An implicit non-ordinary state-based peridynamics with stabilised correspondence material model for finite deformation analysis

Abstract

This paper is devoted to the development of a stabilised implicit non-ordinary state-based peridynamics approach. We propose a geometrically nonlinear implicit approach focusing on quasi-static analyses. Since the construction of the Jacobian matrix is the most time-consuming step in conducting this nonlinear analysis, we formulate an analytical expression based on the equation of motion of non-ordinary state-based peridynamics to ensure optimum convergence of the global residual force. The implicit formulation can adopt fairly large time increments, making it a good choice for analyses of finite deformation. Another important extension presented in this paper is the modification of the correspondence material model to remove zero-energy mode instabilities and reduce the spurious oscillations, as proposed by Silling (2017). The derivative of the additional stabilisation term with respect to displacement is included in the formulation of the Jacobian for the first time. Computational examples of 2D finite deformation problems with a stabilised correspondence model are presented. We assess the effectiveness of different values of the stabilisation parameter, G in terms of the particles’ spacings and horizon sizes for different problems. This allows the non-ordinary state-based peridynamics approach to model material behaviour with greater accuracy where correspondence materials have previously failed due to instabilities. In this paper, a damage model is also proposed, which provides for the first time an implicit approach for the static solution of crack propagation problems for non-ordinary state-based peridynamics. This paper lays the groundwork for non-ordinary state-based peridynamics to be used for a much greater variety of solid mechanics problems than is currently possible and at the same time satisfying the stability condition.