Fractal Dimension Of Fracture Surface Research Articles

Insights from tensile fracture properties and full-field strain evolution of deep coral reef limestone under dynamic loads

Coral reef limestone (CRL) commonly undergoes dynamic tension when underground structures of island reefs encounter impacts, explosions, or seismic activities. Given the complexity of biological pores, the dynamic tensile fracture characteristics of CRL are poorly understood. Therefore, the dynamic tensile fracture behaviors of deep CRL were systematically observed by Split Hopkinson Pressure Bar tests and digital image techniques. Comparing to traditional rocks, the macro-pores near failure band would significantly change cracking path. The failure patterns are dominated by loading rate. The strongly dependence of dynamic tensile strength and dynamic crack initiation toughness on loading rate suggests the two indices overcome the effect of CRL macro-pores under dynamic impacts. At low loading rates, tensile fractures predominantly follow intergranular cracks, whereas transgranular cracks dominate at higher rates. The fractal dimension of fracture surface decreases with increasing crack propagation velocity, loading rate, and dynamic crack initiation toughness. Due to the unique marine sedimentary environment, the mechanical heterogeneity in multiple scales distinguishes CRL from terrestrial rock materials. The insights into underlying mechanisms of dynamic tension provide support to optimization of blasting scheme and stability assessments for island underground engineering.

Digital Fracture Surface Morphology and Statistical Characteristics of Granite Brazilian Tests after Non-Steady-State Thermal Disturbance

With the widespread advent of digital technologies, traditional perspectives in rock mechanics research are poised for further expansion. This paper presents a Brazilian test conducted on granite after non-steady-state thermal disturbance at 25 °C, 200 °C, 400 °C, and 600 °C, with detailed documentation of the damage process and failure response using an acoustic emission (AE) apparatus and a digital image correlation (DIC) system. Subsequently, utilizing point cloud data captured by a three-dimensional (3D) laser scanning system, a digital reconstruction of the failed specimen’s fracture surface was accomplished. The 3D fractal characteristics and roughness response of the digitized fracture surface were studied using the box-counting method and least squares approach. Furthermore, texture information of the digitized fracture surface was calculated using the Gray Level Co-occurrence Matrix (GLCM), and statistical characteristics describing the elevation distribution were analyzed. The results elucidate the influence of thermal disturbance temperature on the mechanical parameters of the specimen, acoustic emission behavior, surface strain field evolution, and digital fracture morphology characteristics. The findings indicate a non-linear degradation effect of temperature on the specimen’s tensile strength, with a reduction reaching 80.95% at 600 °C, where acoustic emission activity also peaked. The rising thermal disturbance temperature inhibited the crack initiation load at the specimen’s center but expanded the high-strain concentration areas and the growth rate of horizontal displacement. Additionally, varying degrees of linear or non-linear relationships were discovered between thermal disturbance temperature and the 3D fractal dimension of the fracture surface, average roughness (Ra), peak roughness (Rz), and root mean square roughness (Rq), confirming the potential of Rsm in predicting the 3D fractal dimension of Brazilian test fracture surfaces. The study of the GLCM of the digitized 3D fracture surface demonstrated a high dependency of its four second-order statistical measures on thermal disturbance temperature. Finally, the statistical parameters of the fracture surface’s elevation values showed a significant non-linear relationship with thermal disturbance temperature, with a critical temperature point likely existing between 400 and 600 °C that could precipitate a sudden change in the fracture surface’s elevation characteristics.

