Strain and stress calculation in bulk magnetostrictive materials and thin films

In this paper, we examine some of the inconsistencies present in the solutions obtained by previous authors for the problem of finding the strains, stresses and energy densities observed in isotropic elastic magnetostrictive materials that are subjected to an arbitrary uniaxial stress, or that are in the form of thin films deposited on non-magnetic substrates. The origin of the inconsistencies is identified and more general and self-consistent solutions using an energy method based on minimization of the Gibbs free energy of the specimens are proposed. The proposed method can be used to solve problems that involve three-dimensional states of stress and can include thermal, piezoelectric, hygroscopic or other types of expansional strains. The advantages over the methods currently used are illustrated by solving a problem in which a film-substrate cantilever is subjected to the action of a point load applied at its free end. Expressions are derived to accurately calculate the strains, stresses and deflections of such a beam, regardless of the relative thickness of the film compared to the substrate. Additionally, the results are used to derive the conditions that define film-substrate cantilevers that show maximum deflection, as required in the case of sensors, or that exert maximum force, as required in the case of actuators.