椭圆曲面上无穷小等距的匹配性与薄壳弹性力学
The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells
摘要：
利用Gamma收敛的概念，在变形的弹性能的尺度类似于$h^β$与$2<β<4$的假设下，讨论了当厚度h收敛到零时，椭圆薄壳的三维非线性弹性能的极限行为.我们建立，对于给定的标度制度，极限理论减少到线性纯弯曲。证明的两个主要内容是：$W^{2,2}$一阶无穷小等距空间中光滑无穷小等距的密度，以及光滑无穷小等距与光滑椭圆曲面上精确等距浸入匹配的结果。
英文摘要：
Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^β$ with $2<β<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.
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