关于锁定材料和梯度多凸性的一个注记
A note on locking materials and gradient polyconvexity
摘要：
我们使用梯度杨措施产生的Lipschitz映射定义的积分泛函，允许达到值$+\infty$，并可以模拟理想的锁定在弹性普拉格在1957年定义的放松。此外，我们证明了弹性材料的能量密度的变分问题的最小值的存在，可以表示在一个函数是连续的变形梯度和凸的梯度的余因子（也可能是梯度的行列式）的相应的变形梯度。我们称相关的能量泛函梯度为多凸的。因此，而不是考虑二阶材料中的变形梯度的二阶导数，只施加较弱的较高的可积性。虽然变形的二阶梯度不包括在我们的模型中，梯度多凸泛函允许一个隐式的一致的正下界的变形梯度的行列式上的封闭的域表示的弹性体。因此，该材料不允许极端的局部压缩。
英文摘要：
We use gradient Young measures generated by Lipschitz maps to define a relaxation of integral functionals which are allowed to attain the value $+\infty$ and can model ideal locking in elasticity as defined by Prager in 1957. Furthermore, we show the existence of minimizers for variational problems for elastic materials with energy densities that can be expressed in terms of a function being continuous in the deformation gradient and convex in the gradient of the cofactor (and possibly also the gradient of the determinant) of the corresponding deformation gradient. We call the related energy functional gradient polyconvex. Thus, instead of considering second derivatives of the deformation gradient as in second-grade materials, only a weaker higher integrability is imposed. Although the second-order gradient of the deformation is not included in our model, gradient polyconvex functionals allow for an implicit uniform positive lower bound on the determinant of the deformation gradient on the closure of the domain representing the elastic body. Consequently, the material does not allow for extreme local compression.
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