Shakedown analysis of a hollow sphere by interior-point method with non-linear optimization

Abstract

The shakedown safety domain of porous materials is investigated in this work by a direct numerical method based on Melan’s theorem. Considering the critical loading path of the load domain instead of the whole history, the statical shakedown condition is transformed to a large size optimization problem, of which the objective function is the shakedown limit factor, with the discretization of a three dimensional model composed of von Mises matrix. As in limit analysis of the hollow sphere model, we consider pure hydrostatic and deviatoric loads to produce the reference elastic stresses in the fictitious elastic body, for the purpose of capturing the effects of compressive/tensile and shear effects. By the use of a non-linear optimizer IPOPT based on the interior-point method, the proposed optimization problem is solved efficiently, providing not only the shakedown limit load factor, but also the distribution of residual stresses for the limit state. The established method is applied to axisymmetric and non-axisymmetric cyclic loadings, as well as the two independent variable loadings, in order to discover the influence of the loading condition and the solid matrix characteristics on the shakedown safety domain of porous medium (hollow sphere). The obtained results are illustrated and fully compared with reference solutions from literature and with the one from the incremental step-by-step FEM computations. The present method has been accessed and validated by the comparative study.