非欧弹性力学中膜极限的黎曼逼近
A Riemannian approach to the membrane limit in non-Euclidean elasticity
摘要：
非欧几里得或不相容弹性是预应力材料的弹性理论，其基于将弹性体建模为黎曼流形。本文导出了薄非协调体薄膜极限的降维降阶模型。通过推广经典的降维技术的黎曼设置，我们能够证明一个一般定理，适用于弹性体的任意尺寸，任意细长尺寸，和任意度量。极限模型是指定义在欧几里得空间的极限子流形的浸入上的积分泛函的极小化。极限能量仅取决于浸入的一阶导数，并且对于帧无关模型，仅取决于在子流形上诱导的所得拉回度量，即，没有弯曲贡献。
英文摘要：
Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e., there are no bending contributions.
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