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Composites Science and Technology 51 (1997) 573-586

0 1997 Elsevier Science Limited

ELSEVIER Printed in Northern Ireland. All rights reserved

PII: SO266-353X(97)00018-3 0266-353X/97/$17.00

CALCULATION OF EFFECTIVE TRANSVERSE ELASTIC MODULI OF FIBER-REINFORCED COMPOSITES BY

NUMERICAL HOMOGENIZATION

Pericles S. Theocaris,” G. E. Stavroulakisb & P. D. Panagiotopoulos’

“Nutional Academy of Athens, PO Box 77230, 175 IO Athens, Greece hTechnical University of Crete, Greece

“Aristotle University of Thessaloniki, Greece

(Received 9 September 1996; revised 27 November 1996; accepted 11 December 1996)

Abstract 1 INTRODUCTION

Effective transverse elastic moduli for fiber-reinforced

composites are calculated here by a numerical

homogenization approach. The effects of fiber

placement (staggering) and of weak-fiber and strong- matrix composites on the effective moduli, both of

which are not very effectively treated by classical methods, are specifically investigated. Comparisons with classical, analytical approaches are included. 0 1997 Elsevier Science Limited

Keywords: composites, numerical homogenization,

effective moduli

Although detailed numerical modeling of the mechanical behavior of a composite structure is in principle

possible by means of the finite-element or the boundary-element method, the effort required for its realization is disproportionally high. Moreover, effective, low-cost models permit us, through parametric investigations, to capture the essentials of the structural behavior of a composite and are indispens- able for design purposes. Effective modulus theories are developed for this purpose. It must be underlined here that the mechanical behavior of a composite structure may deviate from that predicted from its constituents, or from the prediction of a mixture rule based on the properties of its components. NOTATION

8(‘), k = III, I = 1,2,3

E,, ij = 1,2,3

E,, &

E LL~, h-c

Uf

\/ij

V In, Vf

\/LLc, yTTc

d(‘), k = &II, I = 1,2,3

Deformation vectors for the two- dimensional unit cell k subjected to the unit loading case (1) (see Fig. 1 for definitions) Elastic moduli for the two- dimensional anisotropic elasticity Isotropic elastic modulus of matrix and fiber, respectively Longitudinal and transverse elastic moduli for a transversely isotropic, fiber-reinforced composite

Fiber volume fraction of the composite Poisson’s ratios for the two- dimensional anisotropic elasticity Isotropic Poisson’s ratio of matrix and fiber, respectively Longitudinal and transverse Poisson’s ratios for a transversely isotropic, fiber-reinforced composite Stress vectors for the two-dimensional unit cell k subjected to the unit loading case (1) (see Fig. 1 for definitions)

573

This is especially true if an innovative composite design is studied. Weak-fiber/strong-matrix fiber- reinforced composites, used as models to represent the mechanical behavior of porous materials where the inclusion phase may be either a void or a soft material, constitute classes of newly considered composites which belong to this category. Even completely unexpected mechanical behavior may be found, as is the case with re-entrant-corner foams which exhibit a negative Poisson’s ratio.‘-”

The purpose of this paper is to study the effective transverse elastic moduli of some classes of fiber- reinforced composite materials by numerical homogenization techniques and finite-element modeling of representative cells and to compare the results with more classical, analytical approaches.’ In particular, the effect of microstructure-induced anisotropy is investigated, a point which is not effectively treated by analytical methods.

Classical effective modulus predictions are usually based on the consideration of a single fiber cross-section surrounded by a matrix layer and embedded in a continuum, which disposes the required effective elastic properties. Analytical expressions

574 P. S. Theocaris et al.

Fig. 1. Elements of the numerical homogenization technique for a cell. Case I, homogeneous cell; case II, cell of a real structure; cases (l), (2) and (3), unit prestresses.

for the stress and strain fields of this model and appropriate boundary conditions lead to the required estimates.738 This approach has been improved by the introduction of the mesophase concept in previous studies.‘.’ Roughly speaking, a number of variables have been introduced into the model previously outlined in order to capture the mesophase between the fiber and the matrix. The change of the mechanical properties between the two constituents is no longer abrupt but follows a given law. Thus, the mechanical behavior of the interface layer is more accurately taken into account in the overall mechanical moduli.

Nevertheless, complicated fiber cross-sections (e.g. elliptic ones) make the study of the analytical models prohibitively complicated. In addition, mechanical interaction between closely spaced fibers and the effect of the way they are arranged in the cross-section of the composite cannot be taken into account by analytical methods. Computer simulation permits us to study these effects.

