1996 Three Dimensional Modeling of Woven

8/12/2019 1996 Three Dimensional Modeling of Woven

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Composites Science and Technology 56 (1996) 209-223@ 1996 Published by Elsevier Science Limited

ELSEVIER 0266-3538(95)00111-5Printed in Northern Ireland. All rights reserved

0266-3538/96/$15.00

THREE-DIMENSIONAL MODELING OF WOVEN-FABRICCOMPOSITES FOR EFFECTIVE THERMO-MECHANICALAND THERMAL PROPERTIES

A. Dasgupta, R. K. Agarwal & S. M. BhandarkarCAL CE Elect roni c Packaging Research Cent er, Un iv ersit y of M aryl and, Col lege Park, M aryl and 20742, U SA

AbstractThi s paper present s effect i ve t hermo-mechanical andt hermal pr opert ies of plai n-w eave fabric-reinforcedcomposite laminates obtained from micromechanicalanalyses and a two-scale asymptotic homogenizationt heory. A unit cell , enclosing t he characteristi c peri odicrepeat pat tern in t he fabri c w eave, i s isolat ed andmodeled. The ort hot ropi c t ensors for effect i ve mechan-i cal sti fi ess, coefi cient of t herma l expansion andt hermal conducti vi ty are obtai ned by numeri callysolvi ng appropri ate mi croscale boundary val ue prob-l ems (BVPs) i n t he uni t cell by the use oft hree-di mensi onal fi ni t e el ement analy ses. Furt her,analy t ical models consisti ng of seri es-parall el t hermalresist ance netw ork s are developed i n order, t o obt ainort hotr opic t hermal conduct iv it y. The numeri cal andanaly t ical models are expli citl y bused on t he propert iesof the constituent materials and three-dimensionalfeatures of the w eave sty le. Resul t s obt ai ned fr om t hemodels are compared w it h experi mental val ues andw it h models avail able in the li terature. N on-l i nearmechanical constit uti ve behavi or due to resinstress/str ain non-l inearit y and to tr ansverse yarndamage under in-plane uni-axial loads are alsoi nvest i gated. Parametri c studi es are conduct ed t oexamine t he effect of vary ing fi ber vol ume fractions onthe effective thermal properties.Keywords: fabric composite, plain-weave, effectiveproperties, stiffness, coefficient of thermal expansion,thermal conductivity, analytical model, finite elementanalysis, homogenization

NOTATIONa-eABVPC

Geometric parameters for characterizingfabric weave styleVector of normalized characteristic lengthsin the microscaleBoundary value problemTensor of microscale constitutiveproperties

FgGKMF1LP

4RSstTUVVvsX0r6E4

h4Pu;

Forcing function in homogenization theoryWeight functions for nonuniform boundaryconditionsGradient of dependent variable in homog-enization theoryKnappe and Martinez-Freire (Ref. 9)Vector of characteristic lengths in themicroscaleVector of characteristic lengths in themacroscaleFunction relating macroscale effective pro-perties to the constituent properties andthe microscale geometryHeat flux vectorThermal resistanceArc length of the yarnShear strength [eqn (38)jTimeTemperatureDisplacement vectorVolume fraction of fibers in compositeDependent variable in homogenizationtheoryVishnevskii and Shelenskii (Ref. 10)Tensile strength [eqn (38)](Volume averaged) effective propertyRadius of the arc defining a cross-sectionTime lag in responseTensor of mechanical strainMacroscale orthonormal basis vector(= Lx Y Zl =LQ r]2 7731)Radius of yarn undulationSmall parameter proportional to themicroscaleMicroscale orthonormal basis vector(= Lx Y zl =L& 6 531)Response function tensorTensor of mechanical stressVolume of microscale domainMacroscale domain

Subscriptsa Axial (longitudinal) direction along fibers

209


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210 A. Dasgupta, R. K. Agarw al, S. M . Bhandarkar

f FiberF Fill yarn (along y axis)r Resint Transverse direction (orthogonal to fiber

axis)Wx> Y

Warp yarn (along x axis)Axes along warp and fill directions,respectively, of the woven fabricY (Resin impregnated) yarn

2 Axis along laminate out-of-plane directionSuperscripts+ Vector quantities-n Second-order tensor quantityFourth-order tensor quantityI,, Real part of a complex quantityImaginary part of a complex quantity

1 NTRODUCTIONWoven-fabric composites constitute an important classof materials with a wide variety of applicationsranging from advanced aerospace structures andelectronic printed circuit boards to recreational andcommercial equipment. The reinforcement consists offabric weaves of multi-directional yarn bundles, whichcan be tailored for superior damage tolerance, forbalanced ply constructions and for better bi-directional dimensional stability during precisionprocessing operations. The effective thermo-mechanical and thermal constitutive properties of thecomposite depend upon (i) weave style, (ii) propertiesof the constituent materials and (iii) fiber volumefraction. Tailoring of the properties requires quantita-tive understanding of the effects of each of thesevariables on the overall properties. The micro-architectural complexities of woven-fabric compositesmake it difficult to analyse their constitutive propertiesby analytical means and numerical models are oftennecessary.Analytical models currently available in theliterature for mechanical properties include models fortwo- and three-dimensional weaves, such as themosaic model,r7 fiber-crimp (or fiber undulation)model,~* fiber-bridging model: fiber-inclinationmodek3 energy methods,4-7 slice-array modek8 andelement-array model. All of these models exploit theperiodicity of the woven fabric architecture to isolatea representative unit cell and generate bounds onaverage constitutive constants, based on assumptionsabout the unit cell geometry and its deformation field.While most of the models for two-dimensional weavesuse classical lamination theory to provide reasonablebounds on in-plane extensional and flexural rigidities,they are less successful in computing shear stiffness,Poissons ratio, out-of-plane properties and associatednon-linearities. Analytical models for orthotropic

