Magnetostriction in a hard ferromagnetic thin-film beam-plate theory

Magnetostriction is a remarkable non-linear phenomenon which causes shape (and size) variations of a magnetized ferromagnet in response to different magnetic configurations. Its essence falls in the wide realm of magneto-elastic coupling and it is put to advantage in a wide array of technical applications. In fact, until recent times, nickel magnetostriction was successfully employed in sensing devices to capture acoustic and generally mechanical waves \cite[p.627ff]{Bozorth}. It was indeed by studying a Nickel sample that, in 1842, James Joule discovered magnetostriction. From the theoretical standpoint, magneto-elastic (and electro-elastic) coupling provides an excellent example of a solid endowed with a vector microstructure \cite{Toupin, Brown2, Tiersten}. Yet microstructure, with its extra balance equations, is not the only interesting feature about magneto-elastic coupling. Indeed, microstructure interacts on a global scale, which means that a non-local theory is to be dealt with. Even more significant, owing to the nature of the magnetic phenomenon, the integral terms which account for the non-local character of the magnetic action may be improper or even singular (i.e. Cauchy-type), inside the material body (they are indeed potentials). From this property stems a number of important consequences, ranging from the need of two vector fields to fully describe the magnetic action inside magnetized matter (often named magnetic induction and magnetic field) to a substantial change in the character of the tension vector-to-unit normal relation \cite{Brown2}.Although much study has been devoted to magneto-elastic coupling \cite{Brown1, Moonbook, Pao, Ogden}, its treatment provides outstanding difficulties both from the theoretical and from the numerical standpoint. For the most part, available solutions cover the case of the so called para- or dia-magnetic materials \cite{Moon,Wallerstein}, whose simplified magnetic constitutive law is responsible for different and conflicting force expressions \cite{Nobili,Zhou}. The greatest source of trouble is the non-local nature of the interaction mixed with the potential-type nature of the integrals, which sets powerful numerical methods at a loss. In this contribution, a hard ferromagnetic thin-film beam-plate theory is described, whose greatest asset lies in leading to a system of ordinary nonlinear integro-differential equations, which may be tackled through some carefully deployed numerical techniques \cite{Nobili2}.The model was originally developed in \cite{Nobili} for longitudinal elongations. The key step lays in matching the Euler--Bernoulli constrained kinematics with Maxwell equations in a layout where the magnetization vector is bound to follow the beam-plate axis (latent microstructure). Such treatment allows both the mechanical and magnetic problem to be reduced to a one-dimensional form. This outcome is obtained at the expense of letting the magnetization vector vary only along the longitudinal co-ordinate, which approximation, although severe from the domain-theory standpoint, is justified in a beam-theory spirit (and it is consistent with the traditional magnetic circuit approach). Furthermore, it allows involved constitutive laws to be dispensed with, whose major drawback lays in the difficulty of carrying through experimental results to numerically evaluate coupling coefficients