A viscoplasticity theory applied to proportional and non-proportional cyclic loading at small strains

The small strain, isotropic, Viscoplasticity theory Based on Overstress (VBO) is modified so that the experimentally observed, complex cyclic hardening behavior under proportional and non-proportional loading with fixed as well as variable strain amplitude can be reproduced. This is accomplished by formulating a growth law for the isotropic, rate independent stress, a scalar valued state variable of VBO. The non-proportionality measure employed for modeling the effects of loading path, amplitude and prior history is Tanakaʹs fourth order tensor [Eur. J. Mech. Solids 13 (1994) 155]. Numerical experiments show the responses in step-up and down two amplitude tests and include further hardening after an increase in the strain amplitude. The differences in the responses to proportional and non-proportional loading including circular, square, one step and two steps are demonstrated. History dependence of the hardening is depicted. The cross hardening behavior, sudden increase in the stress level which is followed by softening after the path change are modeled well.