The paper establishes exact lower bound on the effective elastic energy of
two-dimensional, three-material composite subjected to the homogeneous,
anisotropic stress. It is assumed that the materials are mixed with given
volume fractions and that one of the phases is degenerated to void, i.e. the
effective composite is porous. Explicit formula for the energy bound is
obtained using the translation method enhanced with additional inequality
expressing certain property of stresses. Sufficient optimality conditions of
the energy bound are used to set the requirements which have to be met by the
stress fields in each phase of optimal effective material regardless of the
complexity of its microstructural geometry. We show that these requirements are
fulfilled in a special class of microgeometries, so-called laminates of a rank.
Their optimality is elaborated in detail for structures with significant amount
of void, also referred to as high-porosity structures. It is shown that
geometrical parameters of optimal multi-rank, high-porosity laminates are
different in various ranges of volume fractions and anisotropy level of
external stress. Non-laminate, three-phase microstructures introduced by other
authors and their optimality in high-porosity regions is also discussed by
means of the sufficient conditions technique. Conjectures regarding
low-porosity regions are presented, but full treatment of this issue is
postponed to a separate publication. The corresponding "G-closure problem" of a
three-phase isotropic composite is also addressed and exact bounds on effective
isotropic properties are explicitly determined in these regions where the
stress energy bound is optimal.Comment: Added section 4.3 and figures 9-11. Minor editorial changes for the
improvement of clarit