WOVEN FABRIC COMPOSITES: DEVELOPMENTS IN ENGINEERINGBOUNDS, HOMOGENIZATION AND APPLICATIONS
Peter W. Chung*and
Kumar K. Tamma^
Department of Mechanical EngineeringUniversity of Minnesota
111 Church St. SEMinneapolis, MN 55455
Ph: (612) 625-1821Fax: (612) 624-1398
Abstract
First, methods are described for approximating upperand lower bounds for the elastic stiffness tensor for gen-eral woven fabric composites. Well accepted minimumenergy principles are briefly presented to establish thefoundation for practical finite element procedures fordetermining these bounds. Secondly, comparisons offour common homogenization procedures are shown:the strain energy balance method, the plate approxi-mation method, a direct approach via area averaging,and asymptotic expansion. The boundary conditionsare applied so as to obtain the well known Rule ofMixtures for a unidirectional uniaxial specimen. Inattempting to consolidate much of the existing knowl-edge of structural constitutive models for WFCs, thisresearch seeks to summarize and compare various ho-mogenization methods via finite element analyses. Andthird, some illustrative applications are presented.
Introduction
To avoid excessively large computational models whenanalyzing complex geometric composite structuressuch as those made from woven fabric composites(WFCs), numerous approaches have been attemptedtowards homogenization of macroscopic propertiesfrom the constituent properties. The relevant associ-ations and relationships via such homogenization pro-cedures is and continues to be of query to researchersin this field.
The development of WFC technology hinges on afull understanding of the microstructure and the inte-gration of micro/macro structural analysis. The associ-ated mechanics problem of characterizing the multiplescales, as depicted in Figure 1, is the greatest obstacleto unrestricted implementation of WFCs. The interme-diate length scale between the micro and macroscale,the so-called microstructural scale or the unit cell, is
the key to merging the details of the microscopic withthe macroscopic. How the so-called unit cell is definedvaries amongst researchers and is an important facetin investigations of the constitutive behavior.
The earliest efforts attempted to predict bounds forspherical and unidirectional composites composed ofisotropic phases with identifiable unit cells.*~4
In areas related to WFCs, prediction of constitu-tive relations has been attempted by laminate platemethods,5"9 finite element mechanics procedures,10'11
and mathematical asymptotic expansion homogeniza-tion.12'13
Following existing efforts,1'4 an approach for esti-mating practical bounds for the effective elastic prop-erties of WFCs is presented. Bounds on the effectiveproperties may, in most instances, be more appropriatefor design and analysis of composites instead of an ap-proach where a single set of properties is defined. Also,calculations and comparisons are made for the mosaicand crimp woven fabric approximations using four keyfundamental homogenization methods: strain energybalance, plate approximation, direct method via areaaveraging, and asymptotic expansion. These four fun-damental approaches underlie many of the techniquesavailable today. The mosaic and crimp models can beretro-validated readily with earlier analytical develop-ments,5 and they are the test cases for which finite ele-ment models were created in this paper. Finally, somesimple applications are shown to demonstrate .the in-corporation of microscopic properties into macroscopicproblems.
Generalized Repetitive Unit Cell (RUG)
Homogenization, as used here, is performed on a pe-riodic unit cell. The concept of a unit cell dates backto as early as the first significant work in estimatingconstitutive models of composites,1 and more recentlyunderstanding of the constitutive elastic behavior has
led some to model the fabric at a more realistic 3Dlevel.12
Minima Theories
Significant contributions to the development of con-stitutive models for composite materials appears firstfor unidirectional (UD) composites with isotropicphases.1'3'4'14 These seminal developments present in-tricate and sophisticated methods of predicting all ofthe elastic constants via minima principles. Use of theminima principles for determining bounds for the elas-tic properties was demonstrated by Paul15 where theupper bound is called a Reuss bound and the lowerbound is called the Voight bound. Unfortunately, themethods for UD composites are incapable of fully de-scribing WFCs. The earlier methods are restricted toUD composites while present WFCs, which employ un-dulating tows and resin pockets, are clearly far fromUD. Homogenization, as it is used in the present con-text, refers to any method by which a complex multi-phase material is replaced by an orthotropic homoge-neous material of identical external geometry.
