Particulate Microstructures

In most of the cases involving random composite materials, the physical properties of the media vary not only with positionxbut also from sample to sample. In general, for the complete determination of the location of the phases and hence the microstructure, we need to specify the functionsχ^(r)(x) for all x in Ω. However, this is not possible in most of the cases of random composites and thus the description of such materials can be achieved via n−point correlation functions, withn being a finite integer number. On the other hand, these random systems can be considered to be statisti-cally uniform (Willis, 1982), which implies that then−point correlation functions are insensitive to translations. For such media, it is usual to make an assumption ofergodic type, which yields that local configurations occur every any one specimen with the frequency with which they occur over a single neighborhood in an ensemble of specimens. Thus, ensemble averages can be replaced by volume

averages, and hence the one-point statistics simply provide information about the volume fraction of the phases. For instance, the porosity is defined in terms of the distribution functions byf =hχ⁽²⁾i.

On the other hand, the two-, three-, or n-point statistics provide information about the relative position of the phases in Ω. In the present study, we make use of homogenization methods (Hashin and Shtrikman, 1963; Willis, 1977; Willis, 1978; Ponte Casta˜neda and Willis, 1995) that involve information up to two-point statistics, although there exist theories that make use of three-point statistics (Beran, 1965; Kr¨oner, 1977; Milton, 1982; Willis, 1981), which are fairly complicated and will not be used here. To begin with, Willis (1977) proposed a general description of the microstructure in terms of the two-point correlation function, which can exhibit, for example, isotropic or ellipsoidal symmetry in the space. This formulation allowed the author to obtain bounds on the effective response of the composite, while explaining the Hashin-Shtrikman (1963) variational principle for isotropic composites in a rigorous mathematical way.

(a) (b)

Figure 2.1: Representative volume element of a “particulate” porous medium. Ellipsoidal voids distributed randomly with “ellipsoidal symmetry.” The solid ellipsoids denote the voids, and the dashed ellipsoids, their distribution.

In a later work, Willis (1978) considered the case of a composite that comprises a matrix phase in which are embedded inclusions of known shapes and orientation. This description represents a

“particulate” microstructure and is a generalization of the Eshelby (1956) dilute microstructure in the non-dilute regime. It is important to note that this representation is a subclass of the above-mentioned more general description (Willis, 1977) of a two-phase heterogeneous material, whose phases are simply described by the two-point correlation function. More specifically, a “particulate”

composite consists of inclusions with known shapes and orientations, whereas the two-point correlation function provides information about the distribution of the centers of the inclusions. For instance, Fig.2.1shows schematically a representation of a “particulate” microstructure consisting of ellipsoidal inclusions (solid lines) with known shapes and orientations distributed randomly in a matrix phase with ellipsoidal symmetry (dashed ellipsoids). In the original work, Willis (1978) considered that the shape and orientation of the inclusions is the same with the shape and orientation of the distribution function, as shown in Fig. 2.1a. However, due to the heterogeneity in the strain-rate fields in the

“particulate” composite, the shape of the inclusions and the shape of their distribution function

is expected, in general, to change in different proportions during the deformation process. In this connection, Ponte Casta˜neda and Willis (1995) generalized the study of Willis (1978) for particulate microstructures by letting the shapes of the inclusions and their distribution to have different shapes and orientation, as shown in Fig.2.1b, and therefore to evolve in a different manner.

( )³

n

( )² ( )¹ n

n a1

a2

a3

Representative ellipsoidal void ( )³

e

( )¹

e

( )²

e

Figure 2.2: Representative ellipsoidal void.

