On the computation of the rank-one convex hull of a function (Q2706468)

The paper addresses the problem of minimization of the functional \(I(u)=\int_\Omega\varphi(\nabla u(x)) dx\) over functions satisfying a non-homogeneous Dirichlet boundary condition, with \(\varphi:{\mathbb R}^{2\times 2}\to{\mathbb R}\) possibly non-quasiconvex and \(\Omega\subset{\mathbb R}^2\). This is related with a minimum of elastic energy stored in a homogeneous elastic crystalline solid loaded by a hard device, if a symmetry is considered to reduce the situation to a 2-dimensional model. The author's aim is to compute (an approximation of) the rank-one convex envelope, which is related with minimizing sequences with a special (possibly hierarchically) laminated pattern. The proposed algorithm optimizes basically the homogeneous Young measure (i.e., a probability measure) being supported on pair-wise rank-one connected matrices; more precisely, those matrices are the leaves of a binary tree whose root is a macroscopical deformation gradient and each of the two branches differs by a rank-one matrix. The efficiency of the algorithm is demonstrated on three different situations of \(\varphi\): Kohn's example of minimum of two quadratic functions, Ericksen-James frame-indifferent potential related with shape-memory alloys crystallizing in cubic/tetragonal configurations, as, e.g., InTl, and finally Tartar's example with 4 non-rank-one-connected point wells. First- and second-order laminates are calculated.