A three dimension lattice-spring model with rotational degree of freedom and its application in dynamic crack propagation

Abstract

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model (3D-LSM) containing the rotational degree of freedom. Using the finite element method (FEM) and share grid, the spring stiffness coefficient is obtained with a rigorous mathematical logic. This 3D-LSM contains the nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of the system. Wave velocity in elastic solid and dynamic fracture of concrete l-specimen are calculated by both 3D-LSM and elastic theory. Results indicate that this model can present the behavior of the material in the elastic stage. By introducing the energy balance principle, 3D-LSM can simulate dynamic brittle fracture both in mesoscopy and macroscopy. This model is used to analyze the evolution law of crack propagation path and propagation velocity in brittle materials under dynamic tension of shock wave. The results show that the stress concentration area at the crack tip changes during the crack propagation, showing a "butterfly shape" before the crack initiation and a "swept wing shape" after the crack initiation. After the crack initiation, the crack tip velocity jumps instantly and then increases continuously, corresponding to the expansion of a single crack. Then the tip velocity decreases slightly and increases to the maximum crack propagation velocity, corresponding to the crushing stage before bifurcation. After that, the crack growth rate will oscillate significantly, which corresponds to the bifurcation stage of the crack.