A total-variation surface energy model for thin films of martensitic crystals (Q1599131)

The authors build a thin-film model for martensitic crystals, which differs from that introduced by \textit{K. Bhattacharya} and \textit{R. D. James} [J. Mech. Phys. Solids 47, No.~3, 531-576 (1999; Zbl 0960.74046)], as they start from the energy functional \(\kappa \int_{\Omega _{h}}|D\left( \nabla u\right) |dx\), which represents the total variation of the deformation gradient inside the domain \(\Omega _{h}=S\times \left( -h/2,h/2\right) \). The deformation is supposed to be known on the lateral boundary \(\Gamma _{h}=\gamma \times \left( -h/2,h/2\right) \). Assuming that the plane domain \(S\) is smooth enough and using the properties of functions having bounded variation, the authors prove the existence of a minimizer of this model. They then let \(h\) go to 0 and derive a limit surface energy of this thin-film model which is similar to that of Bhattacharya and James. A finite element approximation of this thin-film model is finally presented.