Numerical homogenization techniques for the derivation of effective elastic moduli are based on an exact model of a representative periodic cell of the composite structure. This cell, with all the details of the matrix, the fiber and the interface behavior, is

Calculation of effective transverse elastic moduli 575

modeled by a finite-element procedure or, rather rarely in the literature, by the boundary-element method. A number of appropriate stress analysis problems are considered by respecting the periodicity relationships. The effective moduli are computed by using the results of these analyses to adjust the parameters of a reference, mostly anisotropic, elastic solid. Recent contributions in this area, which also include hints for the effective numerical realization of the procedure into existing general-purpose codes, and extensions to non-linear problems include Refs 10

and 11. In this paper a method introduced previously5 for

transversely isotropic elastic bodies, which is based on a finite-element model of a representative periodic cell, is used for the solution of the homogenization problem. The effective transverse elastic moduli of fiber-reinforced composites with a regular, or irregular, but periodic arrangement of the fibers are derived numerically. Both strong-matrix and strong-

fiber composites are considered. The results are compared with the predictions of classical approaches. A discussion of the relative merits of the two

approaches is also included.

2 ELEMENTS OF NUMERICAL HOMOGENIZATION

A relatively simple technique, easily implementable within general-purpose finite-element programs, for the solution of the numerical homogenization problem is described in what follows. The complete theory and more refined methods can be found in the specialized

literature.“,l’ This method is used for the numerical

investigations presented later in this paper. For both discrete (e.g. beam-like) and continuous (e.g. composites) periodic structures, an equivalent homogeneous model can be constructed by using the homogenization technique. For the discretization of the unit cell, quadrilateral, isoparametric, two- dimensionsal elasticity finite elements have been used, in connection with a Gauss 2 X 2 numerical integration technique for the calculation of the stiffness matrices. Details can be found in every book on finite-element analysis; see, for instance, Ref. 12.

Let us assume a representative unit cell of a periodic structure, which, for simplicity, is assumed to be two-dimensional (see Fig. 1). Let the unit cell be orthogonal with dimensions equal to I, and l2 along the two coordinate axes and let it occupy the area Q with boundary I. The boundary is composed of the complementary and non-overlapping parts 11, 12, I; and I; (i.e. I1 uI’~u I’; ur;= r, r, ur, =0 etc.). A unit cell of the real structure (case II in Fig. 1) and a unit reference cell with the same dimensions and with the sought homogeneous elastic moduli (case I in

Fig. 1) are considered. The cells I and II are subjected to the three unit prestresses:

Case (1): (Ti = 1, (T2 = 0, (T3 = ri> = 221 = 0

Case (2): (T,=o, uz=l, a3=2,2=22,=0 (1)

Case (3): (T1 = 0, (72 = 0, (T3 = ri2 = 221 = 1

as is shown in Fig. 1. The solution of cell I for these loading cases is

based on engineering mechanics relations, owing to the assumption that the dimensions of the periodic cell are small with respect to the dimensions of the structure.

A typical size of the finite-element discretization for

the examples presented in this paper is 400 nodes, which leads to systems of about 800 linear equations in 800 unknowns. The periodicity boundary conditions assumed by the numerical homogenization method are introduced into the finite-element model by the multipoint constraint technique, i.e. additional equations are introduced into the finite-element system of equations so as to enforce the kinematic periodicity relationships. The number of these additional equations is small with respect to the size of the problem and depends on the loading case for which the unit cell is solved and on the homogenization technique used. For instance, here, for loading cases (1) and (2) (see eqn (1) and Fig. 1) two additional equations are introduced which ensure that boundary Ii (resp. I,) remains straight after deformation. More complicated numerical homogenization models would require the introduction of additional or different

periodicity conditions (see, for instance, the recent results reported in Refs 10, 13 and 14).

For cell II a finite-element method is employed for

the solution of the above static problem, eqn (1). In addition, the periodicity restraints are taken into account in the problems described above: for cases (1) and (2) the displacements on boundaries Ii and I; along the horizontal direction 1 are the same; for cases (1) and (2) the displacements on boundaries I,and I; along the vertical direction 2 are the same; and for case (3), boundaries I,, I;, r2 and I; remain compatible with the neighboring elements and therefore may be assumed to be straight owing to the kinematic periodicity conditions holding at the boundaries of each unit cell. The periodicity conditions on the boundary displacements of the unit cell are considered by the multipoint constraints option of the finite-element method.

The essence of the energy-based numerical homogenization method is that the parameters of the homogeneous cell I are appropriately chosen, so that it has the same deformation energy as the cell of the real structure (cell II), if both are subjected to the same deformation patterns which respect the periodicity assumptions, i.e. they are periodic for the whole structure.


