  This work presents a new adaptive approach for the numerical simulation of a
phase-field model for fractures in nearly incompressible solids. In order to
cope with locking effects, we use a recently proposed mixed form where we have
a hydro-static pressure as additional unknown besides the displacement field
and the phase-field variable. To fulfill the fracture irreversibility
constraint, we consider a formulation as a variational inequality in the
phase-field variable. For adaptive mesh refinement, we use a recently developed
residual-type a posteriori error estimator for the phase-field variational
inequality which is efficient and reliable, and robust with respect to the
phase-field regularization parameter. The proposed model and the adaptive
error-based refinement strategy are demonstrated by means of numerical tests
derived from the L-shaped panel test, originally developed for concrete. Here,
the Poisson's ratio is changed from the standard setting towards the
incompressible limit $\nu \to 0.5$.
