Numerical analysis of a microstructure for a rotationally invariant, double well energy

The laminated microstructure observed in martensitic crystals can be modeled by energy minimizing sequences of deformations for a rotationally invariant (or frame-indifferent), double well energy density [1, 2, 10]. The deformation gradients of energy minimizing sequences oscillate between energy wells across layers (with width converging to zero) so that the effective energy density becomes the relaxed energy density [2, 12]. We present error estimates for the minimization of the energy ∫_Ω φ(▽υ(x)) dx where the energy density φ(A) is a rotationally invariant, double well energy density by a general class of approximation methods for the deformation in L², the weak convergence of the deformation gradient, the convergence of the microstructure (or Young measure) of the deformation gradient, and the convergence of nonlinear integrals of the deformation gradient.