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.

本文关注的是非线性复合材料的Hashin-Shtrikman型（VHS）变分界的最优性，首先由PonteCastañeda（1991a）通过适当优化的线性比较复合物（LCC）的相应HS界获得。为简单起见，该问题在具有不可压缩的各向同性基体相和二维微结构的多孔粘塑性材料的环境中得到解决，该材料经受平面应变载荷条件。尽管很特殊，但这种情况（表现出无限的异质性对比和可压缩的宏观响应）有望完全代表更通用的三维多孔材料以及更通用的两相，有序复合材料。因此，表明VHS范围 实际上，它们是针对具有统计各向同性的微观结构的非线性复合材料类别而衍生的，实际上是在具有各向异性的非线性响应但具有各向同性的线性响应的较大的微观结构类别中实现的。通过对有限阶非线性层压板的有效势进行精确的变分表示，表明存在一定的施加宏观应力值，为此，有限阶层压板（闭孔多孔）微结构达到了线性HS界限也达到非线性VHS限制。在达到VHS约束的有限秩层压材料的屈服面的理想塑性极限下，可获得明确的结果。尤其是，