Reformulation of dynamic crack propagation using the numerical manifold method

Since the advent of finite element methods, the dynamic response analysis of solids and structures follows such a route without exception. Firstly the spatial discretization is carried out and the system of second order ordinary differential equations with the degrees of freedom as the unknown functions of time is derived, which is called the semi-discrete scheme. Then the temporal discretization is performed to the system of ordinary differential equations and the system of algebraic equations, referred to as the fully-discrete scheme, is obtained. This route has been working well for most problems, where, the meshes deform continuously and, in all the time steps, all the degrees of freedom are valid and the number of them keeps invariant. In the simulation of crack propagation, however, even the number of degrees of freedom varies with crack propagation and those degrees of freedom associated with crack tips become meaningless after the crack tips move away. While this causes no difficulties in linear static solutions, it is not readily handled in time-dependent solutions, leading to the transfer issue of degrees of freedom. Opposite to the conventional order of discretization, in this study the temporal discretization is put prior to the spatial discretization. In this way, all the degrees of freedom are valid only within the current time step. The transfer issue of degrees of freedom is accordingly resolved elegantly. The implementation of the proposed procedure is in the framework of the numerical manifold method, illustrated by some typical examples, where compressed and sheared cracks are involved with frictional contact.