Fractal Crack Research Articles

Morphological characteristics and cleaning effects of collapsing cavitation bubble in fractal cracks

When a cavitation bubble collapses in a crack, due to the lack of symmetry, spatial uniformity, and obvious correlations during the collapse process, it is difficult to effectively explore the interaction mechanism between the bubble and crack walls. In this paper, a combination of numerical simulation and experimental approaches are used to explore the collapse process of a cavitation bubble in a fractal crack. The numerical model is based on the pseudopotential multi-relaxation-time lattice Boltzmann method (LBM). The experiment platform including an underwater pulsed discharge device combined with a high-speed camera system. Moreover, the morphological analysis method based on Minkowski functionals is used to quantitatively depict the morphological features in this paper. The validity of the numerical model is qualitatively verified by the experimental platform, whereas the influence of the complexity and geometric features of the crack wall on the cavitation bubble collapse process is quantitatively studied by LBM simulation and Minkowski functionals. The research findings indicate that the complexity and geometric features of the crack wall markedly influence the collapse time and behavior of the cavitation bubble. Specifically, the collapse time of the cavitation bubble increases with the increasing complexity of the crack wall. Moreover, the maximum jet direction of cavitation bubble toward the wall is closest to the cavitation bubble. Furthermore, as the fractal dimension increases, the intensity of the pressure wave and jet acting upon the crack wall increase while the radiation range decreases. Cavitation cleaning can be applied to arbitrary complex solid surfaces in various environments.

Effects of Pore Geometry and Saturation on the Behavior of Multiscale Waves in Tight Sandstone Layers

AbstractGeometric heterogeneities in tight reservoir rocks saturated with a fluid mixture may exhibit different scale distribution characteristics. Conventional models of rock physics based on poroelasticity, which usually consider single‐scale pore structure and fluid patches, are inadequate for describing elastic wave responses. A major challenge is to establish the relationship between the wave response at different spatial scales and frequencies. To address this problem, three sets of observational data over a wide frequency range were obtained from a tight oil reservoir in the Ordos Basin, China. Ultrasonic measurements were made on eight sandstone samples at partial oil‐water saturation at 0.55 MHz. Data from six borehole measurements and seismic profiles were acquired and analyzed at about 10 kHz and 30 Hz, respectively. Analysis of the cast thin sections shows that dissolution pores and microcracks generally develop, with fractal dimensions of the pores ranging from 2.45 to 2.67 for the samples with porosities between 5.1% and 10.2%. Compressional wave velocity and attenuation were estimated from the observed data. The results show that the velocity dispersion from seismic to ultrasonic frequencies is 10.02%, mostly occurring between sonic and ultrasonic frequencies. The attenuation is stronger at higher oil saturation. The relationships between velocity, attenuation, and wavelength were established and can be used for further forward modeling and seismic interpretation studies. A partial saturation model has been derived based on effective differential medium theory and a double double‐porosity model, assuming that the medium contains fractal cracks and fluid patches. The effects of scale and saturation on wave responses are prevalent. Modeling results consistent with observed data show that the radii of cracks and fluid patches range from 0.1 μm to 2.8 mm, affecting ultrasonic, acoustic, and seismic attenuation. The multiscale data and proposed model quantify the relationship between fracture and fluid distributions and attenuation and could be useful for upscaling to the reservoir scale. The study helps improve the understanding of seismic wave propagation in partially saturated rocks, which has potential applications in seismic exploration, hydrocarbon production in reservoirs, and CO2 sequestration in aquifers.

Characterization of Ductile Crack Propagation by Fractal Energy Dissipation Rate

Because of its geometry dependence and loss of physical meaning, the incremental crack resistance curve cannot characterize ductile fractures with large crack extensions and plastic deformations. Therefore, the energy dissipation rate R is employed to overcome these deficiencies, even though specimen size effects still exist. In the study, considering the fractal crack path and concomitant plastic dissipation in the fractal domain, a scale-invariant energy dissipation rate, γp*, is proposed in the context of renormalization group theory. Some experiments in the literature have validated this approach. The fitted fractal energy dissipation rate is independent of the specimen size and initial crack length; moreover, as the specimen size increases, progressive fractality vanishing is found consistently with geometrical multifractality.

