Conditions for convexity of the isotropic function of the second-rank tensor

Vladimir M. Sadovskii

Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia

Abstract: For a scalar function, depending on the invariants of the second-rank tensor, condition of convexity and strong convexity are obtained with respect to the components of this tensor in an arbitrary Cartesian coordinate system. It is shown that if a function depends only on the four invariants: three principal values of the symmetric part of a tensor and modulus of pseudovector of the antisymmetric part, these conditions are necessary and sufficient. A special system of convex invariants is suggested to construct potentials for the stresses and strains in the mechanics of structurally inhomogeneous elastic media, exhibiting moment properties.

Keywords: isotropic tensor function, convexity, invariants, nonlinear elasticity, plasticity.

UDC: 517.17+539.37

Received: 18.09.2010
Received in revised form: 25.10.2010
Accepted: 10.12.2010