OVERALL VISCOELASTIC RESPONSE OF RANDOM FIBROUS COMPOSITES WITH STATISTICALLY UNIFORM DISTRIBUTION OF REINFORCEMENTS

Transcription

1 OVERALL VISCOELASTIC RESPONSE OF RANDOM FIBROUS COMPOSITES WITH STATISTICALLY UNIFORM DISTRIBUTION OF REINFORCEMENTS M. Šejnoha a,1 and J. Zeman a a Czech Technical University, Faculty of Civil Engineering Department of Structural Mechanics Thákurova 7, Prague 6 Abstract An accurate representation of time dependent response of polymeric composite systems with disordered microstructure is developed within the framework of classical 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 modeling 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 geometry of the fiber arrangement and specific boundary conditions. In the modeling framework, the complicated real microstructure is replaced by a material representative volume element consisting of a small number of particles, which statistically resembles the real microstructure. Periodic distribution of such unit cells is considered 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

2 present time the applications are limited to microstructures, which are sufficiently described by the two-point probability function. The presented results support applicability 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 composites 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 polymers 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

3 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 applied 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 microstructure. 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 effective thermoelastic properties of fiber tow assuming statistically uniform distribution of 3

4 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 aspects 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 principles 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 formulated 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

5 the form J(t, τ) = R(t, τ) = M µ=1 M µ=1 1 D µ (τ) {1 exp [y µ(τ) y µ (t)]}, E µ (τ) exp [y µ (τ) y µ (t)], (1) where y µ (t) = (t/θ µ ). Retardation times Θ µ must satisfy certain rules necessary for the success of calculation [?]. Functions D µ and E µ are usually obtained by fitting the creep or relaxation functions via Eqs. (1) using the method of least squares. Also note that representation (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 M σ(x) = σ µ (x), (2) µ=1 where σ µ, called hidden stress, represents the stress in the µ th Maxwell unit, which satisfies the differential constitutive equation σ µ (x) + ẏ µ σ µ (x) = E µ (x) L(x)( ɛ(x) ɛ 0 (x)), ẏ µ (t) = E µ(t) η µ (t). (3) L µ (x, t) = E µ (x, t) L(x) is the instantaneous stiffness matrix of a linear elastic isotropic material at the material point x. The initial strain vector ɛ 0 i 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 t i. Next, suppose that at the beginning of the i th interval t i 1, t i, the stress vector σ µ (x, t i 1 ), µ = 1, 2,..., M, is known. Further assume that functions E µ (x, τ) = E µ (x, t i t i /2) remain constant within a given time interval. Then, integrating Eq. (3) with respect to time over t and using Eq. (2) 5

6 σ σ η E µ η µ η M 8 E η µ µ M σ (a) M σ (b) Fig. 2. Rheological models: (a) Maxwell chain, (b) Kelvin chain. leads to the following incremental form of the local constitutive equation [?] σ i (x) = L i (x) ( ɛ i (x) µ i (x)), (4) where the current increment of local eigenstrain reads µ i (x) = ɛ 0 i (x) + ɛ i (x). (5) The vector ɛ i (x) in Eq. (5) then represents the time dependent deformation and ɛ 0 i (x) is the increment of initial strain. When admitting only thermal and creep effects their respective forms are ɛ 0 i (x) = m(x) θ i, ɛ i (x) = [ L(x) ] 1 Ê i (x, t) M ( ) 1 e y µ σµ (x, t i 1 ), (6) µ=1 where m(x) lists the coefficients of thermal expansion. The stiffness Êi for the i th interval is determined by M Ê i (x, t) = E µ (x, t i t i /2) ( ) 1 e yµ / yµ. (7) µ=1 6

7 R - Findley s relation R - Dirichlet series expansion R(t-τ) [MPa] t-τ [days] Fig. 3. Relaxation function. Finally recall that the stress σ µ (x, t i ) at the end of the i th interval depends solely on the stress σ µ (x, t i 1 ) found at the beginning of the i th interval such that σ µ (x, t i ) = σ µ (x, t i t)e yµ + E µ (x, t i t i /2)λ µ L(x) ( ɛ i (x, t i ) ɛ 0 i (x, t i ) ).(8) Providing the material is age dependent, the coefficients E µ 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) = 1 a + b(t τ) n, (9) can be employed to fit the experimental data. For the selected epoxy resin the model data are a = , b = , n = 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. (9 2 ) by Eq. (1 2 ) is plotted in Fig. 3. 7

8 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 composite 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 substitute, 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 statistical 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

