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1
AKIMOTO, Sho, Hiroshi ISODA, Masahiro MATSUDA, and Takahiro TSUCHIMOTO. "RESISTING PERFORMANCE AND FRACTUAL BEHAVIOR BY SILL EMBEDMENT DURING EARTHQUAKES." Journal of Structural and Construction Engineering (Transactions of AIJ) 81, no. 722 (2016): 747–55. http://dx.doi.org/10.3130/aijs.81.747.
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Hutton, D. M. "The (Mis)Behaviour of Markets: A Fractual View of Risk, Ruin and Reward20082Benoit B. Mandelbrot and Richard L. Hudson. The (Mis)Behaviour of Markets: A Fractual View of Risk, Ruin and Reward. Profile Books Ltd, 2006. 348 pp., ISBN: 1‐86‐197790‐5 Paperback £9.99." Kybernetes 37, no. 1 (February 15, 2008): 190. http://dx.doi.org/10.1108/03684920810851087.
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Moreles Vázquez, Uriel Octavio. "Los fractales." Acta Universitaria 13 (September 1, 2003): 19–22. http://dx.doi.org/10.15174/au.2003.249.
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Knowles, Asante James. "Fractal Philosophy: Grounding the Nature of the Mind with Fractals." NeuroQuantology 17, no. 8 (August 25, 2019): 19–23. http://dx.doi.org/10.14704/nq.2019.17.8.2799.
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Kertész, János. "Fractal fracture." Physica A: Statistical Mechanics and its Applications 191, no. 1-4 (December 1992): 208–12. http://dx.doi.org/10.1016/0378-4371(92)90529-y.
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COX, B. L., and J. S. Y. WANG. "FRACTAL ANALYSES OF ANISOTROPIC FRACTURE SURFACES." Fractals 01, no. 03 (September 1993): 547–59. http://dx.doi.org/10.1142/s0218348x93000575.
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Natural surfaces of rock fractures often have anisotropic asperity distributions, especially for shear fractures or faults. The asperity distributions could be treated as self-affine fractals with directional dependent scaling in the plane of the rock surfaces. Different fractal analyses (divider, slit-island, variogram) are applied to surface distributions of asperity data (topography): (1) a granitic fracture from the Stripa mine in Sweden; (2) a faulted and geothermally altered fracture from Dixie Valley, Nevada, USA. The cutoff patterns (indicator maps) of the granitic fracture show a radial pattern, while those of the faulted fracture show a very anisotropic stretched pattern of shapes. Different cutoff patterns of the same surface generally yield the same fractal dimension with the slit-island technique. The slit-island technique assumes that the cut-off patterns are self-similar in the plane of the surface, with the perimeter versus area analyzed for the entire population of contours, regardless of aspect ratio. We measure the variance in the two coordinate directions as a function of perimeter/area ratio for the anisotropic fracture from Dixie Valley to determine a self-affine scaling ratio for the slit-island analysis. We compare this ratio with anisotropy ratios obtained from simulated flow models based on channeling of flow through the largest openings. The possible applications of fractal analyses to both the geometry and flow are evaluated.
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Liu, Wei Fu, Shuang Long Liu, and Li Xin Sun. "Metamorphic Buried Hill Fracture Reservoir of Characterization and Evaluation." Applied Mechanics and Materials 522-524 (February 2014): 1303–6. http://dx.doi.org/10.4028/www.scientific.net/amm.522-524.1303.
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Paleostress value, fractural porosity and fractural fractal dimension could be used to demonstrate the basement rock reservoir of buried hill in oilfields. The paleostress value determined the development of the reservoir fractures. Fractured types are recognized and fractural porosity is calculated by using bilateral logging data. Then the fractural fractal dimension is calculated by using seismic data to predict the distribution of fractured zone. Finally based on paleostress value, fractural porosity and fractural fractal dimension, fractured reservoir of buried hill is evaluated by using fuzzy mathematics.
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Pavičić, Ivica, Ivan Dragičević, Tatjana Vlahović, and Tonći Grgasović. "FRACTAL ANALYSIS OF FRACTURE SYSTEMS IN UPPER TRIASSIC DOLOMITES IN ŽUMBERAK MOUNTAIN, CROATIA." Rudarsko-geološko-naftni zbornik 32, no. 3 (June 26, 2017): 1–13. http://dx.doi.org/10.17794/rgn.2017.3.1.
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Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (November 30, 2015): 24–37. http://dx.doi.org/10.12737/14417.
