An approximate analytical solution of the Lame problem for a hollow sphere is obtained for isotropic compressible elastoplastic materials under large elastic and plastic deformations. This solution generalizes the known solution by R. Hill [1] to the case when not only plastic but also elastic deformations are large. To describe elastic deformations, nonlinear constitutive relations previously proposed by the authors are used, which reduce to the Murnaghan equation of state in the case of purely volumetric deformation. To describe plastic deformations, two models of ideal plasticity theory are used: a model based on the Mises yield criterion and the associated flow rule, and a model based on the Drucker–Prager yield criterion and a non-associated flow rule. For these classes of materials, approximate analytical solutions of the Lame problem under large elastic and plastic deformations are obtained. Nonlinear effects and effects due to plasticity are investigated. One of these effects is that there is a limiting pressure that can be applied to the inner boundary of the hollow sphere. Another effect is that the solution for given pressure can be not unique. The detailed analysis is performed for the case in which the hollow sphere is expanded under the action of pressure applied to its internal boundary.
