Stabilized state-based peridynamics for elasticity emanating from constrained Lagrangian

Abstract

State-based peridynamics for elasticity paves the way to bridge the local theory with the inherent capabilities of general elasticity with non-local effects. However, the original non-ordinary state-based peridynamics with correspondence materiel models suffers from high-frequency oscillations due to zero-energy mode instability. To address this issue, we start from the fundamental Lagrangian with the introduction of a constraint potential energy that minimizes the numerical error and suppresses the zero-energy mode oscillations. A stabilized non-ordinary state-based peridynamics is then derived from this constrained Lagrangian that forms an integral–differential–algebraic system. In specifics, the equation of motion for particles is described by integral-differential equations while the constraint suppressing the zero-energy mode instability is expressed as algebraic equations (zero-error constraint). An energy–momentum conserving implicit time-stepping scheme for this integral–differential–algebraic system is then derived. End-to-end 2D and 3D simulations, including scalar wave propagation, crack path, and large deformation problems, are presented to show that the proposed method is promising for these applications in terms of accurately capturing the crack path and large deformation, efficiently eliminating the numeral oscillations, conserving the total energy for long duration simulations, and preserving the zero-error constraint.