A new numerical model for simulation of flow on rough fracture

Hydraulic fracturing is an important means to develop unconventional oil and gas. The fractures produced by hydraulic fracturing have the characteristics of a rough surface, high tortuosity, and strong randomness. The fluid migration law in rough fractures is more complicated, which leads to the change of the law of conductivity in fractures becoming more complex. Meanwhile, it is difficult to get certain regularity for the fracture surface obtained by direct fracturing. As a result, it is an arduous task to explore the influence of fracture roughness on conductivity by experimental means. Therefore, it is a feasible idea to use simulation to study the flow conductivity in rough fractures, which can be calculated by Darcy's seepage equation. In previous studies, quantitative characterization of rough fractures was rarely studied, and the three-dimensional Darcy seepage model of rough fractures was lacking too. In this paper, the rough surface profile and waveform are analyzed by analogy, and the height of the rough fracture surface, as well as the phase and wavelength of the waveform, were taken as the control parameters to generate the rough fracture surface. Then, the fractal dimension was used to characterize the roughness of the fracture surface. Coupled with the rough fracture model and the Darcy flow model, a three-dimensional Darcy flow model of rough fracture was established. Next, a new parameter, equivalent permeability, was defined to quantify and characterize the fracture permeability affected by roughness. Subsequently, the simulation results were verified by the experimental data. Finally, the effects of fracture surface fractal dimension, fracture width, proppant distribution, shear slip, and tortuous angle on the flow field were studied. This study provides a new model for the simulation of a rough fracture flow field, which provides important insight for understanding the properties of rough fracture flow fields.

Study on the Difference of Fracture Extension in Hydraulic Fracturing of Marine and Continental Shales Based on Optical Scanning Technology

After hydraulic fracturing, the geometric characteristics of rock morphology is a crucial means for evaluating the effectiveness of fracture stimulation in enhancing production. In order to quantitatively analyze the surface undulations of marine and continental shale morphology further following hydraulic fracturing, a method for calculating the three-dimensional fractal dimension of fracture surfaces based on optical scanning technology is proposed. This method involves the acquisition of point cloud data using a 3D surface scanner. The obtained data are subsequently subjected to smoothing processes, followed by the reconstruction of the three-dimensional representation of the fractures. The box-counting dimension algorithm is employed to calculate the fractal characteristics of post-fracture morphology. The research results indicate that marine shale, due to its higher proportion of brittle minerals such as quartz and calcite, predominantly exhibits vertically oriented longitudinal fractures, perpendicular to the minimum horizontal stress. The average initiation pressure is 8% higher compared to continental shale, with an average fractal dimension of 2.24397. In contrast, continental shale, characterized by its high clay content and the development of natural fractures and bedding planes, is more prone to capturing hydraulic fracture features, resulting in predominantly single transverse shear fractures. The average fractal dimension is 2.087065, which is 7% lower than that of marine shale. These research findings offer a certain degree of guidance for the optimization of fracturing process parameters for different types of reservoirs.

Mixed mode fracture parameters and fracture characteristics of diorite using cracked straight through Brazilian disc specimen

In this study, the cracked straight through Brazilian disc (CSTBD) specimen was applied to study the I/II mixed mode static fracture of diorite. The fracture load and fracture toughness for different prefabricated crack angles β were determined. The fracture and morphological characteristics were analyzed using digital image correlation (DIC), scanning electron microscope (SEM) and 3D topography scanning techniques. In addition, the key fracture parameters such as fracture energy, crack mouth opening displacement (CMOD) and fracture process zone (FPZ) were characterized, and a fracture energy calculation method considering the primary crack growth was proposed. The results show that the fracture mechanical behavior and fracture trajectories are controlled by the crack angle. At the millimeter measurement scale, the fractal dimensions of fracture surfaces do not vary with crack angle. Further SEM observation shows that intergranular fracture and transgranular fracture are the dominate micro fracture modes of fracture surfaces under mode I and mode II loading, respectively. With the increase of crack angle, the fracture energy first increases and then decreases, while the allowable CMOD decreases monotonically and even appears negative when β ≥ 45°. Based on the displacement gradient method, the calculated width of FPZ for the mode I fracture specimen at peak load is 0.5–1.4 mm and the length is 2.66 mm. Finally, the stress-based and strain-based fracture criteria were modified. It is found that the proposed EEMTSN criterion has the best prediction effect on the mixed mode fracture toughness of diorite.