If the parameters which define the mechanical behavior of cell I (e.g. the elasticity constants) are gathered up in the design vector a, the numerical homogenization method can be described by the following identification problem:

Find Q as a solution of the optimization problem:

miAnd 1 i W, fl (e(‘),a) - ‘fl (e(‘)))2 .i i=l 1

(2) in I”

Here, Aad is the admissible set for the material parameters of the homogenized cell; i runs over all independent periodic deformation patterns, e”), which are considered (i.e. the three cases of Fig. 1); wi are appropriate weights, which transform the multi- objective optimization problem into a classical one, with a cost function as in eqn (2); superscript I or II stands for the quantities of cell I or II, respectively;

and n,, is the internal energy of the considered structure.

In general, eqn (2) can be solved by numerical optimization techniques. In the sequel we shall describe a simple procedure, which is based on the optimality criteria method, for the solution of a certain class of problems, eqn (2). This method avoids the formulation and the solution of large-scale optimization problems, and, if it is applicable, it is considered to be suitable for structural analysis applications.”

Let us assume for simplicity here that all w, terms are equal to unity. Moreover, the homogenized unit cell I is assumed to obey the classical orthotropic elasticity relationships, i.e. we have that (see, for example, Ref. 16):

e=

El

v21

E2 E2 = KP (3)

0 0

The design vector ck is chosen as: (Y=

[a, (~2 (~3 atIT = [l/E, - v12/E, l/E2 1/G,21T.

The internal energy is expressed by

i(i’=l aWe’ dQ

in R

for all i = III, j = 1,2,3, where R is the area of the considered cell. For simplicity we assume here that A,, = R4.

Vi2

El

Under the above assumptions eqn (2) becomes:

{(a’(‘)T(cu)e’(‘)((U) _ a”(‘)“e”(‘))2

+ (u’WT(a)e’W(Cy) _ uW2)Te”(292

+ (uJ(3T(a)e~(~)(a) _ a”(%‘e”(‘))2) dQ} (4)

For the assumed unit prestresses, eqn (l), and the elasicity relationships, eqn (3), we get:

el i(1) = cu,a:(l) = at

e;(l) = Q~(T, i(I) = cy 2

el '(2) = cy2a;'2' = (y? (5) i(2) - e2 - QIfl?

i(2) = cy 3

ei(') = ff4cr3 l(3) = cy

1

with all other components equal to zero. Equations (5) and (1) written for cell I are used in eqn (4).

Moreover, the virtual work equality for cell II reads:

I aW)TeW) dQ =

I S’lC’)Trl’lO) dI, j = 1,2,3 (6)

R I'

for all given unit prestresses of eqn (1) (i.e. S”(‘) = 1 on I,, S”(‘) = 0 on I2 and I;, etc.).

Finally, the optimality conditions for eqn (4) are written by means of eqn (5):

Find cy,, C+ and Q, such that:

I (a4 _ u”‘3Te”“‘) dQ 2 = 0 (7c) $1 J

In eqns (7) the derivatives aa,/&, (i = 1,3,4) are obviously all equal to unity. However, they are retained in eqns (7) in order to indicate the order of differentiation of eqn (4) yielding eqns (7). Indeed, eqn (7a) is derived from eqn (4) by a differentiation with variable (Y,, whereas eqns (7b) and (7~) come from eqn (4) by differentiating it with respect to (Ye or (Ye, respectively. For 0~~ one should either use an additional loading case, or directly use the results with eqn (3) as has been done here.

By using eqn (6) the area integrals are transformed into boundary integrals. Finally, we have:

(8)

For the calculation of Poisson’s ratios, v,~ and v2,, we could proceed analogously. Here, instead, we use directly the basic elasticity relationship, eqn (3), to correlate the deformations of the unit cell (II) subjected to the unit prestresses, cases (l), (2) and (3), together with the values extracted previously.15

3 REVIEW OF CLASSICAL EFFECTIVE MODULUS THEORIES

For the analytical determination of the effective elastic moduli in unidirectional fiber-reinforced composite

Calculation of effective transverse elastic mod&i 577

materials, approximate equations based on the theory of elasticity and consideration of a simplified unit cell of the composite are used. From the development of this analytical model the effective static and dynamic elastic properties of the composite are calculated.