9 (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 decomposition 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 ɛ(x) = E + ɛ (x), (11) where the fluctuation part ɛ (x) must vanish upon volume averaging. This requirement is fulfilled for the present boundary conditions since ɛ ij (x) = 1 ɛ ij(x) d = 1 2 ( u i (x)n j + u j(x)n i ) d( ) = 0, (12) due to periodicity of u (x) (the same displacement on opposite sides of the unit cell). In 9

10 Eq. (12) and 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 averaging to arrive at the desired macroscopic response. To proceed, we first write the principle of virtual work (Hill s lemma) in the form δɛ T σ = δe T Σ. (13) Next, substituting Eq. (4) into the above expression yields δe T L i (x) ( E + ɛ (x) ɛ 0 i (x) ɛ i (x) ) + δɛ (x) T L i (x) ( E + ɛ (x) ɛ 0 i (x) ɛ i (x) ) = δe T Σ, (14) where subscript i now links Eq. (14) with the current time increment. Since δe and δɛ (x) are independent, the preceding equation can be split into two equalities δe T ( L i (x) E + L i (x) ( ɛ i (x) ɛ 0 i (x) ɛ i (x) ) ) δe T Σ = 0 δ(ɛ ) T (x)l i (x) E + δ(ɛ ) T (x)l i (x) ( ɛ i (x) ɛ 0 i (x) ɛ i (x) ) = 0. (15) A standard finite element approach is usually adopted to solve the above system of equations. We proceed by introducing a set of C 0 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 ɛ (x) = B(x) r, δɛ T = (δr) T B T (x). (16) Inserting these formulae in Eqs. (15) provides the following incremental form of algebraic equations K 11 K 12 E K 21 K 22 r i i Σ + F 0 = f 0 i. (17) 10

11 Individual matrices listed in Eq. (17) are written as K 11i = 1 L i (x) d, K 12i = K T 21i = 1 L i (x)b(x) d, (18) K 22i = 1 B T (x)l i (x)b(x) d, and components of the right-hand side vector are F 0 i = 1 f 0 i = 1 L i (x) ( ɛ 0 i (x) + ɛ i (x) ) d, B T (x)l i (x) ( ɛ 0 i (x) + ɛ i (x) ) d. (19) Finally, eliminating the fluctuating displacements vector r i from Eq. (17) readily provides the incremental form of the macroscopic constitutive law Σ i = D i E i + Λ 0 i, (20) where D i = ( ) K 11 K 12 K 1 22 K T 12, i Λ0 i = f 0 i + ( K 12 K 1 22 f 0). (21) i When prescribing the overall strain only the system of equations (17) reduces to (K 22 ) i r i = (K 21 ) i E i + f 0 i. (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

12 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 elements 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 statistically homogeneous distribution of reinforcements. Here, we limit our attention to two-phase fibrous composites with fibers aligned along the x 3 direction. The morphology of such a system can be conveniently described by the two-point probability function S rs. As suggested by Drugan & Willis [?] this function, when combined with the Hashin-Shtrikman variational 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 descriptors (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

13 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 χ r (x, α), which is equal to one when point x lies in the phase r within the sample α and equal to zero otherwise 1 x D r (α) χ r (x, α) = 0 otherwise. (23) The symbol D r (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 χ f (x, α) and χ m (x, α) are related by χ m (x, α) + χ f (x, α) = 1. (24) With the aid of function χ r, the general n-point probability function S r1,...,r n is given by [?]. S r1,...,r n (x 1,..., x n ) = χ r1 (x 1, α) χ rn (x n, α). (25) Thus, S r1,...,r n gives the probability of finding n points x 1,..., x n randomly thrown into the media located in the phases r 1,..., r n. 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 ergodic 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 χ r (x, α) given by 1 χ r (x, α) = lim V V V χ r (x + y, α)dy, (26) 13

14 is independent of α and identical to the ensemble average, χ r (x) = S r = χ r (x) = c r, (27) where c r is the volume fraction of the r th 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 S mm (x 1, x 2 ) = S mm (x 12 ), (28) where x ij = x j x i. In the context of a representative volume element (RVE: a material element which effectively samples all microstructural configurations) the one-point probability function S r and the two-point probability function S rs 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 statistically 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 S r and S rs. Note that for an ergodic and periodic microstructure the two-point probability function S rs receives the following form S rs (x) = 1 χ r (y)χ s (x + y)dy, (29) where is the size of the RVE (the micrograph area). It is worthwhile to mention that only the Fourier transform of function S rs given by S rs (ξ) = 1 χ r(ξ) χ s (ξ), (30) 14

