An Example of Microstructure with Multiple Scales

Abstract

This paper studies a vectorial problem in the calculus of variations arising in the theory of martensitic microstructure. The functional has an integral representation where the integrand is a nonconvex function of the gradient with exactly four minima. We prove that the Young measure corresponding to a minimising sequence is homogeneous and unique for certain linear boundary conditions. We also consider the singular perturbation of the problem by higher-order gradients. We study an example of microstructure involving infinite sequential lamination and calculate its energy and length scales in the zero limit of the perturbation.