Implicit constitutive relations for nonlinear magnetoelastic bodies

Abstract

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

On stretch-limited elastic strings

“…In particular, it includes the simple case of the strain expressed as a function of the stress, which appears more consistent with Newtonian causality than standard constitutive relations since force causes displacement. Implicit constitutive relations have found a wide range of applications in the modelling of electro- and magneto-elastic bodies [5,6], fracture in brittle materials [7,8], gum metal [9] and many other materials (see [10] for further references).…”

Longitudinal shock waves in a class of semi-infinite stretch-limited elastic strings

“…More precisely, if we define elasticity as the incapability of dissipation, then elastic bodies can be modeled by a wider class of relations between the stress variable and strain variable than those where the stress is a function of strain. This generalized notion of elasticity has had numerous applications in the modeling of electro and magneto-elastic bodies [6, 7], fracture in brittle materials [8, 9], gum metal [10], and many other materials (see the recent review [11] for further references).…”

Flexible identification procedure for thermodynamic constitutive models for magnetostrictive materials

“…However, systematic procedures for identification of such models based on experimental data have not been reported in details so far. It is particularly mentioned in [32] that the given equations are ‘too general to be used to correlate with experimental data as there are so many material functions that depend on numerous invariants.' In this paper, we focus on identification of thermodynamic constitutive models based on the theory of invariants.…”

“…The applications have included both isotropic [20,22] and anisotropic [26] solids as well as granular materials [27][28][29][30][31]. In [32], an extensive review of earlier magneto-elastic constitutive models is given, and implicit constitutive equations based on 21 scalar invariants are derived for bodies under large deformations. These equations are then simplified for small fields and applied for solving magneto-elastic boundary value problems for slabs and an annulus.…”