Analytical solution to the magneto‐elastoplastic problem for functionally graded spherical shells under inner pressure

Abstract As composite material techniques mature, the study of functionally graded (FG) structures under the multi‐field coupling has become increasingly important. For FG spherical shells, the inhomogeneous distribution of stress fields leads to permanent plastic deformation when external force is applied. In addition, Lorentz force from constant magnetic fields can significantly affect the mechanical properties and mechanical behaviors of functionally graded materials (FGM). In this study, the elastoplastic states of FG spherical shells include four deformations, such as pure elasticity, partial plasticity from the internal or external surface, and complete plasticity. In this paper, elastic modulus and yield criterion of FG spherical shells are hypothesized to follow the power‐law distribution in the thickness direction of spherical shells, and analytical solutions of displacement and stress for various elastoplastic deformation states are proposed, and the critical loads of transition between these various states are determined. By comparing results with the existing work, the correctness of the analytical solutions is verified. In addition, the effects of inner pressure, FG parameter, and external magnetic fields on elastoplastic deformation states and stress, displacement distributions in FG spherical shells are further analysed. The results show that the plastic deformation of FG spherical shells may occur on the internal or external surface subjected to the effect of external magnetic fields and inner pressure. As the FG parameter gradually increases, the first yielding position moves from the internal to the external surface for spherical shells. Further, the external magnetic field has the effect of decreasing the critical inner pressure value that causes FG spherical shells to yield, making the spherical shell more susceptible to plastic deformation.