A semi-Lagrangian constitutive correspondence framework for peridynamics

Lagrangian formulations, in which material models are established upon a mapping from a deformed domain back to an undeformed reference configuration, have traditionally been employed in most previous developments of the peridynamic theory. In this paper, a semi-Lagrangian constitutive framework for peridynamics is proposed, in which peridynamic material point interactions depend only on their current properties (e.g. position and stress values) in the deformed configuration. A nonlocal version of the velocity gradient is proposed to determine the Cauchy stress rate, using local constitutive theories, as an intermediate quantity in computing peridynamic bond forces. It is shown that a pure state-based peridynamic approach would involve material instabilities. A bond-associated formulation is presented to benefit from both bond-based and state-based views, resulting in an enhanced stability. A bond-associated, correspondence damage modeling is introduced to incorporate classical failure criteria in a convenient way. The proposed framework opens up new pathways in modeling extreme events, where excessive material deformation and damage make the Lagrangian formulation unsuitable. Capabilities of the new theory is showcased in a handful of illustrative scenarios.