Correlation between anisotropic fractal dimension of fracture surface and coercivity for Nd-Fe-B permanent magnets

In this work, SEM images analysis and fractal dimension calculations are performed on the fracture surfaces for N50-type with low coercivity, 30 EH-type with high coercivity, and isotropic Nd-Fe-B permanent magnets. It is discovered for the first time that the difference between the fractal dimensions of fracture surface parallel and perpendicular to the c-axis D∥ and D⊥ for N50-type magnet with low coercivity is large, and the difference between the fractal dimensions of D∥ and D⊥ for 30 EH-type magnet with high coercivity is small. However, there is almost no difference in the fractal dimension of D∥ and D⊥ for isotropic magnet. The correlation between anisotropic fractal dimension of the fracture surface and coercivity is revealed. The difference between the fractal dimension of fracture surface D∥ and D⊥ not only reflects the difference in micro morphology, but also serves as an important performance parameter to characterize the coercivity of the magnet, which provides new insights for studying the relationship between the magnetic properties and the microstructure of permanent magnetic materials.

Transversely isotropic creep behavior of phyllite and its influence on the long-term safety of the secondary lining of tunnels

Phyllite, which is a low-grade metamorphic rock with well-developed foliations, is encountered frequently during the construction of tunnels in western China. First, uniaxial compressive creep tests were performed to study the time-dependent behavior of phyllite specimens from Zhegu mountain tunnel. Then, a numerical approach was put forward based on the particle discrete element method to describe the typical creep behavior of phyllite, including its decay, steady, accelerated creep stages, and its transverse isotropy. Finally, the numerical model was adopted to investigate the failure process of secondary tunnel lining in the phyllite stratum. The following results were obtained: Ai et al. (2014) (1) The creep behavior of phyllite was affected significantly by the weak plane-loading angles and water content. Its creep strength decreased as the water content increased, and the maximum and minimum values occurred at θ = 90° and 30°, respectively; Carrillo et al. (2016) (2) The fractal dimension of fracture surface for θ = 0° was the smallest due to the shear and tensile failure along foliations. For each angle, fractal dimension increased with the increase of water content; Chen et al. (2018) (3) Micro-cracks were initiated at the initial loading stage and propagated during the entire loading process at θ = 0°, while they appeared at a higher stress level after specimens had accumulated sufficient energy at θ = 30° or 90°; Chen et al. (2019) (4) The evolution of micro-cracks was affected jointly by the inclination angle of foliations and the geo-stress field, and the cluster of cracks shows evidently asymmetric features. The results revealed the transversely isotropic creep behavior of phyllite and proposed a method which can aid more accurate predictions for design and construction with respect to geotechnical engineering structures related to layered rock mass.

Fractal Dimension Analysis of Three-Point Bending Concrete Test Specimens

The paper deals with the analysis of the fractal dimension of fracture surfaces of concrete specimens tested in a three-point bending configuration. Fifteen representative specimens were chosen out of a bigger set for the fractal dimension analysis. Their fracture surfaces were scanned by 2D optical profilometry and analysed by the FracDiM software created in the Java programming language. The resulting values of fractal dimensions for specimens of three different sizes are presented.

Forward prediction of early‐time spontaneous imbibition of water in unsaturated rock fractures

AbstractSpontaneous imbibition is a capillary‐driven phenomenon in which a wetting fluid displaces a nonwetting fluid from voids without any increase in external pressure. This study compared forward predictions of early‐time displacement of air by water within Mode I fractures in 14 low‐porosity rock cores using a new fractal model with those based on an established parallel plate model. Spontaneous imbibition was measured using dynamic neutron radiography, along with independent determinations of equilibrium contact angle, fracture aperture width, and fracture surface fractal dimension, Ds. The predicted uptake curves generally agreed with the experimental data. However, both models overpredicted the height of the wetting front at any given time. This overprediction may be due to lateral losses of wetting fluid to the matrix by spontaneous imbibition through fracture surfaces. The predictions of the fractal model were consistently closer to the observed values than those of the parallel plate model, and for the best cases, their upper and lower confidence intervals bounded the data points. In 12 out of 14 cases, the RMSD for the fractal model was less than that for the parallel plate model. A paired t test indicated that, on average, the RMSD for the fractal model was significantly lower than that for the parallel plate model. This statistically improved prediction can be attributed to the retardation of predicted uptake achieved through the introduction of a fracture surface roughness parameter (i.e., Ds) in the fractal model.

