Discontinuous Bifurcation Analysis of the Gurson Model for Ductile Void Growth and a Local Material Failure Model.

The Gurson constitutive equation and variations of it have been used extensively to model ductile damage as represented by the growth in porosity. The model has been widely used in large numerical simulations of dynamic impact and penetration. Because the growth of porosity causes softening there is always a possibility that a discontinuous bifurcation will exist at a critical value of porosity. The criterion for a discontinuous bifurcation is identical to that of loss of ellipticity and, consequently, a well-posed problem becomes ill posed once a discontinuous bifurcation exists. Since the Gurson model has been, and is continuing to be, used extensively in numerical solutions for a wide range of technically complex problems, it is often considered to be not feasible to confirm well-posedness by demonstrating convergence with mesh refinement. Therefore, an analytical criterion for loss of ellipticity is of considerable value for computational purposes. The analysis demonstrates that for a particular form of the Gurson model loss of ellipticity may occur at small strains and values of porosity. The inherent implication is that numerical solutions that do not include a check for loss of ellipticity, or that do not include a convergence study, may not be valid solutions