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Fractals are geometric objects, each part of which is
similar to the whole object, so that if we take a part and increase
its size to the size of the whole object, it would be impossible to
notice a difference. In other words, fractals are sets having scale
invariance. In mathematics, they are associated primarily with
non-differentiable functions. The concept of "fractal" (from the
Latin "Fractus" meaning «broken») had been introduced by Benoit
Mandelbrot (1924–2010), French and American mathematician,
physicist, and economist. Mandelbrot had found that seemingly
arbitrary fluctuations in price of goods have a certain tendency to
change: it turned out that daily fluctuations are symmetrical with
long-term price fluctuations. In fact, Benoit Mandelbrot applied
his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number
of fractal curves and flat objects have been created; and methods
for their application have been developed. From geometrical point
of view, the most interesting fractals are "Koch snowflake" and
"Pythagoras Tree". Two classes of analogues of the volumetric
fractals were created with modern three-dimensional modeling
program: "Fractals of growth” – like Pythagoras Tree, “Fractals of
separation” – like Koch snowflake; the primary classification was
developed, their properties were studied. Empiric data was processed
with basic arithmetic calculations as well as with computer software.
Among other things, for fractals of separation the task was to create
an object with an infinite surface area, which in the future might
acquire great importance for the development of the chemical and
other industries.
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Guo, Li Hui, and Wan Qiang Hu. "Fractal Research in Metal Material Fracture Invalidation." Advanced Materials Research 703 (June 2013): 8–11. http://dx.doi.org/10.4028/www.scientific.net/amr.703.8.
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Though there are a lot of rules and models for metal material fracture, it makes lots of troubles for engineering application, because of the lacking of reasonable description of fracture principle and change law. This article discusses some basic problems of applying fractal geometry to fracture research, mainly including sectional fractal characteristics and measurement, the relationship of fractal dimension and fracture toughness, and some kinds of metal fractal models. At last, application of fractal theory in metal fracture is stated.
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Sahu, Ajay K., and Ankur Roy. "Clustering, Connectivity and Flow Responses of Deterministic Fractal-Fracture Networks." Advances in Geosciences 54 (November 27, 2020): 149–56. http://dx.doi.org/10.5194/adgeo-54-149-2020.
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Abstract. It is well known that fracture networks display self-similarity in many cases and the connectivity and flow behavior of such networks are influenced by their respective fractal dimensions. In the past, the concept of lacunarity, a parameter that quantifies spatial clustering, has been implemented by one of the authors in order to demonstrate that a set of seven nested natural fracture maps belonging to a single fractal system, but of different visual appearances, have different clustering attributes. Any scale-dependency in the clustering of fractures will also likely have significant implications for flow processes that depend on fracture connectivity. It is therefore important to address the question as to whether the fractal dimension alone serves as a reasonable proxy for the connectivity of a fractal-fracture network and hence, its flow response or, if it is the lacunarity, a measure of scale-dependent clustering, that may be used instead. The present study attempts to address this issue by exploring possible relationships between the fractal dimension, lacunarity and connectivity of fractal-fracture networks. It also endeavors to study the relationship between lacunarity and fluid flow in such fractal-fracture networks. A set of deterministic fractal-fracture models generated at different iterations and, that have the same theoretical fractal dimension are used for this purpose. The results indicate that such deterministic synthetic fractal-fracture networks with the same theoretical fractal dimension have differences in their connectivity and that the latter is fairly correlated with lacunarity. Additionally, the flow simulation results imply that lacunarity influences flow patterns in fracture networks. Therefore, it may be concluded that at least in synthetic fractal-fracture networks, rather than fractal dimension, it is the lacunarity or scale-dependent clustering attribute that controls the connectivity and hence the flow behavior.
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Cai, Wantong, Guoping Cen, and Haifu Wang. "Fracture Surface Fractal Characteristics of Alkali-Slag Concrete under Freeze-Thaw Cycles." Advances in Materials Science and Engineering 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/1689893.
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Fractal theory is introduced in fracture surface research of alkali-slag concrete (ASC) under freeze-thaw cycles; crack distribution of ASC fracture surface and freeze-thaw damage zone were calculated. Through fractal analysis of ASC sample fracture surfaces, relevance between section fractal dimension and fracture toughness and relationship between material composition and section fractal dimension are clarified. Results show that the specimen’s cracks before freeze-thaw extend along force direction gently, and there are more twists and turns after freezing and thawing; the fractal dimension D also grows from 1.10 to 1.33. SEM internal microcracks’ D of ASC internal microstructure after freezing and thawing is 1.37; 0 to 300 times ASC fractal dimension under freezing and thawing is between 2.10 and 2.23; with freeze-thaw times increasing, ASC fracture toughness decreases and fractal dimension increases, the fractal dimension and fracture toughness have a good linear relationship, and the fractal dimension can reflect the toughening effect of ASC. It is very feasible to evaluate ASC fracture behaviour under freezing and thawing with the fractal theory. Fractal dimension generally increases with activator solution-slag (A/S for short) or slag content. The greater the amount of A/S or slag content, the lower the dimension.