A body of volume V bounded by its surface S issubject to applied displacements u* on Su and trac-tions T* on St where S = Su U St. In linear elasticity,Hooke's Law provides a relationship between the stressaij and the strain Ey
__ .»-Y /I \
where Cijki is the elastic stiffness tensor.The principles of minimum potential Ue and com-
plementary Uff energies are summarized as16
Ue = f W(eij)dV - I T*UidA (2)Jv Js,
Ua = f W(atj)dV - I Tm\dA (3)Jv Jsu
where W is the strain energy density. The next sec-tion describes a procedure to estimate an upper boundfor the elastic stiffness tensor for a general anisotropicheterogeneous specimen.
Upper BoundConsider a homogeneous body V with surface S withthe associated homogenized properties in the stiffnesstensor C^ki- The body is subject to displacementboundary conditions on its surface so as to simulate auniform strain condition. The strain energy is definedby
- / ^Jv 2 (4)
where ft is the strain energy associated with the exactsolutioney for the homogeneous body. The bars (~) de-note averaged terms that result from the homogeneousassumption.
It can be shown readily that
U - Uh = i / aij (ui - Ui) dA (5)2 7s
where Ui are the displacements on the surface of theheterogeneous body and Ui is for the homogeneousbody. Thus, the energies are equivalent when
Ui — u* on S (6)
Thus, by invoking the principle of minimum potentialenergy
< I -Cijki£ijJv *
(7)
Then, by Clapeyron's equation,17 the strain energy isequal to the work done on the body.
In theory, determination of a rigorous upper boundfor anisotropic materials is straightforward. However,in practice, to predict such bounds for a complex mi-crostructure is difficult. Whitcomb11 introduced a pro-cedure that is suitable for estimating an upper boundfor the tensile elastic properties. Though not a rigorousbound, it may be suitable in most practical instancessince it applies an average uniform strain via displace-ment conditions on the boundary. We will demonstratelater by example a case in which it does not provide agood upper bound.
The shear properties are more difficult to de-fine. Sun and Vaidya18 employ a mixed trac-tion/displacement procedure with an energy-work bal-ance approach to determine the shear modulus. How-ever, such a procedure may not produce a good upperbound due to the mixed boundary conditions. Fol-lowing the methodology of uniform average strain, anapproach is defined for determining shear properties.
Strain Energy Balance MethodWe propose the strain energy balance method11 as ameans of estimating an upper bound to the elastic con-stants of a general 3D orthotropic homogeneous body.The basic premise is to equate the strain energy to thework performed, and through this equality the homog-enized orthotropic moduli can be determined. Givenall key geometric modeling parameters all of the or-thotropic elastic material properties (E, v, G) can befound. The method is general enough to be applicableto spherical composites and the like.
Tensile Properties. Three distinct load cases areemployed for determining the effective tensile proper-ties. In each load case, a uniform displacement bound-ary condition is applied to one side and the remainingfive sides are held fixed but allowed to move in the
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directions parallel to that side. Then, the resulting re-action forces can be combined to form three fictitiousloading scenarios using the method of superposition.Equations governing the scenarios are combined in theform of Hooke's Law from which the elastic constantscan be extracted. To illustrate, the stress-strain rela-tionship in the x-direction is given as
E,
f-E, E,
ax0 (8)
where ay and az are zero since the boundaries in thesedirections are free in the new fictitious scenario. Thestrain in this scenario is defined by ex = exi. Aftersome algebraic manipulations, the following equationresults
where
and
ax =
y2ZlPx = -—-
(9)
(10)
(11)
The constants ax and /3X act as Poisson-like terms inthe new fictitious scenario.
The homogenized stresses are found by dividing thereaction forces by the appropriate area. Thus,
XlP =0 = F l
0 = Ftl
axFX2
axFZ2
- (3XFX3
- P*Fy3
-P*FZ3
(12)(13)(14)
Equations (12)-(14) comprise a set of three simul-taneous linearly independent equations where the un-knowns are P, ax, and /3X. All of the reaction forcesare obtained for each load case. Then, by equating thestrain energy to the total work gives
Pu0
elV (15)
A similar set of equations exist for the remaining twoequivalent load conditions.
Shear Properties. The problem of determiningshear properties of an unknown material has been dis-cussed for many years in the literature.18"21 It ap-pears that the primary difficulty in developing an un-derstanding of the shear behavior of complex media isin the inability to validate with experiment. Some rele-vant issues such as the approximation of uniform shearstrain through the body and improper shear bound-ary conditions are discussed in another paper.18 The
approach developed in the present work is differentfrom those developed elsewhere11'18 by using a puredisplacement-controlled scheme.