15 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)ɛ(x) + λ(x), σ(x) = L 0 ɛ(x) + τ (x), (31) where L(x) is the local stiffness matrix and L 0 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 ( E T Σ (τ λ) T (L L 0 ) 1 (τ λ) 2τ T E ɛ T τ λ T L 1 λ ) d, (32) where the fluctuation part ɛ of the local strain ɛ is provide by (see [?]) ɛ (x) = ɛ(x) E = ɛ 0(x x ) (τ (x ) τ ) d(x ). (33) 15

16 ɛ 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 λ r (x) = λ r 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 n τ (x, α) = τ r χ r (x, α), n λ(x, α) = λ r χ r (x, α). (34) r=1 r=1 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 [ ] δrs (L r L 0 ) 1 c r A rs τ s = Ec r + (L r L 0 ) 1 λ r c r, r = 1,..., n, (35) s=1 where the microstructure-dependent matrices A rs do not depend on x and are provided by, see [?], A rs = ɛ 0(x x ) [S rs (x x ) c r c s ] d(x ) = 1 (2π) 2 ɛ 0(ξ ) S 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

17 n n L = L 0 + c r T rs c s, (38) r=1 s=1 n n λ = c r T rs c s (L s L 0 ) 1 λ s. (39) r=1 s=1 T rs 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 = A mm, K r = L r L 0, T rs = K r ( cf K m + c m K f c f c m A 1) [ K f + K m K r + δ rs (1 c r )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 σ i = L i E i + λ i, (41) and the current increment of λ and the instantaneous stiffness matrix L m of the matrix phase attain the forms λ i = n n r=1 s=1 c r T rs c s (L s L 0 ) 1 i (L s ) i ( ɛ 0 s + ɛ s )i, s = m (L m) i = Êi L m, (42) whereas the fiber phase is assumed elastic. Eqs (41) and (42) drive the solution of a viscoelastic 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

18 in the form ɛ(x) = M(x)σ(x) + µ(x), ɛ(x) = M 0 σ(x) + γ(x), (43) where M(x) now stands for the local compliance matrix and M 0 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 ( Σ T E (γ µ) T (M M 0 ) 1 (γ µ) + 2γ T Σ + σ T γ ) d, (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 1 0 (γ(x) γ ). (45) The operator [σ 0 (γ γ )] can be identified with the operator [ɛ 0 (τ τ )] when replacing γ for τ and σ 0 for ɛ 0 and suitably modifying the boundary term to reflect the traction boundary conditions. Specific forms of tensors ɛ 0 and σ 0 can be found, e.g., in [?]. Assuming only a piecewise uniform distribution of phase polarization strain γ r (x) = γ r and the phase eigenstress vector µ r (x) = µ r provides the trial fields for γ and µ at any point x located in the sample α in the form, recall Eq. (34), n γ(x, α) = γ r χ r (x, α), n µ(x, α) = µ r χ r (x, α). (46) r=1 r=1 We now proceed, in analogy with Section 4.2, by introducing Eqs. (45) and (46) into functional (44), evaluating an ensemble average of U γ and finally taking variation with respect to γ r to get 18

19 n s=1 { [ ] } δrs (Mr M 0 ) 1 + M 1 0 cr B rs M 1 0 c r c s γs = Σc r + (M r M 0 ) 1 c r µ r, 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 γ r. To continue, recall Eq. (36) and write the microstructure dependent matrices B rs in the form B rs = 1 (2π) 2 σ 0(ξ ) S rs(ξ )dξ. (48) Finally, after symbolic inversion of Eq. (47) we arrive at the macroscopic constitutive law, see also Eqs. (37) (39), ɛ = MΣ + µ, (49) where the effective compliance matrix M and the macroscopic eigenstrain vector µ read n n M = M 0 + c r R rs c s, (50) r=1 s=1 n n µ = c r R rs c s (M s M 0 ) 1 µ s. (51) r=1 s=1 For a two-phase composite medium the matrices R rs assume the form B = c f c m M 1 0 B mm, N r = [ (M r M 0 ) 1 ] + M 1 1 0, R rs = N r ( cf N 2 + c m N 1 c f c m B 1) [ N f + N m N r + δ rs (1 c r )B 1]. (52) When solving a viscoelastic problem we replace Eq. (49) by its incremental counterpart such that ɛ i = M i Σ i + µ i, (53) 19

20 and finally the current increment of µ and the instantaneous compliance matrix M m of the matrix phase are expressed as µ i = n n r=1 s=1 ( ) c r R rs c s (M s M 0 ) 1 i ɛ 0 s + ɛ s, s = m (M m) i = 1Êi M m, (54) i 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 E A E T G T ν A a b n [GPa] [GPa] [GPa] [GPa] 1 [GPa] 1 fiber matrix Table 1 Material properties of T30/Epoxy system. the x 3 -axis directed along the fiber direction is selected. Therefore, the macroscopic stress and strain fields written in contracted notation are Σ = {Σ 11, Σ 22, Σ 12, Σ 33 } T and E = {E 11, E 22, E 12, E 33 } 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