Fractal Clustering as Spatial Variability of Magnetic Anomalies Measurements for Impending Earthquakes and the Thermodynamic Fractal Dimension

Several studies focusing on the anomalies of one specific parameter (such as magnetic, ionospheric, radon release, temperature, geodetic, etc.) before impending earthquakes are constantly challenged because their results can be regarded as noise, false positives or are not related to earthquakes at all. This rise concerns the viability of studying isolated physical phenomena before earthquakes. Nevertheless, it has recently been shown that all of the complexity of these pre-earthquake anomalies rises because they could share the same origin. Particularly, the evolution and concentration of uniaxial stresses within rock samples have shown the generation of fractal crack clustering before the macroscopic failure. As there are studies which considered that the magnetic anomalies are created by lithospheric cracks in the seismo-electromagnetic theory, it is expected that the crack clustering is a spatial feature of magnetic and non-magnetic anomalies measurements in ground, atmospheric and ionospheric environments. This could imply that the rise of multiparametric anomalies at specific locations and times, increases the reliability of impending earthquake detections. That is why this work develops a general theory of fractal-localization of different anomalies within the lithosphere in the framework of the seismo-electromagnetic theory. In addition, a general description of the fractal dimension in terms of scaling entropy change is obtained. This model could be regarded as the basis of future early warning systems for catastrophic earthquakes.

Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization

The seismo-electromagnetic theory describes the growth of fractally distributed cracks within the lithosphere that generate the emission of magnetic anomalies prior to large earthquakes. One of the main physical properties of this theory is their consistency regarding the second law of thermodynamics. That is, the crack generation of the lithosphere corresponds to the manifestation of an irreversible process evolving from one steady state to another. Nevertheless, there is still not a proper thermodynamic description of lithospheric crack generation. That is why this work presents the derivation of the entropy changes generated by the lithospheric cracking. It is found that the growth of the fractal cracks increases the entropy prior impending earthquakes. As fractality is observed across different topics, our results are generalized by using the Onsager’s coefficient for any system characterized by fractal volumes. It is found that the growth of fractality in nature corresponds to an irreversible process.

ON ESTIMATING PLASTIC ZONES AND PROPAGATION ANGLES FOR MIXED MODE I/II CRACKS CONSIDERING FRACTAL EFFECT

This paper mainly investigated the influence of the fractal dimension of mixed mode I/II crack on its near-tip plastic zones and propagation angles. The near-tip distortional strain energy density for mixed mode I/II crack is modified by considering its fractal dimension. Using the Von Mises yield condition, the plastic zones at the fractal crack tip are evaluated. The effects of the fractal dimension of crack on the near-tip plastic zones are analyzed qualitatively. A modified criterion minimizes the fractal distortional strain energy density to predict the propagation direction of mixed mode I/II crack. Based on this, the effects of fractal dimension on the mixed mode I/II crack propagation direction at different inclination angles are discussed quantitatively. In addition, the modified criterion is compared to the experimental data in the literature, as well as the classical maximum tangential stress criterion and minimum strain energy density criterion. The results indicate that the fractal dimension of crack is an important factor in evaluating the crack-tip plastic zones and predicting the crack propagation angle, and the modified fractal criterion is more accurate and effective in predicting the crack propagation angles.