Details of this approach, first proposed by Hashin and Rosen, are given in Ref. 8. Note that this model considers perfect adhesion between the matrix and the fiber and a sharp interface between the two phases, with discontinuous change of the elastic material properties in this zone. An improvement of the existing models, which covers accurately the effect of the mesophase between the matrix and the fiber, has been proposed and studied previously.h,y

According to this model, a cylinder of radius r, representing the inclusion is surrounded by a coaxial cylindrical cell extending between r = r, and r = vi, where Ar, = (ri - rt) is the thickness of mesophase. The elastic modulus Ei and Poisson’s ratio Yi of the mesophase layer vary with Ar, according to a negative power law, in order to accommodate the differences in moduli, E, and E,, and Poisson’s ratios, vf and v,, of the inclusion and the matrix. The thickness of the

mesophase was determined through differential scanning calorimetry measurements of the jumps appearing in the respective curves of the heat capacity of the models versus temperature for unit-cell specimens either with inclusions, AC;, or with matrix

material only, AC’:. In this way the mesophase coefficient h is determined by the relationship:

AC’ A=I-L

AC;

Then, the thickness of the mesophase is found from the simple relationship:

(rf + ATi)* _ l = Auf

6 1 - uy

Knowing now the thickness of the mesophase and the limits of the variation of the moduli, Ef and E,, and Poisson’s ratios, Y( and v,, and by evaluating the

negative exponent n of the appropriate power law of variation of the moduli inside the mesophase, we evaluate the variation of Ei = f (ri) and Yi = f (rJ inside the mesophase. In this study we simplified the procedure of defining these curves by assuming a parabolic variation of Ei and Vi between these limits. This approximation was found to be more than satisfactory.

Since these models gave the transverse moduli and Poisson’s ratios for the unit cell, the longitudinal elastic modulus and the Poisson’s ratio are given by the mixture law, as modified by the assumption of the mesophase. The Poisson’s ratio vLTc is computed from relation (11) of Ref. 9. Furthermore, by assuming that V T-,-c = YLTc. the transverse elastic modulus, ETc, is calculated from relation (10) of Ref. 9. In a more

general context the latter quantities can be measured experimentally.

As a general comment we observe here that the analytical models used assume only circular fiber cross-sections and do not take into account the interaction between adjacent fibers. Thus the effect of the fiber placement (staggering) can be studied only by numerical homogenization techniques as is done in the next section of this paper.

4 NUMERICAL EXAMPLES

A number of parametric investigations is presented where the effect of fiber placement and fiber shape on the effective transversal elastic moduli of fiber- reinforced composites is investigated. Comparisons with the predictions of analytical models are included. Emphasis is placed on the effect of induced anisotropy in the effective properties, a point which cannot be captured effectively by the analytical models.

4.1 Orthogonal placement of fibers with circular cross-section, equidistant in the two directions: effect of volume ratio We consider an orthogonal placement of circular fibers in the composite cross-section, as shown in Fig. 2, with H = N. For various fiber volume fractions the homogenized transverse elasticity modulus of the composite is given in Figs 3-6 for E,/E, = 0.1, 5, 10 and 100, respectively.

4.2 Orthogonal placement of fibers with circular cross-section: induced anisotropy due to different distances in the two directions We consider again the orthogonal placement of circular fibers in the composite cross-section of Fig. 2. The horizontal dimension of staggering, H, is kept fixed, the vertical dimension, N, changes and the fiber cross-section is kept fixed. So the variation of the volume fractions shown in Fig. 7 is considered. In the sequel the effective transverse elastic modulus in the two directions is calculated by numerical homogenization and the results are compared with those of the analytical models. It is of interest to point out that while both analytical models used for comparison with the results derived by the homogenization numerical method cannot capture the induced anisotropy in the composite arising from the variation of spacing of the arrays of fibers in the two orthogonal directions, the homogenization method was found to be very sensitive to this factor, yielding different values for the transverse modulus, E,, of the composite in the horizontal (Ehl) and the vertical directions (Eh2) of the cross-section.

For various values of N/H ratio and for parametric values of the ratio E,/E, = 0.1, 5, 10 and 100, the homogenized transverse elasticity modulus of the composite is given in Figs 8-11, respectively. It is clear from these figures that for all parametric values of the


A I

B

'1'

N/2 b

x D C

I4 H/2

u

Fig. 2. Configuration of the periodic fiber-reinforced composite.

