On the optimality of the variational linear comparison bounds for porous viscoplastic materials

Abstract

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.

Introduction

The problem of determining the effective behavior of porous viscoplastic materials has been of great interest for quite some time. In part, this is due to their central role in ductile fracture (Rice and Tracey, 1969), and has led to a huge literature on the subject, dating back to the estimates of Gurson (1977) for the yield function of porous ideally plastic materials. But porous materials are also important in many other application, including topology optimization (Bendsoe and Kikuchi, 1988) and metal foams and other cellular materials (Evans et al., 1998).

In this work, the interest is on rigorous bounds for the effective flow potential of porous viscoplastic solids (see, for example, Ponte Castañeda and Suquet, 1998). Non-trivial bounds for such porous materials with statistically isotropic microstructures were given by Ponte Castañeda and Willis (1988), making use of a generalization of the variational principles of Hashin and Shtrikman (1963), due to Willis (1983) and Talbot and Willis (1985). Improved bounds for the same class of porous materials were obtained by Ponte Castañeda (1991a) by means of a variational principle where the trial fields are the moduli of a linear comparison composite (LCC), in combination with the Hashin-Shtrikman bounds for LCCs with statistically isotropic microstructures. It was then shown that this “variational linear comparison” bound of the Hashin-Shtrikman type (VHS) could also be derived by the Talbot-Willis method (Willis, 1991, Willis, 1992), as well as by means of the Hölder and Jensen inequalities (Suquet, 1992, Suquet, 1993). As pointed out by Ponte Castañeda, 1991a, Ponte Castañeda, 1992b, the LCC variational method has the distinguishing advantage that it can be used in combination with any bound for LCCs within a given class of microstructures to generate corresponding bounds for nonlinear composites; in particular, tighter nonlinear bounds were generated by Ponte Castañeda (1992b) by means of the bounds of Beran (1965) and Milton and Phan-Thien (1981) for LCCs with isotropic two- and three-point statistics.

An important question concerning bounds for composites with given classes of microstructures is whether they are optimal—i.e., whether there exist specific microstructures within the class that attain the bounds for every possible loading condition. Alternatively, there is the possibility that there are microstructures that attain the bounds only for special loading conditions; in this case, the bounds are said to be sharp. Thus, for composites with linear-elastic properties, it has been shown that the Hashin–Shtrikman bounds for microstructures with prescribed volume fractions and statistically isotropic two-point correlation functions are optimal at least for the case when there are only two isotropic, well-ordered phases (which is the case for a porous linear-elastic material). This was first accomplished for the effective or homogenized bulk modulus by means of the composite-sphere assemblage microstructure (Hashin, 1962), and for the effective shear modulus by means of sequentially laminated microstructures (of finite rank) by Francfort and Murat (1986) (see also Milton, 1986; Tartar, 1985).

In the context of the optimality of the Hashin–Shtrikman bound for the shear modulus, it is important to emphasize that the finite-rank laminates attaining the bound (although having isotropic macroscopic behavior) are not statistically isotropic and are, therefore, not members of the class of microstructures (with statistically isotropic two-point statistics) for which the bounds were initially derived (see Willis, 1977). However, bounds have been derived for two-phase composites with more general classes of microstructures by means of other methods, which, somewhat unexpectedly, reduce to the Hashin–Shtrikman bounds when the overall response of the composites is assumed to be isotropic. Thus, building on earlier work by Lurie, Cherkaev, 1984, Lurie, Cherkaev, 1986 and Tartar (1985) and Murat and Tartar (1985) in electrostatics, Milton (1990b) (see also Milton, 1990a) made use of the “translation” method to show that the Hashin–Shtrikman bounds for the shear modulus also hold for two-phase composites with well-ordered phases in prescribed volume fractions having overall isotropic response—but without the assumption of statistical isotropy. (Related results were obtained earlier by Milton and Kohn, 1988, using a generalization of the Hashin–Shtrikman variational principles.) This larger class of microstructures (not requiring statistical isotropy) includes the finite-rank laminates attaining the Hashin–Shtrikman bounds for the shear modulus. This apparent redundancy in bounds for linear-elastic composites with two isotropic, well-ordered phases arises from the fact that such bounds are known to depend only on certain reduced two-point correlation functions (Avellaneda, 1987), which in turn are special cases of the H-measures of Tartar (1990) (see also Allaire and Maillot, 2003; Milton, 2002).

