Some basis-free expressions for stresses conjugate to Hill’s strains through solving the tensor equation AX+XA=C

Abstract

In the present paper, some new and compact basis-free expressions are derived for representing stresses conjugate to the Lagrangean Hill's strain measures. Solving a tensor equation in the form of AX + XA = C, a rather simple general expression is obtained for the conjugate stresses in the case of distinct principal stretches. The result decomposes the conjugate stresses additively into two parts, such that one part is coaxial with the right stretch tensor while another part is orthogonal to it. An expression for the case of double-coalescent principal stretches is also obtained. As an example of the general results, simple expressions are obtained for the stress conjugate to the logarithmic strain.

Basis free expressions for the stress rate of isotropic elastic materials in the cases of coalescent principal stretches

“…In order to avoid calculation of the eigenvalues and corresponding eigenvectors of tensors, the invariant or basis-free representation of tensor valued functions has great advantages over component form representation, such that many researchers have obtained some basis-free results for tensor quantities in the field of nonlinear continuum mechanics. As some examples, the works of Wang and Duan (1991), Xiao (1995), Hoger (1986), Hoger (1987, Man and Guo (1993), Dui et al (1999), Asghari and Naghdabadi (2007) and Asghari et al (2008) can be cited. Specifically, basis-free expressions for the stress rates can be found in Marsden and Hughes (1993) and Dui and Chen (2004).…”