Ductile Fracture: Constrained Strong Ellipticity Condition and Non-Smooth Problems

Abstract

Ductile crack formation and growth is caused by the inability of a material to withstand mechanical constraints. Either porosity growth, shear band formations, particle cracking and second phase nucleations will precede ductile damage and macroscopic fracture. Continuum models are inadequate for some stages of constitutive behavior because they cease to be applicable when strong ellipticity is lost. In fact, these stages must allow dissipation in sets of Lebesgue measure zero, such as the Barenblatt's cohesive theory, which is based on a zero resulting stress intensity factor and a cusp-shaped crack tip. Moreover, this dissipation mechanism introduces non-smoothness since the cohesive traction-separation law is a set of non-smooth equations. Usually, these can form a complementarity problem only if strong simplifying assumptions are made.

From both the theoretical and numerical perspectives, satisfactory simulations of two-dimensional ductile fracture (including crack intersections and coalescence) are seldom seen in the literature. Often, the strong ellipticity condition is applied without consideration of discretization constraints. Ad-hoc approaches are usually used for the cohesive dissipation, without regard for their non-smooth character.

One of the reasons for this, besides the need of some non-trivial geometric calculations and their respective linearization in the implicit case, is that typically high aspect-ratio elements arise, increasing the solution error and the condition number. This can be solved by full remeshing but it becomes costly for real problems and may induce spurious diffusion. Localized tip remeshing has been, therefore, also limited. Lately, local enrichment methods such as the Stong Discontinuity Approach and global methods such as XFEM have been very popular, but are only suited to simple academic problems, and mainly for single crack growth predictions.