Isotropic invariants of a completely symmetric third-order tensor

Marc Olive (1) , Nicolas Auffray (2)

1 I2M - Institut de Mathématiques de Marseille
2 MSME - Laboratoire de Modélisation et Simulation Multi Echelle

In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be impose on tensor functions. To that aim representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the XXth century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. To explicitly construct this basis, the link that exists between the O(3)-action on harmonic tensors and the SL(2,C)-action on the space of binary forms is exploited. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

Domaines: Mécanique des solides [physics.class-ph]; Physique mathématique [math-ph]

Marc Olive, Nicolas Auffray. Isotropic invariants of a completely symmetric third-order tensor. Journal of Mathematical Physics, 2014, 55 (9), pp.1.4895466. ⟨10.1063/1.4895466⟩. ⟨hal-00960463⟩