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The implicit function theorem is used to obtain some necessary and sufficient conditions, for which an implicit constitutive relation can be solved in terms of the stresses and strains, in the sense of expressing, for example, the strains in terms of the stresses or vice versa. For an isotropic body some exact solutions for the strains in term of the stresses are presented, using the theory of tensor equations of the form AX + XA = H. For the exact solutions considered, conditions for the existence and the uniqueness are given. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]