Toward Dynamic Phase‐Field Fracture at Finite Strains

ABSTRACT We investigate the evolution of dynamic phase‐field fracture in the finite‐strain setting, extending our previous work in the small‐strain viscoelastodynamic regime. The elastodynamic equations are coupled with a dissipative damage evolution for the phase‐field variable . The material response is described with a polyconvex stored energy density , where denotes the gradient of the deformation, its cofactor, and its determinant. This ensures compatibility with the principles of nonlinear elasticity. A fully discrete time‐staggered approximation scheme is proposed, along with associated stability of discrete solutions. We present compactness results and analyze the convergence of the discrete approximations. While convergence of the phase‐field variable and the compatibility of the kinematic variables can be demonstrated, the identification of the limit stress in the momentum balance remains open. To address this, two strategies are outlined: an extension of the classical (weak) framework using generalized Young or defect measures, and an alternative formulation via energetic‐variational solutions that avoids the explicit measure construction. Partial results on existence and the structure of the limit system are discussed.