Inserted: 2 oct 2018
Last Updated: 7 feb 2019
Journal: Communications on Pure and Applied Mathematics
We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits ($\Gamma$-limits) as $h\to 0$ in the film's thickness $h$. Firstly, we extend the prior results Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the ``non-oscillatory'' case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the ``oscillatory'' case), extending the results of Agostiniani-Lucic-Lucantonio, Schmidt to arbitrary metrics and higher order scalings. We also show the effective energy quantisation in terms of scalings as a power of $h$ and discuss the scaling regimes $h^2$ (Kirchhoff), $h^4$ (von K\'arm\'an) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in Lewicka. Thirdly, we prove the coercivity inequalities for the singular limits at $h^2$- and $h^4$- scaling orders, while disproving the full coercivity of the classical von K\'arm\'an energy functional at scaling $h^4$.