A FRACTAL PERSPECTIVE ON FRACTURE INITIATION AND PROPAGATION OF RESERVOIR ROCKS UNDER WATER AND NITROGEN FRACTURING

This paper mainly investigated the fractal characteristics of crack propagation in hydraulic fracturing and the effect of fractal dimension of induced fractures on the rock breakdown pressure. A series of fracturing experiments with water and nitrogen (N[Formula: see text] as the fracturing fluids were conducted on coal, red sandstone and shale. The injection pressure curves, rock breakdown pressure and macro- and micro-morphologies of the induced fractures were analyzed. A new fractal crack initiation criterion was developed to evaluate the rock breakdown pressure in hydraulic fracturing. The validity of the new criterion was verified by comparing the theoretical and experimental rock breakdown pressures. Further, the difference between the water fracturing and N2 fracturing was analyzed and the influence of crack fractal effect on the rock breakdown pressure was discussed by considering different initial permeabilities of rocks, injection rates and viscosities of fracturing fluids. The results show: (1) N2 fracturing may reduce the rock breakdown pressure and induce more tortuous macro-fractures with larger fractal dimension; (2) ignoring the actual fractal dimension of induced fractures would cause the prediction values of the rock breakdown pressure to be larger; and (3) the influence degree of crack fractal dimension on the rock breakdown pressure is related to the initial permeability of rocks, injection rate and viscosity of fracturing fluids. Therefore, we suggest the crack fractal dimension is an important factor that should not be ignored in designing the fracturing operations.

Crack propagation rate of hydrogen-induced cracking in high sulfur-containing pipelines

Hydrogen-Induced Cracking (HIC) is one of the main failure models of pipeline steels that are utilized to gather pipeline networks and long-distance pipeline networks within high sulfur-containing gas fields. This phenomenon is mainly caused by the combined effects of hydrogen and stress after the hydrogen atoms or ions in the transmission medium entering into the steel. This can produce a stress corrosion effect on the materials and thus lead to a decrease of fracture ductile or degradation of other properties. In this paper, a micro-fracture shear band fractal model of HIC has been proposed and the mechanism of HIC has been studied. Both the shape of the fracture process zone and the step length of hydrogen-induced crack propagation were integrated, thus building expressions that describe the fractal rate of each step of HIC. Based on this, the fractal crack propagation rates of HIC in pipelines 16 Mn and 20 steel were calculated under certain working conditions and H2S concentrations. Our study adds valuable insight into the life prediction of pipeline failure under the HIC environment.

Ubiquitiformal Crack Extension in Quasi-Brittle Materials

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.

Shape Evolution of Unstable, Flexural Cracks in Brittle Materials

In this manuscript, the crack-shape evolution of unstable, flexural cracks was modeled using a continuum mechanics approach applied to dynamic, brittle, and elliptical cracks. The crack-shape evolution was first computed analytically for both flat and fractal cracks and subsequently compared to experimental values inferred from fractographic features present on silicate glass, gallium arsenide, and silicon single crystals fractured in bending. The trends predicted by the analytical model were consistent with the fractographic observations prior to the onset of crack-tip instabilities. It was determined that the effect of the surface roughness was to slow down the crack and to decrease the local crack-front radius of curvature which, in flexural cracks, led to crack shapes elongated along the free surface direction. This work represents the first attempt to predict the crack-shape evolution in rough, brittle samples fractured in bending and to understand the role played by the surface roughness during fast, unstable propagation.

СВЯЗЬ ФРАКТАЛЬНОЙ РАЗМЕРНОСТИ ПОВЕРХНОСТИ РАЗРУШЕНИЯ БЕТОНА С ХАРАКТЕРИСТИКАМИ МЕХАНИКИ РАЗРУШЕНИЯ

As it was found previously, the concrete fracture surface formed from tensile force is described by fractal geometry methods. It is shown thatthe fractal dimension value is related to the tensile stress gradient φ_i, to the aggregate size and, as shown earlier, does not depend on the strength of concrete. Moreover, the fractal dimension depends on the size of the sample only until its size reaches a value to which linear fracture mechanics is applicable. The stress intensity factor is related to the fractal dimension, and both characteristics are related to the aggregate size. A connection for the critical stress intensity factor K_Ic^f(l,φ_i) characterizing the crack resistance of the material in nonlinear fracture mechanics with the crack size l and the specimenis proposed. The stress intensity factor for a fractal crack K_Ic^f(l,φ_i) can be used to calculate structures using nonlinear fracture mechanics.

