Mixed moving least-squares method for shakedown analysis

The elastic shakedown theory can be used to analyze and design structures under loads that change over time. Methods that use mixed interpolations provide simultaneous information regarding the kinematics and the equilibrium without the necessity of postprocessing. Meshless methods, as suggested by the name, do not require mesh generation or node connectivity and are an alternative to classical numerical methods. However, no current method exists that combines a mixed approximation of the kinematics and equilibrium with meshless methods for an elastic shakedown problem. In this paper, a discretization of a mixed variational principle for the elastic shakedown is presented based on the moving least-squares method, where stress and velocity approximation functions are used. The optimization problem, which comes from the mixed variational principle, determines the load factor and the residual stress and velocity fields. The stresses and velocities are approximated using linear and quadratic polynomial functions, respectively. The solution to the problem is obtained through a nonlinear algorithm based on Newton’s method. The results obtained for the load factor and the residual stress field approach those obtained from the analytical solution for the tested examples.