This contribution deals with the analysis of models for phasefield fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

The notion of dynamic fracture with continuum‐kinematics‐based peridynamics is presented in this work. A geometrically precise version of peridynamics called continuum‐kinematics‐based peridynamics adds surface‐ or volume‐based interactions to the traditional peridynamic bonds, accurately capturing the finite deformation kinematics. The point families produced from the horizon of the material points are used to construct the surfaces and volumes taken into account for these non‐local interactions. In continuum kinematics‐based peridynamics, the traditional bond‐stretch damage technique is insufficient for fracture. Due to the loss of strength in the internal force densities of the material points, it is now extended to the surface‐ and volume‐based interactions by new failure factors. Numerical examples demonstrate that the proposed approach effectively manages crack propagation, impact damage, and spontaneous crack initiation under dynamic loading circumstances with large deformations. When the results are compared to phase‐field calculations, there is a remarkable agreement concerning the damage patterns.

In this contribution we present analytical results on a model for dynamic fracture in viscoelastic materials at small strains that have been obtained in full depth in [1]. In the model, the sharp crack interface is regularized with a phase‐field approximation, and for the phase‐field variable a viscous evolution with a quadratic dissipation potential is employed. A non‐smooth penalization prevents material healing. The viscoelastic momentum balance is formulated as a first order system and coupled in a nonlinear way to the non‐smooth evolution equation of the phase field. We give a full discretization in time and space using a discontinuous Galerkin method for the first‐order system. We discuss the existence of discrete solutions and, with the step size in space and time tending to zero, their convergence to a suitable notion of weak solution of the system. Eventually, we provide a numerical benchmark and compare it with simulation results found in [2].

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