简化板壳杆理论的黎曼方法
A Riemannian Approach to Reduced Plate, Shell, and Rod Theories
摘要：
本文导出了$n$维非线性弹性体沿$k$维方向为细长体的降维极限理论。出发点是把弹性体看作是一个$n$维黎曼流形，以及一个不一定等距的$W^{1,2}$ -浸入在$n$维欧几里得空间中.平衡配置是最小化诱导度量和内在度量之间的平均差异的沉浸。降维极限理论将弹性体看作是一个$k$维的黎曼流形，以及一个等距的$W^{2,2}$ -浸入$n$维的欧几里得空间和在法线方向上的线性数据。平衡配置最小化取决于线性数据的平均共变导数的泛函。降维极限是使用$Γ$ -收敛方法获得的。极限包括板、壳和杆理论的特殊情况。它同样适用于“标准”弹性力学和“不相容”弹性力学，因此包括所谓的非欧几里德板、壳和杆理论。
英文摘要：
We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally reduced limit theory views the elastic body as a $k$-dimensional Riemannian manifold along with an isometric $W^{2,2}$-immersion in $n$-dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionally-reduced limit is obtained using a $Γ$-convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to "standard" elasticity and to "incompatible" elasticity, thus including as particular cases so-called non-Euclidean plate, shell, and rod theories.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
