  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
\cite{gibiansky1987microstructures,GibianskyCherkaev1997} and Chenchiah and
Bhattacharya \cite{Bhattacharya2008}. In the present paper we demonstrate that
the energy is minimized by that 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.
