An implicit non-ordinary state-based peridynamics for large deformation solid mechanics problems

The numerical simulation of the cracking process remains one of the most significant challenges in solid mechanics. Compared classical approaches, peridynamics(PD) has some attractive features because the basic equations remain applicable even when singularities appear in the deformation. Numerical time-integration plays a big role in any computational framework and unlike explicit time-integration, implicit time-integration methods can be much more efficient because of the ability to adopt fairly large time increments, making it a suitable option for PD analyses of large deformation problems. The objective of this thesis is to propose an implicit non-ordinary state-based peridynamics (NOSB PD) approach focusing on quasistatic analyses with large deformation mechanics. Firstly, the use of the adaptive dynamic relaxation (ADR) method as a solution strategy for quasi-static analyses with large deformation mechanics is discussed. Next, an analytical expression of the Jacobian matrix based on the equation of motion of NOSB PD is formulated to ensure optimum convergence of the global residual force. To address some instability issues in the existing “corresponding material” model, caused by zero-energy modes instability, recent approaches proposed by Silling (2017) are used to control the spurious deformation modes. An additional stabilisation term with respect to displacement is included in the derivatives for Jacobian formulation. This allows a more accurate NOSB PD approach to model material behaviour where correspondence materials have previously failed due to instability. Finally, to validate the proposed methodology, several numerical examples of 2D damage problems model using a stabilised correspondence model are verified, and suggestions are made for future implementation. The novelty of this thesis lies in providing theoretical development and numerical implementation of an implicit non-linear NOSB PD focusing on quasi-static analyses with large deformation mechanics. Findings from this thesis will interest researchers working in numerical methods, along with those solving discontinuous solid mechanics problems.