On the modeling of magneto-mechanical effects in solids

The paper develops a thermodynamically-consistent approach to magnetostriction. This is performed by following two different approaches depending on whether a three-dimensional or a one-dimensional setting is considered. In the three-dimensional case the symmetry condition required by the balance of angular momentum results in the need of appropriate variables in the constitutive equations. These variables prove to be Euclidean invariant and comprise the so-called Lagrangian fields usually adopted in the literature. The consequences of the second law of thermodynamics are then determined for a solid described by the temperature, the deformation gradient, and the magnetic field. With this background the magnetostriction is modeled for linear or nonlinear magnetic laws. Next a one-dimensional setting is addressed mainly in connection with available experimental data. The symmetry condition becomes ineffective and hence the classical Eulerian fields are used. Based on the relations established through the thermodynamic consistency a detailed set of constitutive equations, for magnetization and strain, is established. These equations are set up so as to fit the experimental data from a one-dimensional sample under tensile stresses and magnetic fields. • Constitutive properties of magnetoelastic solids are investigated by allowing for non-monotonic properties. • In a three-dimensional setting Lagrangian fields are applied so that balance equations are satisfied. • A detailed description of magnetostriction is obtained in a one-dimensional setting by separation of variables. • The modeling of the solid fits the data by means of elementary functions both for compressive and tensile stresses. • The use of elementary functions emphasizes the occurrence of the Villari effect.