Dynamic particle difference method for the analysis of proportionally damped system and cracked concrete beam

A new dynamic particle difference method (PDM) for the simulation of a proportionally damped system subjected to dynamic load and the fracture simulation of cracked concrete beam is presented. The dynamic PDM utilizes only node model without involving any mesh or grid structure to take advantage of the merits of strong formulation; it remarkably accelerates computational speed owing to the avoidance of numerical integration and also sophisticatedly handles awkward topological change due to the crack growth in node model. The proportional damping is successfully implemented in the dynamic PDM by adding both mass and stiffness proportional terms to the equation of motion and the constitutive equation, respectively. It provides extra stability in the dynamic fracture simulation by eliminating erroneous oscillations. Governing partial differential equation is straightforwardly discretized in time by using the central difference method. However, a novel modification is devised in the Newmark method implementation where the transient equations and update formulae for kinematic variables are newly formulated; this modification appropriately corrects the period and amplitude of kinematic variable responses. The stability and accuracy of the developed methods are verified by solving various dynamic problems involving transient loading. Furthermore, the fracture process of the cracked concrete beam under the impact loading is successfully simulated and the dynamic energy release rates are effectively evaluated during the simulation.