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Fan, Cunhui, Qirong Qin, Dongfeng Hu, Xiaolei Wang, MengYue Zhu, Wei Huang, Yuxi Li, and Muhammad Aqeel Ashraf. "Fractal characteristics of reservoir structural fracture: a case study of Xujiahe Formation in central Yuanba area, Sichuan Basin." Earth Sciences Research Journal 22, no. 2 (April 1, 2018): 113–18. http://dx.doi.org/10.15446/esrj.v22n2.72250.
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The reservoir structural fractures have excellent fractal characteristics, as well as self-similarities. Based on the fractal theory, the surface fractal characteristics of faults and the fractal characteristic of fractures in the core of the Xujiahe Formation in the Fault-Fold Belt of the central Yuanba area were studied, and a quantitative relationship was set up between them. Based on the fractal characteristics of faults, predictions were made of the favorable fracture zone, which provides a new idea for the research of fracture, as well as offers theoretical references for exploring the fracture development zone during oil-gas exploration. The results show the following: the seismic value of reflection fault fractal dimension of the Xujiahe Formation is 1.5284; the correlation coefficient R2 is bigger than 0.9901; the capacity dimension linear regression correlation coefficient of the fracture in core of the Xujiahe Formation is bigger than 0.98; the fractal dimension D can well reflect the fault and fracture development degree, as well as the complexity of the fracture system; it can quantitatively calculate the density of the fracture of the reservoir in the area; the areas of capacity dimension bigger than 1.45 are the fracture development zones in the Fault-Fold Belt of the central Yuanba area; the oil and gas enrichment degree is high; the areas with the fractal dimension value between 0.95 and 1.45 are the fracture relatively-developed zones; the fractal dimension with values smaller than 0.95 are the lack of fracture areas.
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Rossal Aragón, Juan Enrique. "Caracterización clínica, epidemiológica y radiológica de pacientes con fractura de extremo distal de radio." Revista Ciencia Multidisciplinaria CUNORI 4, no. 2 (October 26, 2020): 22–27. http://dx.doi.org/10.36314/cunori.v4i2.124.
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La fractura de radio es la principal de las fracturas del antebrazo en los pacientes que consultan a los servicios de emergencia principalmente por accidentes o caídas. Se tuvo como objetivo caracterizar clínica, epidemiológica y radiológicamente a los pacientes con fractura de extremo distal de radio ingresados en el servicio de traumatología del Hospital Nacional de Chiquimula siendo un estudio descriptivo retrospectivo en donde se revisaron expedientes de 246 pacientes que fueron ingresados durante el periodo de enero de 2015 a diciembre de 2019 en el servicio de traumatología del hospital Nacional de Chiquimula. Se determinó que la edad de 23-32 años predominó en un 35%. El sexo masculino muestra diferencia con un 73% en comparación con el sexo femenino. 75% de los casos no tuvo ningún antecedente de fractura previa. Se observó que la mayoría de casos provienen de zona urbana con un 65%. Un 75% se fracturó de lunes a viernes con un 46% en horario de noche. La causa principal por la que ocurren las fracturas fue accidentes en motocicleta en un 44%. Así mismo, el miembro superior derecho fue el más afectado en un 86% y el tratamiento brindado fue quirúrgico en un 53%. Se determinó que la situación de la fractura fue de tipo alineada con el 76%. De las fracturas completas, la fractura transversal fue más frecuente en un 35% y de la clasificación de Fernández, la fractura tipo I con un 47% del total de casos estudiados fue la más observada.
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Kononov, Dmitry, Svetlana Gubenko, Igor Ivanov, and Sergey Urushev. "Using fractal characteristics to analyze the development of whole-rolled wheel destruction." MATEC Web of Conferences 329 (2020): 02009. http://dx.doi.org/10.1051/matecconf/202032902009.
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Purpose: Determination of mechanical characteristics of wheel steel by analyzing the development of destruction of solid-rolled wheels. Methods: Determination of the crack resistance of all-rolled wheels by testing samples from wheel steel for off-center tension with an edge crack, fractal parametrization of the fracture relief by processing the fracture photo using the R/S analysis method for fracture sections, which allows determining both the total fractal dimension of the fracture (the average for all sections), and in a specific area where there are changes in the fracture morphology. Construction of the dependence of the crack resistance parameter on the fractal dimension. Results: The possibility of analyzing wheel steel fractures by studying their fractal dimension is determined. A method of fractal parametrization of the wheel steel fracture relief is developed, which allows to obtain the value of the fractal dimension in different directions of crack growth. The fractal dimension of wheel steel is determined for different hardness values (after annealing and after thermal improvement). The dependence of the fractal dimension on the ductility characteristics of wheel steel is revealed: the lower the ductility characteristics (elongation, relative contraction) and, accordingly, the higher the hardness, the lower the value of the fractal dimension. The dependence of the fractal dimension on the coefficient of crack resistance of wheel steel KIC is determined, which makes it possible to predict the crack resistance of wheel steel depending on the fractal dimension of the fracture, as well as to judge the nature of the fracture – brittle, viscous or fatigue. Practical significance: The use of fractal fracture parameterization avoids time-consuming tests when determining the mechanical characteristics of wheel steel, as well as to determine the mechanical properties in the case when other methods are no longer applicable (small sample sizes).