The important lesson to be drawn from this observation is that the Hashin–Shtrikman bounds for linear-elastic composites with two isotropic, well-ordered phases hold not only for composites with statistically isotropic microstructures, but also for composites with more general microstructures (including sequential laminates), so long as these microstructures still result in overall isotropic behavior. This fact has huge implications for the variational bounds for nonlinear composites obtained by means of the Hashin–Shtrikman bounds for the corresponding LCCs. Thus, in spite of their original derivation (which assumed statistical isotropy), it is clear that the VHS bounds of Ponte Castañeda (1991a) also hold for this more general class of microstructures that are not statistically isotropic—and optimality of the bounds must be considered in the context of this much larger class of microstructures having isotropic overall linear response. Consequently, in this work, we investigate the optimality of the VHS bounds for the special, but important case of porous viscoplastic composites by considering a certain type of sequentially laminated microstructures that will be shown to attain the VHS bounds at least for some specific loading conditions—thereby demonstrating that the VHS bounds are sharp.

The results of this paper will shed light on why prior attempts to show optimality of the VHS bounds failed. For example, composite-sphere microstructures were found not to attain the VHS bounds for porous viscoplastic materials, at least for hydrostatic loadings (Michel, Suquet, 1992, Ponte Castañeda, 1991b). In addition, building on the work of deBotton and Ponte Castañeda (1992) for simple nonlinear laminates, Ponte Castañeda (1992b) first attempted to use sequentially laminated microstructures for nonlinear composites, but quickly concluded that finite-rank laminates could not be isotropic and therefore could not be used to show optimality of the VHS bound. Later deBotton and Hariton (2002) made use of high-rank laminates and showed a tendency towards isotropic behavior. deBotton (2005) and Idiart (2008) were the first to generate estimates for sequentially laminated composites of infinite rank with (transversely) isotropic properties. However, such isotropic laminated microstructures were also found not to attain the VHS bounds; in particular, they could not attain the VHS bounds for porous materials, even for purely deviatoric loadings. In fact, as will be shown, the extremal microstructures attaining the VHS bound turn out to have the most anisotropic macroscopic response in the class, while both the composite-sphere assemblages and infinite-rank laminates have isotropic response. For completeness, it should be noted that there are special cases, such as the case when one of the phases is linear and the other is nonlinear, for which the VHS bounds have been shown to be attained by sequentially laminated microstructures (when the nonlinear material plays the role of the inclusion phase) that are either statistically anisotropic (Ponte Castañeda, 1991c), or statistically isotropic (Idiart, 2013).

In this work, the choice is made to focus on porous viscoplastic materials with incompressible matrix phase because it is precisely for this case when the VHS bounds were found to perform the worst when compared to exact estimates for composite-sphere or sequentially laminated microstructures, especially for hydrostatic loading conditions. However, for simplicity, the choice is also made to work with two-dimensional (2-D) microstructures for which the results are easier to compute and understand. The results of the paper are organized as follows. Section 2 provides the basic definition of the 2-D composites and their effective response under plane-strain loading. Section 3 describes the special class of composites known as finite-rank laminates in the linear context, and discusses their optimality as it pertains to the associated linear bounds. Next, Section 4 provides the framework for computing bounds for porous composites, as well as exact results for nonlinear finite-rank laminates, using the method of Ponte Castañeda (1992a). Section 5 shows explicitly that the VHS bounds are sharp for both purely deviatoric and more general plane-strain loading conditions, while Section 6 presents the results in the ideally plastic limit and compares the yield surfaces and effective hydrostatic flow stresses of the extremal finite-rank laminates to the bounds obtained using the VHS method. Finally, some general conclusions are drawn.

We close this section by introducing the notation that will be used throughout this work. We fix the standard Cartesian basis {e₁, e₂}, with respect to which vectors with Cartesian components b_i are represented by bold letters b, while second-order tensors with Cartesian comnents A_ij are represented by bold italic letters A. Here i, j range from 1 to 2. The real 2-D space is denoted by ${\mathbb{R}}^2,$ and endowed with the scalar product u · v, and norm ${\left|\mathbf{u}\right|}^2=\mathbf{u}·\mathbf{u}$. Similarly, we define the inner product between two second-order tensors via $\mathit{A}·\mathit{B}=\mathrm{tr}\left(\mathit{A}{\mathit{B}}^T\right),$ which induces the norm ${\parallel \mathit{A}\parallel }^2=\mathit{A}·\mathit{A}$. The second-order identity tensor will be denoted by I. Given two vectors a, b, the dyadic product a⊗b is defined to be the second-order tensor with Cartesian components a_ib_j.