thermal constitutive properties are almost nonexistent.Knappe and Martinez-Freire (KMF) have usedseries-parallel thermal resistance network models toobtain the effective orthotropic thermal conductivityof non-woven cross-ply laminates. The KMF modelignores (i) fiber crimp, (ii) yarn cross-sectionalgeometry and (iii) interactions with the orthogonalyarns and with the resin. Vishnevskii and Shlenskiihave developed a two-dimensional thermal resistancenetwork model for woven-fabric composites whichaccounts for fiber crimp but ignores effects (ii) and(iii) listed above.Numerical finite element analyses have beenattempted by some investigators to arrive at effectivemechanical properties., These studies use mosaic-like idealizations of two- and three-dimensionalharness-weave architectures. A detailed finite elementstudy by Yoshino et aZ.12 is restricted to twodimensions and does not derive effective compositeproperties. Ishikawa and Chou have used the fiberundulation model to provide a method for analysingthe knee behavior in the mechanical response ofplain-weave composites under in-plane uni-axial loadsdue to failure of yarns transverse to the loadingdirection. They assume uniform strain distribution inthe yarn and consider transverse failure strengths ofthe yarn for the analysis. Whitcomb13 presents athree-dimensional finite element study which utilizes afairly coarse mesh and does not exploit the geometricsymmetries of the unit cell. Whitcomb does not deriveeffective mechanical properties in his report. Nonumerical studies are available for effective thermalproperties of plain-weave fabric-reinforcedcomposites.This study reviews a two-scale asymptotic homoge-nization scheme,r4 which has been successfully appliedto periodic composites5 and specifically, to woven-fabric composites,627 and presents a micromechanicalanalysis of the periodic unit cell using the finiteelement method and thermal resistance networks,based on three-dimensional simulation of plain-weavefabric composites. The effective orthotropic tensorsfor stiffness, coefficient of thermal expansion andthermal conductivity are obtained using appropriateaveraging schemes. The effective orthotropic non-linear mechanical properties of the composite areobtained by considering non-linear constitutivestress/strain relationships for the matrix. The kneebehavior observed in the mechanical behavior due todamage under in-plane axial loads, is analysed byusing a fully three-dimensional Tsai-Hill criterion topredict transverse yarn failure. Effective thermalconductivity of the laminate is obtained using bothfinite element analysis, as well as three-dimensionalthermal resistance network models. Parametric studiesare conducted to examine the dependence of theeffective thermal conductivities on the fiber volume


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Modeli ng of w oven-fabri c composit es 211fraction. Results obtained from the models arecompared with predictions from existing models in theliterature and with experimental results.

2 HOMOGENIZATION SCHEMEThe homogenization scheme used here is a two-scaleasymptotic technique for computing the effectivemacroscopic properties of heterogeneous media withperiodic microstructure.4 The two length scales are: amacroscale representing either the characteristiclength of the composite structure, or the associatedloading and boundary conditions (whichever issmaller); and a microscale representing the periodicmicrostructure in the heterogeneous composite. Thehomogenization process uses asymptotic techniques toexpress the macroscopic variations of dependentvariables in terms of the microscale variations of thegeometric and material variables. This transforms thegoverning differential equation into a series of coupledboundary value problems (BVPs) of successivelyhigher orders, defined in the microscale. The solutionof each successive BVP acts as a non-homogeneousterm in the next higher order BVP. The boundaryconditions on the unit ceil are determined fromsymmetry. periodicity and loading conditions. Thistechnique is exploited here for obtaining accurateeffective (macroscale) properties of composite lamin-ates by explicitly modeling the microscale architectureand interactions through simulation of the unit cell.Let the macroscale region of interest in awoven-fabric-reinforced laminate be defined bycharacteristic lengths Z+ (i = 1 to 3) in the domain Q(d) with orthonormal basis vector G = Lq, Q ~1 andboundary JQ. Let the microscale be defined bycharacteristic lengths li (i = 1 to 3) in the domain ofthe unit cell with orthonormal basis vector z = I[, &&A. Let the ratios 1,/L, be denoted as A i & where AiMaxis normalized to the value 1 and A is a small parameterindicating the ratio of the microscale to the macro-scale. The microscale $ is now expanded homotheti-tally to the same scale as the macroscale ij using themapping & = rl i /(Aih) (i = 1 to 3). The central schemein the two-scale asymptotic method is to express thedependent variable, V(?j, 4, t ) (displacement vector,3, V and scalar temperature, T, in a thermalproblem), as an asymtotic series, with weak varia-tions in 4 and strong (but periodic) variations in 4:

3(ij,&t) = W(ij $ t) + AP(ij, g t)3+ A~*](ij,~,t) + . (1)

where V(G,& t) are periodic in 2 with periods oflength scale 1, and t is time. Thus the constitutiveproperty, e, of interest (either the mechanical stiffnessor thermal conductivity in this study) is assumed to be

uniform in the macroscale ?j but periodic in themicroscale 5 having period li. Treating G and g asindependent variables, the gradient operator 0, in thistwo-scale domain can be written as:

0; = (a/a77)j + h-(A-a/ag)j (2)The gradient of the dependent variable, v, can nowbe written as:

Gij=[(:),+(& )JKi,i=lto3) (3)Substituting eqn (1) into eqn (3) and separatingterms according to powers of A, we get:

+ [ (~)/P)i+ &)i(m]+A[(),(Vl)i&)jVIIy] .. (4)

Equation (4) is written in compact notation as:G, = ; Gj;j + Gj; + AG$ + . .m = c A - @ (5)n =, ,

where($ ?I =4 for Iz = 1 t co

To avoid singularities in the gradient Gji as A-+0, G$]must vanish in eqn (5). Thus, eqn (4) implies that

