关于任意形状薄壳非线性弹性泛函低能临界点收敛性的注记
A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry
摘要：
我们证明了三维非线性弹性泛函的临界点的小厚度$h$壳和周围的中间表面$S$的任意几何形状，收敛$h\to 0$的临界点的冯卡门功能$S$，最近在\cite{lemopa1}。该结果扩展了\cite{MuPa}中的陈述，该陈述是针对$S\subset\mathbb{R}^2$时的板的情况导出的。我们进一步证明了静态平衡方程（形式上是与弹性泛函相关联的欧拉-拉格朗日方程）的弱解的相同收敛结果。如果三维变形的弹性能尺度如$h^4$，外部体力尺度如$h^3$，则收敛保持不变。
英文摘要：
We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness $h$ and around the mid-surface $S$ of arbitrary geometry, converge as $h\to 0$ to the critical points of the von Kármán functional on $S$, recently derived in \cite{lemopa1}. This result extends the statement in \cite{MuPa}, derived for the case of plates when $S\subset\mathbb{R}^2$. We further prove the same convergence result for the weak solutions to the static equilibrium equations (formally the Euler- Lagrange equations associated to the elasticity functional). The convergences hold provided the elastic energy of the 3d deformations scale like $h^4$ and the external body forces scale like $h^3$.
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