  The favored phase field method (PFM) has encountered challenges in the finite
strain fracture modeling of nearly or truly incompressible hyperelastic
materials. We identified that the underlying cause lies in the innate
contradiction between incompressibility and smeared crack opening. Drawing on
the stiffness-degradation idea in PFM, we resolved this contradiction through
loosening incompressible constraint of the damaged phase without affecting the
incompressibility of intact material. By modifying the perturbed Lagrangian
approach, we derived a novel mixed formulation. In numerical aspects, the
finite element discretization uses the classical Q1/P0 and high-order P2/P1
schemes, respectively. To ease the mesh distortion at large strains, an
adaptive mesh deletion technology is also developed. The validity and
robustness of the proposed mixed framework are corroborated by four
representative numerical examples. By comparing the performance of Q1/P0 and
P2/P1, we conclude that the Q1/P0 formulation is a better choice for finite
strain fracture in nearly incompressible cases. Moreover, the numerical
examples also show that the combination of the proposed framework and
methodology has vast potential in simulating complex peeling and tearing
problems
