The nonlinear theory of magnetoelasticity and the role of the Maxwell stress: a review

This paper reviews macroscopic aspects of the theory of magnetoelastostatics, starting with a brief summary of the experimental and theoretical contributions leading to the development of the current state-of-the-art. It offers some different perspectives than hitherto, with incompressible materials being the main concern. The use of the so-called total energy (density) functions is highlighted along with their associated total stress tensors and succinct forms of the constitutive equations. The symmetry of the total Cauchy stress tensor, which incorporates the non-symmetric Maxwell stress within the material, is emphasized and it is noted that the use of such a Maxwell stress, often appearing in the literature, is thereby avoided. The theory is illustrated for some simple prototype boundary-value problems, specifically the homogeneous deformation of an infinite slab of magnetoelastic material in the presence of a magnetic field and the non-homogeneous extension and inflation of an infinitely long circular cylindrical tube in the presence of either an axial or a circumferential magnetic field.