Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

¹ Università di Verona, Strada le Grazie 15, 37134 Verona, Italy.
² SISSA, via Bonomea 265, 34136 Trieste, Italy.

^* Corresponding author: virginia.agostiniani@univr.it

Received: 27 July 2017
Accepted: 6 September 2018

Abstract

We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x′. At the same time, the spontaneous strain is h-close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ³, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.