A non-linear programming approach to kinematic shakedown analysis of composite materials

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non-linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non-linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non-linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non-linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials