Special features of numerical simulation for elastic-plastic buckling of hemispherical shells under loading with the rigid indenter

The three-dimensional problem of elastic-plastic deformation and buckling of the hemispherical shells under contact with the rigid bodies is considered. Current Lagrangian formulation is applied to describethe shell deformation. The equation of motion is derived from thebalance of possible work powers. The ratios of the flow theory with isotropic hardening are used as the equations of state. Contact interaction of a rigid body and shell is modeled based on non-penetration conditions. We use a 8-node isoparametric finite element with multilinear form functions for a digitization of the constitutive system of equations. The solution of the problem under given boundary and initial conditions is based on the moment scheme of finite element method and an explicit cross type finite-difference scheme of time integration. Calculations of elastic-plastic deformation and buckling of hemispherical steel shell, located on the stationary plate at quasi-static indentation by nondeformable indenter, are done. Indenter has a form of a cylindrical shell with longitudinal notches in the contact area. Number of notches is varied in the calculations. A numerical study of the influence of the indenter form on the level of plastic deformation and the value of the critical load is carried out. As shown by the results of calculations, significant local deformations of hemispherical shell in the contact area with the indenter in the process of loading are formed. It is characterized by large displacements and rotation angles of finite element as a rigid body. The reliability of the results of numerical solution of the problem is confirmed by a good agreement with the experimental data of other authors. It is shown that the application of moment scheme of finite element method to determine the rate of strain and stress in the local basis is justified at low shear deformations and large angles of rotation torque by using explicit cross type finite-difference scheme of time integration with small time steps.