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Peridynamics (PD), is a non-local adaptation of classical continuum mechanics (CCM). In PD, each material point \(\mathbf X\) interacts with its neighbors inside a finite interaction zone \(B_\delta (\mathbf X)\) with the length \(\delta \), see Fig. 1. This type of non-local interaction principle is also seen in molecular dynamics (MD) [30] and smoothed particle hydrodynamics (SPH) [31, 32] simulations. The important feature of PD fracture modeling is that the interaction between the intact material and fractured material is modeled implicitly through a non-local field equation that remains the same everywhere in the computational domain. Material damage and fracture phenomena are captured through a force versus strain constitutive model. Because of this, PD is able to capture fracture as an emergent phenomena arising from the non-local equation of motion. This contrasts with classic fracture theory where, off the crack, the elastic interaction is modeled by the equation of elastodynamics and the fracture set is a free boundary with motion coupled to elastodynamics through a physically motivated kinetic relation. Other recently developed non-local models exhibiting emergent behavior include the Cucker Smail equation, where swarming behavior emerges from leaderless flocks of birds [33,34,35,36].
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Second, dynamic brittle fracture in glassy materials was studied in [378, 379]. In this study, a phase-field model [186], a discontinuous-Galerkin implementation of PD [380], and a meshfree discretization of PD [38] are used in the geometry shown in Fig. 10. For the PD simulation, a nodal spacing \(h=\)0.1mm and a horizon \(\delta =\)0.5mm were used. For the discontinuous-Galerkin implementation of PD a non-uniform mesh with average element size h of 0.1mm and a horizon \(\delta \) of 0.5mm was used. For the phase-field model, the nodal spacing was \(h=\)0.3mm and the length scale parameter \(l_0\) was 0.6mm. Note that the authors did some \(\delta \)-convergence study in [379], however, we only report the finest resolution here. Fore more details, we refer to [379, Section 5.2]. For all three implementations, the crack angle after branching, the time of crack branching, and position of the crack branching were compared with the experimental results. In this study various discretization parameters were studied, however, we report the discretization parameters corresponding to the best agreement with the experimental data. First, the value for the meshfree discretization is presented, followed by the value for the discontinuous-Galerkin discretization, and the value for the phase-field model last. The relative errors for the crack angle are: -0.21, -0.35, and -0.51, respectively. The relative errors -0.06 for the event of crack branching in time are the same for all simulations. The relative errors for the crack branching position are: 0, -0.12, and 0, respectively. 2ff7e9595c
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