Kinematic shakedown analysis under a general yield condition with non-associated plastic flow

A nonlinear, purely kinematic approach with the finite element implementation is developed to perform shakedown analysis for materials obeying a general yield condition with non-associated plastic flow. The adopted material model can be used for both isotropic materials (e.g. von Misesʹs, Mohr–Coulomb and Drucker–Prager criteria) and anisotropic materials (e.g. Hillʹs and Tsai-Wu criteria) with both associated and non-associated plastic flow. Nonlinear yield criterion is directly introduced into the kinematic shakedown theorem without linearization and instead a nonlinear, purely kinematic formulation is obtained. By means of mathematical programming techniques, the finite element model of shakedown analysis is formulated as a nonlinear programming problem subject to only a small number of equality constraints. The objective function corresponds to plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a structure can then be obtained by solving the minimum optimization problem. A direct, iterative algorithm is proposed to solve the resulting nonlinear programming problem, where a penalty factor based on the calculation of the plastic dissipation power is used to overcome the numerical difficulty caused by the non-differentiability of the objective function in elastic areas. The calculation is entirely based on a purely kinematical velocity field without calculation of stresses. Meanwhile, only a small number of equality constraints are introduced into the nonlinear programming problem. So the computational effort is very modest. Numerical applications prove that the developed algorithm has a very good numerical stability and computational efficiency. The proposed approach can capture different plastic behaviours of materials and therefore has a very wide applicability.