  This paper addresses the modeling of fracture in quasi-brittle materials
using a phase-field approach to the description of crack topology. Within the
computational mechanics community, several studies have treated the issue of
modeling fracture using phase fields. Most of these studies have used an
approach that implies the lack of a damage threshold. We herein explore an
alternative model that includes a damage threshold and study how it compares
with the most popular approach. The formulation is systematically explained
within a rigorous variational framework. Subsequently, we present the
corresponding three-dimensional finite element discretization that leads to a
straightforward numerical implementation. Benchmark simulations in two
dimensions and three dimensions are then presented. The results show that while
an elastic stage and a damage threshold are ensured by the present model, good
agreement with the results reported in the literature can be obtained, where
such features are generally absent.
