Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Abstract

We study the noneuclidean (incompatible) elastic energy functionals in the description of prestressed thin films at their singular limits (Γ‐limits) as h → 0 in the film's thickness h. First, we extend the prior results to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the “nonoscillatory” case). Second, 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 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 quantization in terms of scalings as a power of h and discuss the scaling regimes h2 (Kirchhoff) and h4 (von Kármán) 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 nonoscillatory setting described in Lewicka (2018). Third, we prove the coercivity inequalities for the singular limits at h2‐ and h4‐ scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling h4. © 2019 Wiley Periodicals, Inc.