E,/E, ratio from 0.1 to 100 the isotropy of the composite remains practically unchanged for values of the N/H ratio between 0.85 and 1.00. By contrast, when N/H varies between 0.5 and O-85, we experience a rather rapid increase in the anisotropy of the composite. Figure 12 presents the variation of the anisotropy of the composite plates for different arrangements of the fiber arrays in an orthogonal formation and for values of the sides of the orthogonal array varying between N/H = O-5 and N/H = 1.0. Four typical ratios of the modulus of the fibers (E,) to that of the matrix (E,) were considered correspond- ing to the above-cited values. It is clear from these plots that for weak-fiber composites (E,/E,, = 0.10)

the anisotropy of the composite increases rapidly as the NIH ratio diminishes, tending to NIH = O-50. For strong-fiber composites, with EJE, > 1.0 the increase of the anisotropy of the plate is milder than for weak-fiber composites, it passes through a minimum appearing at a value of E&E, = 10.0 and then, for higher values of this ratio, a continuous increase of the transverse anisotropy of the plate is established for increasing values of the EJE, ratio.

4.3 Ellipsoidal fibers: effect of anisotropy The effect of using fibers with non-circular cross- section is investigated in this last set of parametric investigations. The cross-section of the composite and

Calculation of effective transverse elastic mod&i 579

a2 0.3 0.4 0.5 0.6 03 0.8

qlv, -

Fig. 3. Transverse elastic modulus of the composite for various fiber volume contents. Comparison of the numerical homogenization (E,,), the mesophase model (E,) and the Hashin-Rosen model (I&) (15, = 100.0 MPa, E, =

10.0 MPa).

the unit cell analyzed by the finite-element-based numerical homogenization method are shown in Fig. 13. The volume fraction of the fibers is vr = 0*558. Both matrix and fiber materials are considered to be linearly elastic. For the matrix we consider a material with E, = 100 MPa and Y, = 0.36. For the fibers we have investigated the range of elasticity modulus

between Ef = 10.0 and Ef = 10 000 MPa, with Poisson’s ratio varying between vr = 0.46 and 0.20.

Since for soft materials with Ef = 10 MPa the Poisson’s ratio should be high, it is taken as an upper limit for this quantity with vrO = O-46. In contrast, for hard materials with Ef = 10 000 MPa the Poisson’s ratio should be low and it is taken as vlod = 0.20. It was further assumed that the variation of Poisson’s ratio in the intermediate zone of modulus varies according a sigmoidal symmetric curve with the two extreme regions horizontal and a smooth variation following the variation of the respective moduli, Ef. 6,17.‘x The dimensions H = 100 and N = 8.5 of the representative unit cell, as indicated in Fig. 13, were selected in such a way as to minimize the anisotropy

50011111 0.0

VI/V, -

Fig. 4. Transverse elastic modulus of the composite for various fiber volume contents. Comparison of the numerical homogenization (Eh), the mesophase model (EJ and the Hashin-Rosen model (EHR) (E, = 100.0 MPa, E, =

500.0 MPa).

0.2 0.3 0.4 0.5 0.6 O.? 08 VfIV, -

Fig. 5. Transverse elastic modulus of the composite for various fiber volume contents. Comparison of the numerical homogenization (Eh), the mesophase model (E,) and the Hashin-Rosen model (EHR) (E, = 100.0 MPa, E, =

1000 MPa).

induced by the shape orientations and the properties of non-cylindrical inclusions, with the intent to study exclusively the influence of the geometric arrangement of the inclusion arrays on the anisotropy of the composite.

The homogenized transverse elasticity moduli of the equivalent orthotropic elastic body along the two principal orthotropy directions of Fig. 14 and for various EJE, ratios are given in Figs 15(a), 16(a) and 17(a). Thus Fig. 15(a) presents the variation of transverse elastic modulus, ET, of the composite given by the two orthogonal components Ehl and Eh2 along the two principal stress directions of the plate for a composite with the modulus of the elliptical fiber material, E, = Ef, varying between 10 and 100 MPa while the modulus of the material of the matrix,

En = E,, was taken always equal to 100 MPa.

Similarly, Figs 16(a) and 17(a) give the same variation of the anisotropic transverse elastic moduli of composites with the same matrix but with stronger fibers than the matrix, whose elastic moduli vary either between 100 and 1000 MPa for Fig. 16(a), or

0.2 0.3 0.S 0.5 06 0.7 0.0 v f/v,-

Fig. 6. Transverse elastic modulus of the composite for various fiber volume contents. Comparison of the numerical homogenization (EJ, the mesophase model (E,) and the Hashin-Rosen model (EHR) (E,=lOO.O MPa, E,=

10 000 MPa).


0.21 0.5 0.6 0.7 08 0.9 1.0

N/H--

Fig. 7. Variation of the fiber content versus the ratio of H and N parameters defining the arrangement of the fibers.

between 1000 and 10000 MPa for Fig. 17(a). The values predicted by the Hashin-Rosen theory (HR) and by the mesophase theory with u, = 0.001 are also given in the same figures. Note that both latter models consider fibers with circular cross-sections and do not take into account the effect of fiber staggering (placement). Thus, the effect of anisotropy cannot be studied by these models.

