Phase transformations surfaces and exact energy lower bounds

The paper investigates two-phase microstructures of optimal 3D composites that store minimal elastic energy in a given strain field. The composite is made of two linear isotropic materials which differ in elastic moduli and self-strains. We find optimal microstructures for all values of external strains and volume fractions of components. This study continues research by Gibiansky and Cherkaev (1987), Gibiansky and Cherkaev (1997) and Chenchiah and Bhattacharya (2008). In the present paper we demonstrate that the energy is minimized by laminates of various ranks. Optimal structures are either simple laminates that are codirected with external eigenstrain directions, or inclined laminates, direct and skew second-rank laminates and third-rank laminates. These results are applied for description of direct and reverse transformations limit surfaces in a strain space for elastic solids undergoing phase transformations of martensite type. The surfaces are computed as the values of external strains at which the optimal volume fraction of one of the phases tends to zero. Finally, we compare the transformation surfaces with the envelopes of the nucleation surfaces constructed earlier for nuclei of various geometries (planar layers, elliptical cylinders, ellipsoids). We show the energy equivalence of the cylinders and direct second-rank-laminates, ellipsoids and third-rank laminates. We note that skew second-rank laminates make the nucleation surface convex function of external strain, and they do not correspond to any of the mentioned nuclei.