Figures 2 depict the displacement conditions alongthe boundaries used to simulate a shear condition. Thesubsequent reaction forces along the directions of theapplied displacements, then, serve as equivalent shearforces. Hence,
2 (Pi6i(16)
where i and j are used to simplify notation and repre-sent the shear mode in either the xy, yz, or xz direc-tions. Due_to the_assumption for orthotropicjiomoge-nization, Gxy = Gyx,Gyz = Gzy, and GX2 = G:x. Theshear strain, 7^ is given by
Dj
where 5's are the applied displacement values and Djand DJ are the relevant dimensions of the unit cell.
Lower BoundConsider^ again a homogeneous body V bounded by asurface 5 with the associated homogenized propertiesin the stiffness tensor Cijki- The body is subject totraction boundary conditions on its surface so as tosimulate a uniform stress condition. The strain energyis given by
-rS (18)
The assumption of a homogeneous body allows us todetermine the homogenized stress field exactly fromthe prescribed tractions on the boundary by using thedefinition for tractions a^nj = f1,.
The requirement for obtaining a lower bound esti-mate to the stiffness tensor is that a uniform stress isfirst applied.15'16 Three sets of tractions are applied tothe unit cell. Each set of tractions corresponds to oneload case. The tractions in the first set are applied inthe x-direction while zero tractions are applied in they and z-directions. This loading scheme is permutatedfor the other two load cases.
Stress Energy Balance MethodWhitcomb et al.22 discuss an alternative method to thestrain energy balance approach. However, no mentionof its relevance to a Voight bound is given. For sakeof discussion, the method by which a uniform tractionis applied to a finite element model for calculating theresulting nodal displacements will be called the stressenergy balance method. The nodal displacements canthen be multiplied to the effective nodal force vectorto give the total work. In short, the procedure is the
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complement of the strain energy balance approach dis-cussed earlier.
Upper and Lower Bounds: 3D WFCA 3D WFC unit cell was constructed to determine itselastic properties (see Figure 3). The unit cell is 35%by overall volume E-glass/epoxy composite. The fiberbundles contain a local volume fraction of 65% E-glass.The properties are shown in Table 1. The fiber bun-dles are transversely isotropic with the x-axis directedalong the line of material symmetry. The 3D upperand lower bound stiffness terms for the unit cell werecomputed and are shown in Table 2.
Comparison of VariousHomogenization Techniques
A comparison of four homogenization techniques isconsidered next to assess their differences and sim-ilarities. The techniques considered are: (i) thestrain/stress energy balance method, (ii) the plateapproximation method, (Hi) the geometric averagingmethod, and (iv) the asymptotic expansion technique.The comparisons give good estimations that show howexisting methods produce bounds similar to Reuss andVoight bounds. The comparison is performed for IDWFC approximations that were originally developedby Ishikawa and Chou5 called the Mosaic and Crimpmodels. The numerical procedure employed was pre-sented before5 and is called informally as the plate ap-proximation method. The uniform stress assumptiondue to the plate approximation makes this method sim-ilar to a stress balance approach, hence a lower bound.Also demonstrated and compared are the strain energybalance method, a more conventional, albeit less so-phisticated area averaging approach, and the so-calledasymptotic expansion homogenization method.
Strain/Stress Energy BalanceA detailed description of the approach and applicabil-ity to a repetitive unit cell was described earlier. The3D procedure can be degenerated for 2D problems.
Plate ApproximationThe origin of the plate approximation method is inthe analytical procedure that predicts the plate elasticstiffness or compliance tensors (A, B, and D) due toIshikawa and Chou.5 The Kirchoff-Love plate assump-tions, however, restrict from determining propertiesthrough the plate's thickness since negligible strainsin the transverse direction are inherently assumed.
The mosaic and crimp models were developed basedupon classical laminate plate theory to provide the ex-tensional, coupling, and bending stiffness (or compli-ance) tensors. The plate approximation homogeniza-tion technique is, in turn, based upon the mosaic andcrimp analytical models. The methodology can be un-
derstood readily by the plate equation
Mi D tj(19)
The goal here is to apply either strains or resultantsand calculate the work conjugate via finite elementanalysis. From this, the plate stiffness (or compliance)terms can be evaluated. In the present analysis, a com-bination of a uniform strain and uniform stress condi-tion is applied for reasons described later.
Only the in-plane modulus is used for comparisonpurposes. Other properties such as shear moduli andPoisson's ratios calculated for the mosaic and crimpmodels do not correlate well with analogous propertiesdetermined from more realistic and complicated mod-els.
