Existence and stability for a visco-plastic material with a fractional constitutive equation

The well posedness of the evolutive problem for visco-plastic materials represented by two different fractional constitutive equations is proved. We show that, for these materials, we can observe permanent deformations. So that, as it is usual in plasticity, when the stress goes to zero, then the strain assumes a constant non zero behavior. Moreover, we prove the compatibility of our models with the classical laws of thermodynamics. For the second model, described through a fractional derivative with an exponential kernel, we obtain the exponential decay of the solutions by means of the semigroup theory.