21 Σ 11 [MPa] Σ12 [MPa] E 11 [%] (a) E 12 [%] (b) (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, respectively (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 assuming 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 contrast 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

22 5.0e e e e-02 Overall strain E e e e-03 2-fibers Unit cell 5-fibers Unit cell 10-fibers Unit cell 20-fibers Unit cell Hexagonal array Overall strain E e e e-02 2-fibers Unit cell 5-fibers Unit cell 10-fibers Unit cell 20-fibers Unite cell Hexagonal array 2.5e-03 (a) 8.0e-03 (b) Fig. 6. Overall response Unit cell model: creep test. 3.0e e+00 Overall stress Σ 11 [MPa] 2.5e e e+01 2-fibers Unit cell 5-fibers Unit cell 10-fibers Unit cell 20-fibers Unit cell Hexagonal array Overall stress Σ 12 [MPa] 5.0e e e e+00 2-fibers Unit cell 5-fibers Unit cell 10-fibers Unit cell 20-fibers Unit cell Hexagonal array 1.0e+01 (c) 3.0e+00 (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 properties 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

23 4.0e-03 Overall strain E e e e e-03 5-fibers Unit cell 10-fibers Unit cell 20-fibers Unit cell Hexagonal array 1.5e-03 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 simple 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. 3.0e e-03 Σ 22 due to ε e-03 ε 11 due to Σ 11 Matrix strain Stress [MPa] 2.5e e+01 Σ 11 due to ε 11 Matrix stress Overall stress Fiber stress Strain 5.0e e e-03 ε 22 due to Σ 22 Overall strain 2.0e-03 Fiber strain 1.5e+01 (a) 1.0e-03 (b) Fig. 9. Overall and local response Unit cell model: (a) relaxation test, (b) creep test. The remaining set of results (Figs ) 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 (L 0, M 0 ). To this end, we draw the readers attention to Fig. 10. First, recall Table 1 and notice that the matrix moduli 23

24 3.0e e-03 Overall stress S 11 [MPa] 2.5e e e+01 H-S Upper bound H-S Estimate H-S Lower bound Unit cell Overall strain E e e e e-03 H-S Upper bound H-S Estimate H-S Lower bound Unit cell 1.0e+01 (a) 2.5e-03 (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 L 0 and M 0 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 L 0 and M 0 (e.g., L 0 = 1(L 2 f + L m )). (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 noteworthy 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

25 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 L 0 and compliance M 0 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. 5.0e e e e-02 Overall strain E e e e x 716 pixels 488 x 358 pixels 244 x 358 pixels 122 x 84 pixels Overall strain E e e e x 716 pixels 488 x 358 pixels 244 x 179 pixels 122 x 84 pixels 2.5e-03 (a) 8.0e-03 (b) Fig. 12. Overall response Hashin-Shtrikman principle: creep test. 3.0e e+00 Overall stress Σ 11 [MPa] 2.5e e e x 716 pixels 488 x 358 pixels 244 x 179 pixels 122 x 84 pixels Overall stress Σ 12 [MPa] 5.0e e e e x 716 pixels 488 x 358 pixels 244 x 179 pixels 122 x 84 pixels 1.0e+01 (a) 3.0e+00 (b) Fig. 13. Overall response Hashin-Shtrikman principle: relaxation test. 25

26 3.0e e-03 Stress [MPa] 2.5e e+01 Σ 11 due to ε 11 Σ 22 due to ε 22 Matrix stress Overall stress Fiber stress Strain 6.0e e e e-03 ε 11 due to Σ 11 ε 22 due to Σ 22 Matrix strain Overall strain 2.0e-03 Fiber strain 1.5e (a) 1.0e-03 (b) Fig. 14. Overall and local response Hashin-Shtrikman principle: (a) relaxation test, (b) creep test. 3.0e e-03 Stress [MPa] 2.5e e+01 UC H-S: LB Matrix stress Overall stress Fiber stress Strain 6.0e e e e-03 Unit cell H-S: UB Matrix strain Overall strain 2.0e-03 Fiber strain 1.5e+01 (a) 1.0e-03 (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

27 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 principles 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 reinforcements. 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

28 principles further extended to reflect the presence of eigenstresses and eigenstrains in the formulation of macroscopic equations. An efficient procedure for evaluation of required materials statistics, already examined in [?], was implemented in connection with binary images of a real microstructure. The results plotted in Figs 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 applicability 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