X-RAY MICRO-TOMOGRAPHY CHARACTERIZATION OF VOIDS CAUSED BY THREE-POINT BENDING IN SELECTED ALKALI-ACTIVATED ALUMINOSILICATE COMPOSITE

This paper deals with the pilot characterization of a special alkali-activated aluminosilicate composite composed of waste brick powder, brick rubble and a solution of potassium water glass. Fracture tests were conducted on the specimens via three-point bending and fracture parameters were evaluated. Selected specimen was investigated using micro-tomography to supplement the results with visual information about the inner structure of this newly designed material before and after the mechanical loading. Tomographic measurements and image processing were conducted for a qualitative and quantitative assessment of changes in the internal structure with an emphasis on the calculation of porosimetric parameters and visualization of the fracture surface. Fractal dimension of fracture surface was estimated.

Experimental investigation on mode I fracture characteristics of granite after cyclic heating and cooling treatments

Rock in the geothermal reservoir is frequently subjected to cyclic heating and cooling, and fracture characteristics are closely related to the stable operation of Enhanced Geothermal Systems. To examine impacts of cyclic heating and cooling treatments on mode I fracture characteristics of granite, fracture tests of semi-circular bend specimens were conducted, acoustic emission (AE) events were monitored, and morphology characteristics of fracture surfaces were analyzed. Meanwhile, the open porosity was measured, and the scanning electron microscope (SEM) was applied to identify variations of microstructure in granite affected by cyclic treatments. Failure pattern and modifications of microstructure in treated granite specimens were discussed. Results indicated that cyclic treatments can lead to serious degradation in granite, manifested as the increase of open porosity, the decrease of fracture toughness and cumulative AE counts with both increasing temperature and cycles. The smoother post-peak stage in load-displacement curves and the prolonged active period of AE events of granite specimens after cyclic treatments implied the enhancement of ductility. Fractal dimensions of fracture surfaces exhibited evidently negative correlation with the brittleness index, which demonstrated that the enhanced ductility promotes the generation of rougher fracture surfaces. The failure pattern of treated granite specimens revealed that the propagation path of the main crack is strongly affected by the thermal cracks. SEM observations verified the increased, extended and deepened micro-cracks in granite after cyclic treatments, which can be attributed to the comprehensive effects of heating, rapid cooling, thermal cycling, and water-induced weakening.

SPONTANEOUS IMBIBITION OF A WETTING FLUID INTO A FRACTURE WITH OPPOSING FRACTAL SURFACES: THEORY AND EXPERIMENTAL VALIDATION

Spontaneous imbibition (SI) is a capillary-driven flow process, in which a wetting fluid moves into a porous medium displacing an existing non-wetting fluid. This process likely contributes to the loss of fracking fluids during hydraulic fracturing operations. It has also been proposed as a method for an enhanced recovery of hydrocarbons from fractured unconventional reservoirs. Numerous analytical and numerical approaches have been employed to model SI. Invariably, these idealize a fracture as the gap formed between parallel flat surfaces. In reality, rock fracture surfaces are rough over multiple scales, and this roughness will influence the contact angle and rate of fluid uptake. We derived an analytical model for the early-time SI behavior within a fracture bounded by parallel impermeable surfaces with fractal roughness assuming laminar flow. The model was tested by fitting it to experimental data for the SI of deionized water into air-filled rock fractures. Twenty cores from two rock types were investigated: a tight sandstone (Crossville) and a gas shale (Mancos). A simple Mode I longitudinal fracture was produced in each core by compressive loading between parallel flat plates using the Brazilian method. Half of the Mancos cores were fractured perpendicular to bedding, while the other half were fractured parallel to bedding. The two main parameters in the SI model are the mean separation distance between the fracture surfaces, [Formula: see text], and the fracture surface fractal dimension [Formula: see text]. The [Formula: see text] was estimated for each core by measuring the geometric mean fracture aperture width through image analysis of the top and bottom faces, while [Formula: see text] was estimated inversely by fitting the SI model to measurements of water uptake obtained using dynamic neutron radiography. The [Formula: see text] values ranged from 45[Formula: see text][Formula: see text]m to 190[Formula: see text][Formula: see text]m, with a median of 93[Formula: see text][Formula: see text]m. The SI model fitted the height of uptake versus time data very well for all of the rock cores investigated; medians of the resulting root mean squared errors and coefficients of determination were 0.99[Formula: see text]mm and 0.963, respectively. Estimates of [Formula: see text] ranged from 2.04 to 2.45, with a median of 2.24. Statistically, all of the [Formula: see text] values were significantly greater than two, confirming the fractal nature of the fracture surfaces. Future research should focus on forward prediction through independent measurements of [Formula: see text] and extension of the existing SI model to late times (through the inclusion of gravity) and fractures with permeable surfaces.

