subcells is about 0.8% for a 0.2 volume fraction, 2.3% for a 0.5 volume fraction, and 4.4% for a 0.7 volume fraction. With 10,000 subcells, the corresponding errors are: 0.3%, 0.8% and 1.6%. The fact that the error is larger for higher volume fractions means, of course, that more random subcells are needed to achieve transversely isotropic behavior at higher volume fractions. This is probably due to the fact that the 'randomness' of the subcell pattern is higher for lower volume fractions. It is also conceivable that this behavior could be improved by improving the random number generation scheme.

1.20 ~ 1.18

~

1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 0.98 0.96 0.94 0.92

5 CONCLUSIONS

0.90 0.88

0.86

10

Figure 5

100

1000 1000 NUMBER OF SUBCELLS

E33/E22 vs number of subcells

ff 1.6o

. . . . . . . .

I

. . . . . . . .

I

. . . . . . . .

1.5o 1.40 [

© FIBERVOLUMEFRACTION : 0.2 <> FIBERVOLUMEFRACTION:O.5

~

\ ] / , n ERvoL Ac oN:07 1.30

1.20

1.10 Q

©

1.00

0.90

, 10

Figure 6

,

~

,

~,,I

,

100 Ratio of E22/(1 + v23) to

~

, , ,,,,I , , , , . . . . tl 1000 10000 NUMBER OF SUBCELLS

G23 vs number of subcells

that this error is higher for higher fiber volume fractions. The error produced by GMC for an array of 576 subcells or more is of the order of one percent. For a 100 x 100 subcell array (10,000 subcells) the error is practically zero (0.01%). From Figure 6 and Table 4, it can be observed that the ratio ofE22/2(1 + v23) to G23 also converges to 1.0 but at a slower rate than the ratio of the Young's moduli. Furthermore, it converges from above. This means that GMC tends to slightly overestimate the quantity E22/2(1 + v23). GMC also tends to underestimate (323. This suggests that an average between E22/2(1 + v23) and G23 is probably a better estimate of the transverse shear modulus for any given discretization of the repeating cell.

174

A sparse implementation of the generalized method of cells (GMC) has been presented and tested in connection with problems that require a large number of subcells. It is shown that the sparse implementation is up to 34 times faster than previous dense implementations when used to predict the effective elastic properties of complex microstructure composites. Furthermore, the new implementation makes it possible to solve problems with a degree of detail that was not possible before. Cell discretizations with up to 10,000 subcells were utilized in the present study. It was also shown that GMC correctly models the asymptotic behavior of the engineering properties of composites with a large number of random inclusions. This was demonstrated by observing that the values of the effective engineering properties of the composites studied tend to those corresponding to transversely isotropic behavior when the number of subcells is sufficiently large. The new implementation of GMC makes it possible to investigate the elastic response of advanced composite materials with complex microstructures such as multi-phased composites. These include composites with fibers of different properties, different sizes, different shapes, and nonuniform fiber spacings, such as composites with macroscopically periodic, but locally functionally graded microstructures. ACKNOWLEDGEMENTS The author is grateful to Professor Marek-Jerzy Pindera for his encouragement and support and for the many enlightening discussions and suggestions that made this work possible. The useful comments of the reviewers are also greatly appreciated. REFERENCES 1

2

3

Aboudi, J. 'Mechanics of Composite Materials--A Unified Micromechanical Approach'. Elsevier, Amsterdam, New York, 1991 Paley, M. and Aboudi, J. Micromechanical analysis of composites by the generalized method of cells. Mechanics of Materials, 1992, 14, 127 Wilt, T.E. On the finite element implementation of the generalized method of cells micromechanics constitutive model. Technical Report NASA CR 195451, NASA, 1995

Modeling complex microstructure using GMC: C. E. Orozco 4

5

Sankurathri, A., Baxter, S. and Pindera, M.J. The effect of fiber architecture on the inelastic response of metal matrix composites with interfacial and fiber damage. In 'Proceedings of the Symposium on Damage and Interracial Debonding in Composites,' New Orleans, 1996 Arnold, S.M., Pindera, M.J. and Wilt, T.E. Influence of fiber architecture on the elastic and inelastic response of metal matrix composites. Technical Report NASA-TM 106705, NASA, 1995

6 7 8

Curtin, M.E. 'Introduction to Continuum Mechanics.' Academic Press, 1981 Hill, R. Theory of mechanical properties of fibre-strengthened materials--i, elastic behavior. Mechanics and Physics of Solids, 1964, 12, 199 Aboudi, H. and Pindera, M.J. Micromechanics of metal matrix composites using the generalized method of cells model (gmc) user's guide. Technical Report NASA-CR 190756, NASA, 1992

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