Surface energy of coal particles under quasi-static compression and dynamic impact based on fractal theory

The crushing energy of coal plays an important role in mining, gas extraction, occurrence of outburst accidents, and coal pulverisation engineering. To calculate the crushing energy of coal, a fractal model considering the crack area was established. The fractal fragmentation distribution and the fractal crack distribution, with their fractal dimensions (Df and Ds) calculated from the particle size distribution, were used to calculate the new specific surface area (SSA) of coal after crushing. Two different experimental methods, quasi-static compression on a single particle (strain rate: 0.0005–0.001 s−1) and dynamic impact on particles (strain rate: 300 s−1), were used to crush coal particles to particle size of 2–3 mm. The experimental results show that Df for coal particles under quasi-static compression (QSC) and dynamic impact (DI) are 1.44–1.85 and 1.9–2.74, respectively. Df increases significantly with the crushing ratio, and there is a limit value, which is defined as Ds. Ds measured by QSC and DI were 1.87 and 2.75, respectively. The SSA of a single particle obtained by the proposed fractal model increases with the decrease in the particle size, which can be highly consistent with the BET SSA by fitting. The surface energy of coal particles under QSC and DI calculated by the traditional theory, which does not take into account the crack area, is 240.10 J/m2 and 180.92 J/m2, respectively, independent of the crushing ratio. The surface energy calculated by the fractal model is 3.55 J/m2, taking into account the frictional heat.

Finite element cohesive fracture modeling of asphalt mixture based on the semi-circular bending (SCB) test and self-affine fractal cracks at low temperatures

Thermal cracking is one of the most common distresses for asphalt pavement constructed in the cold regions. In order to address this issue, the combined use of fracture mechanics-based tests and simulation is a solid option. First, asphalt mixture samples are prepared based on the German standard and the low temperature strength are measured by Semi-Circular Bending (SCB) test and the Uniaxial Creep (UC) test at three different temperatures: −6, −12 and −18 °C. Next, experimental test results are used to perform the cohesive zone (CZ) modeling by using a two-dimensional finite element (FE) simulation. As a new approach, the CZ is modeled along a self-affine crack path, which allows performing a simulation closer to reality. The FE results provide a comprehensive understanding of the mechanism of crack initiation and propagation while keeping the computational time within a reasonable level.

Fractal Characteristics of Transverse Crack Propagation on CRTSII Type Track Slab

Objectives of this study are to examine influence of the surface crackles fractal characteristics of CRTSII track slab on its propagation velocity and stress intensity factor (SIF) and to improve the service performance of track slab. In this paper, fractal dimension D of transverse crack on the surface of CRTSII track slab was calculated by image processing, digital image technology, and box-counting method. Based on the fractal structure of transverse crack on track slab surface, a fractal transverse crack propagation model was established. The impact of fractal effect on the surface crack propagation velocity and SIF of CRTSII track slab was analyzed quantitatively. The fractal dimension D of transverse crack is 1.0652 on the surface of CRTSII track slab. The actual measured crack propagation velocity V0 on the surface of the track slab can be corrected as the true crack propagation velocity V. When the V0/Cr=0.899<1, the SIF K′L1.0652,t,V of transverse dynamic fractal crack propagation on the surface of track slab has been equal to 0, and the V0 is always lower than Rayleigh wave velocity. In addition, the true attenuation rate of SIF for CRTSII track slab surface transverse dynamic crack propagation is faster than the actual measured value.