Two-dimensional finite element meshes of UD, mo-saic, and crimp geometries were created and analyzedin plane stress for varying ng, a parameter that de-notes "weaviness". Weaviness refers to the number ofharness in a particular weave. The UD graphite/epoxycomposite mesh, as indicated in Figure 4a, representsthe ng = oo case. Tabulated in Table 3 are the 2Dmaterial properties for the fill and warp regions. ThePoisson's ratios for the warp regions are actually thetow properties rotated 90°, and since they are trans-versely isotropic in 3D, they are taken to be isotropic in2D. Values for ng used were 2, 3, 4, 5, 10, and oo, eachwith 50% volume fraction. The finite element modelsmust also account for rotation of the material axes asdepicted in Figure 5.
The 2D finite element solution provides informationabout the midplane strains, curvatures, and resultants.Thus, solving for the plate stiffness terms gives
+ "£J(20)
Finally, the modulus is determined by invoking theorthotropic material assumption and the inverse ofEqn.(19)
- Ill-(I-;.2
t(21)
where An is determined from Eqns.(20), v is foundfrom Ai2/Au, and t is the height of the 2D model(mosaic or crimp).
Area Averaging: Direct MethodArea averaging is a direct stiffness or flexural approachwhereby boundary conditions are applied to enforce
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stresses or strain, and Hooke's Law is used to deter-mine the constitutive properties. Averaging refers tothe technique of taking area (in 2D) or volume (in'3D) averaged quantities. It is demonstrated here thatfor simple UD composites, the geometrically averagedmodulus coincides exactly with the well known Rule ofMixtures given by
Ec = EfVf + EmVm (22)where E is the longitudinal modulus, V is the volumefraction of the constituent, and the subscripts c, /, andm denote the composite, fiber, and matrix components,respectively. The plate approximation method assumesa combination of both uniform strain and stress so asto provide reasonable comparisons with the other twomethods as ng —>• oo.
The stresses and strains are element averaged quan-tities defined by
Vjy(23)
(24)
Consider the x-equation from the first load case
where (~) signifies effective properties, and brackets ( )are the element averaged quantities. The x-equationsfrom the remaining two load cases can be used to de-termine the Poisson's ratios in terms of the geomet-rically averaged stresses. Then, the modulus can bedetermined solely in terms of geometrically averagedstresses and strains. A similar procedure is used fordetermining all other elastic constants such as shearmoduli and the y and z-direction tensile moduli.
Asymptotic ExpansionThe asymptotic expansion homogenization approachhas been studied for many years.23 More recently Hoi-lister and Kikuchi13 and Lukkassen et al.24 and oth-ers12 have employed the method using finite elementsfor complex problems.
The unit cell is defined in a domain Y which is peri-odic in a global domain X. The coordinate systems aredefined respectively as j/j and £j and the microstructurescaling is characterized by j/j = ^"-, where e is the sizeof the periodic cell. Following Bensoussan, Lions andPapanicolaou23 the method is based on the asymptoticexpansion of the displacement solution for the problem
A(u\ ~ ft in n (26)(27)u\ is subject to conditions on <9Q
Consider x and y as independent variables. Then bychain rule the differential operator is equivalent to
(30)
where
8
d
A3 = ~
Substituting (30) and (31) into (27) and separatingcommon factors of e yields
= 0
A3u r =
(32)
(33)
(34)
Equation (32) indicates that u^ is independent of y,or in other words, it is the macroscale displacement.Equation (33) gives the auxiliary equation
(35)
By superposition, u'1' can be defined up to an addi-tive constant of x(whose value can be set to zero byappropriate boundary conditions on Y). This gives
,(°)(36)
Finally, the effective properties are determined by sub-stituting (36) into (34) and noting that a unique solu-
(f)\tion to u\~ exists if and only if
= 0 (37)
This indicates that the effective property of the unitcell in Y is given by
UlJ ~W\JY Vy ^ dVkdY (38)
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A 2D finite element program was written to per-form these calculations to determine the effective prop-erties of rectangular shaped unit cells. A verificationwas initially performed and compared with a resultfound in24 for the transverse properties of a UD fibercomposite (see Figure 6). The fiber (Ef = 70.0zl09
and Vj = 0.20) and the matrix (Em = 3.5zl09 andvm — 0.35) are both isotropic. Subsequently, applica-tion to the 2D representations of the UD, mosaic, andcrimp models was attempted.