PO(fj,&t) = 3[O(fj,,) (6b)i.e. vlol does not depend on the microscale.15In a mechanical problem, the vector t isdisplacement 3, and its gradient G, is strain E (asecond order tensor), given as:

for y1= 1 to m

In a thermal problem, P is a scalar field oftemperature T, and the gradient G is a vector aT:


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212 A. Dasgupta, R. K. Agarw al, S. M . BhandarkarThe response ii, of a material with microscale

property e, due to the gradients G, is given by anappropriate constitutive law:iG,ZJ - 8) = Q&r - W(G,Z,t) (9)

where S is the time lag in the response, which hasbeen included in the constitutive material property forcomputational convenience. The response prepresents stress Cr n a mechanical problem and heatflux 4 in a thermal problem. For lossy materials, C isoften expressed as:

qg,t - 8) = eyc$,t> - iP(&t,s) (Wwhere c is the stored energy (capacitive) term and cis the dissipative part. In a mechanical problem, erepresents the storage modulus or elastic constitutiveproperty and e represents the dissipative componentin a viscoelastic problem. In a steady state thermalproblem, e is set to zero.Substituting eqn (5) into eqn (9), the asymptoticexpansion of the response p is obtained analogous tothe gradient as:

where+ . .m = cp-lpj,l (114n=O

pad = ~ ~ C ~ Wb)Therefore, by the arguments given in eqns (6b) and(9):

PI = 0 (12)without loss of generality.15The response function in eqn (9) must satisfy aconservation law at all times, t:

fL,+F=(j (13)where 3 is a forcing function. In a mechanicalproblem, the forcing function is the body force densityand in a thermal problem it is the heat source density.Considering cases with no body force or heat source,in the region of interest, we have a homogeneousgoverning differential equation:

9.p = 6 (14)Substituting eqn (11) into eqn (14) and arranging inascending powers of A, we obtain:

Because eqn (15), by definition, has to hold for all

values of A, it follows that each term of the seriesshould vanish identically. This leads to a series ofBVPs in successively higher orders of A, representingsuccessively higher modes of perturbation of thedependent variable at the macroscale, due tomicroscale heterogeneities and boundary conditions.In practice, higher order terms can be dropped if A issufficiently small, i.e. if the macroscale is sufficientlylarger than the microscale of the unit cell. Let usconsider first the 0(X2) BVP:

(16)Equation (16) is trivially satisfied in view of eqn (12).In other words, the solution to the 0(X2) BVPdepends only on the macroscale loads and does nothave variations at the microscale.

Substituting this result into the next higher orderBVP (O(h-)):

we get:(17)

(18)where ptll is given by eqn (11).

Equation (18) describes the microscale variations ofPI,, with the macroscale variation of p0 acting as thenon-homogeneous load term. Solving eqn (18) eitheranalytically or numerically gives the solution for cr.Neglecting the higher order BVPs, effective properties(c) of the medium are obtained from the volume-averaged gradients of vl, as follows:

.+($l) = pl(fj,t) = (Q $= (e)(G(i-j,t)) (19)

where the macroscopic gradient is given as:

(20)Further, (pt]) is continuous, leading to:

(21)The next and higher order BVPs lead to similarhomogenized BVPs, giving higher order approxima-tions for the effective property. For sufficiently smallA, however, the solution of O(h-) BVP provides anadequate estimate of the volume-averaged property

(0.In the present paper, only non-lossy materials are ofinterest and the equations for (e) are obtained from


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Modeli ng of w oven-fabri c composit es

eqns (7)-(20) by replacing (c) with (c). Forconvenience of further discussions, the microscalecoordinates g = L& 6 &A are set to Lx y zl and themacroscale coordinates +j = Lql n2 ~1 are set toLx Y 21.

3 UNIT CELL MODELThe microscale domain consists of the unit cellenclosing the smallest periodic repeat volume in thelaminate. The in-plane dimensions of the unit cell aredetermined by the yarn pitch along the warp and filldirections for a plain-weave reinforced laminate. Theout-of-plane dimension is determined by the thicknessof a single lamina, assuming that the laminate isconstructed by stacking similar laminae. The amountof stagger between the adjacent plies determines theboundary conditions on the top and bottom faces, asdiscussed in Section 4. The vertical faces of the unitcell are faces of periodicity. In the analyses that followit is assumed that: (a) the laminate is not onlyorthotropic (with the material principal in-plane axesof the laminate coincident with the warp and filldirections), but also that the warp and fill directionsare identical, giving the laminate a balanced-plyconstruction and cubic symmetry for (e); (b) yarnfiber bundles impregnated with resin can be modeledas transversely isotropic unidirectional compositeswith material principal axes aligned tangential to theyarn axis; and (c) contact between dissimilar materials

is perfect, i.e. displacements and tractions arecontinuous and thermal contact resistance is neg-ligible. Transversely isotropic properties of the yarnare obtained first by analytical means, and then used

213

in the unit cell numerical and analytical modelspresented in Section 4.The finite element model of a unit cell for theplain-weave fabric composite is shown in Fig. 1. Themodel uses eight-noded, three-dimensional, linear,

isoparametric brick elements, along with some wedgeand tetrahedral elements at the yarn/matrix inter-faces. This mesh utilizes a total of 2208 elementswhich lead to a total of 2751 nodes with 169 on thetop and bottom faces and 257 on side faces of the unitcell. Automatic mesh generation capability is de-veloped for varying values of the dimensionalparameters a-e shown in Fig. 1. This enables themodel to be used to study the dependence of laminateproperties on microstructural geometry and volumefractions. The dimensions of the yarn bundles andresin regions in Fig. 1 are not to scale, for the purposeof clarity. However, realistic dimensions measuredfrom micrographs of typical printed wiring board(PWB) cross-sections are used for the actualcomputations.-* Since these dimensions usually varyfrom point to point in the micrograph, average valuesare used in the numerical model, so that the unit cellof Fig. 1 is in effect an average unit cell. As anapproximation, the minor differences in the warp andfill yarn geometry are ignored. Also, parameter b is

2t

Unit Cell

Fig. 1. Unit cell for plain-weave fabric-reinforced composite showing dimensional parameters.


