  Modeling crack initiation and propagation in brittle materials is of great
importance to be able to predict sudden loss of load-carrying capacity and
prevent catastrophic failure under severe dynamic loading conditions.
Second-order phase-field fracture models have gained wide adoption given their
ability to capture the formation of complex fracture patterns, e.g. via crack
merging and branching, and their suitability for implementation within the
context of the conventional finite element method. Higher-order phase-field
models have also been proposed to increase the regularity of the exact solution
and thus increase the spatial convergence rate of its numerical approximation.
However, they require special numerical techniques to enforce the necessary
continuity of the phase field solution. In this paper, we derive a fourth-order
phase-field model of fracture in two independent ways; namely, from Hamilton's
principle and from a higher-order micromechanics-based approach. The latter
approach is novel, and provides a physical interpretation of the higher-order
terms in the model. In addition, we propose a continuous/discontinuous Galerkin
(C/DG) method for use in computing the approximate phase-field solution. This
method employs Lagrange polynomial shape functions to guarantee
$C^0$-continuity of the solution at inter-element boundaries, and enforces the
required $C^1$ regularity with the aid of additional variational and interior
penalty terms in the weak form. The phase-field equation is coupled with the
momentum balance equation to model dynamic fracture problems in hyper-elastic
materials. Two benchmark problems are presented to compare the numerical
behavior of the C/DG method with mixed finite element methods.
