Calculus of Variations and Geometric Measure Theory

V. Agostiniani - A. Lucantonio - D. Lučić

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

created by luči on 07 Jun 2017
modified by lučić on 26 Jun 2020


Accepted Paper

Inserted: 7 jun 2017
Last Updated: 26 jun 2020

Journal: ESAIM: COCV
Year: 2017
Doi: 10.1051/cocv/2018046

ArXiv: 1706.00629 PDF


We rigorously derive a Kirchhoff plate theory, via $\Gamma$-convergence, from a three-di\-men\-sio\-nal 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 $\mathbb{R}^3$, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from a $x'$-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.