Third-order elastic coefficients and logarithmic strain in constitutive and finite element modelling of anisotropic elasticity

TL;DR

This paper examines how the choice of strain measure significantly affects third-order elastic coefficients and corrects formulas for tensor derivatives crucial for finite element analysis in anisotropic elasticity.

Contribution

It demonstrates the dependency of TOECs on strain measures and provides corrected formulas for tensor derivatives used in finite element modeling.

Findings

TOECs vary greatly with different strain measures

Corrected formulas improve accuracy of tangent stiffness calculations

Highlights importance of strain measure choice in elastic property estimation

Abstract

Third-order elastic coefficients (TOECs) have been measured experimentally and tabulated with pretty good accuracy since the middle of the previous century. In the classical acoustic measurement method the recalculation of instantaneous stiffness change onto TOECs is based on the use of Green strain. In recent calculations performed by means of atomistic and quantum methods many different strain measures are employed. In result, quite different sets of TOECs can be obtained for the same material. In this paper, it is shown how dramatically the coefficients obtained depend on the choice of strain measure. The known formulas for calculation of the second derivative of a tensor-valued function of tensor variable are corrected. The formulas are essential for the correct analytic calculation of the tangent stiffness matrix in finite element method, among others.