An implicit method in non-ordinary state-based peridynamics for quasi-static simulations

Peridynamics is a particle-based continuum model that represents a new approach to handle problems involving discontinuities, such as damage and fracture. To date, the majority of research in peridynamics has used explicit time stepping formulations. However, one of limitation of the approach arises from the time step size limitation, and in the case of quasi-static loading, it becomes challenging to obtain solutions in a reasonable run time. This paper focuses on applying implicit time stepping approaches a non-ordinary state-based peridynamics formulation for quasi-static problems using an implicit method. A 2D stabilised method using the non-ordinary state-based peridynamics formulation with correspondence model for finite deformation is developed here. The paper is restricted to two-dimensional finite deformation analysis with linear elastic material behaviour. However, the adopted non-ordinary state-based formulation allows for general continuum constitutive models. Equilibrium is obtained at each quasi-static time (or load) step through a Newton-Raphson solution procedure, including the derivation of an analytical expression for the consistent Jacobian matrix to ensure optimum convergence rates. The correspondence approach, which is a subclass of the non-ordinary state-based peridynamics, permits constitutive models from local theory to be used for the modelling of long-range forces. However, previous studies show the standard correspondence model suffers from zero-energy mode instability. Hence, this study also considers the effectiveness of the stabilised parameter in removing zero-energy mode instability. Computational examples verify the ability of the modified peridynamics model including the impact of the particles spacing and horizon sizes on the choice of the stabilisation parameter on eliminating the zero-energy modes