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XU, PENG, CUIHONG LI, SHUXIA QIU, and AGUS PULUNG SASMITO. "A FRACTAL NETWORK MODEL FOR FRACTURED POROUS MEDIA." Fractals 24, no. 02 (June 2016): 1650018. http://dx.doi.org/10.1142/s0218348x16500183.
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The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.
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Xie, Yan Shi, Kai Xuan Tan, Liang Chen, Wei Huang, Kai Guang Hu, and Xiu Cai Wang. "Fractal Character of Structural Control on Uranium Mineralization in South China." Applied Mechanics and Materials 229-231 (November 2012): 2597–600. http://dx.doi.org/10.4028/www.scientific.net/amm.229-231.2597.
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South China is the most important uranium producer in the country. Most uranium deposits in south China are strictly controlled by NE-NNE trending regional fracture structure. Fractal analyses on spatial distribution of uranium deposits and regional fracture structure in south China have been done in this paper. It indicates that the spatial distribution of both uranium deposits and regional fracture structure in south China show fractal character. The fractal dimension D=1.4142 for the spatial distribution of regional fracture structure is significantly greater than the critical value and it indicate a higher ripening degree in the fracture structure evolution and an advantages to fluid flow and uranium mineralization. The fractal dimension D=1.0527 for the spatial distribution of uranium deposits in south China show a lower complexity than regional fracture structure. The fractal spatial distribution of uranium deposits in south China is the result of the evolution of the fractal fracture structure system.
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IKESHOJI, TOSHITAKA, and TADASHI SHIOYA. "BRITTLE-DUCTILE TRANSITION AND SCALE DEPENDENCE: FRACTAL DIMENSION OF FRACTURE SURFACE OF MATERIALS." Fractals 07, no. 02 (June 1999): 159–68. http://dx.doi.org/10.1142/s0218348x99000189.
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The fractal dimension of fracture surfaces obtained within brittle-ductile transition regime is evaluated at various observation scales. Fracture surfaces are generated by the tensile fracture test. The brittle-ductile transition is accomplished by using the round-notched bar specimens with various notch radii, which cause the variation in stress triaxiality. The specimens are manufactured from mild steel, steel and cast-iron bar. The fracture model is identified according to the observation through scanning electron micrographs. The fractal dimension for ductile fracture surfaces is almost constant despite variations in observing scale and changes in stress triaxiality. Meanwhile, the fractal dimension on brittle fracture surfaces shows the different values for macroscopic and microscopic observing scales. This transition-like scale dependence of fractal dimension for brittle fracture surfaces is considered to reflect such a characteristic of the fracture i.e. its specific length in microscopic fracture mechanism. The existence of transition in fractal dimension with observing scale is considered to be an index, used to distinguish the ductile fracture surface from the brittle fracture one.
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LV, WEIFENG, GUOLIANG YAN, YONGDONG LIU, XUEFENG LIU, DONGXING DU, and RONG WANG. "EFFECT OF FRACTAL FRACTURES ON PERMEABILITY IN THREE-DIMENSIONAL DIGITAL ROCKS." Fractals 27, no. 01 (February 2019): 1940015. http://dx.doi.org/10.1142/s0218348x19400152.
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The fracture has great impact on the flow behavior in fractured reservoirs. Fracture traces are usually self-similar and scale-independent, which makes the fractal theory become a powerful tool to characterize fracture. To obtain three-dimensional (3D) digital rocks reflecting the properties of fractured reservoirs, we first generate discrete fracture networks by stochastic modeling based on the fractal theory. These fracture networks are then added to the existing digital rocks of rock matrixes. We combine two low-permeable cores as rock matrixes with a group of discrete fracture networks with fractal characteristics. Various types of fractured digital rocks are obtained by adjusting different fracture parameters. Pore network models are extracted from the 3D fractured digital rock. Then the permeability is predicted by Darcy law to investigate the impacts of fracture properties to the absolute permeability. The permeability of fractured rock is subject to exponential increases with fracture aperture. The relationship between the permeability and the fractal dimension of fracture centers is exponential, as well as the relationship between permeability and the fractal dimension of fracture lengths.
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Murillo, Bernardo, Christian Antonio Allende Nores, and Orlando Rodríguez. "Incidencia de diagnóstico y tratamiento de la osteoporosis en pacientes con fractura de radio distal. [Diagnosis and treatment incidence of osteoporosis in patients with distal radius fractures]." Revista de la Asociación Argentina de Ortopedia y Traumatología 84, no. 2 (May 2, 2019): 99–104. http://dx.doi.org/10.15417/issn.1852-7434.2019.84.2.664.
