Fractals surround us in so many different aspects of life. Since the term is becoming more widely used we wanted to create the definitive guide to understanding what Fractals are, why Fractals are important, and how Fractals impact our lives. This Ultimate Guide to Fractals will address common questions like: What is a Fractal? How do fractals work? What are Fractals used for? and much more.
While Fractals surround us in so many different ways, there are physical limitations as to how deep we can go in examining the fractals seen in the physical world. Eventually if we zoom in far enough we will see individual molecules and no longer be able to see the fractal pattern. In computers the story is a bit different. Computers have unlocked our ability to explore fractals at an incredibly detailed level, and because fractals are derived from mathematical equations, we can explore these shapes at an infinity deep level.
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Fractals have a significant impact on how we think about information and data management. If you consider the way information is organized and structured, it originates with some kind of overarching category, take this webpage as an example. The overarching category is about fractals. From there, the information is broken down into individual sub categories or topics, like fractals in Nature, fractals in Computer Systems and Fractals in Math. Each of these topics also has their own set of fragmentation into similar or related sub categories of ideas. While fractals for information are finite, you can see a similar branching tree pattern in the way the information is structured. Eventually the information will be broken down into the smallest possible modular component possible about the subject. Approaching information and organization of content using fractal theory is called the Hanby Iternal Information Theory. John Byron Hanby IV developed the Hanby Iternal Information Theory in an effort to better organize and categorize information in a systematic way that could optimize processing, management, and creation of content using the information.
The crack propagation behavior of rock during compression involves complex mechanisms. Describing the growth behavior of a large number of cracks with conventional mechanical models is a major challenge. Therefore, in this work, we propose a new method to describe crack growth behavior by considering crack bodies as free voxels that can expand and coalesce within a rock sample according to certain rules. Specifically, we first propose a crack growth model that quantitatively describes the crack growth ratio and crack growth rate, which are integrally related to the loading rate, internal friction angle, cohesion, initial porosity, and confining stress. Second, to avoid the complex analytical process of the traditional mechanical model in solving the propagation directions of multiple cracks, we introduce a method for determining the crack growth directions of shearing failure based on the colony growth assumption. This method defines the crack propagation direction as a synthetic vector of the inertial direction, the attractive direction, the Coulomb direction, and the edge direction. Moreover, a new mathematical description method of fracture energy and plastic energy is proposed to calculate the crack growth at each time step. The simulation results show that our crack growth model for shearing failure agrees well with the experimental results and explains the fracture behavior and transformation law of cracks to some extent.
The evolution process of cracks within the rock, such as initiation, propagation and coalescence, within the rock has long been a research focus and difficult problem. Rock fragmentation caused by engineering disturbance under high confining stress poses a concern to downstream operations like excavation and haulage as a result of the rising need for deep excavating resources. Therefore, it is inevitable and essential to analyze fracture expansion patterns under high-stress situations. In actual underground works such as tunneling and mining, the surrounding rock is inherited with numerous tiny defects such as joints and faults. Due to the complex heterogeneous stress state in deep underground works, the surrounding rocks are prone to fracture due to its inherent defects, leading to severe engineering disasters such as rock stripping, ejection and zonal disintegration2,3,4. A variety of abnormal phenomena in deep surrounding rock have been observed due to the increasing exploration depth of the underground work. For instance, deep rocks possess higher crack expansion rate and energy storage and therefore are more prone to burst under excavated disturbance5. Furthermore, the damage mode of rock will transform from brittle to ductile when increasing the depth6. It is concluded that during excavation, fractures are prone to initiate from the primary cracks due to non-uniformity energy accumulation around the crack tips7. Additionally, rock failure in deep depth is closer to shearing failure pattern rather than splitting pattern because the high surrounding pressure environment compresses the existing fractures and causes the fractures to propagate in a direction more inclined to slip along the shear surface. Therefore, the study of shearing fracture behavior under deep surrounding rocks is of great value for the prevention and control of deep underground works.
Many research efforts have been made in seeking the intrinsic and fundamental laws of fracture through different research tools, such as Infrared radiation monitoring (IRR)8, acoustic emission (AE) method9, strain field monitoring10 and equivalent transparent material simulation11. As an effective non-contact monitoring method, infrared radiation detection has been widely used in the study of rock damage and fracture mechanisms12,13. Sun et al. in 2017 found that infrared thermography can classify the compression process of rock samples into three stages before failure, i.e., the compaction stage, the elastic stage and the plastic stage. Based on the three-stage signals obtained by infrared radiation, Jiawang et al. in 2021 conducted true triaxial compressive tests on preprocessed sandstones under infrared monitoring and concluded that high confining stresses promote the development of cracks from the initial crack tip to the horizontal direction7. Research has been conducted to verify the correlation between the AE signals and rock failure that the initiation, closure, propagation, coalescence of the crack are reflected as different typical AE amplitude and frequency-spectrum14,15. Scholars also found that the location and propagation path of cracks can be obtained from the AE signal sources, which indicates that the crack propagation path is relatively consistent with the Mohr Coulomb Criterion16,17, namely, the orientation of the fracture is consistent with the rupture angle. Strain-field monitoring is another frequently-used method normally utilizing Digital Image Correlation (DIC) to observe the evolution of cracks on the rock surface, which has been proven useful for deformation-related strain heterogeneity in rock masses18,19. IRR, AE method and strain field monitoring are non-contact ways to capture the indirect signals to analyze the fracture extension. However, these methods cannot visualize the process of the inner development of the cracks, thus, many scholars conducted experiments with equivalent transparent material such as polymethylmethacrylate and photopolymer, from which equivalent "rock samples" made with carefully designed cracks are embedded11,20,21,22, allowing a well observation of the crack evolution during the compressive test. However, due to the limitations of equipment, this method can only simulate the fracture behavior of macroscopic cracks by omitting the existence of microstructure.
In recent years, very limited studies have considered mathematical models to explain or simulate crack propagation under loading, and most scholars prefer to use the finite element method (FEM) or discrete element method (DEM) for numerical simulation of crack development. Yan et al. built a coupled thermo-mechanical model based on a combined finite-discrete element method (FDEM) to simulate the thermal cracking of rocks. This model combined the heat conduction equation and fracture mechanism to perform prediction of the crack extension33. However, this study did not provide insight into the fracture propagation mechanism, but only analyzed the correlation between fracture growth and indirect thermal indices. Ehsan et al. conducted an in-depth study of the fracture mechanism of heterogeneous anisotropic rocks using the extended finite element method (XFEM), and the results showed that this formulation has great potential for estimating the stress intensity factor (SIF) and estimating the fracture extension trajectory34. Li et al. also proposed an optimized model using a grain-based finite-discrete element method (GB-FDEM) according to the calibration results of benchmark experiments including uniaxial compression test, confined compression test and Brazilian disc test, which focus on the fracture process of brittle rocks and the damage behavior due to crack initiation and propagation prior to peak stress35. The aforementioned studies provide a good theoretical basis for the analysis of crack evolution but currently, most related research results are based on FEM or DEM which is not a direct tool to simulate crack extension from the view of cracks.
The proposed crack growth model defines the ratio of cracks \(C\) as the volumetric ratio of the fractured voxels transformed from rock skeletons and provide an analytical solution for \(C\) based on the Mohr Coulomb Criterion.
Our model also defines the propagation direction of each crack as a resultant vector governed by four factors, namely, the inertial direction, the attractive direction, the Mohr Coulomb direction and the edge attraction direction.
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