Comparative Results
A full comparative study of the homogenizationschemes was conducted using the meshes shown in Fig-ures 4. The study compares only the in-plane Young'smodulus because of the limitations inherent in usingthe plate approximation. Figures 7 show the moduli forthe mosaic and crimp models and compare the resultsfrom the four homogenization schemes: strain energybalance, plate approximation, and their correspondinggeometric averages, and asymptotic expansion.
As mentioned earlier, it is evident in 7a that theupper bound approximation provided by the strain en-ergy balance approach is poor. In fact, the excep-tional asymptotic expansion result appears to provide ahigher in-plane modulus. This indicates that a uniformaverage strain assumption does not produce the desiredeffect of a uniform microstructural strain. A true uni-form microstructural strain was imposed in calculatingthe analytical upper bound using mosaic model platetheory. In general, however, the strain energy balanceapproach is still capable of providing a reasonable up-per bound estimate as is evident in Figure 7b.
Comparative QuantitativeAnalysis, Validations and Applications
The purpose of this section is to demonstrate applica-tion of the unit cell properties in general macroscopicanalysis employing only the energy balance approachfor purposes of illustration. In general, one cannot di-rectly include details of the microstructure when per-forming macroscopic analysis.
Several examples are presented for applying thestrain energy balance homogenization method to re-alistic engineering situations: (i) a comparison of thefull 3D elastic properties for a UD glass-epoxy compos-ite to results available from literature, (ii) a compari-son between the earlier bound estimates for a typical3D WFC and single property estimates for similar mi-crostructures found elsewhere, and (Hi) a macroscopicplate with a hole structural finite element analysis.
Verification of Strain Energy Balance:UD ComparisonsA 3D mesh of a UD composite unit cell, made predomi-nantly of brick elements, was used to predict the elastic
properties of a 60% fiber volume fraction AS4/3501-6UD graphite/epoxy composite. The mesh is shown inFigure 8. In table 6 are the UD composite elastic con-stants via the strain energy balance method. Theyshow strong agreement with another source18 whichare also favorably compared in that investigation withother theoretical and experimental results.
3D Weave In-Plane Moduli ComparisonsShown in Table 2 were the proposed bound estimateson the stiffness terms for a typical WFC. A priori es-timates of the elastic properties are also present else-where in the literature. Unfortunately, every investi-gation seems to employ a unique unit cell geometry.Parameters such as the crimp angle8 and thickness-yarn width ratio9 have been investigated in the pastto understand the influence on the elastic properties ofthe unit cell.
Only qualitative comparisons can be made withother results in the literature due to the variations ofweave types studied. A parameter is used to removethe geometrical dependence amongst the scattered re-sults. The local fiber volume fraction (V/) and theoverall fiber volume fraction (Vf) can be combined togive the tow volume fraction (V<) of the overall unitcell. The matrix volume fraction is then 1 - Vt. Thus,the unit cell is comprised of two components: the ma-trix and the impregnated tows. Assuming these com-ponents have distinct sets of effective properties, a pa-rameter is defined by
E
-i(39)
which may be thought of as a Rule of Mixtures esti-mate of a 0/90 cross-ply laminate with an equivalentoverall fiber volume fraction to the unit cell. Tableshows the results as well as those from other investiga-tions. Without full reproduction of the finite elementmeshes used, more exact comparisons cannot be made.All results are for plain weave fabric reinforced com-posites. The upper and lower bounds in the first rowrepresent the estimated values from stress and strainbalance calculations. It is evident that not every resultin the literature agrees with these bounds. However,the number of results that indeed agree, despite differ-ences in the unit cell microstructure, is significant.
Orthotropic Plate with a HoleThe problem of a plate with a circular hole is treatedfor layered WF and laminate composites of various con-stituent materials (see Figure 9).
Finite element analysis using layered composite el-ements is employed and comparisons are shown withapproximate theoretical results.25
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To make rational comparisons between the twotypes of composites, the WF lamina is given a thick-ness twice that of the UD lamina to simulate a warp-like and fill-like 0°/90° layering in a single weave. Theoverall volume fractions of fiber are the same for boththe weave and the UD lamina. Stacking sequences of[±45]2s is used in the UD laminate and [±45]s for theweave laminate.
Plate Geometry and Mesh. The geometry of theplate in the present investigation is depicted in Figure10 where, due to symmetries, only the first quadrant isconsidered. The y-direction stresses are computed us-ing theory along the line AB. The finite element meshesused in the plate with a hole problem are depicted inFigures 11. The plate thickness is 0.01m and thick-nesses of the lamina in the UD laminate are 0.00125mand in the weave laminate are 0.0025m.
