Hencky’s logarithmic strain and dual stress–strain and strain–stress relations in isotropic finite hyperelasticity

It has been known that the Kirchhoff stress tensor (tau) and Hencky’s logarithmic strain tensor h may be useful in formulations of isotropic finite elasticity and elastoplasticity. In this work, a straightforward proof is presented to demonstrate that, for an isotropic hyperelastic solid, the just-mentioned stress–strain pair (tau) and h are derivable from two dual scalar potentials with respect to each other. These results establish a simple, explicit dual formulation of isotropic finite hyperelasticity. As a result, they supply a complete solution to the problem of finding out the inverted stress–strain relation for isotropic hyperelastic solids, raised by J.A. Blume [Int. J. Non-linear Mech. 27 (1992) 413]. Moreover, an explicit form of such an inverted hyperelastic stress–strain relation is derived in terms of the powers I, (tau) and (tau)^2.