On the effects of material strength in dynamic fracture: A phase-field study

Over the past seven years, full-field analyses of a wide range of classical as well as modern quasi-static fracture experiments on nominally elastic brittle materials -- ranging from hard ceramics to soft elastomers -- have repeatedly identified the material strength surface as one of the key material properties that governs not only the nucleation of cracks, but also their propagation. Central to these analyses are the results generated by the Griffith phase-field fracture theory with material strength introduced in [21,23,20]. The first of two objectives of this paper is to extend this theory to account for inertia, this for the basic case of isotropic linear elastic brittle materials. From an applications point of view, the theory amounts to solving an initial-boundary-value problem comprised of a hyperbolic PDE coupled with an elliptic PDE for the displacement field $\mathbf{u}(\mathbf{X},t)$ and the phase field $d(\mathbf{X},t)$. A robust scheme is presented to generate solutions for these equations that is based on an adaptive finite-element discretization of space and an implicit finite-difference discretization of time. %At every time increment $t_m$, the resulting discretized equations are solved separately in a staggered manner for $\mathbf{u}(\mathbf{X},t_m)$ and $d(\mathbf{X},t_m)$ by means of Newton-Raphson schemes. The second objective is to illustrate the descriptive and predictive capabilities of the proposed theory via simulations of benchmark problems and experiments. These include problems involving fracture nucleation from large pre-existing cracks, such as the classical Kalthoff-Winkler experiments, as well as problems involving fracture nucleation within the bulk, such as the dynamic Brazilian fracture experiments.