Implicit elasticity presents the general response of materials without imposing assumptions at the fundamental level. A popular implausible assumption of continuum mechanics is that the reference configuration is stress-free, since residual stress is ubiquitous in Nature. This paper develops large and small deformation implicit elasticity frameworks using residually stressed reference configurations. The general forms of constitutive relations, in finite deformations, are obtained by or of all the associated tensors to the same (Eulerian or Lagrangian) configuration. These general forms are used to study the relationship between “residual stress and material symmetry” for implicit elasticity. Further, we use a stress-free body, which is implicit elastic, to exactly determine the response of an initially stressed reference configuration. A number of such exact implicit relations are presented for residually stressed reference configurations, which are further simplified through interesting tensor analysis. The simplified implicit relations directly evaluates strain from a given Cauchy stress and residual stress tensor. One of these constitutive relations are employed for investigating the finite inflation of a residually-stressed, thick sphere. Finally, a small deformation implicit theory is attained by linearizing the developed relations for small strain and small rotation. To represent the small from a stressed reference, we need to invert a fourth order tensor. The closed-form inverse is determined in a new approach presented in the paper.

中文翻译：

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代表残余应力参考的隐式弹性


隐弹性表现了材料的一般响应，而无需在基本层面上强加假设。连续介质力学的一个流行的令人难以置信的假设是参考配置是无应力的，因为残余应力在自然界中无处不在。本文使用残余应力参考配置开发了大变形和小变形隐式弹性框架。在有限变形中，本构关系的一般形式是通过相同（欧拉或拉格朗日）配置的所有相关张量获得的。这些一般形式用于研究隐弹性的“残余应力和材料对称性”之间的关系。此外，我们使用无应力体（隐式弹性体）来精确确定初始应力参考配置的响应。针对残余应力参考配置提出了许多此类精确的隐式关系，并通过有趣的张量分析进一步简化了这些关系。简化的隐式关系直接评估给定柯西应力和残余应力张量的应变。其中一种本构关系用于研究残余应力厚球体的有限膨胀。最后，通过对小应变和小旋转的发展关系进行线性化，获得了小变形隐式理论。为了表示重音参考中的小值，我们需要反转四阶张量。封闭式逆是通过本文提出的一种新方法确定的。