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Working on quasi-static phase-field fracture modeling in nearly incompressible solids for crack propagation is a challenging task. To avoid arising locking effects therein, a mixed form for the solid displacement equation is developed, resulting in two unknowns: a displacement field and a hydro-static pressure variable. In order to fulfil an inf-sup condition, stable Taylor-Hood elements are employed for the displacement-pressure system. The irreversibility condition of the crack evolution is handled by help of a primal-dual active set method. To get both a sharper crack and reasonable computational costs, adaptive meshes are used based on a predictor-corrector scheme. The crack paths from the numerical simulations are compared on the experimentally observed crack paths in carbon black filled ethylene propylene diene monomer (EPDM) rubber strips. The punctured EPDM strips with a hole and a given notch at different heights are stretched till total failure.