Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture

Gradient damage models used in phase-field approaches to brittle fracture are characterised by material softening and instabilities. We present novel numerical techniques for the bifurcation and stability analysis along quasi-static evolution paths as well as practical tools to select stable evolutions. Our approach stems from the variational approach to fracture and the theory of rate-independent irreversible processes whereby a quasi-static evolution is formulated in terms of incremental energy minimisation under unilateral constraints. Focusing on the discrete setting obtained with finite elements techniques, we discuss the links between bifurcation criteria for an evolution and stability of equilibrium states. Key concepts are presented through the analytical solution of a two-degrees-of-freedom model featuring a continuum family of bifurcation branches. We introduce numerical methods to (i) assess (second-order) stability conditions for time-discrete evolutions subject to damage irreversibility, and (ii) to select possible stable evolutions based on an energetic criterion. Our approach is based on the solution of a coupled eigenvalue problem which accounts for the time-discrete irreversibility constraint on damage. Several numerical examples illustrate that this approach allows us to filter out unstable solutions provided by standard (first-order) minimisation algorithms as well as to effectively compute stable evolution paths. We demonstrate our purpose on a multifissuration problem featuring complex fracture patterns, to show how the eigenvalue analysis enables to compute and retrieve morphological properties of emerging cracks.