Shakedown of porous material with Drucker-Prager dilatant matrix under general cyclic loadings

Abstract

This paper is concerned with the shakedown limit states of porous ductile materials with Drucker-Prager matrix under cyclically repeated loads. Using the hollow sphere model and Melan’s shakedown theorem based on time-independent residual stress fields, a macroscopic fatigue criterion is derived for the general conditions of cyclic loads. First, the case of the hollow sphere subjected to pure hydrostatic loading is studied and the limit states of collapse by fatigue or by development of mechanism are derived. Then, the general case involving shear effects with any arbitrary cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The key idea is in two steps: (i) the choice of appropriate trial stress and trial residual stress fields and (ii) then maximizing the size of the load domain in the spirit of the standard lower shakedown theorem. The new macroscopic shakedown criterion depends on the porosity, the friction angle, Poisson’s ratio, the two stress invariants of the effective stress tensor and the sign of the third one. Together with the limit analysis-based yield criterion corresponding to the sudden collapse by development of a mechanism at the first cycle, it defines the safety domain of porous materials subjected to cyclic load processes. Interestingly, it is found that the safe domain is little sensitive to variations of the friction angle, however, it is considerably reduced compared to the one under monotonic loads obtained by limit analysis. Finally, a comparative study between the analytical results and numerical predictions performed by micromechanics-based finite element simulations is conducted for different porosities and friction angles.