Stabilized mixed formulation for phase-field computation of deviatoric fracture in elastic and poroelastic materials

In the numerical approximation of phase-field models of fracture in porous media with the finite element method, the problem of numerical locking may occur. The causes can be traced both to the hydraulic and to the mechanical properties of the material. In this work we present a mixed finite element formulation for phase-field modeling of brittle fracture in elastic solids based on a volumetric-deviatoric energy split and its extension to water saturated porous media. For the latter, two alternative mixed formulations are proposed. To be able to use finite elements with linear interpolation for all the field variables, which violates the Ladyzenskaja-Babuska-Brezzi condition, a stabilization technique based on polynomial pressure projections, proposed and tested by previous authors in fluid mechanics and poromechanics, is introduced. We develop an extension of this stabilization to phase-field mixed models of brittle fracture in porous media. Several numerical examples are illustrated, to show the occurrence of different locking phenomena and to compare the solutions obtained with different unstable, stable and stabilized low order finite elements.