Abstract
We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Padé 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 Padé 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.
Highlights
Among different strain measures in the finite-deformation theory, the logarithmic (Hencky) strain possesses special properties that have led to a variety of applications in solid mechanics, e.g. [1,2,3,4,5]
We examine the applicability of Padé approximants in finite-element modelling of Hencky-type hyperelasticity problems with an emphasis on the anisotropic elasticity
We have shown that T(0), the stress conjugate to the logarithmic strain H = E(0), can be computed, i.e. evaluated numerically, using the automatic differentiation (AD) technique and the related closed-form representation of the matrix logarithm [11]
Summary
Among different strain measures in the finite-deformation theory, the logarithmic (Hencky) strain possesses special properties that have led to a variety of applications in solid mechanics, e.g. [1,2,3,4,5]. The Padé approximation method is a general approach that can be used to approximate arbitrary tensor functions It seems that the only related applications in computational mechanics concern the tensor logarithm [21,22] and the tensor exponential [21,22,23,24]. We examine the applicability of Padé approximants in finite-element modelling of Hencky-type hyperelasticity problems with an emphasis on the anisotropic elasticity. For this purpose, the performance of Padé approximants is evaluated against the truncated Taylor series, and the closed-from representation of the matrix logarithm [11] is considered as a reference for the comparison.
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