The transverse Poisson’s ratios given by the finite-element-based numerical homogenization theory, and for the same parametric investigation used previously, are given in Figs 15(b), 16(b) and 17(b). Regarding the Poisson’s ratios and their anisotropic values, as derived from the numerical solution of the homoganization procedure, these can be only compared with the averaged values of this material parameter as derived by application of the law of mixtures, v,; this quantity does not consider the influence of the anisotropy of the plate, thus yielding an average value for the transverse Poisson’s ratio of the plate. More elaborate cross-sections of fibers. which may even lead to negative values of the homogenized Poisson’s ratios, have been investigated previously.’

0.5 0.6 0.7 0.8 0.9 1.0

N/H -

Fig. 8. Transverse elastic modulus of the composite for various N/H ratios. Comparison of numerical homogenization (E,,, . E,,J, the mesophase model (I!?,) and the Hashin-Rosen model (EHR) (E, = 100.0 GPa, E, =

10 MPa).

N/H-

Fig. 9. Transverse elastic modulus of the composite for various N/H ratios. Comparison of numerical homogenization (E,,,, E,,J, the mesophase model (E,) and the Hashin-Rosen model (Em<) (E,,, = 100.0 MPa, E, =

500 MPa).

It is clear from Figs 15-17 that some amount of anisotropy in the composite plate is developed as we recede from the isotropic plate where E, = E,, and the elastic modulus of the inclusions, E, = E,, coincides with the elastic modulus, Err = E,,,, of the matrix. This state is indicated in Fig. 15 for EI/EII = 14 as well as in Fig. 16 for the same abscissa.

For the soft inclusions represented in Fig. 15 there is a rather important development of anisotropy expressed by the ratio E,,/E,, because of the softness of the inclusion material which does not exceed a value of Eh2/Ehl = 0.62 and v2,/v12 = 0.62. On the contrary, for the composite with a strong elastic inclusion represented in Figs 18-20, the anisotropy ratios E,,/E,,, = v~,/v,~ do not exceed a value of 0.75, indicating that there is not much influence on the

transverse anisotropy of the plate as the elastic modulus of its inclusions is increased, especially for high values of E,IE,, exceeding about 50. Indeed, in this region of variation of the modulus of the fibers, the induced anisotropy in the plate remains practically unchanged.

N/H h

Fig. 10. Transverse elastic modulus of the composite for various N/H ratios. Comparison of numerical homogenization (Eh,, EhZ), the mesophase model (E,) and the Hashin-Rosen model (E,,,,) (E,,, = lOO.OMPa, E, =

1000 MPa).


400

a : 300

2 200

100 0.5 0.6 0.7 0.8 0.9 1.0

N/H-

Fig. 11. Transverse elastic modulus of the composite for various N/H ratios. Comparison of numerical homogenization (E,,,, EhZ), the mesophase model (I?,) and the Hashin-Rosen model (I&) (E, = 100.0 MPa, Ef=

10 000 MPa).

5 RESULTS

A numerical homogenization technique, which is simple enough to be applied by means of general- purpose finite-element software, has been used in this paper to investigate the effect of the shape of the inclusion cross-section and the placement on the overall (effective) transverse elastic mechanical properties of fiber-reinforced composites. These

effects cannot be studied by means of the classical

homogenization approaches, which are based on analytical considerations, either without or with the

05 06 03 08 0.9

N/H -

Fig. 12. Variation of the orthotropy of the transverse elastic modulus E.r, defined by the ratio E,,/E,,, versus the N/H ratio, for parametric values of the elastic moduli of the

constituent phases.

interface assumption, as they have been introduced by Hashin and Rosen’ and Theocaris,6*9 respectively.

For an orthogonal placement of fibers of circular cross-section and for equal horizontal and vertical

n +I H w 85X2 100

85

100 w *

Fig. 13. Cross-section of a composite with arrays of parallel elliptic fibers in a staggered arrangement.


Fig. 14. Finite-element model of the analyzed unit cell

dimensions of the periodicity cell, as shown in Fig. 2 (with N = H), the effect of fiber-to-matrix elastic modulus for various fiber volume fractions has been investigated. For weak-fiber and strong-matrix composites the deviation of the predicted transverse elastic moduli is rather large; the predictions of the numerical homogenization method and of Theocaris’ mesophase model lead to larger values than those given by the Hashin-Rosen model (see Fig. 3). Nevertheless, one should mention that this case (i.e. the weak-fiber and strong-matrix composite) has not been included in the assumption used for the derivation of the classical theories, so their use here may be considered to be an extrapolation outside the range of their validity.

