Reduced polynomial invariant integrity basis for in-plane magneto-mechanical loading

Abstract

The description of the behavior of a material subjected to multi-physics loadings requires the formulation of constitutive laws that usually derive from Gibbs free energies, using invariant quantities depending on the considered physics and material symmetries. On the other hand, most of crystalline materials can be described by their crystalline texture and the associated preferred directions of strong crystalline symmetry (the so-called fibers). Moreover, among the materials produced industrially, many are manufactured in the form of sheets or of thin layers. This article has for object the study of the magneto-mechanical coupling which is a function of the stress $\sigma$ and the magnetization M. We consider a material with cubic symmetry whose texture can be described by one of three fibers denoted as $\theta$, $\gamma$ or $\alpha$ ' , and which is thin enough so that both the stress and the magnetization can be considered as in-plane quantities. We propose an algorithm able to derive linear relations between the 30 cubic invariants I k of a minimal integrity basis describing a magneto-elastic problem, when they are restricted to in-plane loading conditions and for different fiber orientations. The algorithm/program output is a reduced list of invariants of cardinal 7 for the {100}-oriented $\theta$ fiber, of cardinal 15 for the {110}-oriented $\alpha$ ' fiber and of cardinal 8 for the {111}-oriented $\gamma$ fiber. This reduction (compared to initial cardinal 30) can be of great help for the formulation of low-parameter macroscopic magneto-mechanical models.