An improved efficient implicit solution strategy for elastic cracking simulation based on ordinary state-based peridynamics

Abstract

Peridynamics is a non-local theory based on integral equations to simulate cracks or fracture behaviors. So far, most solution techniques use explicit algorithms with a small timestep, however, leading to low efficiency and accuracy. Conventional implicit strategies, such as Newton-Raphson, can alleviate those disadvantages because the timestep is no longer restricted to a very small value and cumulative error can be eliminated. Although conventional implicit strategies (e.g., Newton-Raphson) can get rid of computation limits with a somehow larger timestep, additional computational efforts are consumed for the formation and decomposition of a high-dimensional global stiffness matrix in the case of a large-scale model. For this reason, a reduced-order Newton-Raphson strategy was proposed in which the global non-linear iteration analysis was replaced with a reduced-order one. In this case, only the small capacitance matrix related to broken bonds needs to be formed and decomposed rather than the global stiffness matrix. Enlightened by this, this study proposed a modified solution strategy for the cracking simulation of elastic bodies based on ordinary state-based peridynamics. In specific, the inversion of the secant stiffness matrix is updated in each nonlinear iteration, during which the capacitance matrix related to new broken bonds is solved. Once the maximum number of the new broken bonds in the iteration is limited, the capacitance can therefore be constrained in a low-dimensional space. Three practical examples using the proposed implicit solution strategy shown in the study demonstrated that the size of the matrix operation can be significantly reduced using the proposed method and accordingly, the computational efficiency is improved.