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214 A. Dasgupt a, R. K. Agarw al, S. M . Bhandarkarassumed to have non-zero values for modelingconvenience. Each yarn bundle is assumed to have alenticular cross-section (formed by two arcs) whichremains constant along the length of the yarn.Further, each yarn is assumed to lie along a splinewhich lies in the vertical plane. The fiber volumefraction of the laminate, which is equal to the fibervolume fraction of the unit cell, is written in terms ofcell parameters as:

2As(c + d)2(4a + b + 2e)1 224

where A is half of the yarn cross-sectional area, s isthe arc length of the yarn and up( is the fiber volumefraction in the yarn as opposed to vf, which is the totalfiber volume fraction in the woven fabric laminate.The area A is computed as follows:A = r2{cosf1[y) - fsin[2cos-i(y)]) (22b)

where I is the radius of the arc defining thecross-section (Fig. 2) given by:

c2 t a2r=- 2a (22c)The arc length s is written as:

S = K COS- [K - (a + 0%)

K

where K is the radius of yarn undulation, shown in Fig.2. K is given in terms of weave parameters as:(c + d) + (a + 0.5b)2

K= 2(a + 0.S) (22c)The transversely isotropic yarn thermo-mechanicalproperties for a given un are computed using a

generalized self-consistent version of Mori-Tanakamethods,*20 from the properties of single fiber andplain resin. The transversely isotropic yarn thermalconductivities are computed using Claytons semi-

radius, r radius, K

f

:e2.ab2ae

Fig. 2. Schematic diagram of unit cell showing yarngeometrical parameters.

empirical equation.8~21~22t is important to note thatin the fabric composite, the yarn principal materialdirections change continuously along the longitudinalyarn direction due to yarn curvature. In the finiteelement model, the principal material directions arecomputed at each element centroid.

The finite element solution for the model in Fig. 1requires considerable computational effort, especiallyfor non-linear problems. Therefore, an attempt ismade to reduce the model size by further exploitingthe symmetries and antisymmetries of the geometriesand loads of the unit cell. As a first orderapproximation, the warp and fill yarns are assumed tohave the same cross-sectional geometry and pitch.This not only introduces additional symmetries in theunit cell, but reduces the number of independentconstants that need to be computed. Since materialproperties in the X and Y directions are now thesame, the unit cell has cubic symmetry. Thus, only sixof the nine elastic constants, two of the three CTEsand two of the three thermal conductivies need to becomputed for complete characterization. Note thatthis is different from assuming transverse isotropy inthe plane of the composite. Suitable exploitation ofthe symmetries and antisymmetries, with respect tothe load and boundary conditions about the unit cellmid-planes, allow the use of only the half and quartercells (Fig. 3) for generating all the effective properties.

4 MICROSCALE BVP AND EFFECTIVEPROPERTIESThe microscale BVP formulated in Sections 2 and 3within the unit cell of plain-weave composites isanalysed in this section to obtain effective propertiesat the macroscale. Computations are performed andresults presented for uniform boundary conditions,because only O(K) BVP is of interest when solvingeffective bulk properties.4.1 Thermo-mechanical properties4.1.1 Finite element solutionA homogeneous deformation boundary condition,compatible with a uniform strain field is applied to theboundaries of the unit cell. In other words, pol(+j) ineqn (1) is chosen such that Glol is independent of q.In the stacking sequence of the laminate, it is assumedthat each lamina is displaced by one unit cell lengthfrom the lamina below it, along the two in-plane axes.The symmetry of the unit cell and the assumption ofstaggered laminae form the basis of the boundaryconditions used to generate the thermo-mechanicalproperties from different loading conditions.

The symmetry/antisymmetry of the material micro-structure about the origin and the orthogonal planesformed by the axes are simulated by enforcing


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Modeli ng of w oven-fabri c composites 215

Fig. 3. Half and quarter cells showing material symmetries/antisymmetries.

symmetries in the solution for the dependent variableover different portions of the quarter (or half) cell.Symmetry of the solution in a finite element model isachieved through coupling of nodal degrees offreedom. In the thermo-mechanical problem, thedegrees of freedom are nodal displacements. There-fore, implementation of these coupled displacementboundary conditions requires the presence ofsymmetrically/antisymmetrically located nodes in thequarter and half cells. The automatic mesh generationsoftware is tailored to ensure that this condition issatisfied.The accuracy of loads and boundary conditions onthe unit cell are verified by ensuring that the strainenergy contribution from the dominant deformationmode is at least two orders of magnitude larger thanthe contributions from all other modes for eachparticular load case. Each load case then generates anindependent equation for determining the constitutiveconstants. A total of four load cases are used for thestiffness properties: two extensional load cases alongthe in-plane and out-of-plane directions for theextensional stiffness and Poissons ratios; and twoshear load cases for the in-plane and out-of-planeshear stiffnesses. One thermal strain load case is usedfor both in-plane and out-of-plane CTEs. As anexample, the boundary conditions on the quarter cellfor in-plane extensional loads (for computing E,(= EY), ~XY, vxz = vyz)) are discussed in detailbelow.A uniform strain field (1 PE) is applied in the xdirection and boundary conditions are applied on thefaces of the quarter cell, allowing free Poissonscontractions in the y and z directions. The boundaryconditions are given below. On the external faces the?[I($) field is assumed to be:All nodes on face x = --x0 have uniform u = -xg( 10e6)

y = -y. have uniform uz = +zo have uniform w (23)

