A variational asymmetric phase-field model of quasi-brittle fracture: Energetic solutions and their computation.

Abstract

We derive the variational formulation of a gradient damage model by applying the energetic formulationof rate-independent processes and obtain a regularized formulation of fracture. The model exhibits dif-ferent behaviour at traction and compression and has a state-dependent dissipation potential whichinduces a path-independent work. We will show how such formulation provides the natural frameworkfor setting up a consistent numerical scheme with the underlying variational structure and for the deriva-tion of additional necessary conditions of global optimality in the form of a two-sided energetic inequal-ity. These conditions will form our criteria for making a better choice of the starting guess in theapplication of the alternating minimization scheme to describe crack propagation as quasistatic evolutionof global minimizers of the underlying incremental functional. We will apply the procedure for two- andthree-dimensional benchmark problems and we will compare the results with the solution of the weakform of the Euler-Lagrange equations. We will observe that by including the two-sided energetic inequal-ity in our solution method, we describe, for some of the benchmark problems, an equilibrium path whendamage starts to manifest, which is different from the one obtained by solving simply the stationarietyconditions of the underlying functional.