Lower and upper shakedown bounds for fatigue limit in two phase materials

Abstract

The goal of this study is to find the ‘safe’ long term behavior of elasto-plastic structural materials subjected to fluctuating load (shortly ‘fatigue limit’). The materials whose fatigue limits are checked are: (a) materials reinforced with unidirectional stiff fibers; (b) materials with dilute amount of inclusions; (c) materials with minute porosity. To reach this goal we employ two different shakedown theorems: Melan’s static shakedown theorem (1936) as the lower bound and Koiter’s kinematic shakedown theorem (1960) as the upper bound. The solutions to the lower and upper bounds for the prescribed stress amplitude, (Δσ _th)^l,b. and (Δσ ^tu.b._th , are expressed in a rigorous form with three parametric entities: (i) the volume fraction of the second phase in the base matrix phase, f, (ii) the quality of the ‘bond’ between the two phases ‘m’, (iii) the magnitude of the residual stress, ‘p’, pre-existed in the material. The deviation between the two bounds represents the ‘uncertainty’ in our knowledge of the actual safe/unsafe state, where the materials fail to withstand the alternating load. It is shown that in the considered three types of materials, at certain amount of residual stresses, the safe load amplitudes based on shakedown analysis are indeed higher than their corresponding elastic limits (at least by 5%, 10% and 25% respectively). The apparent advantage of using shakedown bounds to predict the safe/unsafe loading amplitude is that no prior information on the actual complex failure mechanisms is required and no empiricism is needed. However, empirical data which were found in the open literature are ‘falling’ satisfactorily between the computed dual bounds.