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Introducción: Las fracturas de radio distal en pacientes mayores son un indicador de osteoporosis. El objetivo de este estudio fue determinar el diagnóstico y el tratamiento de la osteoporosis en pacientes con fractura de radio distal, operados en nuestra institución, entre 2012 y 2014.Materiales y Métodos: Los datos se obtuvieron de entrevistas telefónicas a 41 pacientes mayores, operados por fracturas de radio distal. Las variables evaluadas fueron: sexo, edad, enfermedades asociadas, tabaquismo, fracturas previas, tratamiento antiosteoporótico previo o posterior a la fractura en cuestión, especialidad del médico que solicitó el tratamiento y realización o no de densitometría mineral ósea.Resultados: La muestra incluyó a 41 pacientes (32 mujeres). Veintiséis contaban con una densitometría antes de la fractura (15 con osteoporosis), 11 habían sufrido una fractura por osteoporosis, y sólo 7 recibían tratamiento antiosteoporótico cuando ocurrió la fractura. Luego de la cirugía, solo 4 de ellos continuó con el tratamiento. Se encontró cierta asociación entre una densitometría patológica y la presencia de diabetes tipo 2, no así con otras comorbilidades. La tasa de incidencia anual de osteoporosis, calculada entre todos los pacientes que se atendieron en nuestra institución, en 2014, fue alrededor del 1%. Los traumatólogos solicitaron el 1,5% de todas las densitometrías prescritas dicho año.Conclusiones: Este estudio sugiere que los traumatólogos que se desempeñan en nuestra institución tienen nula o poca participación en la prevención secundaria de la osteoporosis; por esta razón, se consideraría necesario un protocolo de prevención de fracturas secundarias a la osteoporosis. Abstract Introduction: Distal radius fractures in elderly patients are an indicator of osteoporosis. The aim of this study was to determine osteoporosis diagnosis and treatment in patients with distal radius fractures treated surgically at our institution between 2012 and 2014.Methods:Information of 41 patients who had surgical intervention for distal radius fracture was obtained through telephones interviews. Several variables evaluated: age, sex, smoking, associated pathologies, previous fractures, preoperative and postoperative anti-osteoporotic treatments, specialty of the physicians that indicated antiosteoporotic treatment, and bone mineral density (BMD) studies performed.Results: The study included 41 patients (32 female).Twenty-six had a BMD performed before the fracture (15 evidenced osteoporosis), 11 had had previous fractures secondary to osteoporosis. Only 7patients were under anti-osteoporotic treatment to the moment of the fracture. After surgery, only 4 of the patients continued with the treatment. Pathological BMD had certain degree of associationwith the presence of Diabetes (type 2), but not with other comorbidities. The annual incidence rate of osteoporosis, calculated using all patients attended at our institution in 2014, was about 1%. Orthopedic surgeons indicated only 1.5% of the total number of BMDs prescribed that year.Conclusion: Our study suggests that there is poor prevention by orthopedic surgeons of secondary osteoporotic fractures, which is why a national prevention protocol for fractures secondary to osteoporosis would be considered necessary.
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CAI, JIANCHAO, WEI WEI, XIANGYUN HU, RICHENG LIU, and JINJIE WANG. "FRACTAL CHARACTERIZATION OF DYNAMIC FRACTURE NETWORK EXTENSION IN POROUS MEDIA." Fractals 25, no. 02 (April 2017): 1750023. http://dx.doi.org/10.1142/s0218348x17500232.
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Fracture network and fractured porous media as well as their transport properties have received great attentions in many fields from engineering application and basic theoretical researches. Fracture will dynamically extend in length and aperture to form complex fracture network under some external conditions such as percussion drilling, wave propagation, desiccation and hydrofracturing. The complexity of fracture network can be well quantitatively characterized by fractal dimension. In this work, the dynamic characterization of fracture network extension in porous media under drying process is measured by the improved box-counting technique, and fractal dimensions of fracture network are respectively related to drying time, average aperture, moisture content and fracture porosity. The fractal dimension increases exponentially with drying time and average aperture, and decreases with moisture content in the form of power law. Specially, the fractal dimension is approximatively increased with porosity in the form of linearity in a narrow porosity range. The transport capacity of fracture network, described by seepage coefficient, is also related to the fractal dimension with drying time in the form of exponential function. The presented fractal analysis of fracture network could also shed light on the hydrofracturing application in subsurface unconventional oil and gas reservoirs.
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Sheng, Guanglong, Farzam Javadpour, Yuliang Su, Jinghua Liu, Kunjie Li, and Wendong Wang. "A Semianalytic Solution for Temporal Pressure and Production Rate in a Shale Reservoir With Nonuniform Distribution of Induced Fractures." SPE Journal 24, no. 04 (April 22, 2019): 1856–83. http://dx.doi.org/10.2118/195576-pa.
