Mathematics

In this talk I will present a dimensionally reduced Kirchoff-like model, rigorously derived (via -convergence) from a 3D model that describes the finite elasticity of a thin heterogeneous sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends both on the thickness variable and on the planar variable . The case in which the spontaneous strain is -close to the identity, where  is the small parameter quantifying the thickness, will be considered. The 2D limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into , with the corresponding energy penalizing deviations of the curvature tensor  associated with a deformation from -dependent target curvature tensor. The discussion on the 2D minimizers will be provided in the case where the target curvature tensor is piecewise constant. We will also see that the family of energy functionals considered in this talk is relevant from the viewpoint of applications to shape morphing materials, especially in the modeling of swelling-induced shape changes in heterogeneous thin gel sheets. This is a joint work with V. Agostiniani and A. Lucantonio.