Numerical modeling of the processes of deformation and buckling of multilayer shells of revolution under combined quasi-static and dynamic axisymmetric loading with torsion

A two-dimensional formulation and method for numerical solution of problems of deformation and loss of stability of multilayer elastoplastic shells of rotation under quasi -static and dynamic axisymmetric loading with torsion have been developed. The defining system of equations is written in a Cartesian or cylindrical coordinate system. Modeling of the process of deformation of shell layers is carried out on the basis of hypotheses of solid mechanics or the theory of Timoshenko-type shells, taking into account geometric nonlinearities. Kinematic relations are written in speeds and formulated in the metric of the current state. The elastic -plastic properties of the shell are described by the generalized Hooke's law or the theory of plastic flow with nonlinear isotropic hardening. The variational equations of motion of the shell layers are derived from the three-dimensional balance equation of the virtual powers of the work of continuum mechanics, taking into account the accepted hypotheses of the theory of shells, either a plane deformed state or a generalized axisymmetric deformation with torsion. Modeling of the contact interaction of shell layers is based on the condition of rigid gluing or the condition of non -penetration along the normal and sliding along the tangential. To solve the governing system of equations, a finite -difference method and an explicit time integration scheme of the "cross" type are used. The method was tested on the problem of buckling of a three -layer cylindrical shell with elastoplastic load -bearing layers of aluminum alloy D16T and an elastic filler under quasi -static and dynamic loading by hydrostatic pressure, linearly increasing with time. The problem was solved in two versions: all three layers were modeled as a finite element of a continuous medium, or the load -bearing layers were modeled as shell elements, and the filler as elements of a continuous medium. The results of calculations using the two models are in good agreement with each other in terms of ultimate pressures and modes of buckling.