Title: Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets,
Abstract: 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.