Calculus of Variations and Geometric Measure Theory
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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 02 Oct 2018


Accepted Paper

Inserted: 7 jun 2017
Last Updated: 2 oct 2018

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


We present a rigorous derivation of a non-linear plate theory, via $\Gamma$-convergence, from a three-dimensional 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 on both the thickness variable and on the planar variable $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D limiting energy constrains the deformation of the mid-plane of the plate to be an isometric immersion, and penalizes deviations of its curvature from an $x'$-dependent target curvature tensor. A discussion on the minimizers of the 2D limiting model is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply our plate theory to the modeling of swelling-induced shape changes in heterogeneous, thin gel sheets.


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