论文标题
通过伽马收敛降低尺寸
Dimension reduction through Gamma convergence for general prestrained thin elastic sheets
论文作者
论文摘要
我们通过分析材料的厚度趋向于0 $ 0来研究薄膜具有残留应变的薄膜，因为$γ-$限制的限制趋于0 $0。$我们首先要扩展先前的结果\ cite \ cite {bhattacharya2016plates} \ cite \ cite {agostinianiianiianiianiianiy2018heterogeniel} \ cite {schmidt2007plate}，到更宽的电影类别，其prestrain取决于中板和横向变量。我们的$γ-$收敛结果的ANSATZ使用特定类型的皱纹，该皱纹构建在Monge-Ampere方程的异国情调解决方案上，该方程是通过凸集成\ cite \ cite {lewicka2017convex}构建的。我们表明，$γ-$限制的表达在残留应力对合适子空间的正交投影方面具有自然的解释。我们还表明，在某些情况下，某种类型的皱纹现象对于匹配$γ-$限制的下限是必要的。 These results all assume a prestrain of the same order as the thickness;我们还讨论了为什么通过考虑较大时会发生什么会发生什么是很自然的。
We study thin films with residual strain by analyzing the $Γ-$limit of non-Euclidean elastic energy functionals as the material's thickness tends to $0.$ We begin by extending prior results \cite{bhattacharya2016plates} \cite{agostiniani2018heterogeneous} \cite{lewicka2018dimension} \cite{schmidt2007plate}, to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our $Γ-$convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration \cite{lewicka2017convex}. We show that the expression for our $Γ-$limit has a natural interpretation in terms of the orthogonal projection of the residual strain onto a suitable subspace. We also show that some type of wrinkling phenomenon is necessary to match the lower bound of the $Γ-$limit in certain circumstances. These results all assume a prestrain of the same order as the thickness; we also discuss why it is natural to focus on that regime by considering what can happen when the prestrain is larger.