For increasing strong-fiber and weak-matrix composites and for increasing values of the ratio of the

fiber-to-matrix elasticity modulus, the predictions of the three approaches are comparable (see Figs 4-6). One may observe that the transverse elasticity

E, = 100 MPa

(b)

Fig. 15. Variation of (a) the transverse elastic modulus, E,, and (b) the Poisson’s ratio, Y, for fiber moduli E, = Ef varying between 10 and 100 MPa with E, = E,, = 100 MPa, as derived from the numerical homogenization method (E,,,, IT,,*), from the

Hashin-Rosen model (EHK), and from the mesophase model (I?,).


E, = 100 MPa

"0 2.0 4.0 6.0 8.0 lo.0 (a) Wh-

E, = 100 MPa

(b) Wh- Fig. 16. Variation of (a) the transverse elastic modulus, ET, and (b) the Poisson’s ratio, Y, for fiber moduli EI = E, varying between 100 and 1000 MPa with E, = E,, = 100 MPa, as derived from the numerical homogenization method (I&, E,,), from

the Hashin-Rosen model (EHR), and from the mesophase model (EJ.

modulus predicted by the numerical homogenization approach lies between the values given by the mesophase concept (which are greater) and the the Hashin-Rosen model (which are smaller). For relatively small ratios of fiber-to-matrix elasticity modulus (Fig. 4) the results of the homogenization approach are very close to the results of the mesophase concept. On the other hand, for relatively large ratios of fiber-to-matrix elasticity modulus (Fig. 6) the predictions of the Hashin-Rosen model, up to an E,/E, ratio approximately equal to 0.65, correlate well with the numerical homogenization approach. Nevertheless, one should take into account that the mesophase volume fraction of the latter model has been approximated and kept constant and equal to ui = 0.001 throughout this investigation. Thus the variation of the elastic modulus within the constant

mesophase layer yields larger moduli in the results shown in Figs 5 and 6. However, this point can be improved, if a better value for this quantity (the mesophase volume fraction) is available (possibly obtained by exprimental technique@).

On the other hand, for an orthotropic placement of fibers with circular cross-sections and for different horizontal and vertical dimensions of the periodicity cell (see Fig. 2 with N # H), the effect of induced anisotropy on the overall transverse elastic moduli has been investigated. Note that both classical theories used cannot capture the induced anisotropy, thus they give equal effective elasticity moduli for the two material orthotropy directions for all N/H ratios.

We first consider the variation of the predicted elastic moduli for the two orthotropy directions with respect to the values of the fiber-to-matrix elasticity


/ I I I Eln= 100 MPa I

0 20 40 60 80 100

(4 E Ih w

E, = 100 MPa

0 20 40 60 80 100

@I wh-

Fig. 17. Variation of (a) the transverse elastic modulus, E,, and (b) the Poisson’s ratio, Y, for fiber moduli E, = E, varying between 1000 and 10 000 MPa with E,, = E,, = 100 MPa, as derived from the numerical homogenization method (E,,,, E,,,).

from the Hashin-Rosen model (E,,,). and from the mesophase model (E,).

modulus (from Figs 8-11). The ratios (E,,, - E,,,)/E,, for N/H = 0.5 are equal to 243%, 181%, 26% and 67% for E,IE, ratios equal to 0.1, 5.0, 10.0 and 100.0, respectively. On the other hand, for N/H = 0.75, the above ratios are equal to 8.3%, 25%, 6.6% and 14% for E,/E, equal to 0.1, 5.0, 10.0 and 100.0, respectively (see also Fig. 12).

It seems that the relationship considered is rather complicated and no clear conclusions can be drawn at present. Definitely, for an orthogonal placement (e.g. for N/H = O-5) instead of square, and for large differences in the elasticity moduli of the fiber and the matrix (weak-fiber and strong-matrix composites or strong- fiber and weak-matrix ones), this deviation is larger.

Concerning the comparison of the predicted transverse elastic moduli with those given by the

classical theories, we observe that the mean value of the numerical homogenization moduli in the two orthotropy directions is best approximated by the mesophase model for weak-fiber and strong-matrix composites (see Fig. 8) or for strong-fiber and weak-matrix composites with moderate fiber-to-matrix elasticity ratios (e.g. Fig. 9 for EJE, = 0.5). For strong-fiber and weak-matrix composites with greater fiber-to-matrix elasticity ratios (e.g. Figs 10 and 11) the mean value of the predicted elasticity moduli is best approximated by the Hashin-Rosen model. Nevertheless, the same remark as previously is valid here: the mesophase volume is again kept constant for this parametric investigation, while a precise determination of its value would possibly reduce the discrepacy in the results.