On the inner faces, the following two sets of boundaryconditions are imposed:

All nodes on x axis have u = w = 0y axis have u = w = 0z axis have u = u = 0 (24)

Nodes on the planes x = 0 and y = 0 satisfy boundaryconditions given below, which couple the surfacedisplacements on the lower two quadrants of themidplanes to the displacements of the upper two:

u(0, -4, Y) = -u(O, -9, -r)up, 4, r>=u(O, 4, -r>~(0, -4, Y) = -w(O, -9, -Y) andu(-p, 0, r) = +p, 0, -y)u(-p, 0, r) = -u(- p, 0, -r )w (-p, 0, r) = -w (-p, 0, -r) (25)

In the notation used here, u(p,q,r) is the displace-ment of the node located at the coordinate (X =p,y = q, z = r). The effective in-plane extensionalmoduli and Poissons ratios are calculated using eqns(19)-(21) as: .x=EySJ

xx

E()Y A-y= -&

where (&I) and (.8]) are the stresses and strainsaveraged over the volume of the unit cell. Thesuperscript [l] indicates it is the solution of order


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216 A. Dasgupta, R. K. Agarw al, S. M . BhandarkarO(A-r) BVP. Equation (21) is implemented in finiteelement analysis using Gauss-Legendre quadrature.23The boundary conditions for the five other load casesare developed along similar lines and are discussed indetail elsewhere.6,19,24

4.2 Thermal conductivityThe microscale boundary value problem for thermalconduction is solved both numerically by using finiteelement methods and analytically through thermalresistance network models. Equations (19)-(21) aresolved in a quarter of the unit cell to obtain effectivethermal conductivities.4.2.1 Finite element solutionIn case-l temperature, and in case-2 heat fluxboundary conditions are applied on the facesorthogonal to the conduction axis with the four lateralfaces insulated to realize periodicity boundaryconditions. The temperature boundary conditionsyield an upper bound estimate of the thermalconductivity and flux boundary conditions the lowerbound. The difference between the bounds is used asan indicator of the adequacy of the mesh density.

The unit cells in adjacent plies are assumed to bestaggered by half of the unit cell length along x as wellas y axes. This leads to insulated top and bottom facesof the unit cell for in-plane conduction. However, forout-of-plane conduction, a uniform temperature orflux is assumed on the top and bottom faces for the0(X) problem.The conduction axis for in-plane conduction kx( = ky) is the x axis, and either temperature or fluxboundary conditions are imposed with respect to it. Inboth cases symmetry/antisymmetry of the material,geometry and loading about the vertical center planes,xy and yz, exist within the unit cell of a balancedplain-weave fabric-reinforced laminate. The symmetryis utilized in reducing the domain of analysis to thequarter cell without loss of generality. The reduceddomain requires a combination of temperatureboundary conditions and heat insulation on the innerfaces. The boundary conditions on the quarter cell forinplane heat conduction along the x: axis (or y axis)are discussed below as an example.

For case-l, boundary conditions are listed below toapply a vlol ( = Tlol for this problem) linearlydependent on +j. The superscript [0] indicates thenon-homogeneous term in the asymptotic series.

1. A uniform temperature Tlol = TI is prescribedon the external face normal to the conduction xaxis of a quarter cell at x = -(c + d).2. A uniform temperature Tiol = To s prescribed atall points along the axis (O,y,O), along with thefollowing constraint:

T(O,y,z) + T(O,y, - z) = 2 = 0 (27)

Points along (x,0,0) axis are assigned normalheat flux component qpl = 0, along with thefollowing temperature constraint:

T( - x,O,z) = T( - x,0, - z) (28)The remaining three lateral faces y = (c + d),and z = *(2a + 0.3~ + e) are assigned normalheat flux component q ] = 0, to simulateinsulated surface.

Heat flux boundary conditions in case-2 haveuniform heat flux qF1 = q. applied to the external facex = -(c + d) and the conditions 2-4 listed for case-l.The effective in-plane thermal conductivity is com-puted as:

kx= -$ (29)dX

where (qi ]) and (~?T[]/ax) are volume averaged heatflux and temperature gradient, in the order O(A-)BVP, respectively. In finite element analyses, thevolume averages are computed using Gauss-Legendrequadrature.23 The boundary conditions and computa-tion for the out-of-plane thermal conductivity kZ areperformed along similar lines and are discussedelsewhere.73244.2.2 Analytical solutionTwo different thermal models are proposed, one forin-plane and another for out-of-plane effectivethermal conductivity, of a plain-weave fabric-reinforced laminate. In both models, the domain ofthe microscale boundary value problem is a quarter ofthe unit cell. Equivalent thermal resistances areselectively assembled in series and parallel to formthree-dimensional networks in order to simulate theactual heat conduction through the domain.The non-homogeneous term T[]($ is assumed tolead to a uniform gradient dTLol. The effectivelaminate thermal conductivity along the conductionaxis i , ki, is computed from the effective thermalresistance along that axis, Ri, as follows:

(30)The superscript [l] indicates it is the solution of where1 s the unit cell dimension along the conduction axis,and A is the area of the quarter cell cross-sectionorthogonal to the conduction axis. The effectivethermal resistance is obtained by numerically integrat-ing the thermal resistance of a suitably chosendifferential element through the quarter cell. Weaveparameters such as yarn cross-sectional geometry andcrimp are explicitly modeled by the integrationscheme. Due to the complexity of the geometry, the