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Summary The network of induced fractures and their properties control pressure propagation and fluid flow in hydraulically fractured shale reservoirs. We present a novel fully fractal model in which both the spacing and the porosity/permeability of induced fractures are distributed according to fractal dimensions (i.e., fractal decay of fracture density and the associated porosity/permeability away from the main induced fracture). The fractal fracture distribution is general, and handles exponential, linear, power, and uniform distributions. We also developed a new fully fractal diffusivity equation (FDE) using the fractal distribution of fractures and their properties. We then used, for the first time, the semianalytic Bessel spline scheme to solve the developed diffusivity equation. Our proposed model is general and can capture any form of induced-fracture distribution for better analysis of pressure response and production rates at transient- and pseudosteady-state conditions. We compared the unsteady-state and pseudosteady-state pressure responses calculated by our fully fractal model with former models of limited cases: uniform fracture spacing and uniform porosity/permeability [conventional diffusivity equation (CDE)]; variable fracture spacing and uniform porosity/permeability [modified CDE (MCDE)]; and uniform fracture spacing and fractal porosity/permeability distribution (FPPD). We used these models to match and predict the production data of a multifractured horizontal gas well in the Barnett Shale. Our results showed that the fractal distribution of fracture networks and their associated properties better matches the field data. Uniform distribution of induced-fracture networks underestimates production rate, especially at early time.
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Liu, Xiaoli, Tao Liang, Sijing Wang, and Kumar Nawnit. "A Fractal Model for Characterizing Hydraulic Properties of Fractured Rock Mass under Mining Influence." Geofluids 2019 (December 20, 2019): 1–17. http://dx.doi.org/10.1155/2019/8391803.
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In this paper, two basic assumptions are introduced: (1) The number and length distribution of fractures in fractured rock mass are in accordance with the fractal law. (2) Fluid seepage in the fractures satisfies the cubic law. Based on these two assumptions, the fractal model of parallel seepage and radial seepage in fractured rock mass is established, and the seepage tensor of fracture network which reflects the geometric characteristics and fractal characteristics of fracture network under two kinds of seepage is derived. The influence of fracture geometry and fractal characteristics on permeability is analyzed, and the validity and accuracy of the model are verified by comparing the calculated results of the theoretical model and physical model test. The results show that the permeability coefficient K of fracture network is a function of the geometric (maximum crack length Lmax, fractured horizontal projection length L0, diameter calculation section porosity Φ, fracture strike α, and fracture angle θ) and fractal characteristics (fracture network fractal dimension Df and seepage flow fractal dimension DT). With the increase of fractal dimension Df, the permeability coefficient increases. With the increase of DT, the permeability coefficient decreases rapidly. And the larger the Df (Df>1.5), the greater the change of permeability coefficient K with DT.
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Sander, Evelyn, Leonard M. Sander, and Robert M. Ziff. "Fractals and Fractal Correlations." Computers in Physics 8, no. 4 (1994): 420. http://dx.doi.org/10.1063/1.168501.
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Cahn, Robert W. "Fractal dimension and fracture." Nature 338, no. 6212 (March 1989): 201–2. http://dx.doi.org/10.1038/338201a0.
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Wnuk, Michael P., and Arash Yavari. "Discrete fractal fracture mechanics." Engineering Fracture Mechanics 75, no. 5 (March 2008): 1127–42. http://dx.doi.org/10.1016/j.engfracmech.2007.04.020.
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Chelidze, T., and Y. Gueguen. "Evidence of fractal fracture." International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 27, no. 3 (June 1990): 223–25. http://dx.doi.org/10.1016/0148-9062(90)94332-n.
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Liu, Guannan, Dayu Ye, Feng Gao, and Jishan Liu. "A Dual Fractal Poroelastic Model for Characterizing Fluid Flow in Fractured Coal Masses." Geofluids 2020 (June 22, 2020): 1–13. http://dx.doi.org/10.1155/2020/2787903.
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In the process of coalbed methane exploitation, the fracture and pore structure is the key problem that affects the permeability of coalbed. At present, the coupling effect of fracture and pore structure and in situ stress is seldom considered in the study of coal seam permeability. In this paper, the fractal seepage model is coupled with coal deformation, and the adsorption expansion effect is considered. A multifield coupling model considering the influence of matrix and fracture structure is established. Then, the influence of pore structure parameters of main fracture on macropermeability is analyzed, including (1) fractal dimension of fracture length, (2) maximum fracture length, (3) fractal dimension of throat diameter, and (4) fractal dimension of throat bending. At the same time, the simulation results are compared with the results of Darcy’s uniform permeability model. The results show that the permeability calculated by the proposed model is significantly different from that calculated by the traditional cubic model. Under the action of in situ stress, when the porosity and other parameters remain unchanged, the macropermeability of coal is in direct proportion to the fractal dimension of coal fracture length, the fractal dimension of throat diameter, and the maximum fracture length and in inverse proportion to the fractal dimension of coal throat curvature.
