Static deformation of a spherically anisotropic and multilayered magneto-electro-elastic hollow sphere

Abstract

In this paper, we present an analytical solution on the general static deformation of a spherically anisotropic and multilayered magneto-electro-elastic (MEE) hollow sphere. We first express the general solution in each layer in terms of the spherical system of vector functions where two transformations of variables are also proposed to achieve the analytical results. The spherical system of vector functions can be applied to expand any vector as well as scalar function, and it further automatically separates the static deformation into two independent sub-problems: The LM-type and N-type. The LM-type is associated with the spheroidal deformation and is coupled further with the electric and magnetic fields. The N-type is associated with the torsional deformation and is purely elastic and independent of the electric and magnetic fields. To solve the multilayered spherical problem, the propagation matrix method is introduced with the propagation matrix being simply the exponential matrix for each layer. By assuming the continuity conditions on the interface between the adjacent spherical shells, the solution can be simply propagated from the inner surface to the outer surface of the layered and hollow MEE sphere so that specific boundary value problems can be solved. As numerical examples, a three-layered sandwich hollow sphere with different stacking sequences under different boundary conditions is studied. Our results illustrate the influence of the stacking sequences while showing the effectiveness of the proposed method.