A simplified gradient-enhanced damage model based on energy limiters for crack propagation under time-dependent loading

This paper presents the development and investigation of a simplified energy limiter-based nonlocal damage model for dynamic crack propagation in brittle media. The key idea underlying the proposed model is that crack growth under impact loading is primarily influenced by the tensile component of the strain tensor. Consequently, the energy-based damage-driving term is simplified to a strain-based counterpart, which is integrated using the first principal strain. This simplification leads to a model that is not only easier to implement but also more effective in capturing dynamic crack propagation compared to the original theory. In addition, the computational framework incorporates an energy limiter-based gradient damage formulation with a damage threshold, enabling natural crack initiation and propagation while significantly reducing spurious damage. One of the distinctive features of the proposed approach is the treatment of the nonlocal crack field as a primary unknown, alongside displacements. This allows the use of identical shape functions for both fields within the finite element analysis, enhancing consistency and computational efficiency. Consistent with classical continuum damage mechanics, the model can accurately simulate arbitrary and complex multiple crack paths, including three-dimensional (3D) crack propagation. Furthermore, to provide a more efficient numerical framework under time-dependent loading conditions with complex crack patterns, an explicit dynamic fracture algorithm is employed. This algorithm utilizes the central difference method, the row-sum technique for mass lumping, and a consistent procedure for updating the kinematic and damage-related terms. The advantages and modeling capabilities of the proposed strain-based gradient-enhanced damage formulation are demonstrated through representative numerical examples of dynamic fracture under shear, tension, and compression loading scenarios.