We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Pade approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Pade approximants are favourably compared to an alternative approximation method employing the truncated Taylor series. As an application, Hencky-type hyperelasticity models are considered, in which the elastic strain energy is expressed in terms of the Hencky strain, and of our particular interest is the anisotropic energy quadratic in the Hencky strain. Finite-element computations are carried out to examine performance of the Pade approximants of tensor logarithm in Hencky-type hyperelasticity problems. A discussion is also provided on computation of the stress tensor conjugate to the Hencky strain in a general anisotropic case.

中文翻译：

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张量对数的 Padé 近似在 Hencky 型超弹性中的应用

我们表明，可以使用张量（矩阵）对数的 Pade 近似值以直接的方式和高精度逼近对数 (Hencky) 应变及其导数。与采用截断泰勒级数的替代近似方法相比，Pade 近似的精度和计算效率是有利的。作为一个应用，考虑了 Hencky 型超弹性模型，其中弹性应变能用 Hencky 应变表示，我们特别感兴趣的是 Hencky 应变中的各向异性能量二次方程。进行有限元计算以检查张量对数的 Pade 近似值在 Hencky 型超弹性问题中的性能。