Determination of the Dynamic Propagation Toughness of Marble Based on a Fractal Propagation Model and an Experimental–Numerical–Analytical Method

Precracked chevron-notched Brazilian disc specimens were impacted by a split Hopkinson pressure bar to determine the dynamic propagation toughness of marble. The dynamic expansion process of the crack was measured by a crack propagation gauge glued ahead of the specimen’s crack tip. A fractal crack propagation model was introduced to modify the universal function, and the dynamic propagation toughness of marble was calculated using an experimental–numerical–analytical method. The results show that, when the crack speed is in the range from 0.42 to 0.54 CR, both the fractal dimension and the dynamic propagation toughness increase with the crack propagation speed and with an approximately linear relationship.

FRACTAL CHARACTERISTICS OF CRACK PROPAGATION IN COAL UNDER IMPACT LOADING

This paper studied the fractal characteristics of crack propagation in coal containing beddings under impact loading condition. Split Hopkinson pressure bar (SHPB) system was applied to determine the prepared notched semi-circular bending specimens. The high-speed camera was used to record the propagation characteristics of cracks. The image processing method and fractal dimension calculation software are combined to further analyze the effects of bedding and loading rate on the fractal characteristics of crack propagation in coal. The experimental results presented that the presence of bedding has a remarkable impact on the crack propagation. The crack velocity of coal samples with [Formula: see text] bedding angle is of the maximum, however, the crack velocity of samples with [Formula: see text] bedding angle is of the minimum. Bedding angles also have obvious influence on the fractal dimension of cracking path, the bedding angle of [Formula: see text] being the largest, and those of [Formula: see text] and [Formula: see text] having the intermediate values, while those of [Formula: see text] and [Formula: see text] being the smallest. Several points of instantaneous fractal crack velocity are close to the Rayleigh wave velocity ([Formula: see text]). The crack velocities of coal specimens with bedding angles of [Formula: see text], [Formula: see text] and [Formula: see text] are prone to the high value.

Correlation between fragment shape and mass distributions in impact disruption

We propose that the shape of impact fragments reflects their fragmentation mechanisms; the fragmentation process that generates smaller fragments (fractal crack bifurcation) produces the shapes frequently observed in the previous studies, and those that generate larger fragments (spallation, random tessellation, and geometrical effects) produce flatter fragments. Fragment shape analyses derived from hypervelocity impact experiments in a variety of mass distribution ranges qualitatively support this view.

A cohesive zone model for self-similar fractal crack propagation

Cracks in the nature have been proven to be fractal by many experiments. Despite the fractal fracture mechanics has been developed by many researchers, fractal geometry still has few applications in analysis of engineering structures. One of the reasons is that previous studies are somehow inconvenient when apply, e.g. to the finite element method. This study proposes a Cohesive zone model (CZM) for self-similar fractal crack propagation of material interfaces. The determination of the CZM parameters and the simulation of fractal crack propagation are developed by replace the fractal crack with the equivalent smooth crack. The fractal dimension has effects on both the crack extension resistance and the max traction stress. As shown by the simulation of a DCB specimen, the fractal dimension also affect the ultimate load and the crack propagation process. It is shown that it is possible to predict the propagation of fractal cracks without considering geometric modeling of the crack topology by our cohesive zone model.

Dynamic cracking resistance of metallic materials during rapid propagation of a self-similar crack

A model is proposed to determine the dynamic cracking resistance KID of metals and alloys for the case of a rapidly moving fractal or self-affine crack. The values of this characteristic correlate with the fractal dimension Df of the future contour of a crack surface profile. KID is lower or higher than KIC depending on the fractal dimension.

The Design and Study of Fractal Graph Based on Visual C++

The effect of fractal theory describes roughness and irregular geometric shapes in the nature or in non-linear system. In recent years fractal theory is extensively applied to many fields, such as mathematics, physics, chemistry, geography, earthquake and astronomy, computer science. In this paper, Visual C++ language is used to simulate fractal trees, snowflakes and cracks because of good use interface and strong computer programming technology. The parameter of the fractal graphics, such as recursive degree, length and rotation angle, can be modified by using the parameter dialogue box in order to obtain various fractal graphs. This paper focuses on the algorithm study of fractal simulation graphs.