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.

中文翻译：

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横向应变变化的薄膜尺寸减小：振荡和非振荡情况

我们在描述预应力薄膜的奇异极限（Γ极限）为h  →0时薄膜厚度h中描述了非核素（不相容）弹性能函数。首先，我们将先前的结果扩展到取决于中板和横向变量的任意不相容性度量标准（“非振荡”情况）。其次，我们分析了更一般的不相容性类别，其中低阶项的横向相关性不一定是线性的（“振荡”情况），将的结果扩展到任意度量和高阶缩放。我们通过多项式张量空间上适当曲率形式的投影来展示两种情况之间的联系。我们还显示了按比例缩放的有效能量量化为h的幂，并讨论了缩放比例h ²（Kirchhoff）和h ⁴一般情况下（vonKármán）以及所有可能的（偶数幂）保形指标制度，从而为随后对Lewicka（2018）中所述的非振荡设置进行完整分析铺平了道路。第三，我们证明了奇异极限在h ²和h ⁴标度阶上的矫顽力不等式，同时证明了经典vonKármán能量函数在h ⁴标度上的完全矫顽力。©2019 Wiley Periodicals，Inc.