OVERALL VISCOELASTIC RESPONSE OF

RANDOM FIBROUS COMPOSITES WITH

STATISTICALLY UNIFORM

DISTRIBUTION OF REINFORCEMENTS

M. Šejnoha a,1 and J. Zeman a

aCzech Technical University, Faculty of Civil EngineeringDepartment of Structural Mechanics

Thákurova 7, 166 29 Prague 6

Abstract

An accurate representation of time dependent response of polymeric composite

systems with disordered microstructure is developed within the framework of clas-

sical homogenization methods. A graphite fiber tow impregnated by an epoxy resin,

Fig. 1(a), is just an example of such systems. The investigation is focused on mod-

eling issues pertinent to random, non-periodic, material systems, while the loading

conditions are left to those promoting the linear viscoelastic deformation only. Two

different approaches are examined. The first approach assumes a well defined ge-

ometry of the fiber arrangement and specific boundary conditions. In the modeling

framework, the complicated real microstructure is replaced by a material represen-

tative volume element consisting of a small number of particles, which statistically

resembles the real microstructure. Periodic distribution of such unit cells is consid-

ered and the finite element method is called to carry out the numerical computation.

The theoretical basis for the second approach are the Hashin-Shtrikman variational

principles. The random character of the fiber distribution is incorporated directly

into the variational formulation employing certain statistical descriptors. At the

Preprint submitted to Elsevier Preprint 6 August 2001

present time the applications are limited to microstructures, which are sufficiently

described by the two-point probability function. The presented results support ap-

plicability of both methods to the description of viscoelastic behavior of the selected

material system.

Key words: Unit cell, variational principle, Maxwell chain model, eigenstrain

(a) (b)

Fig. 1. Geometry of fiber bundle and distribution of fibers within transverse plane section.

1 Introduction

Reliability, ease of processibility and relatively low cost offered by the polymer matrix com-

posites contribute to their continuous rise in both mechanical and civil engineering industry.

An adequate description of the deformation behavior of such systems is therefore in order.

The time dependent response of polymers when subject to sustained loading is well known

and has been under thorough study since their introduction. Inventory of contributions to

analysis of polymer systems includes [?,?, among others]. The deformation behavior of poly-

mers is generally quite complex ranging from viscoelastic regime at very low stresses, over the

non-linear viscoelastic deformation at moderate stresses up to the yield behavior at hight

1 Corresponding author.

2

stresses. Such deformation mechanisms are also found in the polymer matrix composites.

Examples include polymer matrix systems reinforced by aligned fibers, whiskers or fabrics

typically supplied in form of laminated plates or woven tubes. Constitutive modeling of such

systems usually assumes linearly elastic response of reinforcements while the matrix phase

may undergo an inelastic or time dependent deformation. e.g., [?,?].

Apart from proper selection of the constitutive model, the overall response of composite

structures is highly influenced by their microstructure configuration. Estimates of the overall

response of composite materials are often derived on the basis of periodic microstructures,

but such are difficult to realize in practice. Instead, a highly disordered microstructures arise

in most practical applications, see Fig. 1(b). Therefore, recognizing a random nature of such

systems is a key step to a successful modeling of composite materials and structures. This

viewpoint is also adopted in the present contribution. In particular, we focus on an accurate

description of microstructure configuration and its effect on the overall response of polymer

composites under sustain loading, while admitting only such loading conditions which result

in a liner viscoelastic response of the matrix phase. A detailed description of more complex

material behavior of polymer systems is in progress and will be presented in a forthcoming

paper.

Two classical computational techniques in the analysis of composites are reviewed and ap-

plied to modeling of random composites. The first approach hinge on the existence of a

periodic unit cell having the same statistics up to two-point correlations as the real mi-

crostructure. A thorough study on this subject can be found in [?]. Note that regardless of

their origin, the present formulation is applicable to any composite systems which can be

represented by periodic unit cells. The second approach builds upon the well known Hashin-

Shtrikman (H-S) variational principles [?]. The interested reader may also consult the work

of [?,?], for generalization of these principles to treat random composites. Evaluation of ef-

fective thermoelastic properties of fiber tow assuming statistically uniform distribution of

3

fibers is presented in [?]. A strong attention is paid to an efficient evaluation of certain

microstructure dependent matrices, given in terms of the two-point probability functions,

which arise in the solution procedure based on the H-S principles. The most important as-

pects of both procedures are revisited here, but extended to account for the distribution of

local eigenstrain or eigenstress fields, which enter the constitutive equations to accommodate

creep or relaxation effects.

The present article is organized as follows. Section 2 outlines formulation of incremental

form of local constitutive equations describing the linear viscoelastic response on the basis of

generalized Maxwell chain model. Their introduction into a specific form of homogenization

methods treating spatially periodic microstructures is addressed in Section 3. An alternate

procedure to predict the time dependent response of random composites via the H-S princi-

ples is presented in Section 4. The results for the selected material system are summarized

in Section 5.

2 Local constitutive equations

The present section provides a brief overview of local constitutive equations describing the

viscoelastic response of homogeneous and isotropic materials. Such equations can be formu-

lated either in the integral form, or in the differential form. The differential form, that is more

convenient for numerical implementation, can be derived by converting the integral equations

into a rate-type form and by subsequent integration under certain simplifying assumptions.

