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The mathematical foundations of anelasticity: existence of smooth global intermediate configurations
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences ( IF 3.5 ) Pub Date : 2021-01-06 , DOI: 10.1098/rspa.2020.0462
Christian Goodbrake 1 , Alain Goriely 1 , Arash Yavari 2, 3
Affiliation  

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

中文翻译:

非弹性的数学基础:存在光滑的全局中间配置

非线性非弹性的核心工具是变形张量的乘法分解,假设变形梯度可以分解为弹性和非弹性张量的乘积。它通常由中间配置的存在来证明。然而,一般来说,这种配置不能存在于欧几里得空间中,而且这种假设的数学基础并不令人满意。在这里,我们从变形梯度的乘法分解开始,推导出全局中间配置存在的充分条件。我们表明这些全局配置在等距之前是独一无二的。我们检查将这些配置等距嵌入到高维欧几里得空间中的结果,并构造反映这些嵌入的变形梯度的乘法分解。例如,对于一系列径向对称变形,我们构建所得中间配置的等距嵌入,并明确计算残余应力场。
更新日期:2021-01-06
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