Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case

TL;DR

This paper extends the analysis of thin film elasticity with incompatible prestrain to more general cases, exploring oscillatory and non-oscillatory incompatibilities, different scaling regimes, and establishing coercivity properties of the limiting energies.

Contribution

It generalizes previous models to arbitrary incompatibility metrics, analyzes oscillatory effects, and connects different regimes via curvature projections, advancing the understanding of thin film elastic limits.

Findings

Extended results to arbitrary incompatibility metrics depending on midplate and transversal variables.

Analyzed oscillatory incompatibilities with higher order scalings, generalizing prior work.

Proved coercivity inequalities at specific scalings and showed non-coercivity of classical von Kármán energy.

Abstract

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 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 $h^4$.