Abstract

Through suitable changes of variables for a typical problem in hyper-elasticity, either in the reference or deformed configurations, one can setup and analyze versions of the same problem in terms of inner or outer maps or variations. Though such kind of transformations are part of the classical background in the Calculus of Variations, we explore under what sets of hypotheses such versions can be shown to have minimizers and be equivalent to the standard form of the problem. Such sets of hypotheses lead naturally to some distinct poly-convex energy densities for hyper-elasticity. Likewise we explore optimality in either of the two forms through a special way to generate one-parameter families of feasible deformations, feasibility including injectivity and non-interpenetration of matter.

中文翻译：

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超弹性的内部和外部版本

摘要

通过对超弹性典型问题的变量进行适当的改变，无论是在参考配置还是变形配置中，人们都可以根据内部或外部映射或变化来设置和分析同一问题的版本。尽管此类变换是变分法经典背景的一部分，但我们探讨了在哪些假设集下，此类版本可以被证明具有最小化并等效于问题的标准形式。这些假设自然会导致超弹性的一些独特的多凸能量密度。同样，我们通过一种特殊的方式探索两种形式中的任意一种的最优性，以生成可行变形的单参数族，可行性包括物质的单射性和非互穿性。