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Zhang, Z. Z. "Fractal Dimension of Fracture Surface in Rock Material after High Temperature." Advances in Materials Science and Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/468370.
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Experiments on granite specimens after different high temperature under uniaxial compression were conducted and the fracture surfaces were observed by scanning electron microscope (SEM). The fractal dimensions of the fracture surfaces with increasing temperature were calculated, respectively. The fractal dimension of fracture surface is between 1.44 and 1.63. Its value approximately goes up exponentially with the increase of temperature. There is a quadratic polynomial relationship between the rockburst tendency and fractal dimension of fracture surface; namely, a fractal dimension threshold can be obtained. Below the threshold value, a positive correlativity shows between rockburst tendency and fractal dimension; when the fractal dimension is greater than the threshold value, it shows an inverse correlativity.
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XU, PENG, HAICHENG LIU, AGUS PULUNG SASMITO, SHUXIA QIU, and CUIHONG LI. "EFFECTIVE PERMEABILITY OF FRACTURED POROUS MEDIA WITH FRACTAL DUAL-POROSITY MODEL." Fractals 25, no. 04 (July 25, 2017): 1740014. http://dx.doi.org/10.1142/s0218348x1740014x.
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As natural fractures show statistically fractal scaling laws, fractal geometry has been proposed and applied to model the fracture geometry and to study the hydraulic properties of fractured porous media. In this paper, a fractal dual-porosity model is developed to study the single-phase fluid flow through fractured porous media. An analytical expression for effective permeability of fractured porous media is derived, which depends on the fractal dimension and fracture aperture. The effect of fractal dimensions for fracture aperture distribution and tortuosity, the ratio of minimum to maximum fracture apertures and fracture fraction on the effective permeability have been discussed. In addition, a power law relationship between the effective permeability and fracture fraction is proposed to predict the equivalent hydraulic properties of fractured porous media. Compared with empirical formulas for effective permeability, the present fractal dual-porosity model can capture the statistical characteristics of fractures and shed light on the transport mechanism of fractured porous media.
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Wen, Hua. "Quantitative Identification of Reservoir Fracture with the Variable Scale Fractal Technique in Su53 Gas Field, Ordos Basin." Advanced Materials Research 1010-1012 (August 2014): 1723–26. http://dx.doi.org/10.4028/www.scientific.net/amr.1010-1012.1723.
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Typical tight gas reservoirs in the Su53 Gas Field, Ordos Basin, are the demonstration region for deploying horizontal wells. In order to provide foundation for evaluating the volume fracturing adaptability of horizontal wells, through using the amplitude difference data between deep investigate double lateral resistivity (Rd) and shallow investigate double lateral resistivity (Rs), and other conventional logging data, in combination with the response characteristics of fracture in the logging curve, reservoir fracture was quantitatively identified with the variable scale fractal technique, the vertical heterogeneity of fracture distribution was analyzed, and the relationship between fractal dimension value of fracture and initial deliverability of gas well was researched. The results indicate that, this method is applied to quantitatively identify and evaluate the fracture development degree of single-well is feasible; the relationship between fractal dimension value and fracture development degree is positively relative, fractal dimension value is bigger, the fracture is more developed; there is a good corresponding relationship between fractal dimension value and deliverability of gas well, fractal dimension value decreases with a decrease of deliverability.
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Tanaka, M., Y. Kimura, A. Kayama, L. Chouanine, Reiko Kato, and J. Taguchi. "Image Reconstruction and Analysis of Three-Dimensional Fracture Surfaces Based on the Stereo Matching Method." Key Engineering Materials 261-263 (April 2004): 1593–98. http://dx.doi.org/10.4028/www.scientific.net/kem.261-263.1593.
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A computer program of the fractal analysis by the box-counting method was developed for the estimation of the fractal dimension of the three-dimensional fracture surface reconstructed by the stereo matching method. The image reconstruction and fractal analysis were then made on the fracture surfaces of materials created by different mechanisms. There was a correlation between the fractal dimension of the three-dimensional fracture surface and the fractal dimensions evaluated by other methods on ceramics and metals. The effects of microstructures on the fractal dimension were also experimentally discussed.
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Han, Juhong, Dingcheng Huang, Jingyu Chen, and Xiaofang Lan. "Experiment Study and Finite Element Analysis of the Coupling Effect of Steel Fiber Length and Coarse Aggregate Maximum Size on the Fracture Properties of Concrete." Crystals 11, no. 8 (July 22, 2021): 850. http://dx.doi.org/10.3390/cryst11080850.
