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In response to the problem that the standard AT2 phase field model cannot effectively model quasi-brittle failure and the existence of length dependence, a new modified phase field model is presented in this paper. By introducing an additional energy, the competing relationship between elastic strain energy and dissipation energy during fracture is changed. A new crack dissipation functional is established using the energy equivalent approach. By introducing a novel rational degradation function, not only can the strength of material failure be effectively utilized, but the model can also reproduce the cohesive softening relationship. A multi-field finite element method is used to discretize the model governing equations, and the equations are solved by an efficient BFGS monolithic algorithm. Finally, some representative numerical examples are used to analyze the effects of parameters in degradation function, length scale and mesh size on the results. The presented numerical simulation results demonstrate length scale and mesh scale independence, and are in good agreement with the experimental results and previous numerical results. At the same time, the numerical results also exhibit cohesive softening properties similar to the current phase field cohesive zone model. These results verify the robustness and effectiveness of the modified phase field model presented in this paper for simulating quasi-brittle failure and brittle fracture. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical modeling of brittle fracture using a phase field approach. We propose solving the coupled phase field / displacement problem by employing the scaled boundary finite element method, which facilitates the use of hierarchical meshes. An adaptive meshing approach based on this method is summarized. Contrary to existing applications of the scaled boundary finite element method in the context of phase field modeling, scaled boundary shape functions are employed in both staggered and monolithic solution schemes. The proposed methodology is verified considering two-dimensional benchmark problems. Very good agreement with finite element results of the force-displacement curves and crack paths is observed regardless of the solution scheme or meshing strategy. [ABSTRACT FROM AUTHOR]

The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials. [ABSTRACT FROM AUTHOR]

The construction of a homogeneous medium equivalent to a heterogeneous one under quasi-brittle fracture is investigated in the case of non-separated scales. At the microscale, the phase field method to fracture is employed. At the scale of the homogeneous medium, another phase field model either isotropic or anisotropic, depending on the microscale crack length and on the underlying microstructure, is assumed. The coefficients of the unknown phase field model for the homogeneous model are identified through the mechanical response of a sample subjected to fracture whose microstructure is fully described and estimated numerically. We show that the identified models can reproduce both the mechanical force response as well as overall crack paths with good accuracy in other geometrical configurations than the one used to identify the homogeneous model. Several numerical examples, involving cracking in regular lattices of both hard particles and pores, are presented to show the potential of the technique. [ABSTRACT FROM AUTHOR]