Nevertheless, both forms of constitutive equations requires, in general, an introduction of

either creep J(t, τ) (compliance) or relaxation R(t, τ) function. Recall that the compliance

function of a linear viscoelastic material represents the strain at time t due to a unit stress

applied at time τ and kept constant, while the relaxation function represents the stress at

time t due to a unit strain applied at time τ and held constant. To facilitate the numerical

solution these functions are typically approximated by the degenerate (Dirichlet) kernels in

4

the form

J(t, τ) =M∑µ=1

1

Dµ(τ){1− exp [yµ(τ)− yµ(t)]} ,

R(t, τ) =M∑µ=1

Eµ(τ) exp [yµ(τ)− yµ(t)], (1)

where yµ(t) = (t/Θµ). Retardation times Θµ must satisfy certain rules necessary for the

success of calculation [?]. Functions Dµ and Eµ are usually obtained by fitting the creep

or relaxation functions via Eqs. (1) using the method of least squares. Also note that rep-

resentation (1) corresponds to the well known Maxwell and Kelvin chain models displayed

in Fig. 2. Both rheological models are equivalent and can be used to deliver the searched

constitutive equations. As an example, consider the Maxwell chain model and write the local

stress in the form

σ(x) =M∑µ=1

σµ(x), (2)

where σµ, called hidden stress, represents the stress in the µth Maxwell unit, which satisfies

the differential constitutive equation

σ̇µ(x) + ẏµσµ(x) = Eµ(x)L̂(x)(�̇(x)− �̇0(x)), ẏµ(t) =Eµ(t)

ηµ(t). (3)

Lµ(x, t) = Eµ(x, t)L̂(x) is the instantaneous stiffness matrix of a linear elastic isotropic

material at the material point x. The initial strain vector ∆�0i may represent many different

physical phenomena including thermal strains, shrinkage, swelling, plastic strains, etc. A

simple finite difference based integration scheme can be devised to integrate Eq. (3). In

doing so, first subdivide the time axis into intervals of length ∆ti. Next, suppose that at

the beginning of the ith interval 〈ti−1, ti〉, the stress vector σµ(x, ti−1), µ = 1, 2, . . . ,M , is

known. Further assume that functions Eµ(x, τ) = Eµ(x, ti−∆ti/2) remain constant within a

given time interval. Then, integrating Eq. (3) with respect to time over ∆t and using Eq. (2)

5

. . . . . .

1 2 3 M

σ

σ

η

µE

ηMµ

8

µEηµ

. . .

.

M

2

1

σ

σ

η 01

(a) (b)

Fig. 2. Rheological models: (a) Maxwell chain, (b) Kelvin chain.

leads to the following incremental form of the local constitutive equation [?]

∆σi(x) = Li(x) (∆�i(x)−∆µi(x)) , (4)

where the current increment of local eigenstrain reads

∆µi(x) = ∆�0i (x) + ∆�̂i(x). (5)

The vector ∆�̂i(x) in Eq. (5) then represents the time dependent deformation and ∆�0i (x) is

the increment of initial strain. When admitting only thermal and creep effects their respective

forms are

∆�0i (x) = m(x)∆θi, ∆�̂i(x) =

[L̂(x)

]−1Êi(x, t)

M∑µ=1

(1− e−∆yµ

)σµ(x, ti−1), (6)

where m(x) lists the coefficients of thermal expansion. The stiffness Êi for the ith interval

is determined by

Êi(x, t) =M∑µ=1

Eµ(x, ti −∆ti/2)(1− e−∆yµ

)/∆yµ. (7)

6

0 10 20 30 40 50

t-τ [days]

0

500

1000

1500

2000

R(t

-τ) [

MP

a]

R - Findley’s relation

R - Dirichlet series expansion

Fig. 3. Relaxation function.

Finally recall that the stress σµ(x, ti) at the end of the ith interval depends solely on the

stress σµ(x, ti−1) found at the beginning of the ith interval such that

σµ(x, ti) = σµ(x, ti −∆t)e−∆yµ + Eµ(x, ti −∆ti/2)λµL̂(x)(∆�i(x, ti)−∆�0i (x, ti)

).(8)

Providing the material is age dependent, the coefficients Eµ should be determined at every

time step. In the present formulation, however, the time dependent material properties of

the epoxy matrix derived experimentally from a set of well cured specimens are assumed,

[?], so that the material aging can be neglected. If this is the case a simple power law like

formula for the unit creep rate

J(t, τ) = a+ b(t− τ)n, R(t, τ) = 1a+ b(t− τ)n

, (9)

can be employed to fit the experimental data. For the selected epoxy resin the model data are

a = 0.04744, b = 0.002142, n = 0.3526 with (t− τ) given in minutes, [?]. The corresponding

relaxation function R(t, τ) appears in Fig. 3. Ten elements of the Dirichlet series expansion

Eq. (1) uniformly distributed in log(t − τ) over the period of hundred days were assumed.

The fit of the relaxation function Eq. (92) by Eq. (12) is plotted in Fig. 3.

7

Eqs. (4) – (8) will now be used within the framework of individual solution techniques to

introduce the time dependent behavior of the polymer systems into composites. We begin

with simulations of viscoelastic processes in random composites by introducing a statistically

equivalent unit cell in Section 3 and then by following an alternate approach based on H-S

principles in Section 4.

3 Description of viscoelastic behavior via periodic fields

As already outlined in the introductory section, most of heterogeneous materials and compos-

ite systems in particular exhibit random distribution of reinforcing constituents. Analyzing

large samples of such materials, however, is computationally very expensive and cumbersome.