Size Effects on Fracture Parameters of High Alumina Refractories

The randomness of phase compositions and their distribution as well as the micro cracks and pores in refractories result in the dependence of mechanical properties on the sample size. In this work, the size effects on fracture parameters including bending strength, elastic modulus and fracture toughness of high alumina bricks were investigated by testing samples of different sizes according to the national standard in China. The results showed that the smaller samples presented higher strength, while the larger samples possessed greater elastic modulus. Weibull modulus m of bending strength for high alumina bricks calculated for samples of two sizes, each group featured 32 samples, ranged from 11 to 12, which was higher than in other refractories such as low grade high alumina, magnesia-carbon and magnesia-chrome bricks. The scatter of the average strength and its standard deviation became stable when the sample number was more than 20. The fractal dimension of fracture surface for high alumina bricks was inversely and directly proportional to the sample size and the fracture toughness, respectively. The estimated fracture toughness of high alumina bricks was close to window glass and less than alumina and mullite.

Relationship between fractal dimensions of fracture surface and impact toughness of polycaprolactone/nano‐CaCO3 composites

AbstractThe fractal dimensions of the specimen impact fracture surface of nanosized calcium carbonate reinforced polycaprolactone (PCL) composites were determined using a structure‐function method; and the correlation between V‐notched Izod impact strength measured at room temperature and fractal dimensions was studied. It was found that the fractal dimension range of the specimen fracture surface was from about 1.918 to 1.967, and all of the correlation coefficients were higher than 0.9. There was a strong relationship between the fracture surface fractal dimension and the impact strength. It indicated that the specimen fracture surface of the composites had obvious fractal character, the measured impact strength increased with increasing the fracture surface fractal dimension, and the correlation between them obeyed a three‐order polynomial equation. In addition, the toughening mechanisms of the PCL nanocomposites were discussed.

Effect of water absorption ratio on tensile strength of red sandstone and morphological analysis of fracture surfaces

Brazilian disc tests were undertaken on a number of red sandstone samples with different water absorption ratios. The tensile strength of the red sandstone decreases as the water absorption ratio increases. The fracture surfaces of failed red sandstone discs were scanned by Talysurf CLI 2000. With the aid of Talymap Gold software, based on ISO25178, a set of statistical parameters was obtained for the fracture surfaces. The maximum peak height (S p), maximum pit height (S v) and maximum height (S z) of the fracture surfaces exhibited the same decreasing trend with increasing water absorption. S a and S ku values for the fracture surfaces showed a downward trend as the water absorption ratio increased. The fractal dimensions of fracture surfaces were calculated and found to decrease as the water absorption ratio increased. Through analysis of PSD curves, the smallest dominant wavelength was observed to reflect the roughness of the fracture surfaces. Additionally, the results suggest that the roughness of fracture surfaces becomes small as the water absorption ratio increases.

Fractal study on fracture surface of PAN-based carbon fiber

ABSTRACTPAN-based carbon fiber is a type of brittle material whose fracture surface has a structure peculiar to brittle materials. The tensile strength of a carbon fiber is determined by the strength of weak links in the fiber to a large extent. Since fracture surface is a type of weak link, the fracture surfaces of PAN-based carbon fibers were investigated in this article. SEM (Scanning Electron Microscopy) micrographs of selected carbon fibers with different tensile strengths were analyzed on computer and the fractal dimensions of fracture surfaces were obtained on the basis of fractal theories. The relationship between the tensile strength of PAN-based carbon fiber and the fractal dimension of its fracture surface was demonstrated.

Fracture Surface Morphology and Impact Strength of Cellulose/PLA Composites.

