Ubiquitiformal Crack Extension in Quasi-Brittle Materials
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ISSN 1860-2134
Ubiquitiformal Crack Extension in Quasi-Brittle Materials Zhuo-Cheng Ou1
Yi-Bo Ju1
Jing-Yan Li1
Zhuo-Ping Duan1
Feng-Lei Huang1
1
( State Key Laboratory of Explosion Science and Technology, School of Mechatronics, Beijing Institute of Technology, Beijing 100081, China)
Received 31 July 2019; revision received 21 May 2020; Accepted 26 May 2020 c The Chinese Society of Theoretical and Applied Mechanics 2020
ABSTRACT 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.
KEY WORDS Ubiquitiformal crack, Complexity, Quasi-brittle materials, Random distribution of tensile strength, Weibull distribution parameters
1. Introduction Following the pioneering work of Mandelbrot et al. [1], and with abundant evidence showing that real fracture surfaces of various kinds of materials do exhibit some fractal geometric characteristics [2–5], a new fracture mechanics branch—fractal fracture mechanics—has been developed over the past few decades, which has been used to describe crack extension paths or fracture surface tortuosity for heterogeneous quasi-brittle materials [6–10]. However, as fractals are infinitely complex self-similar patterns, fractal crack has a singularity in its dimensional measure. This renders some mathematically inherent difficulties in dealing with real-world fracture problems. First, the boundary value problem is undefinable for a fractal crack since the normal of a fractal crack surface cannot be defined; thus, the path of fractal crack growth cannot be predicted. To remedy the theory, some researchers [11, 12] tried to use the fractional calculus to solve such a boundary value problem, but no satisfactory solution is available to date, because of its mathematical complexity. Wnuk and Yavari [13] introduced the correspondence principle, which defines an equivalent smooth blunt crack to represent the fractal crack, but such geometrical or physica
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