Peridynamic differential operator and its applications

The nonlocal peridynamic theory has been proven extremely robust for predicting damage nucleation and propagation in materials under complex loading conditions. Its equations of motion, originally derived based on the principle of virtual work, do not contain any spatial derivatives of the displacement components. Thus, their solution does not require special treatment in the presence of geometric and material discontinuities. This study presents an alternative approach to derive the peridynamic equations of motion by recasting Navier’s displacement equilibrium equations into their nonlocal form by introducing the peridynamic differential operator. Also, this operator permits the nonlocal form of expressions for the determination of the stress and strain components. The capability of this differential operator is demonstrated by constructing solutions to ordinary, partial differential equations and derivatives of scattered data, as well as image compression and recovery without employing any special filtering and regularization techniques.