论文标题
板的新霍克人模型
A Neohookean model of plates
论文作者
论文摘要
本文涉及板（平面域）的高弹性变形，这些变形最小化了新胡克型能量。特别是，我们研究了J.M. Ball在他的开创性论文“ Sobolev功能的全球可逆性和物质的互穿”中引入的存储能量功能。所考虑的映射是Sobolev同构及其较弱的限制。从C. B. Morrey的意义上讲，它们是单调的。采用单调sobolev映射的一个主要优点在于能量最小变形的存在。但是，注射率不可避免地会丢失，因此一个明显的问题是：参考配置的最大子集是哪些最小变形仍然具有射态的最大子集？该子集具有完整度量的事实应与全局不可逆转的概念进行比较，该概念涉及变形配置的子集。在这方面，我们提出了一种cantor型结构，以表明分支集及其图像都可能具有正区域。我们方法的另一种新颖性在于允许沿边界弹性变形，称为无摩擦问题。
This article is about hyperelastic deformations of plates (planar domains) which minimize a neohookean type energy. Particularly, we investigate a stored energy functional introduced by J.M. Ball in his seminal paper "Global invertibility of Sobolev functions and the interpenetration of matter". The mappings under consideration are Sobolev homeomorphisms and their weak limits. They are monotone in the sense of C. B. Morrey. One major advantage of adopting monotone Sobolev mappings lies in the existence of the energy-minimal deformations. However, injectivity is inevitably lost, so an obvious question to ask is: what are the largest subsets of the reference configuration on which minimal deformations remain injective? The fact that such subsets have full measure should be compared with the notion of global invertibility which deals with subsets of the deformed configuration instead. In this connection we present a Cantor type construction to show that both the branch set and its image may have positive area. Another novelty of our approach lies in allowing the elastic deformations be free along the boundary, known as frictionless problems.
