Explicit dual stress-strain and strain-stress relations of incompressible isotropic hyperelastic solids via deviatoric Hencky strain and Cauchy stress

Abstract

It is demonstrated that of all finite strain measures there is one and only one able to realize fully uncoupled separation of the volumetric and the isochoric deformation in a natural, additive manner. This unique measure is Hencky’s logarithmic strain, and its remarkable property is utilized to establish dual stress-strain and strain-stress relations for isotropic, incompressible hyperelastic solids. It is indicated that the deviatoric Hencky strain and the deviatoric Cauchy stress are derivable straightforwardly from two dual elastic potentials with respect to each other. Further, the three basic invariants of Hencky’s strain naturally reduce to two in the case of incompressibility, and these two invariants may be used to express the isotropic elastic potential. This results in simple, explicit dual representations for the foregoing dual stress-strain and strain-stress relations. In particular, it is shown that the Cauchy-Green deformation tensor can be explicitly derived from a complementary potential as function of the basic invariants of the deviatoric Cauchy stress, which automatically satisfies the hyperelasticity condition, the isotropy condition, and the incompressibility condition.