Rankine Plastic Model with Softening
This is a very simple model with strength degradation, which exhibits a certain analogy with the rotating crack model. In the rotating crack model the inelastic strain has the same principal directions as the stress tensor, while in Rankine plasticity this holds for the inelastic strain increments.
The formulation is classical. The compliance tensor is fixed, and so we write
where the only primary internal variable is taken to be єр. The loading function is taken to be the Rankine yield criterion,
a, – ф(е“) < 0 . (8.5.46)
where <7, is the largest principal strain, £p the equivalent plastic uniaxial strain, and</> is a function defining the evolution of the strength. This means that the inelastic strain occurs when the major principal stress attains the instantaneous strength. The flow rule simply states that the inelastic strain takes place in the direction of the maximum principal strain, i. e.:
єр = p, <g> p, f (8.5.47)
where p, is a unit eigenvector corresponding to the principal stress <7,.
The reader can easily check that for uniaxial tension under monotonic straining, ёр = єр and <7/ = <7. So, the function ф(єр) is nothing more than the curve of stress vs. inelastic strain for uniaxial tension.