Stabilized formulation for phase‐field fracture in nearly incompressible hyperelasticity

This work presents a stabilized formulation for phase-field fracture of hyperelastic materials near the limit of incompressibility. At this limit, traditional mixed displacement and pressure formulations must satisfy the inf-sup condition for solution stability. The mixed formulation coupled with the damage field can lead to an inhibition of crack opening as volumetric changes are severely penalized effectively creating a pressure-bubble. To overcome this bottleneck, we utilize a mixed formulation with a perturbed Lagrangian formulation which enforces the incompressibility constraint in the undamaged material and reduces the pressure effect in the damaged material. A mesh-dependent stabilization technique based on the residuals of the Euler-Lagrange equations multiplied with a differential operator acting on the weight space is used, allowing for linear interpolation of all field variables of the elastic subproblem. This formulation was validated with three examples at finite deformations: a plane-stress pure-shear test, a two-dimensional geometry in plane-stress, and a three-dimensional notched sample. In the last example, we incorporate a hybrid formulation with an additive strain energy decomposition to account for different behaviors in tension and compression. The results show close agreement with analytical solutions for crack tip opening displacements and performs well at the limit of incompressibility.