Although exceptions exist with reference to elastic problems, such an approach inevitably

becomes infeasible when allowing inelastic or time dependent deformations. A viable substi-

tute, which yet provides notion of the local nature of variables of interest such as stresses

and strains, relies on existence of the representative volume element (RVE) defined in terms

of a periodic unit cell with a certain number of particles, which possesses similar statisti-

cal properties as the original material, and therefore it can be considered as its reasonable

approximation. This task requires formulation of an objective function, which relates the

material’s statistics of the real microstructure and the corresponding unit cell. Its minimum

then provides locations of individual reinforcements together with an optimal ratio of the

unit cell dimensions. Solution systems based on evolution strategies were found appealing

in the search for optimal unit cells. A thorough discussion on this subject with reference

to evaluation of the overall thermoelastic properties is given in [?]. The interested reader

may also consult the work by [?] for a comprehensive overview of various useful algorithms

based on evolution strategies applied to this and similar problems commonly encountered in

engineering practice.

The section now proceeds by considering an optimal unit cell similar to one of Fig. 4 that

8

(a) (b)

Fig. 4. Periodic unit cells: (a) 5-fibers PUC, (b) 10-fibers PUC.

statistically resembles the real microstructure, Fig. 1(b). Further suppose that the prescribed

loading conditions produce a uniform distribution of macroscopic strain E or stress Σ fields.

In either case, an increment of the local displacement field u(x) admits the following decom-

position

∆u(x) = ∆E x + ∆u∗(x), (10)

where ∆u∗(x) represents a fluctuation of the local displacement due to the presence of

heterogeneities and is considered being periodic, [?,?, and references therein]. The local

strain increment then assumes the form

∆�(x) = ∆E + ∆�∗(x), (11)

where the fluctuation part ∆�∗(x) must vanish upon volume averaging. This requirement is

fulfilled for the present boundary conditions since

〈∆�∗ij(x)

〉=

1

Ω

∫Ω

∆�∗ij(x) dΩ =1

2Ω

∫∂Ω

(∆u∗i (x)nj + ∆u

∗j(x)ni

)d(∂Ω) = 0, (12)

due to periodicity of ∆u∗(x) (the same displacement on opposite sides of the unit cell). In

9

Eq. (12) Ω and ∂Ω represent the volume and the boundary of the unit cell, respectively.

The goal now becomes the evaluation of local fields within the unit cell and then their aver-

aging to arrive at the desired macroscopic response. To proceed, we first write the principle

of virtual work (Hill’s lemma) in the form

〈δ�T∆σ

〉= δET∆Σ. (13)

Next, substituting Eq. (4) into the above expression yields

δET〈Li(x)

(∆E + ∆�∗(x)−∆�0i (x)−∆�̂i(x)

)〉+〈δ�∗(x)TLi(x)

(∆E + ∆�∗(x)−∆�0i (x)−∆�̂i(x)

)〉= δETΣ, (14)

where subscript i now links Eq. (14) with the current time increment. Since δE and δ�∗(x)

are independent, the preceding equation can be split into two equalities

δET(〈Li(x)〉∆E +

〈Li(x)

(∆�∗i (x)−∆�0i (x)−∆�̂i(x)

)〉)− δET∆Σ = 0

(15)〈δ(�∗)T(x)Li(x)

〉∆E +

〈δ(�∗)T(x)Li(x)

(∆�∗i (x)−∆�0i (x)−∆�̂i(x)

)〉= 0.

A standard finite element approach is usually adopted to solve the above system of equations.

We proceed by introducing a set of C0 continuous shape functions such that ∆u(x) =

N(x)∆r; ∆r stores the components of the increment of nodal displacements. The vector of

local strains then receives, after employing the geometrical equations, its familiar form

∆�∗(x) = B(x)∆r, δ�∗T = (δr)TBT(x). (16)

Inserting these formulae in Eqs. (15) provides the following incremental form of algebraic

equations K11 K12K21 K22

i

∆E∆ri

=

∆Σ + ∆F0

∆f 0

i

. (17)

10

Individual matrices listed in Eq. (17) are written as

K11i =1

Ω

∫Ω

Li(x) dΩ,

K12i = KT21i =

1

Ω

∫Ω

Li(x)B(x) dΩ, (18)

K22i =1

Ω

∫Ω

BT(x)Li(x)B(x) dΩ,

and components of the right-hand side vector are

∆F 0i =1

Ω

∫Ω

Li(x)(∆�0i (x) + ∆�̂i(x)

)dΩ,

∆f 0i =1

Ω

∫Ω

BT(x)Li(x)(∆�0i (x) + ∆�̂i(x)

)dΩ. (19)

Finally, eliminating the fluctuating displacements vector ∆ri from Eq. (17) readily provides

the incremental form of the macroscopic constitutive law

∆Σi = Di∆Ei + ∆Λ0i , (20)

where

Di =(K11 −K12K−122 KT12

)i, ∆Λ0i = −∆f 0i +

(K12K

−122 ∆f

0)i. (21)

When prescribing the overall strain only the system of equations (17) reduces to

(K22)i∆ri = −(K21)i∆Ei + ∆f 0i . (22)

Eqs. (17) and (22) can now be used to run either the creep or relaxation experiments. The

results are discussed in Section 5.