Polylactide (PLA)-based composite materials reinforced with ball-milled celluloses were manufactured by extrusion blending followed by injection molding. Their surface morphology from impact fracture were imaged with scanning electron microscopy (SEM) and investigated by calculating their fractal dimensions. Then, linear regression was used to explore the relationship between fractal dimension and impact strength of the resultant cellulose/PLA composite materials. The results show that filling the ball-milled celluloses into PLA can improve the impact toughness of PLA by a minimum of 38%. It was demonstrated that the fracture pattern of the cellulose/PLA composite materials is different from that of pristine PLA. For the resultant composite materials, the fractal dimension of the impact fractured surfaces increased with increasing filling content and decreasing particle size of the ball-milled cellulose particles. There were highly positive correlations between fractal dimension of the fractured surfaces and impact strength of the cellulose/PLA composites. However, the linearity between fractal dimension and impact strength were different for the different methods, due to their different R-squared values. The approach presented in this work will help to understand the structure–property relationships of composite materials from a new perspective.

Trituration and Fractal Dimension in Homeopathic Pharmacopoeia

For many years now, homeopathy is considered to be an alternative and more physical way of treatment compared to traditional medicine. Though, little has yet been scientifically studied upon the matter. The scope of this paper is to investigate the relationship between the size-shape variations of eight starting Raw Solid Materials (RSM) and the role of trituration in the size-shape of these RSM’s before they are turned into solutions. These materials are used in homeopathy as remedies, after successive grindings, before they are turned into homeopathic solutions. What is more, their fracture surface fractal dimension is investigated. The trituration process will be analyzed, which in homeopathic pharmacology leads to the substances having self-affined fractal structure regardless their type or hardness (metals, minerals, salts, shells, dry plants, etc.), a structure only depending on the size and shape of the powder grains of the starting RSM. The physical meaning of these results is that at the end of the process there is a gradual transformation of the size and shape of the original RSM during the trituration, depending on the physical condition of the materials before grinding them. In this paper, it is proved that, regardless of the starting RSM, the fixed trituration formalism used in Homeopathic Pharmacopoeia leads to a result which depends only on the powder grains size of the starting RSM.

Texture, microstructure, and fractal features of the low-cycle fatigue failure of the metal in pipeline welded joints

Crystallographic texture and fracture features are studied after low-cycle fatigue tests of laboratory specimens cut from the base metal and the characteristic zones of a welded joint in a pipeline after its longterm operation. The fractal dimensions of fracture surfaces are determined. The fractal dimension is shown to increase during the transition from ductile to quasi-brittle fracture, and a relation between the fractal dimension of a fracture surface and the fatigue life of the specimen is found.

Investigation of meso-failure behaviors of Jinping marble using SEM with bending loading system

In this study, the meso-failure mechanism and fracture surface of Jinping marble were investigated by means of scanning electron microscope (SEM) with bending loading system and laser-scanner equipment. The Yantang and Baishan marbles specimens from Jinping II hydropower station were used. Test results show that the fracture toughness and mechanical behaviors of Yantang marble were basically higher than those of Baishan marble. This is mainly due to the fact that Baishan marble contains a large percentage of dolomite and minor mica. Crack propagation path and fracture morphology indicated that the direction of tensile stress has a significant effect on the mechanical behaviors and fracture toughness of Baishan marble. For Yantang and Baishan marbles, a large number of microcracks around the main crack tip were observed when the direction of tensile stress was parallel to the bedding plane. Conversely, few microcracks occurred when the direction of tensile stress was perpendicular to the bedding plane. The presence of a large number of microcracks at the main crack tip decreased the global fracture toughness of marble. The results of three-point bending tests showed that the average bearing capacity of intact marble is 3.4 times the notched marble, but the ductility property of the defective marble after peak load is better than that of the intact marble. Hence, large deformation may be generated before failure of intact marbles at Jinping II hydropower station. The fractal dimension of fracture surface was also calculated by the cube covering method. Observational result showed that the largest fractal dimension of Yantang marble is captured when the direction of tensile stress is parallel to the bedding plane. However, the fractal dimension of fracture surface of Yantang and Baishan marbles with tensile stress vertical to the bedding plane is relatively small. The fractal dimension can also be used to characterize the roughness of fracture surface of rock materials.