Invertibility and Non-Invertibility in Thin Elastic Structures
Invertibility and Non-Invertibility in Thin Elastic Structures
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薄弹性结构中的可逆性和不可逆性
DOI：
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发表时间：
2011
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通讯作者：
Peter Hornung
中科院分区：
文献类型：
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作者：
Peter Hornung
厚度为h的薄膜的非线性弹性能量由函数Eh给出。Friesecke、James和Müller导出了当h → 0时，泛函h−αEh对α ∈ 3的Γ-极限。在这篇文章中，我们研究这些泛函的几乎极小的可逆性，更一般的序列具有等可积的能量密度。我们表明，他们是可逆的几乎无处不在远离薄膜表面附近的薄边界层。此外，我们还得到了这一层的宽度的上界和原像直径的统一上界。我们构造的例子表明，这些界限是尖锐的。特别地，对于所有α ∈ 3，存在局部不可逆的Lipschitz连续低能形变。
The nonlinear elastic energy of a thin film of thickness h is given by a functional Eh. Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h−αEh for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.