AI summaryⓘ
The authors study a way to model cracks in materials using bond-based peridynamics, which doesn't need complex spatial derivatives. They show that using different interaction distances (horizons) across the material can cause problems like unbalanced forces and fake wave reflections. By applying a principle from physics (Lagrange-d'Alembert), they create a new consistent mathematical approach and a method that allows different regions to be simulated with different time steps. Their tests show this method avoids earlier problems, matches detailed simulations, and saves computational work. This helps simulate fractures more accurately and efficiently when the material's properties change across space.
bond-based peridynamicsnon-local modelingspatially varying horizonsvariational formulationLagrange-d'Alembert principleasynchronous variational integratordynamic fracture simulationwave propagationcrack growthvelocity-Verlet method
Authors
Prateek Prateek, Giuseppe Capobianco, Kai Partmann, Kestin Weinberg, Michael Ortiz, Sigrid Leyendecker
Abstract
Bond-based peridynamics provides a non-local framework for modelling fracture without requiring spatial derivatives of the displacement field. However, when spatially varying horizons are used together with non-uniform discretisations, the classical single-horizon bond-based peridynamics formulation leads to asymmetric interactions between material points. These asymmetric interactions violate balance laws and can introduce non-physical artefacts such as ghost forces and spurious wave reflections. In this work, we develop a variational formulation for bond-based peridynamics with spatially varying horizons. Starting from the Lagrange-d'Alembert principle, we derive the governing equations of motion and show that the dual-horizon peridynamics formulation emerges naturally from the variation of the internal energy. Building on this variational structure, we construct asynchronous variational integrators that allow different time step sizes in different regions of the domain. This is particularly useful for dynamic fracture simulations with local refinement, where small time steps are required only near regions of high resolution or expected crack growth. Numerical examples involving wave propagation, a pre-cracked plate under tension, and the Kalthoff-Winkler impact experiment demonstrate that the proposed framework removes spurious reflections caused by non-uniform horizons, preserves physically consistent fracture patterns, and achieves results comparable to uniformly refined simulations. At the same time, the asynchronous variational integrator reduces the number of internal force evaluations compared to the standard velocity-Verlet method. The proposed approach therefore provides a consistent variational foundation and an efficient time-integration strategy for bond-based peridynamic simulations with spatially varying horizons.