A rate-dependent model and its user subroutine for cohesive element method to investigate propagation and branching behavior of dynamic brittle crack

Abstract Modeling the branching behavior of dynamic brittle crack helps reveal dynamic mechanism of brittle geomaterial fragmentation induced by multi crack bifurcation. In this study, the correlation between crack propagation speed and separation strain rate of crack surfaces is derived theoretically and verified based on the Kalthoff plate impact test, showing a strong linear positive proportional relationship. A rate-dependent constitutive law for cohesive element is proposed to predict the direction and branching of dynamic crack. The methodology of compiling the rate-dependent model for cohesive element via the user subroutine VUSDFLD is introduced. The sensitivity of element size, mesh structure, and material parameters to the dynamic branching model are discussed. Four models with different rate-dependent forms are compared with the previous numerical results to find the best one. It is found that the most superior constitutive relationship for cohesive element in modeling crack branching is the model in which only the maximum separation displacement is rate-dependent, and the traction strength keeps constant. As Young’s modulus increases, the position of the first large branching is advanced due to the increase of stress wave speed. The crack branching angle is positively correlated with fracture energy, which is speculated to be closely related to the geometrical nonlinearity of the tested plate and the redistribution of tensile stress. The appearance of fracture process zone of dynamic crack represented by the damage cohesive elements is investigated, based on which the branching mechanism and the number of crack branches are discussed. Unlike static crack, the fracture process zone of dynamic crack represented by the damage cohesive element zone shows a fan shape, of which the radius is also rate-dependent. The extent of the dynamic fracture process zone varies with the crack propagation speed, which controls the generation of subsequent crack branches and the instability of crack direction.