高阶预应变不协调板
Plates with incompatible prestrain of higher order
摘要：
我们研究了不相容预应变薄板的有效弹性行为，其特征在于在参考配置上的黎曼度量$G$。我们假设预应变是“弱”的，即它引起不相容弹性能$E^h$按板的厚度$h$换算，其量级小于$h^2$。我们基本上证明了两个结果。首先，我们建立的$Γ$ -限制的标度能量$h^{-4}E^h$和表明，它包括一个冯卡门的能量，给出的一阶无穷小等距和表面上的容许应变等距浸入$G_{2\times 2}$（即中板上的预应变度量）在$\mathbb{R}^3$。其次，我们证明了在缩放制度$E^h\sim h^β$与$β>2$，有没有其他的限制理论：如果$\inf h^{-2} E^h \to 0$，然后$\inf E^h\leq Ch^4$，如果$\inf h^{-4}E^h\to 0$，然后$G$是可实现的，因此$\min E^h = 0$为每个$h$。
英文摘要：
We study the effective elastic behaviour of the incompatibly prestrained thin plates, characterized by a Riemann metric $G$ on the reference configuration. We assume that the prestrain is "weak", i.e. it induces scaling of the incompatible elastic energy $E^h$ of order less than $h^2$ in terms of the plate's thickness $h$. We essentially prove two results. First, we establish the $Γ$-limit of the scaled energies $h^{-4}E^h$ and show that it consists of a von Kármán-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing $G_{2\times 2}$ (i.e. the prestrain metric on the midplate) in $\mathbb{R}^3$. Second, we prove that in the scaling regime $E^h\sim h^β$ with $β>2$, there is no other limiting theory: if $\inf h^{-2} E^h \to 0$ then $\inf E^h\leq Ch^4$, and if $\inf h^{-4}E^h\to 0$ then $G$ is realizable and hence $\min E^h = 0$ for every $h$.
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