A technique for numerical analysis of nonlinear dynamic deformation and progressive failure of multi-layered metal-plastic shells of revolution is developed with account for their strain-rate dependent strength characteristics. The kinematic deformation model of a layered package is based on the non-classical theory of shells. The geometric dependencies are formulated based on the quadratic version of the nonlinear theory of elasticity. The relationship between stress and strain tensors in a composite macrolayer is established on the basis of Hooke’s law for an orthotropic body combined with the theory of effective modules, while for metal macrolayer, within the flow theory with linear hardening. The process of progressive layer-by-layer failure is described in the framework of the degradation model of stiffness characteristics. The strain rate dependence of stiffness and strength characteristics of composite materials is accounted for. An energetically consistent system of equations of motion for a shell of revolution is constructed using the principle of possible displacements. A numerical method for solving the problem is based on an explicit variational-difference scheme. The adequacy of the proposed technique was considered on the problem of unsteady deformation of a cylindrical shell subjected to pulse pressure simulating an explosion in the shell center.