A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

In the theory of solid-solid phase transitions the deformation of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study Frame indifference, within a linearized setting, requires that W depends only on the symmetric part of ∇u. The potential W is nonnegative and vanishes on two wells, i.e., for d = 2, on two lines in the space of matrices. We determine, for d = 2, the Gamma limit I₀ = Γ− lim _ɛ→0 I_ɛ. The limit I₀u] is finite only for deformations u that fulfill W(∇u)=0 almost everywhere and have sharp interfaces where ∇u has jumps. For these u, I₀u] equals the line integral over the interfaces of a surface energy density.