论文标题
非欧亚人弹性中壳的渐近刚度
Asymptotic rigidity for shells in non-Euclidean elasticity
论文作者
论文摘要
我们考虑弹性壳的典型“拉伸和弯曲”功能。除了指标外，还以参考的第二基本形式模拟了外壳作为D维riemannian歧管。将壳浸入（D+1）维度环境空间中，弹性能解释了诱导的度量标准和第二个基本形式与其参考值的偏差。假设环境空间是恒定的截面曲率的，我们证明，渐近消失的能量的任何沉浸序列都会收敛到壳的等距浸入环境空间中，并具有参考第二基本形式。特别是，如果环境空间是欧几里得空间，则参考度量标准和第二基本形式满足了高斯 - 科达西 - 米纳迪的兼容性条件。该定理可以看作是Reshetnyak的渐近刚性定理的（流动价值）共同含量1的类似物。它还与表面连续性相对于其基本形式的连续性的最新结果有关。
We consider a prototypical "stretching plus bending" functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.
