Universal relations in isotropic nonlinear magnetoelasticity

In this paper we first summarize the basic constitutive equations for (nonlinear) magnetoelastic solids capable of large deformations. Equivalent formulations are given using either the magnetic induction vector or the magnetic field vector as the independent magnetic variable in addition to the deformation gradient. The constitutive equations are then specialized to incompressible, isotropic magnetoelastic materials in order to determine universal relations. A universal relation, in this context, is an equation that relates the components of the stress tensor and the components of the magnetic field and/or the components of the magnetic induction that holds independently of the specific choice of constitutive law for the considered class or subclass of materials. As has been shown previously for the case in which the magnetic induction is the independent magnetic variable, in the general case there exists only one possible universal relation. We show that this is also the case if the magnetic field is taken as the independent variable and that the universal relations resulting from the two cases are equivalent. A number of special cases are found for certain specializations of the constitutive equations. These include some connections between the deformation, the magnetic field and magnetic induction that do not involve the components of the stress tensor. Universal relations are then examined for some representative homogeneous and inhomogeneous universal solutions.