11

4 Overall viscoelastic response via Hashin-Shtrikman principles

Suppose that no information about the nature of local fields is needed, but instead a notion

about the phase volume averages of field variables is sufficient in simulating the behavior

of composite materials. Then, the sometimes time consuming implementation of finite ele-

ments in conjunction with the unit cell analysis can be replaced by more simple averaging

techniques. Such an approach is examined in this section.

To introduce the subject consider an ergodic heterogeneous material system with statisti-

cally homogeneous distribution of reinforcements. Here, we limit our attention to two-phase

fibrous composites with fibers aligned along the x3 direction. The morphology of such a sys-

tem can be conveniently described by the two-point probability function Srs. As suggested

by Drugan & Willis [?] this function, when combined with the Hashin-Shtrikman varia-

tional principles provides sufficient information for deriving bounds on effective properties

of random composites. The material system under present study has been examined in very

details in [?] with reference to evaluation of effective thermoelastic properties of statistically

homogeneous materials using both the primary and dual variational principles of Hashin

and Shtrikman. The remainder of this section outlines implementation of this approach to

solve the viscoelastic problem. It is organized as follows: a review of basic statistical descrip-

tors (Section 4.1), viscoelastic formulation based on the primary H-S principle to run the

relaxation tests (Section 4.2) and viscoelastic formulation based on the dual H-S principle

to study the creep behavior (Section 4.3).

4.1 Review of basic statistical descriptors

To reflect a random character of heterogeneous media it is convenient to introduce the

concept of an ensemble – the collection of a large number of systems which are different

in their microscopical details but identical in their macroscopic details. In the context of

12

quantification of the microstructure morphology, an ensemble represents the collection of

material micrographs taken from different samples of the material. To describe a random

microstructure we introduce a characteristic function χr(x, α), which is equal to one when

point x lies in the phase r within the sample α and equal to zero otherwise

χr(x, α) =

1 x ∈ Dr(α)0 otherwise. (23)

The symbol Dr(α) denotes here the domain occupied by r-th phase in the sample α. For

a two-phase fibrous composite, r = f,m, characteristic functions χf (x, α) and χm(x, α) are

related by

χm(x, α) + χf (x, α) = 1. (24)

With the aid of function χr, the general n-point probability function Sr1,...,rn is given by [?].

Sr1,...,rn(x1, . . . ,xn) = χr1(x1, α) · · ·χrn(xn, α). (25)

Thus, Sr1,...,rn gives the probability of finding n points x1, . . . ,xn randomly thrown into the

media located in the phases r1, . . . , rn. We limit our attention to functions of the order of

one and two.

Analysis of random composites usually relies on various statistical assumptions such as er-

godic hypothesis, spatial homogeneity or isotropy, which may simplify the computational

effort to a great extent. In particular, the ergodic hypothesis demands all states available to

an ensemble of the systems to be available to every member of the system in the ensemble

as well [?]. Then, the spatial or volume average of function χr(x, α) given by

〈χr(x, α)〉 = limV→∞

1

V

∫V

χr(x + y, α)dy, (26)

13

is independent of α and identical to the ensemble average,

χr(x) = Sr = 〈χr(x)〉 = cr, (27)

where cr is the volume fraction of the rth phase. Note that the above assumption is usually

accepted as an hypothesis subject to experimental verification. The statistical homogeneity

assumption means that value of the ensemble average is independent of the position of

coordinate system origin. Then, for example, the two-point matrix probability function reads

Smm(x1,x2) = Smm(x12), (28)

where xij = xj − xi. In the context of a representative volume element (RVE: a material

element which effectively samples all microstructural configurations) the one-point proba-

bility function Sr and the two-point probability function Srs are the same in any RVE (a

micrograph of the material sample) irrespective of its position. Thus only one such sample is

needed for their evaluation. When constructing the RVE we add an additional requirement

with respect to its minimum size. Apart from the above statement we shall require the size

of the RVE to be at least such that there exist two points within the RVE which are statisti-

cally independent. Then, it appears acceptable to consider a periodicity of the selected RVE.

This becomes particularly important when developing an efficient procedure for evaluation

of Sr and Srs. Note that for an ergodic and periodic microstructure the two-point probability

function Srs receives the following form

Srs(x) =1

Ω

∫Ω

χr(y)χs(x + y)dy, (29)

where Ω is the size of the RVE (the micrograph area). It is worthwhile to mention that only

the Fourier transform of function Srs given by

S̃rs(ξ) =1

Ωχ̃r(ξ)χ̃s(ξ), (30)

14

is needed in H-S variational formulation [?]. Note that · now stands for the complex conjugate.

When introducing a binary image of the actual microstructure may evaluate Eq. (30) very

efficiently employing the discrete Fourier transform. See also [?] for further discussion on

this subject. Eqs. (27) and (30) will be now implemented within the framework of the H-

S variational principles. We present only the most important aspects of the formulation

relevant to this study. For a personal review we refer the reader to original papers by [?,?].

Additional references to related work are, e.g., [?,?] and [?].