Calculation of effective transverse elastic mod&i

1.0

0.9

0.8

0.7

0.6

E/E L I II

Fig. 18. Variation of the ratio of the transverse elastic moduli, E,,, and E,,, and Poisson’s ratios, y2, and Y,*, of a composite plate, with fiber inclusions of elliptic cross- section, arranged in a staggered formation, along the principal directions of anisotropy of the plate, versus the ratio of the moduli of the constituent phases of the composite, EJE,,, for E,, = E,, = 100 MPa and E, = E, varying between

10 and 100 MPa.

The combined effect of fibers with non-circular

(here, elliptical) cross-sections and of anisotropic placement of the fibers in the cross-section of the composite has been investigated in the last series of parametric studies (Figs 13-17). The elastic variables of a transversely orthotropic elastic composite are fitted by means of the numerical homogenization technique while, as previously, the classical approaches considered cannot capture the anisotropy. Here the mean value of the two orthotropic elasticity moduli compares better with the predictions of the interface model for the whole range of strong-fiber and weak-matrix composites considered (see Figs 16(a) and 17(a)). For weak-fiber and strong-matrix composites, and for a low Ef/E, ratio, the prediction is better approximated by the Hashin-Rosen model. As previously, the additional flexibility of the interface model deriving from the possibility to assign various interface volume fractions has not been examined.

The forms of variation of the orthotropic Poisson’s ratios lead to predictable conclusions (see Figs 15(b), 16(b) and 17(b)): the material orthotropy manifests itself for all E,/E, ratios, different from the trivial

V.”

0 20 4.0 6.0 8.0 10.0

V+----+

E,‘En- Fig. 19. Variation of the ratio of the transverse elastic moduli, E,, and Eh2, and Poisson’s ratios, v2, and v,,, of a composite plate, with fiber inclusions of elliptic cross- section, arranged in a staggered formation, along the principal directions of anisotropy of the plate, versus the ratio of the moduli of the constituent phases of the composite, EJE,,, for E,, = E,,, = 100 MPa and EI = E, varying between

100 and 1000 MPa.

value 1 (isotropic, homogeneous material I). The mixture law gives a prediction for the overall Poisson’s

ratio which can be accepted (depending on the specific application) for most values of the E,/E, ratio; nevertheless, it cannot capture the material anisotropy either.

As has been shown in a companion paper,’ the shape of the fiber cross-section has a considerable influence on the overall elasticity moduli. In particular, cross-sections with re-entrant corners lead to composite materials with even negative Poisson’s ratios, even if the individual constituents are classical materials.’

Finally, we would like to remark that we tried to investigate the behavior of the proposed numerical homogenization technique and to examine its predictive capability for certain quantities which are given by explicit analytical formulae. Thus, the comparison has been restricted to the variables for which the classical theory used here can be applied and, accordingly, only the relevant plots are presented here. The obvious fact that the method is also capable of establishing the form of variation of the


T 0.9

wz 0.8

” w 0.7

06 -. 0 20 40 60 80 100

E/E L I II

0.67 0 20 40 60 80 100

Er4,--- Fig. 20. Variation of the ratio of the transverse elastic moduli, Ehl and E,,, and Poisson’s ratios, v2, and v,~, of a composite plate, with fiber inclusions of elliptic cross- section, arranged in a staggered formation, along the principal directions of anisotropy of the plate, versus the ratio of the moduli of the constituent phases of the composite, Et/E,,, for E,, = E, = 100 MPa and Et = Et varying between

1000 and 10 000 MPa.

longitudinal shear modulus, Gi2, is not included here because of a lack of adequate means of comparison of these results, since this quantity is in general difficult to evaluate or determine experimentally.

In conclusion, a simple numerical homogenization method, which is based on a general-purpose finite-element software, is proposed and used here for prediction of the effective transverse elastic moduli of fiber-reinforced composites. More complicated numerical homogenization techniques have been proposed in the literature,‘0~13~‘4 which could lead to more refined predictions. Nevertheless, they require specialized software for their application. In view of the other uncertainties involved in the material design problem examined and given that the essential overall characteristics of the composite are modeled by the simple approach which is proposed here, we have avoided in this first investigation extension of our investigation into more complicated numerical homogenization methods.

ACKNOWLEDGEMENTS

The research program presented in this paper was supported by the National Academy of Athens Research Fund under code no. 200/294. The authors acknowledge this generous support. The authors are also indebted to Mrs Anna Zografaki for typing and printing the manuscript, as well as for drawing its

figures.

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