Changing Constitutive Equations in Time and in Space
As stress evolves with time, a new problem is solved for each time step. Suppose that in time step n, rate dependent plasticity is used at Gauss point m. Then in time step n+1, suppose rate independent plasticity is used at Gauss point m. This discontinuity or switch of the constitutive equations does not cause any problems. The reason is that the initial conditions required for each time step are the initial geometry, initial stress, initial strain, the boundary conditions from tn to tn + dt and the constitutive equations in the interior of the time step. If the time step is from I u to Іи + dt, the constitutive equation must be defined only for time t such astn<t <= tn + dt. There is no need to define the constitutive equation at times earlier than tn.
Suppose a rate independent constitutive equation is used at one Gauss point and a rate dependent constitutive equation is used at a neighboring Gauss point. This does not cause any problems in the analysis. At each Gauss point, the analysis gives the Gauss point an initial stress, values of the internal variable such as yield stress or deformation resistance and strain rate and asks for the stress at that point at the end of the time step. Each Gauss point can have its own constitutive equations and it causes no problems and introduces no complexity into the analysis other than providing the capability of switching between constitutive equations and providing the mappings at internal variables required by each constitutive equation such as yield strength to deformation resistance.