4.2 Extended primary Hashin-Shtrikman variational principle

Suppose that the composite body is loaded by prescribed surface displacements resulting in

a uniform macroscopic strain E and by the local eigenstresses λ(x) = −L(x)µ(x). With

reference to the primary Hashin-Shtrikman variational principle the following representation

of local stresses is equivalent

σ(x) = L(x)�(x) + λ(x), σ(x) = L0�(x) + τ (x), (31)

where L(x) is the local stiffness matrix and L0 is the stiffness matrix of a certain homogeneous

reference medium. The vector τ is called the polarization stress. It can be shown that τ

satisfying the equilibrium conditions, constitutive equations and boundary conditions is the

one, which minimizes the functional

Uτ =1

2

∫Ω

(ETΣ− (τ − λ)T(L− L0)−1(τ − λ)− 2τTE − �′Tτ − λTL−1λ

)dΩ, (32)

where the fluctuation part �′

of the local strain � is provide by (see [?])

�′(x) = �(x)−E =∫Ω

�∗0(x− x′) (τ (x′)− 〈τ 〉) dΩ(x′). (33)

15

�∗0(x− x′) is termed the fundamental solution that relates τ to a strain derived under zero

displacement boundary conditions and for which the stress is self-equilibrated.

Substituting Eq. (33) into Eq. (32) and then taking variation with respect to τ provides a

system of algebraic equation for the searched polarization stress τ . To facilitate the solution

we further restrict our attention to a piecewise uniform distribution of polarization stress

τ r(x) = τ r and the eigenstress vector λr(x) = λr within a given phase r. Then, with

reference to Eq. (23), the trial field for τ and eigenstress λ at any point x located in the

sample α are provided by

τ (x, α) =n∑r=1

τ rχr(x, α), λ(x, α) =n∑r=1

λrχr(x, α). (34)

These expressions when plugged into functional Uτ Eq. (32) open a way for evaluation of

its ensemble average Uτ , e.g., [?,?]. Performing variation of Uτ with respect to τ r finally

supplies a set of equations for unknown phase averages of polarization stress τ r

n∑s=1

[δrs(Lr − L0)−1cr −Ars

]τ s = Ecr + (Lr − L0)−1λrcr, r = 1, . . . , n, (35)

where the microstructure-dependent matrices Ars do not depend on x and are provided by,

see [?],

Ars =∫Ω

�∗0(x− x′) [Srs(x− x′)− crcs] dΩ(x′) =1

(2π)2

∫Ω

�̃∗0(ξ′)S̃ ′rs(ξ

′)dξ′. (36)

Formal inversion of Eq. (35) yields the overall constitutive equation in the form

σ = LE + λ, (37)

where the spatially average overall stiffness matrix L and the macroscopic eigenstress vector

λ are provided by

16

L= L0 +n∑r=1

n∑s=1

crTrscs, (38)

λ=n∑r=1

n∑s=1

crTrscs(Ls − L0)−1λs. (39)

Trs then represents individual blocks of the inverse matrix to the left hand side of system (35).

For a two-phase composite medium they can be written in the form

A = Amm, Kr = Lr − L0,Trs = Kr

(cfKm + cmKf − cfcmA−1

) [Kf + Km −Kr + δrs(1− cr)A−1

], (40)

To introduce the thermal and viscoelastic effects we first recall Eq. (4). Then, in analogy

with Eqs. (20) and (22) the incremental form of Eq. (37) becomes

∆σi = Li∆Ei + ∆λi, (41)

and the current increment of λ and the instantaneous stiffness matrix Lm of the matrix

phase attain the forms

∆λi = −n∑r=1

n∑s=1

crTrscs(Ls − L0)−1i (Ls)i(∆�0s + ∆�̂s

)i, s = m→ (Lm)i = ÊiL̂m, (42)

whereas the fiber phase is assumed elastic. Eqs (41) and (42) drive the solution of a vis-

coelastic problem under strain control loading conditions suitable for modeling the stress

relaxation.

4.3 Extended dual Hashin-Shtrikman variational principle

Consider a composite body under prescribed surface tractions which produce a uniform

macroscopic stress Σ. In addition, a distribution of local eigenstrains µ(x) can be introduced

in the present formulation. In analogy with Eq. (33) we write the local constitutive equations

17

in the form

�(x) = M(x)σ(x) + µ(x), �(x) = M0σ(x) + γ(x), (43)

where M(x) now stands for the local compliance matrix and M0 is the compliance matrix of a

homogeneous reference medium. The polarization strain γ, when satisfying the compatibility

condition and the constitutive equation, provides the minimum of the extended dual H-S

functional

Uγ =1

2

∫Ω

(ΣTE − (γ − µ)T(M−M0)−1(γ − µ) + 2γTΣ + σ′Tγ

)dΩ, (44)

where the fluctuation part σ′

of the local stress σ written in terms of polarization strain γ

is, see [?],

σ′(x) = σ(x)−Σ = [σ∗0 (γ − 〈γ〉)]−M−10 (γ(x)− 〈γ〉) . (45)

The operator [σ∗0 (γ − 〈γ〉)] can be identified with the operator [�∗0 (τ − 〈τ 〉)] when replacing

γ for τ and σ∗0 for �∗0 and suitably modifying the boundary term to reflect the traction

boundary conditions. Specific forms of tensors �∗0 and σ∗0 can be found, e.g., in [?].

