  We present an energy-preserving mechanic formulation for dynamic
quasi-brittle fracture in an Eulerian-Lagrangian formulation, where a
second-order phase-field equation controls the damage evolution. The numerical
formulation adapts in space and time to bound the errors, solving the mesh-bias
issues these models typically suffer. The time-step adaptivity estimates the
temporal truncation error of the partial differential equation that governs the
solid equilibrium. The second-order generalized-$\alpha$ time-marching scheme
evolves the dynamic system. We estimate the temporal error by extrapolating a
first-order approximation of the present time-step solution using previous ones
with backward difference formulas; the estimate compares the extrapolation with
the time-marching solution. We use an adaptive scheme built on a residual
minimization formulation in space. We estimate the spatial error by enriching
the discretization with elemental bubbles; then, we localize an error indicator
norm to guide the mesh refinement as the fracture propagates. The combined
space and time adaptivity allows us to use low-order linear elements in
problems involving complex stress paths. We efficiently and robustly use
low-order spatial discretizations while avoiding mesh bias in structured and
unstructured meshes. We demonstrate the method's efficiency with numerical
experiments that feature dynamic crack branching, where the capacity of the
adaptive space-time scheme is apparent. The adaptive method delivers accurate
and reproducible crack paths on meshes with fewer elements.
