Inserted: 2 oct 2018
Last Updated: 2 oct 2018
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 exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. 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 Karman), $h^6$ in the general case, and all possible (even power) regimes for conformal metrics. 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 Karman energy functional at scaling $h^4$.