Goodbrake Christian, Goriely Alain, Yavari Arash
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
Proc Math Phys Eng Sci. 2021 Jan;477(2245):20200462. doi: 10.1098/rspa.2020.0462. Epub 2021 Jan 6.
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.
非线性滞弹性的一个核心工具是变形张量的乘法分解，该分解假设变形梯度可分解为一个弹性张量和一个滞弹性张量的乘积。通常，这一假设通过中间构型的存在得以论证。然而，一般而言，这种构型在欧几里得空间中并不存在，并且该假设的数学基础并不令人满意。在此，我们从变形梯度的乘法分解出发，推导全局中间构型存在的充分条件。我们表明，这些全局构型在等距变换下是唯一的。我们研究将这些构型等距嵌入到高维欧几里得空间的结果，并构建反映这些嵌入的变形梯度的乘法分解。例如，对于一族径向对称变形，我们构建所得中间构型的等距嵌入，并明确计算残余应力场。
