Nonlocal operator method for dynamic brittle fracture based on an explicit phase field model

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In this work, we present a nonlocal operator method (NOM) for dynamic fracture exploiting an explicit phase field model. The nonlocal strong forms of the phase field and the associated mechanical model are derived as integral forms by variational principle. The equations are decoupled and solved in time by an explicit scheme employing the Verlet-velocity algorithm for the mechanical field and an adaptive sub-step scheme for the phase field model. The sub-step scheme reduces phase field residual adaptively in a few substeps and thus achieves a rate-independent phase field model. The explicit scheme avoids the calculation of the anisotropic stiffness tensor in the implicit phase field model. One advantage of the NOM is its ease in implementation. The method does not require any shape functions and the associated matrices and vectors are obtained automatically after defining the energy of the system. Hence, the approach can be easily extended to more complex coupled problems. Several numerical examples are presented to demonstrate the performance of the current method.