We propose a sharp-interface model for a hyperelastic material consisting of two phases. In this model, phase interfaces are treated in the deformed configuration, resulting in a fully Eulerian interfacial energy. In order to penalize large curvature of the interface, we include a geometric term featuring a curvature varifold. Equilibrium solutions are proved to exist via minimization. We then use this model in an Eulerian topology optimization problem that incorporates a curvature penalization. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.
Keywords: curvature varifolds; elasticity; interfacial energy; multi-phase materials; varifolds.