In the following, the discussion is restricted to porous media, however, the foregoing definitions hold for any type of inclusion phase (e.g., fiber reinforced media). As already remarked in the previous paragraph, we need to define microstructural variables that describe the shape and orientation of the voids and their distribution function (Ponte Casta˜neda and Willis, 1995). Nevertheless, the effect of the shape and orientation of the distribution function on the effective behavior of the porous material becomes less important at low and moderate porosities (Kailasam and Ponte Casta˜neda, 1998), due to the fact that the contribution of the distribution function is only of order two in the volume fractions of the inclusions. Since, this work is mainly concerned with low to moderate concentrations of the voids, for simplicity, we will make the assumption, which will hold for the rest of the text, that the shape and orientation of the distribution function is identical to the shape and orientation of the voids, as shown in Fig.2.1a, and hence it evolves in the same fashion when the material is subjected to finite deformations. The basic internal variables characterizing the state of the microstructure are:

1. the volume fraction of the voids or porosity f =V2/V, where V =V1+V2 denotes the total volume, with V1 and V2 being the volume occupied by the matrix and the vacuous phase, respectively,

2. the two aspect ratiosw1=a3/a1, w2=a3/a2 (w3= 1), where 2ai withi= 1,2,3 denote the lengths of the principal axes of the representative ellipsoidal void,

3. the orientation unit vectorsn⁽ⁱ⁾(i= 1,2,3), defining an orthonormal basis set, which coincides with the principal axes of the representative ellipsoidal void.

The above set of the microstructural variables is denoted as

sα={f, w1, w2,n⁽¹⁾,n⁽²⁾,n⁽³⁾=n⁽¹⁾×n⁽²⁾}. (2.16)

A schematic representation of the above-described microstructure is shown in Fig.2.2. In turn, the vectorsn⁽ⁱ⁾can be described in terms of three Euler angles, which correspond to three rotations with respect to the laboratory frame axese⁽ⁱ⁾, such that

n⁽ⁱ⁾=Rψe⁽ⁱ⁾, with Rψ=Rψ3Rψ2Rψ1. (2.17) In this expression,Rψis a proper orthogonal matrix, defined in terms of three other proper orthogonal matrices given by

Rψ1=







1 0 0

0 cosψ1 sinψ1

0 −sinψ1 cosψ1





,

Rψ2 =







cosψ₂ 0 sinψ₂

0 1 0

−sinψ2 0 cosψ2





, (2.18)

Rψ3 =







cosψ3 sinψ3 0

−sinψ3 cosψ3 0

0 0 1





,

whereψ1,ψ2andψ3are three Euler angles, which denote rotation of the principal axes of the ellipsoid about the 1−, 2−and 3−axis, respectively.

Now, the shape and orientation of the voids, as well as the shape and orientation of the two-point correlation function can be completely characterized by a symmetric second-order tensor Z introduced by Ponte Casta˜neda and Willis (1995) (see also Aravas and Ponte Casta˜neda (2004)), which is given in terms of the two aspect ratios and the three orientation vectors shown in Fig. 2.2, such that

Z=w₁n⁽¹⁾⊗n⁽¹⁾+w₂n⁽²⁾⊗n⁽²⁾+n⁽³⁾⊗n⁽³⁾, det(Z) =w₁w₂. (2.19) Note that this assumption could be relaxed by allowing the shapes and orientation of the inclusions and of their distribution functions to be described by two different tensorsZ(Ponte Casta˜neda and Willis, 1995). However, this more general configuration is not adopted here, for the reasons explained in the previous paragraphs.

In this class of particulate microstructures, some special cases of interest could be identified.

Firstly, when w1 = w2 = 1, the resulting porous medium exhibits an overall isotropic behavior, provided that the matrix phase is also characterized by an isotropic stress potential. Secondly, if w1 =w2 6= 1, the corresponding porous medium is transversely isotropic about then⁽³⁾−direction, provided that the matrix phase is isotropic or transversely isotropic about the same direction. For this special configuration of the microstructure, i.e., forw1=w26= 1 with an isotropic matrix phase, several studies have been performed by several authors (Gologanu et al., 1993, 1994, 1997; Leblond et al., 1994; Gˇarˇajeu et al. 2000; Flandi and Leblond, 2005a, 2005b; Monchiet et al., 2007).

Apart from the simple cases discussed previously, it is relevant to explore other types of particulate microstructures, which can be easily derived by appropriate specialization of the aforementioned variablessa (Budiansky et al., 1982). In this regard, the following cases are considered:

ψ

e