NONLINEAR MAGNETOELASTIC DEFORMATIONS by A. DORFMANN (Institute of Structural Engineering, Peter Jordan Strasse 82, 1190 Vienna, Austria) and R. W. OGDEN (Department of Mathematics, University of Glasgow, Glasgow G12 8QW) [Received 17 November 2003. Revise 12 July 2004] Summary In this paper we first summarize, in a simple form, the equilibrium equations for a solid material capable of large magnetoelastic deformations. Such equations are needed for the analysis of boundary-value problems for elastomers endowed with magnetic properties by the embedding of distributions of ferrous particles. The general constitutive law for an isotropic material in the presence of a magnetic field is described and expressed in a compact form, with either the magnetic field or the magnetic induction as the independent magnetic variable. The equations are applied, in the case of an incompressible material, to the solution of representative problems involving circular cylindrical geometry, specifically the helical shear of a circular cylindrical tube, its specializations to axial and azimuthal shear, and the problem of extension and torsion of a solid circular cylinder. For each problem a general formulation is afforded without specialization of the (isotropic) constitutive law, and then specific results are discussed briefly for special choices of such laws. It is noted, in particular, that certain restrictions may be placed on the class of constitutive laws for a considered combination of deformation and magnetic field to be admitted. 1. Introduction The equations for a continuum deforming in the presence of an electromagnetic field are well established, as exemplified by (1, 2). Here we concentrate on the static situation for materials that respond to a magnetic field. This is motivated by recent renewed interest in the subject of electromagnetic continua generated by the development of so-called ‘smart’ materials used, for example, in devices for controlling the damping characteristics of vibration absorbers. Specifically, such materials are elastomers with a distribution of ferrous particles embedded within their bulk. The relevant theory owes much to the work of Brown (3), while a recent account of some aspects of the theory can be found in (4). Kovetz (4) took the magnetic induction vector B as the basic variable, which was also the case with the work of the present authors (5, 6). Alternative formulations based on the use of the magnetic field vector H or the magnetization vector M have been discussed in (7), while the analysis in (8) is based on the use of M as an independent variable. A treatment of universal relations for magnetoelastic solids is contained in (9). In the papers by Dorfmann and Ogden (5, 6) several different formulations of the governing (equilibrium) equations and the constitutive law were discussed. Here, in sections 2 and 3, we provide a new formulation, based on a modified free energy function, that gives the combined Q. Jl Mech. Appl. Math. (2004) 57 (4), 599–622 c  Oxford University Press 2004; all rights reserved. Downloaded from https://academic.oup.com/qjmam/article-abstract/57/4/599/1842336 by guest on 15 June 2020