Numerical Solution of a Virtual Internal Bond Model for Material Fracture

A virtual internal bond (VIB) model is proposed recently in mechanical engineering literatures for simulating dynamic fracture. The model is a nonlinear wave equation of mixed type (hyperbolic or elliptic). There is instability in the elliptic region and usual numerical methods might not work. We examine the artificial viscosity method for the model and apply central type schemes directly to the corresponding viscous system to ensure appropriate numerical viscous term for such a mixed type problem. We provide a theoretical justification of the convergence of the scheme despite the difficulty of the type change. The exact solution of a Riemann problem is used to justify the numerical method for one dimensional cases. We then generalize the method to a two dimensional material with a triangular or hexagonal lattice structure. Computational results for a two dimensional example are given.