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Modeli ng of w oven-fabri c composites 217integration is carried out numerically, using 15 X 1.5point Gauss-Legendre quadrature.23 The thermalinteractions among the mutually orthogonal yarns andthe surrounding resin are modeled by constructingequivalent series-parallel nets of thermal resistances.Thus, this model can be viewed as a piecewise rule ofmixtures.The conduction axis for the in-plane model isarbitrarily chosen to be the x axis, since x and y areidentical directions with respect to geometry of theunit cell. Figure 4(a) shows the orientation of adifferential element chosen for the computation of theeffective in-plane thermal resistance, R,. Also shownin the figure are three surfaces Z,, I, and I3 which areused in the model to partition the quarter cell intofour subdomains El, E2, E3 and Ed. These surfaceshelp simulate the thermal conduction paths bytracking the yarn boundaries and allowing heatexchange mechanisms between the subdomains, asobserved in related finite element simulations.Surfaces Z, and 1, are coincident with the interfacebetween the yarn and the surrounding resin. Thesurface Z2 follows the yarn/resin interface of thelongitudinal yarn, along the conduction axis for x 5 p.At x = p, surface Z, is allowed to ingress into thelongitudinal yarn, as shown in Fig. 4(a). This allowsthe heat flowing through the transverse yarn to

preferentially flow through the longitudinal yarn whenfaced with the high thermal resistance of the resindownstream of x =p. This modeling feature ismotivated by phenomenological observations fromnumerical finite element simulations.7~20 The mag-nitude of the parameter p is obtained in the interval0 sp 5 c, by minimizing the thermal energyz5contained in the quarter cell.

In the integration scheme, the effective in-planethermal resistance R, of the quarter cell, is obtainedby integrating the thermal resistance, dR, of adifferential slice (see Fig. 4(b)) of thickness d_x (seeFig. 4(a)):

R, = [c+d)dR = [-R(x)& (31)The slice resistance is obtained by connecting effectivethermal resistances of subdomains dE,, dE2, dE3 anddE4 in parallel. This scheme assumes as anapproximation that the temperature is uniformthroughout each cross-section. It is convenient forsubsequent development to cluster the resistance ofdE, and dE2 together as dR1, and that of dE3 and dE4as dR2. Hence:

R x) = dR,dR2dR, + dR2

(a)

Transverse/ fill yarn

h

(b)

(32)

Fig. 4. Thermal model for in-plane conduction. (a) Unit cell showing orientation of the differential slice element, (b) view ofthe differential element showing the warp yarn zWIand zW2due to ingress of the plane Z2, and the fill yarn zf, and (c) thermalresistance net comprising an infinite series combination of parallel stacks of resistive elements, each stack representing adifferential element.


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218 A. Dasgupta, R. K. Agarw al, S. M . BhandarkarThe resistances dR, and dR2 are written in integralform in terms of the parameters of the differential

slice shown in Fig. 4(b), as follows:(c+d) 1dR, = [I (gw,zw& + g,z&)dy0 1(c+d)dR, = (gw2zw&x + g&fx + g,znk,)dy I - (33)

where z represents the heights of subdomains dEi,i = l-49 (see Fig. 4(b)) and k represents the thermalconductivities. Subscripts w, f and Y refer tolongitudinal yarn, orthogonal yarn and resin, respec-tively. The g parameters are weight functions obtainedfrom the applied loading at the boundaries and insidethe domain, and are useful for modeling of the effectsof nonuniform boundary conditions and distributedheat flux sources in the domain. In the present study,temperature boundary conditions TIol are assumed tobe uniform perpendicular to the conduction axis.Therefore, the weight functions g(x,y) are set equal toone.

The z(x,y) terms are computed based on the yarncross-section and crimp geometry., zrZ (Fig. 4(b)) isnot an independent variable and is defined as:

zrz= h - (zwl+ zw2) zr - zr/ (34)and kLr and k , are the yarn conductivities along andtransverse to the conduction x axis, respectively, andare given by the tensor transformation:

kl X k, cos* 8 + k, sin 8k, = k, sin* 8 + k, cos* 8 (35)

where 8 is the slope of the yarn axis with respect tothe x axis (see Fig. 4(a)). When the yarn crimp ismodeled by circular arcs of radius K (eqn (22e)), theangle 0(x) (Fig. 2) is given by:

e x) = cos--ICK:()where(Y(X)= K K - X2)+ (37)

A schematic representation of the equivalentthermal resistance network is shown in Fig. 4(c). Thenet comprises of an infinite series of resistive elementsof infinitesimal thickness du, indicated symbolically.Since dE3 is not defined for x 4 p, the resistance ofthe longitudinal yarn in this region is entirely due toelement dE2, i.e. dRwl = dR, and dR,z = 0. A similarthermal resistance model developed for computingout-of-plane thermal conductivity is discussed in detailelsewhere7,8 and is omitted here for brevity.

5 RESULTS AND DISCUSSIONThe numerical and analytical models are used forpredicting the linear thermo-mechanical and thermalproperties of E-glass/epoxy (FR-4) fabric-reinforcedlaminates. Photomicrographs of sample E-glass/epoxy(FR-4) laminates with uF= 0.35, and ufY= 0.65, aredigitized in order to determine the dimensionalparameters of Fig. 1. The fiber volume fraction in theyarn, ufy, is obtained from manufacturers literatureand photomicrographs, while the overall fiber volumefraction in the composite, L+, is obtained from aciddigestion tests. Input material properties for theE-glass fiber and epoxy resin are given in Table 1.5.1 Thermo-mechanical propertiesLinear properties are determined for E-glass/epoxylaminates. Non-linear constitutive behavior of thelaminates due to non-linearity in the stress/strainrelationship of the epoxy matrix and due toprogressive failure of the transverse yarn (kneebehavior) are also presented in this section5.1 I L i near t hermo -mechanical propert i esThe effective yarn properties are computed using theMori-Tanaka method,19~20for ufy = O-65, followed bycomputation of the effective laminate properties asdescribed in Section 4.1. Table 2 shows thenumerically obtained thermo-mechanical properties ofthe E-glass/epoxy laminate and comparison betweenexperimental results and typical values reported in theliterature. The numerical values show good agreementwith the experimental results and fall within theranges reported in the literature. Dependence oflinear properties on fiber volume fraction is discussedelsewhere.1y,25. I .2 Matrix non-l ineari tyThe yarn itself is assumed to remain linear elastic, formodeling convenience. A bilinear stress/strain curve isused for the epoxy resin in the matrix, with initiallinear-elastic properties as in Table 1, an elasticlimit of 70*0MPa, and residual stiffness of 0.18 GPabeyond the elastic limit. Figures 5-7 are plots of thenon-linear thermo-mechanical properties of theE-glass/epoxy laminate obtained from this non-linearmodel. Values for the material constants can beobtained from the slopes of the graphs. The in-plane