Repetitive Unit Cell (RUG). A periodic unit cellfor a plain weave fabric is defined as the smallest re-peating unit in the composite plate. The RUC em-ployed here is defined at the micro/macro length scaleand is the smallest divisible unit at the macro level.Figure 3 shows the finite element model of hexahedralelements and the constituent tow and matrix elements.
Analytical Stress Distribution Equations. Thegeneral stress distribution solution for a uniaxiallyloaded plate containing a circular hole employs thedistinct complex roots of the characteristic equationfor a 2D anisotropic material. This expression wasfound to have a reasonably simplified form by takingthe binomial series expansion of the exact solution.25
The anisotropic stress concentration factor K is definedby terms of the laminate plate extensional stiffnesses
Material Properties. Two composite types are con-sidered: E-glass/Epoxy and SCS-6/Timetal/321-S. Thefirst is an epoxy-based composite with E-glass fibers.The second is a metal matrix composite with carbonfibers and titanium matrix. The constituent materialproperties are listed in Table 8.
The constituent material properties are used in thestrain energy balance method for 3D meshes of UDcomposites with fiber volume fractions of 35% and65%. The UD composite with 35% fiber volume frac-tion is used in the UD laminate study of eight stackedlamina at orientations [±45]2s. The composite valuesfor 65% fiber volume fraction are used for the local towproperties in a WFC such that the overall volume frac-tion average is 35%. The stacking sequence used in theWF laminate is [±45] s since each weave-lamina is twicethe thickness as the UD-lamina. The effective proper-ties computed for the UD composites are indicated inTable 9.
Finally, the pure resin properties from Table 8 andthe 65% tow properties of Table 9 are used in the WFunit cell to calculate the effective properties of WFClamina where the overall fiber volume fractions are 35%and the lamina thickness is twice the thickness of theUD lamina. Using the strain energy balance method,the effective properties are tabulated in Table 10.
Results. The y-stress distributions for a laminatedplate with a circular are shown in Figures 12 for theUD laminates and Figures 13 for the weave laminates.Normal stress comparisons were made along the lineAB (see Figure 10). The theoretical curve comes di-rectly from the extended isotropic solution.25 The fi-nite element results are computed at the centroids ofthe elements that sit along line AB. Figures 14 and 15show the comparisons between theory and finite ele-ment results for the normal stresses.
From the Eqn.(40) and the finite element results,comparisons can also be drawn for the stress concen-tration factor, K. Table (11) shows K for the variouscalculations performed in this analysis.
Concluding Remarks
Estimates for the upper and lower bounds were firstpresented for a typical WF unit cell. Strain energy-work balance principles, common in the study ofbounds on elastic properties, were employed.
Four different homogenization techniques widelyadvocated in the literature were comparatively studiedfor 2D problems with corresponding analytical results.5
Several examples were shown to validate the presentresults with other efforts in the literature. Finite ele-ment analyses of UD and WFC microstructures gaveproperties and estimates for elastic property boundswhich were shown via Table to provide practical, real-istic estimates.
Acknowledgements
The authors are very pleased to acknowledge sup-port in part by Battelle/U.S. Army Research Office(ARO) Research Triangle Park, North Carolina, undergrant numbers DAAH04-96-1-0172 and DAAH-96-C-0086. Also, Dr. Raju Namburu at the U.S. WaterwaysExperiment Station in Vicksburg, MS has been instru-mental in the present efforts and findings. Supportin part by Dr. Andrew Mark, Mr. William Merma-gen, Sr., and Mr. C. Nietubicz of the IMT Computa-tional Technology Activities and the ARL/MSRC fa-cilities is also gratefully acknowledged. Special thanksare also due to the CICC Directorate and the Mate-rials Directorate at the U.S. Army Research Labora-tory, Aberdeen Proving Grounds, Maryland. Otherrelated support in form of computer grants from theMinnesota Supercomputer Institute (MSI), Minneapo-lis, Minnesota is also gratefully acknowledged.
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990American Institute of Aeronautics and Astronautics
"Material set used in current investigation.*Terms in parentheses denote lower bounds, where applicable.e Analytical 2D study for differing weave parameters.^Effect of undulation studied using 2D plate approximation technique."Experimental results.'Varying waviness ratio.'Varying waviness ratio (comparison with11).