Assuming only a piecewise uniform distribution of phase polarization strain γr(x) = γr and

the phase eigenstress vector µr(x) = µr provides the trial fields for γ and µ at any point x

located in the sample α in the form, recall Eq. (34),

γ(x, α) =n∑r=1

γrχr(x, α), µ(x, α) =n∑r=1

µrχr(x, α). (46)

We now proceed, in analogy with Section 4.2, by introducing Eqs. (45) and (46) into func-

tional (44), evaluating an ensemble average of Uγ and finally taking variation with respect

to γr to get

18

n∑s=1

{δrs[(Mr −M0)−1 + M−10

]cr −Brs −M−10 crcs

}γs

= Σcr + (Mr −M0)−1crµr, r = 1, 2, . . . , n. (47)

Eq. (47) represents a system of algebraic equations to be solved for unknown components

of phase averages of the polarization strain γr. To continue, recall Eq. (36) and write the

microstructure dependent matrices Brs in the form

Brs =1

(2π)2

∫Ω

σ̃∗0(ξ′)S̃ ′rs(ξ

′)dξ′. (48)

Finally, after symbolic inversion of Eq. (47) we arrive at the macroscopic constitutive law,

see also Eqs. (37) – (39),

� =MΣ + µ, (49)

where the effective compliance matrixM and the macroscopic eigenstrain vector µ read

M= M0 +n∑r=1

n∑s=1

crRrscs, (50)

µ=n∑r=1

n∑s=1

crRrscs(Ms −M0)−1µs. (51)

For a two-phase composite medium the matrices Rrs assume the form

B = cfcmM−10 −Bmm, Nr =

[(Mr −M0)−1 + M−10

]−1,

Rrs = Nr(cfN2 + cmN1 − cfcmB−1

) [Nf + Nm −Nr + δrs(1− cr)B−1

]. (52)

When solving a viscoelastic problem we replace Eq. (49) by its incremental counterpart such

that

∆�i =Mi∆Σi + ∆µi, (53)

19

and finally the current increment of µ and the instantaneous compliance matrix Mm of the

matrix phase are expressed as

∆µi =n∑r=1

n∑s=1

crRrscs(Ms −M0)−1i(∆�0s + ∆�̂s

)i, s = m→ (Mm)i =

1

ÊiM̂m, (54)

Eqs. (53) and (54) can be used, while assuming the time independent elastic behavior of the

fiber phase, to find the solution to a stress control viscoelastic problem such as creep. The

results for a selected class of problems are presented in the next section.

5 Numerical examples

As an example, consider a two-phase fibrous composite system displayed in Fig. 1(b). As

suggested in [?] this medium can be regarded as ergodic and statistically homogeneous thus

serving as a suitable candidate for the numerical analysis. Both phases are homogeneous

with constant material properties listed in Table 1. The Cartesian coordinate system with

phase EA ET GT νA a b n

[GPa] [GPa] [GPa] [GPa]−1 [GPa]−1

fiber 386 7.6 2.6 0.41

matrix 2.1 2.1 0.75 0.40 0.0474 0.00214 0.3526Table 1Material properties of T30/Epoxy system.

the x3-axis directed along the fiber direction is selected. Therefore, the macroscopic stress

and strain fields written in contracted notation are Σ = {Σ11,Σ22,Σ12,Σ33} T and E =

{E11, E22, E12, E33} T. Recall that the generalized plane strain conditions are assumed. In

the present study, the numerical results are derived for loading applied within the transverse

plane section of the composite aggregate only. Both the strain and stress control loading

conditions are considered in simulations. Fig. 5 illustrates the time variation of the applied

load.

20

11

t [min]

Σ[MPa]

15

11.25

360 720 t [min]

Σ[MPa]

12

15

11.25

360 720

(a) (b)

11

t [min]

E[%]

360 720

0.5

0.375

t [min]

[%]E 12

360 720

0.5

0.375

(a) (b)

Fig. 5. Applied loading: (a,b) creep test, (c,d) relaxation test.

The first set of figures (Figs. 6-9) represents the composite response derived from the unit cell

model. The macroscopic creep behavior is considered first. Fig. 6 shows the time variation of

the overall strain developed under pure tensile and shear stress loading conditions, respec-

tively (Fig. 5a,b). Similar response resulting from the strain loading conditions is plotted in

Fig. 7. Individual results suggest that at least 5-fibers Unit cell should be used in numerical

simulations to provide sufficiently accurate response of the actual composite, Fig. 1(b). Note

that the same characteristic of the present material system was discovered when studying

a pure-elastic behavior of such composites, [?]. However, unlike the results derived assum-

ing pure elasticity, the viscoelastic response of the hexagonal array model slightly deviates

from that found using the statistically equivalent unit cells. Nevertheless, a very low con-

trast in material parameters in the transverse direction promotes the hexagonal array model

as a suitable substitute for more complicated unit cells when the fiber volume fraction is

sufficiently large.

21

0.0 200.0 400.0 600.0 800.0

t [min]

2.5e-03

3.0e-03

3.5e-03

4.0e-03

4.5e-03

5.0e-03O

vera

ll st

rain

E11

2-fibers Unit cell

5-fibers Unit cell

10-fibers Unit cell

20-fibers Unit cell

Hexagonal array

0.0 200.0 400.0 600.0 800.0

t [min]

8.0e-03

1.0e-02

1.2e-02

1.4e-02

1.6e-02

1.8e-02

Ove

rall

stra

in E

12

2-fibers Unit cell

5-fibers Unit cell

10-fibers Unit cell

20-fibers Unite cell

Hexagonal array

(a) (b)

Fig. 6. Overall response – Unit cell model: creep test.