Table 1. E-glass fiber and epoxy resin propertiesProperty E-glass (fiber) Epoxy (matrix)E (GPa) 724 3.4.5Y 0.22 0.37a (&C) 5.4 69.0k (W/m/K) 1.03 0.19


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Modeli ng of w oven-fabri c composit es 219Table 2. Linear thermomechnical properties of E-glass/epoxy fabric-reinforced laminates (fiber volume frac-tion u, = O-35)

Numerical Experimental Literature228E,, E, @Pa) 19.7 18.8 17YV. 0.14 0.14ax, (YYWC) 17.6 15.0 12-16az (/.&C) 54.2 58.1 40-70

stiffnesses, which are governed primarily by the fabric,are affected to a lesser extent by resin non-linearitythan the out-of-plane stiffnesses.5.1.3 Damage-i nduced non-l ineari t yKnee behavior of fabric-reinforced composites is aphenomenon by which the stress/strain curve exhibitsa knee or change in slope when the laminate issubjected to an extensional in-plane load. The knee iscaused by progressive failure of the yarn perpendicu-lar to the load, as shown in Fig. 8. Ishikawa and Chouhave used the fiber undulation model to predict kneebehavior for glass/polyester composites. Classicallamination theory is used with the additionalassumptions that there is no bending-extensioncoupling and the strain in the thickness direction isuniform. Failure of the yarn is governed by a specifiedvalue of breaking strain in the transverse direction.Kimpara et aLz9 have utilized a two-dimensional finiteelement model of the composite cross-section tosimulate knee behavior. Failure of transverse yarnelements is assumed to occur when the maximumprincipal strain in the element reaches a specifiedthreshold value, after which the stiffness of theelement is assumed to drop to 1% of the originalvalue.

_._o5 p&q

29 0.4-H EXJEYI 0.3- /

O-D,\ 0.04Normalstrain

o.ooY . I . 1 . / . I .0.00 0.01 0.02 0.03 0.04 MEllghdng shear strainFig. 5. Non-linear extensional modulus for E-glass/epoxy Fig. 7. Non-linear shear modulus for E-glass/epoxycomposites. composites.

0.00 0.01 Oh Oil3 wNormal strain (appkd)

Fig. 6. Non-linear Poissons ratio for E-glass/epoxycomposites.

In the present simulation, a fully three-dimensionalTsai-Hill criterion3 is used for defining failure of theyarn elements which, unlike previous investigations inthe literature, takes into account all the orthotropicstrength characteristics of a unidirectional orthotropiclamina. The Tsai-Hill failure criterion is defined by:1=(~~+(~)+(~~+(~~+(~)

+p)+ ( --& 2n3( 1 1+ -+--- (T,(T3X21 x22 X23 11 1+ L--+- cTlU2x21 xz2 x23 1 (38)

where u are the stress components and the X and S

0.10 ,I- /0.08-

0. 06-

0. 04-

0.02 -


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220 A. Dasgupta, R. K. Agarw al, S. M . Bhandarkar

Load -

Failure of transverse amperpendidar totoaddirection

- Load

Fig. 8. Schematic diagram describing the knee phenomenonin fabric composites.

are the tensile and shear strengths, respectively, in theprincipal material directions of the yarn. The strengthsused for the analysis (Table 3) are those reported byClements and Moore3 for an E-glass/epoxy unidirec-tional lamina with fiber volume fraction uW = 0.65.The strength S,, is not available in literature and isestimated from X, values by comparing the principalstress concentrations at the fiber/matrix interface forapplied far-field unit tensile and shear loads, usingMori-Tanaka analysis. Both loads are applied in theplane of transverse isotropy. A far-field shear stress ofO-726 is found to give the same stress concentration(of principal stress) at the fiber/matrix interface as afar-field unit tensile stress. Assuming that the failuremechanisms in the two cases are similar, the strengthS,, is approximated as O-726X2. The orthotropicfailure criterion is incorporated into the finite elementanalysis for both the transverse and longitudinal yarns,along the orientation of the local principal materialdirections. Post-failure stiffnesses are assumed to bezero so that the yarn elements are perfectly plasticafter failure defined by eqn (38). A non-linear analysisis performed with incremental tensile in-plane loading.Figure 9 shows the results of the finite elementsimulation. The knee initiates at 0.15% strain andcontinues up to 0.4% strain where all the transverseyarn elements have failed. The post-knee slopes forthe numerical and experimental curves in Fig. 9 are ingood agreement. However, the numerical predictionof the knee occurs earlier than the experimentalknee, suggesting that the yarn strength values used in

Table 3. Orthotropic strengths for unidirectional E-glass/epoxy laminae (u, = O-65)Stength Value (MPa)

X1 1108X, 7.5X, 7.5S12 224S13 224s23 5.45 (0.726X,)

xperimentalBhandarkar, et. al., (1991a)

