If an elastic body is defined as one that does not dissipate energy into heat, the classes of elastic bodies not only include the Green elastic solid, but also some types of implicit constitutive relations recently presented in the literature. In this paper one of such new implicit relations is studied in detail, wherein the energy function depend on the second Piola–Kirchhoff stress tensor and the Green Saint-Venant strain tensor. It is assumed that the function is anisotropic having two directions of anisotropy, thus the case of a transversely isotropic body and an isotropic body are special cases of the above function. Spectral invariants are used and explicit expressions for some second derivatives of the energy function are found. Such second derivatives appear in the implicit constitutive relation.

中文翻译：

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关于一类非线性弹性体的隐式本构关系


如果将弹性体定义为不将能量耗散为热的弹性体，则弹性体的类别不仅包括格林弹性固体，还包括最近在文献中提出的某些类型的隐式本构关系。本文详细研究了这种新的隐式关系之一，其中能量函数取决于第二个 Piola-Kirchhoff 应力张量和 Green Saint-Venant 应变张量。假设该函数是各向异性的，具有两个各向异性方向，因此横观各向同性体和各向同性体的情况是上述函数的特例。使用谱不变量并找到能量函数的一些二阶导数的显式表达式。这样的二阶导数出现在隐含的本构关系中。