PSI - Issue 18

Riccardo Fincato et al. / Procedia Structural Integrity 18 (2019) 75–85 Author name / Structural Integrity Procedia 00 (2019) 000–000

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the viscoplastic term tends to zero. This leads to the unrealistic response that the material can bear an infinite load for high rate loading conditions. To correct this drawback Hashiguchi (2009) modified the definition of the viscoplastic strain rate introducing the aforementioned variable R m . This parameter, at the denominator of Eq. (6), prevents the viscoplastic term to go to zero whenever the time interval dt tends to zero. Computationally, the effect of R m can be seen in the stress-strain curves of Fig. 3a, when fast cyclic conditions are applied at the top of the cubic element. As it can be seen, the conventional viscoplastic model predicts almost an elastic response. In case of faster loading conditions, the response would be purely elastic even for stress state much bigger than the initial macroscopic yield stress (i.e. F 0 ). On the other hand, the DOSS model can predict an irreversible deformation, the magnitude of which can be regulated by the parameter R m . Lower values of R m induce larger viscoplastic contribution, vice versa higher values of R m reduce the generation of irreversible deformation. It is worth mentioning that due to the overstress effect the viscoplastic strain accumulation (see Fig. 3b), and consequently the damage accumulation (see Fig. 3c), continues during the initial part of the unloading until the stress crosses the subloading surface, that is equivalent to say that the viscoplastic strain are generated until the condition R = R d is satisfied. This aspect can be graphically seen in Fig. 3b and c during the inversion of the loading conditions, where the red and green solid curves display a moderate increase before a flat plateau. From a practical point of view, R m can be easily calibrated by minimizing the difference between numerical and experimental stress-strain curves at high loading rate. In general, this parameter should be chosen to be 1 m R  . In the following graphs the parameter s 1 was set to be 10 MPa in order to avoid a fast damage acceleration and allow to complete two loading cycles.

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Fig. 3 Cyclic loading analyses for a fast loading condition: (a) stress-strain curves, (b) cumulative plastic strain, (c) damage evolution.