0.00. 000 0.002 0. 004 0.006 o.ooB c I10Fig. 9. Knee behavior of E-glass/epoxy composites.

the numerical analysis may be lower than the actualvalues. Ishikawa and Chou and Kimpara et aZ.,29 aveobtained knees for glass/polyester composites in therange of 0.2 to 05% strain.5.2 Thermal conductivityLaminate effective in-plane and out-of-plane thermalconductivities are predicted as functions of fibervolume fraction by solving the microscale BVP forbalanced plain-weave fabric reinforcements for E-glass/epoxy. The solution to the problem is obtainedusing the analytical thermal models and three-dimensional steady-state finite element analysisdiscussed in Section 4.2. The difference between thelower and upper bounds obtained from finite elementsimulations is less than 5% for the material systemconsidered, indicating acceptability of the meshdensity. In this paper, the upper bound values arepresented which are obtained by applying temperatureboundary conditions.Predicted conductivities are compared with avail-able experimental results and with predictions byKMF9 for different fiber volume fractions, as shown in

0.5 E-glasslcpoxy

-Io.2 20Fiber vchme~crion, vf ( )

4l

Fig. 10. Thermal conductivities of E-glass/epoxy compositesas a function of fiber volume fraction.


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Modeli ng of w oven-fabri c composit es 221

Fig. 10. Comparison indicates good agreement of thepredicted in-plane as well as out-of-plane conduc-tivities between the proposed analytical models andthe finite element analysis results. The effectivethermal conductivity of the laminate is sensitive to theyarn crimp and its orthotropy ratio, as defined by theratio of fiber axial thermal conductivity, kf,, totransverse conductivity, kP. As shown in Fig. 11, theKMF model prediction shows increasing deviationfrom the proposed analytical model due to increasingeffect of interactions between the warp and fill yarnand due to the yarn crimp. The KMF in-planeconductivity prediction deviates from this study by amaximum of 8% for isotropic fiber axial conductivity,12 times that of the matrix resin conductivity, k,, asshown in Fig. 11(a). The deviation is an increasingfunction of the fiber axial conductivity for fibers withlow transverse conductivity, as indicated by the case of

%

L,& k,= 10II 10 20 30 40 50

Fiber axial conductiwty, k,J k,a>

0v,=28%

7. I -.-,..-0 10 20 30 40Fiber axial conductivity, k,J k,

@IFig. 11. Deviation of KMF model predictions fromproposed analytical model. (a) In-plane thermal conductivityand (b) out-of-plane thermal conductivity.

fiber having an orthotropy ratio of 10. Theout-of-plane conductivity predictions of the KMFthermal model come closer to the experimental,analytical and finite element results of this study thanthe in-plane predictions due to the dominance of resinconductivity. Deviation of the out-of-plane conduc-tivity from KMF model increases with fiber axialconductivity for low orthotropy ratio of 1, as shown inFig. 11(b). The deviation decreases for low fibertransverse conductivity, as indicated for a fiberorthotropy ratio of 10. The model of Vishnevskii andShlenskiil neglects the transverse yarn and thesurrounding resin, resulting in large deviations fromthe finite element analysis results and experimentallyobtained values, as indicated in Table 4.

6 CONCLUSIONSThe effective thermo-mechanical and thermal pro-perties of plain-weave composites are determinedusing a two-scale homogenization theory. The periodicreinforcement pattern, which is inherent in the weaveof fabric composites, has been effectively exploited inhomogenization theory for defining the domain of amicroscale boundary value problem and for determin-ing the periodicity boundary conditions. A three-dimensional finite element model of the microscaleunit cell is developed with automatic mesh generationcapability for accurate simulation of variable weavemicroarchitectures. Appropriate boundary conditionsare defined on the unit cell for determination of acomplete set of orthotropic properties for theplain-weave laminate. This analysis allows simulationof linear and non-linear behavior of the laminate asfunctions of varying fiber volume fractions. Thisprovides valuable information to laminate designerson the variabilities that are expected in laminateperformance due to variations in resin content.Non-linearities modeled in this study are those due tomatrix material non-linearity and due to damage oftransverse yarns. Degradation in the mechanicalbehavior is investigated by analysing knee behaviorof the composite using the Tsai-Hill criterion topredict progressive failure of the transverse yarn.Results show good agreement with experimental dataand with data from the literature.

Non-linearities due to large deformation of the yarncan also be modeled by this approach and aredeferred to a future study. The present results andsimilar models in the literature are valid when themicroscale of the heterogeneities are small comparedto the macroscale of the composite material. Thetwo-scale asymptotic technique discussed in this paperprovides a convenient formalism for extending thisstudy when the micro and macro length scales are nolonger vastly different.

Effective thermal conductivity predictions from


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222 A. Dasgupta R. K. Agarwal S. M . Bhandarkar

Table 4. Comparison of Vishnevskii and Shlenskii (VS) thermal modelresults with experimentally obtained values and with finite element simula-tion results for E-glass/epoxy laminates; values are thermal conductivitiesin W/m/KFiber volume fraction VS FEA Experimentsin laminate, ur

kx, k, k, kx, k, k, kx, k, k,O-28 0.73 0.85 0.36 0.27 0.35 0.28

using the KMF models are strictly applicable to[O/90] laminates, and are within 12% of the currentpredictions that account for the yarn crimp inwoven-fabric composites. This is true for mostfiber/resin materials and low fiber volume fractionsused in the electronic packaging industry. The thermalconduction properties of low conductivity resin andlow fiber volume fraction laminates are dominated bythe resin conductivity.ACKNOWLEDGEMENTSThis work has been funded by the National ScienceFoundation and the 26 members of the CALCEElectronic Packaging Research Center at the Univers-ity of Maryland, USA.

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