Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented.

In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.

In this work, we consider adaptive mesh refinement for a monolithic phase‐field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase‐field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual‐type error estimator for the phase‐field fracture model in each time‐step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle‐point system has three unknowns: displacements, phase‐field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

In the context of phase‐field modeling of fractures in incompressible materials, a mixed form of the elasticity equation can overcome possible volume locking effects. The drawback is that a coupled variational inequality system with three unknowns (displacements, pressure and phase‐field) has to be solved, which increases the overall workload. Efficient preconditioning at this point is an indispensable tool. In this work, a problem‐specific iterative solver is proposed leveraging the saddle‐point structure of the displacement and pressure variable. A Schur‐type preconditioner is developed to avoid ill‐conditioning of the phase‐field fracture problem. Finally, we show numerical results of a pressure‐driven benchmark which to confirm the robustness of the solver.

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