Keywords: numerical stability, softening, visco-plasticity, elasto-plasticity, visco-elasticity, regularization, overlay model, Duvaut-Lions, Maxwell.

The visco-plastic models of Duvaut-Lions and Perzyna are often used to introduce viscosity in inviscid elasto-plastic material laws with softening, in order to reduce the mesh-dependency of Finite Element solutions. The first of these models has the limitation that it must be used in conjunction with an integration algorithm for the inviscid elasto-plastic rate equations, in which the initial stress is used only to compute the trial stress, since it may be outside the yield surface. Usually a backward Euler interpolation is used. Furthermore, an evolution rule must be postulated for the yield surface, in case of hardening/softening plasticity. The second one -the Perzyna model- may not converge to the inviscid solution, when the viscosity goes to zero, in the case of multi-surface yield locus.

Another possibility to solve the same problem consists of using an overlay model, in which the inviscid elasto-plastic model deforms in parallel with a Maxwell-type visco-elastic element. This alternative is specially attractive in the context of displacement-based finite elements, since in this case the problem is strain-driven and, as a consequence, the visco-elastic and the elasto-plastic parts of the material response may be computed separately. This approach has been developed and used by the author [1] and does not suffer the above referred limitations of the models of Duvaut-Lions and Perzyna.

In the computation of the material response, different integration techniques are used for the visco-elastic and for inviscid elasto-plastic elements. However, both are numerically robust: an unconditionally stable and oscillation-free mid-point indirect interpolation scheme for the visco-elastic element and a linear backward Euler approach for the elasto-plastic element. Although the latter model element displays also unconditional numerical stability, when used in hardening plasticity, in the softening case numerical difficulties arise, as a consequence of the strain localization.

After an outline of the main features of the overlay model, the constitutive law of the visco-elastic Maxwell element, as well as the numerical integration technique used in the one-dimensional case, are described.

The analysis of the numerical stability is subsequently performed on the special case of softening elasto-plasticity. To this end, the deformation of a chain of overlay elements under constant elongation rate is computed. By a trial and error procedure the way as the stability limit varies with the main material parameters is investigated. a similar study is also carried out for the case of the Duvaut-Lions model.

The main conclusion, in the case of the overlay model, is that the mesh refinement does not influence the stability limit. This conclusion is then tested in a three-dimensional problem: the simulation of the process of formation of a shear band in a von Mises specimen. It is verified that, also in this case, the maximum time step for numerical stability does not suffer reduction with the mesh refinement.

1

V. Dias da Silva, "A simple model for viscous regularization of elasto-plastic constitutive laws with softening", Communications in Numerical Methods in Engineering, 20, 547-568, 2004. doi:10.1002/cnm.700

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