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e+01

1.5e+01

2.0e+01

2.5e+01

3.0e+01

Ove

rall

stre

ss Σ

11 [M

Pa]

2-fibers Unit cell

5-fibers Unit cell

10-fibers Unit cell

20-fibers Unit cell

Hexagonal array

0.0 200.0 400.0 600.0 800.0

t [min]

3.0e+00

3.5e+00

4.0e+00

4.5e+00

5.0e+00

5.5e+00

Ove

rall

stre

ss Σ

12 [M

Pa]

2-fibers Unit cell

5-fibers Unit cell

10-fibers Unit cell

20-fibers Unit cell

Hexagonal array

(c) (d)

Fig. 7. Overall response – Unit cell model: relaxation test.

Quite different conclusion, however, can be drawn from Fig. 8. The macroscopic response

plotted in Fig. 8 provides evidence that increasing the material contrast leads to a noticeable

difference in the material response derived from the hexagonal array model and statistically

optimal unit cells. To arrive at such a result we simply replaced the actual transversally

isotropic fiber with the isotropic one to attain a higher contrast between the material prop-

erties of the fiber and matrix phases. Also note a possible change in the smallest RVE to

be considered for numerical simulations (10-fibers UC). Additional support for using the

22

0.0 200.0 400.0 600.0 800.0

t [min]

1.5e-03

2.0e-03

2.5e-03

3.0e-03

3.5e-03

4.0e-03

Ove

rall

stra

in E

11

5-fibers Unit cell

10-fibers Unit cell

20-fibers Unit cell

Hexagonal array

Fig. 8. Overall response – Unit cell model: creep test assuming isotropic fiber.

optimal unit cells instead of the hexagonal array model is provided by Fig. 9 suggesting

an anisotropic character of the present medium. Such a result cannot be attained by sim-

ple periodic unit cells. The present approach, which draws on the existence of a unit cell

statistically equivalent to the actual composite system, is therefore preferable.

0.0 200.0 400.0 600.0 800.0

t [min]

1.5e+01

2.0e+01

2.5e+01

3.0e+01

Str

ess

[MP

a]

Σ22 due to ε22Σ11 due to ε11

Overall stress

Matrix stress

Fiber stress

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e-03

2.0e-03

3.0e-03

4.0e-03

5.0e-03

6.0e-03

7.0e-03

Str

ain

ε11 due to Σ11ε22 due to Σ22

Overall strain

Matrix strain

Fiber strain

(a) (b)

Fig. 9. Overall and local response – Unit cell model: (a) relaxation test, (b) creep test.

The remaining set of results (Figs. 10–14) was found when employing the Hashin-Shtrikman

variational principles. Recall that the primary H-S principle might be invoked to simulate

the relaxation response while the dual H-S principle should be called to study the creep

behavior. In both instances, however, the resulting representation of the viscoelastic response

is governed by the selection of a homogeneous comparison medium (L0,M0). To this end, we

draw the readers attention to Fig. 10. First, recall Table 1 and notice that the matrix moduli

23

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e+01

1.5e+01

2.0e+01

2.5e+01

3.0e+01O

vera

ll st

ress

S11

[MP

a]

H-S Upper bound

H-S Estimate

H-S Lower bound

Unit cell

0.0 200.0 400.0 600.0 800.0

t [min]

2.5e-03

3.0e-03

3.5e-03

4.0e-03

4.5e-03

5.0e-03

Ove

rall

stra

in E

11

H-S Upper bound

H-S Estimate

H-S Lower bound

Unit cell

(a) (b)

Fig. 10. Overall response – hexagonal packing: (a) relaxation test, (b) creep test.

are indeed weaker than those of fiber. Therefore, according to [?] we select the matrix phase

to fill individual entries of L0 and M0 providing we search for a lower bound on the relaxation

response (Fig. 10(a)) and an upper bound on the creep response (Fig. 10(b)), respectively.

The fiber phase is then selected to yield estimates on opposite bounds. A number of other

results, contained within the H-S bounds, can be derived when mixing individual phases to

set L0 and M0 (e.g., L0 =12(Lf + Lm)).

(a) (b)

Fig. 11. Idealized binary images: (a)–resolution 488x358 pixels, (b)–resolution 244x179 pixels.

Fig. 10 gives some idea of this approach assuming hexagonal packing of fibers. One notewor-

thy feature of the hexagonal array model is the correspondence of the periodic hexagonal

unit cell with the Mori-Tanaka averaging technique. It is well known that the Mori-Tanaka

24

estimates of the overall response of composites with weak matrices correspond to the lower

and upper bounds derived from the primary and dual H-S variational principles, respectively.

From this point of view one may also judge the results derived from the periodic hexagonal

unit cell (solid lines in Fig. 10a,b). Therefore, in order to make the results found from the H-S

variational principles comparable with those derived previously using the optimal unit cells,

we selected the matrix phase to create the 4x4 homogeneous stiffness L0 and compliance

M0 matrices. Also note that these matrices are kept constant throughout the integration

process. This assumption is acceptable since the instantaneous moduli of the matrix phase

do not vary considerably for the selected time-stepping procedure.

0.0 200.0 400.0 600.0 800.0

t [min]

2.5e-03

3.0e-03

3.5e-03

4.0e-03

4.5e-03

5.0e-03

Ove

rall

stra

in E

11

976 x 716 pixels

488 x 358 pixels

244 x 358 pixels

122 x 84 pixels

0.0 200.0 400.0 600.0 800.0

t [min]

8.0e-03

1.0e-02

1.2e-02

1.4e-02

1.6e-02

1.8e-02

Ove

rall

stra

in E

12

976 x 716 pixels

488 x 358 pixels

244 x 179 pixels

122 x 84 pixels

(a) (b)

Fig. 12. Overall response – Hashin-Shtrikman principle: creep test.

