Deformation of an ellipsoidal ferrogel sample in a uniform magnetic field

The elongation of a ferroelastic material sample (whose initial shape is a sphere or an ellipsoid of revolution) under the action of an external magnetic field is studied in an in approximation of small strains. For a sphere, there is a classical estimate obtained under the assumption that elongating in the direction of the field, it becomes a spheroid and the stress and strain fields remain uniform. In the present calculation, it is assumed that the body is an ellipsoid (a sphere in a particular case) only in the absence of an external field; the shape of the sample in the presence of a field is not specified in advance but is found from the condition of balance of surface forces (elastic and magnetic). For the spherical case, the problem is solved exactly: it is shown, that the contour of the deformed body is described by a third-order algebraic equation. The case where the initial configuration is an ellipsoid of revolution is studied numerically. It is shown that in all versions, the refined solution leads to an appreciable increase in the elongation of the sample compared to the classical estimate.