A nonlinear magnetoelastic tube under extension and inflation in an axial magnetic field: numerical solution

In the context of nonlinear magnetoelasticity theory very few boundary-value problems have been solved. The main problem that arises when a magnetic field is present, as compared with the purely elastic situation, is the difficulty of meeting the magnetic boundary conditions for bodies with finite geometry. In general, the extent of the edge effects is unknown a priori, and this makes it difficult to interpret experimental results in relation to the theory. However, it is important to make the connection between theory and experiment in order to develop forms of the magnetoelastic constitutive law that are capable of correlating with the data and can be used for making quantitative predictions. In this paper the basic problem of a circular cylindrical tube of finite length that is deformed by a combination of axial compression (or extension) and radial expansion (or contraction) and then subjected to an axial magnetic field is examined. Such a field cannot be uniform throughout, since the boundary conditions on the ends and the lateral surfaces of the tube would be incompatible in such circumstances. The resulting axisymmetric boundary-value problem is formulated and then solved numerically for the case (for simplicity of illustration) in which the deformation is not altered by the application of the magnetic field. The distribution of the magnetic-field components throughout the body and the surrounding space is determined in order to quantify the extent of the edge effects for both extension and compression of the tube.