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e+01

1.5e+01

2.0e+01

2.5e+01

3.0e+01

Ove

rall

stre

ss Σ

11 [M

Pa]

976 x 716 pixels

488 x 358 pixels

244 x 179 pixels

122 x 84 pixels

0.0 200.0 400.0 600.0 800.0

t [min]

3.0e+00

3.5e+00

4.0e+00

4.5e+00

5.0e+00

5.5e+00

Ove

rall

stre

ss Σ

12 [M

Pa]

976 x 716 pixels

488 x 358 pixels

244 x 179 pixels

122 x 84 pixels

(a) (b)

Fig. 13. Overall response – Hashin-Shtrikman principle: relaxation test.

25

200.0 400.0 600.0 800.0

t [min]

1.5e+01

2.0e+01

2.5e+01

3.0e+01S

tres

s [M

Pa]

Σ11 due to ε11Σ22 due to ε22

Overall stress

Matrix stress

Fiber stress

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e-03

2.0e-03

3.0e-03

4.0e-03

5.0e-03

6.0e-03

7.0e-03

Str

ain

ε11 due to Σ11ε22 due to Σ22

Overall strain

Matrix strain

Fiber strain

(a) (b)

Fig. 14. Overall and local response – Hashin-Shtrikman principle: (a) relaxation test, (b) creep test.

0.0 200.0 400.0 600.0 800.0

t [min]

1.5e+01

2.0e+01

2.5e+01

3.0e+01

Str

ess

[MP

a]

UCH-S: LB

Overall stress

Matrix stress

Fiber stress

0.0 200.0 400.0 600.0 800.0

t [min]

1.0e-03

2.0e-03

3.0e-03

4.0e-03

5.0e-03

6.0e-03

7.0e-03

Str

ain

Unit cellH-S: UB

Overall strain

Matrix strain

Fiber strain

(a) (b)

Fig. 15. Overall and local response – UC vs. H-S: (a) relaxation test, (b) creep test.

As outlined in Section 4.1, evaluation of the overall response of random composites requires

first selection of a RVE. Binary images of the RVE of the present material system, Fig. 1(b),

which complies with certain requirements discussed in Section 4.1, are displayed in Fig. 11.

The RVEs of Fig. 11 were employed to evaluate the microstructure dependent matrices in

Eqs. (36) and (48). The effect of bitmap resolution on the overall response was examined

first. Results appear in Figs. 12 and 13. Evidently, even low resolution of 244x179 pixels ( ≈

55 pixels per fiber ) provides sufficiently accurate results. This should also hold for combined

loading. Such a result is quite encouraging, particularly if one would like to increase the

26

size of the RVE. Fig. 14 further confirms ability of this approach to model an anisotropic

character of the present material system already suggested by the FEM analysis of optimal

periodic unit cells.

Finally, we bring some comparison between the UC analysis and the H-S variational prin-

ciples plotted in Fig. 15. As expected, recall our previous discussion on H-S bounds, the

relaxation data obtained from 5-fibers periodic UC correlates fairly well with the H-S lower

bound and similarly the creep response of UC is found close to the H-S upper bound. Thus

the applicability of both approaches to simulate the viscoelastic behavior of statistically

homogeneous material systems such as the one under present study is confirmed.

6 Conclusion

Most of the real composite material systems experience a random distribution of reinforce-

ments. In this contribution, two generally accepted approaches in the micromechanics were

reviewed and applied to the analysis of random statistically homogeneous composite systems

undergoing viscoelastic deformation. Formulation of macroscopic constitutive equations was

first outlined within the framework of periodic fields. Random nature of microstructure

configuration was accounted for through various unit cells found such as to represent the

material statistics of real composites. An effect of number of reinforcements in the unit cell

needed for an accurate representation of macroscopic response was examined. In view of the

results presented in [?] and plots displayed in Figs. 6-9 it appears necessary to introduce at

least five fibers in the optimal unit cell to arrive at sufficiently accurate predictions of the

overall behavior of the present material system. In addition, the present results also proved

necessity for an accurate modeling of microstructure configuration particularly when the

contrast between the phase material parameters becomes large.

The second part of this contribution revisited the well known Hashin-Shtrikman variational

27

principles further extended to reflect the presence of eigenstresses and eigenstrains in the

formulation of macroscopic equations. An efficient procedure for evaluation of required mate-

rials statistics, already examined in [?], was implemented in connection with binary images of

a real microstructure. The results plotted in Figs. 12-13 illustrate insensitivity of the solution

procedure to a selected bitmap resolution, which may further increase expected efficiency of

this approach. Final comparison with the FEM analysis shown in Fig. 15 assesses the appli-

cability and supports the use of this method, at least in the range of elastic or viscoelastic

response.

7 Acknowledgement

Financial support was provided by the research project CEZ:MSM 210000003.

28

IntroductionLocal constitutive equationsDescription of viscoelastic behavior via periodic fieldsOverall viscoelastic response via Hashin-Shtrikman principlesReview of basic statistical descriptorsExtended primary Hashin-Shtrikman variational principleExtended dual Hashin-Shtrikman variational principle

Numerical examplesConclusionAcknowledgement