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Based on the concept of ubiquitiform,a ubiquitiformal crack extension model is developed for quasi-brittle materials.Numerical simulations are carried out using the ABAQUS software with the XFEM-based cohesive segments method to determine the ubiquitiformal crack extension path or fracture surface profile of the material under quasi-static loading.Such a ubiquitiformal crack model removes the singularity of a fractal crack;for the latter,the boundary value problem cannot be uniquely defined.In the simulation,the material properties,e.g.,the tensile strength,are assumed to obey the Weibull distribution.The meso-element equivalent method is used to determine the correlation between the Weibull distribution parameters and the aggregate gradation of concrete materials.The numerical results show that the complexities of the ubiquitiformal crack configurations are in good agreement with the previous experimental data.Through the numerical simulation,it is further demonstrated that the complexity of a ubiquitiformal crack is insensitive to the random spatial distribution of the aggregates,but more dependent on the Weibull distribution parameters which reflect the heterogeneity of the concrete.