On the quasiconvex hull for a three-well problem in two dimensional linear elasticity

We provide quantitative inner and outer bounds for the symmetric quasiconvex hull Q^e(𝒰) on linear strains generated by three-well sets 𝒰 in ℝ^2× 2_sym . In our study, we consider all possible compatible configurations for three wells and prove that if there exist two matrices in 𝒰 that are rank-one compatible then Q^e(𝒰) coincides with its symmetric lamination convex hull L^e(𝒰) . We complete this result by providing an explicit characterization of L^e(𝒰) in terms of the wells in 𝒰 . Finally, we discuss the optimality of our outer bound and its relationship with quadratic polyconvex functions.