  A variational modeling framework for hydraulically induced fracturing of
elastic-plastic solids is developed in the present work. The developed
variational structure provides a global minimization problem. While fracture
propagation is modeled by means of a phase-field approach to fracture, plastic
effects are taken into account by using a Drucker-Prager-type yield-criterion
function. This yield-criterion function governs the plastic evolution of the
fluid-solid mixture. Fluid storage and transport are described by a
Darcy-Biot-type formulation. Thereby the fluid storage is decomposed into a
contribution due to the elastic deformations and one due to the plastic
deformations. A local return mapping scheme is used for the update of the
plastic quantities. The global minimization structure demands a
$H($div$)$-conforming finite-element formulation. Furthermore this is combined
with an enhanced-assumed-strain formulation in order to overcome locking
phenomena arising from the plastic deformations. The robustness and
capabilities of the presented framework will be shown in a sequence of
numerical examples.
