We give an analysis of the stability and uniqueness of the simplylaminated microstructure for all three tetragonal to monoclinicmartensitic transformations. The energy density for tetragonal tomonoclinic transformations has four rotationally invariant wells sincethe transformation has four variants. One of these tetragonal tomonoclinic martensitic transformations corresponds to the shearing ofthe rectangular side, one corresponds to the shearing of the squarebase, and one corresponds to the shearing of the plane orthogonal to adiagonal in the square base. We show that the simply laminatedmicrostructure is stable except for a class of special materialparameters. In each case that the microstructure is stable, we deriveerror estimates for the finite element approximation.

Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.

We give results for the approximation of a laminate withvarying volume fractions for multi-well energy minimizationproblems modeling martensitic crystals thatcan undergo either an orthorhombicto monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfythe corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energyfor the approximation of the limiting macroscopicdeformation and the simply laminated microstructure.Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.