SUMMARYUsing 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.

In this paper, a nonlinear numerical technique is developed to calculate the limit load and failure mode of structures obeying an ellipsoid yield criterion by means of the kinematic limit theorem, nonlinear programming theory and displacementbased finite element method. Using an associated flow rule, a general yield criterion expressed by an ellipsoid equation can be directly introduced into the kinematic theorem of limit analysis. The yield surface is not linearized and instead a nonlinear purely kinematic formulation is obtained. The nonlinear formulation has a smaller number of constraints and requires less computational effort than a linear formulation. By applying the finite element method, the kinematic limit analysis with an ellipsoid yield criterion is formulated as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the dissipation power which is to be minimized and an upper bound to the plastic limit load of a structure can then be calculated by solving the minimum optimization problem. An effective, direct iterative algorithm has been developed to solve the resulting nonlinear programming formulation. The calculation is based purely on kinematically admissible velocities. The stress field does not need to be calculated and the failure mode of structures can be obtained. The proposed method can be used to calculate the bearing capacity of clay soils in a direct way. Some examples are given to illustrate the validity and effectiveness of the proposed method.