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The effects of steel fiber length (lf = 30 mm, 40 mm, 50 mm and 60 mm) and coarse aggregate maximum size (Dmax = 10 mm, 20 mm, 30 mm and 40 mm) on fractural properties of steel fiber reinforced concrete (SFRC) was investigated. The results show that the fracture energy (Gf) of SFRC reaches its maximum when Dmax increases to 30 mm, and it increases first and then decreases as lf increases, but it still has a significant increase compared to the control concrete. The Gf ratio increases first and then decreases as the lf/Dmax increases. The Gf of the SFRC fracture surface follows the same trend as the fractal dimension. The rational range of the lf/Dmax is 2.5–4 for the considerable strengthening effect of steel fiber on fracture performances of concrete with the Dmax of 10 mm and 20 mm and 1.5–2.33 for that concrete with the Dmax of 30 mm and 40 mm. The finite element analysis results are compared with the experimental results, and the results show that the fracture process of the finite element model is consistent with the experiment.
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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (August 30, 2020): 961. http://dx.doi.org/10.3390/e22090961.
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The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.
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Chen, Jing, and Lin Mao. "Multi-Fractal Spectrum and their Applications in Metal Fracture Surface Images Feature Extraction." Applied Mechanics and Materials 536-537 (April 2014): 241–44. http://dx.doi.org/10.4028/www.scientific.net/amm.536-537.241.
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Based on Bandelet transform and Multi-fractal, the metal fracture surface image denoising is described in the paper. Using multi-fractal analysis, eight multi-fractal spectrum values which are used as shape characteristic parameter of metal fracture surface are extracted. Experimental results showed that different metal fracture surfaces have different shape characteristics and the same kind of metal fracture surfaces have similar shape characteristics. So the shape characteristic parameter of metal fracture surface can be used to study the fracture surface recognition and as the sample of failure analysis.
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Hu, Bowen, Jianguo Wang, and Zhanguo Ma. "A Fractal Discrete Fracture Network Based Model for Gas Production from Fractured Shale Reservoirs." Energies 13, no. 7 (April 10, 2020): 1857. http://dx.doi.org/10.3390/en13071857.
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A fractal discrete fracture network based model was proposed for the gas production prediction from a fractured shale reservoir. Firstly, this model was established based on the fractal distribution of fracture length and a fractal permeability model of shale matrix which coupled the multiple flow mechanisms of slip flow, Knudsen diffusion, surface diffusion, and multilayer adsorption. Then, a numerical model was formulated with the governing equations of gas transport in both a shale matrix and fracture network system and the deformation equation of the fractured shale reservoir. Thirdly, this numerical model was solved within the platform of COMSOL Multiphysics (a finite element software) and verified through three fractal discrete fracture networks and the field data of gas production from two shale wells. Finally, the sensitivity analysis was conducted on fracture length fractal dimension, pore size distribution, and fracture permeability. This study found that cumulative gas production increases up to 113% when the fracture fractal length dimension increases from 1.5 to the critical value of 1.7. The gas production rate declines more rapidly for a larger fractal dimension (up to 1.7). Wider distribution of pore sizes (either bigger maximum pore size or smaller minimum pore size or both) can increase the matrix permeability and is beneficial to cumulative gas production. A linear relationship is observed between the fracture permeability and the cumulative gas production. Thus, the fracture permeability can significantly impact shale gas production.
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Cossio, M., G. J. J. Moridis, and T. A. A. Blasingame. "A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory." SPE Journal 18, no. 01 (January 24, 2013): 83–96. http://dx.doi.org/10.2118/153715-pa.
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Summary The exploitation of unconventional reservoirs complements the practice of hydraulic fracturing, and with an ever-increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture, and the problem has been approached from different directions in the past 3 decades—starting with the work of Gringarten et al. (1974) for an infinite-conductivity case, followed by contributions from Cinco-Ley et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991), and Blasingame and Poe (1993) for a finite-conductivity case. This topic remains an active area of research and, for the more-complicated physical scenarios such as multiple transverse fractures in ultratight reservoirs, answers are currently being sought. Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure-transient testing, although with an emphasis on the effects of natural fractures in pressure/rate behavior. In this paper, we begin by performing a rigorous analytical and numerical study of the fractal diffusivity equation (FDE), and we show that it is more fundamental than the classic linear and radial diffusivity equations. Thus, we combine the FDE with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semianalytic solution for flow in a finite-conductivity vertical fracture that we name the “fractal-fracture solution (FFS).” This new solution is instantaneous and comparable in accuracy with the Blasingame and Poe solution (1993). In addition, this is the first time that fractal theory is used in fluid flow in a porous medium to address a problem not related to reservoir heterogeneity. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is flexible, and we believe that the same methodology could be extended to different applications. One objective of this work is to develop a fast and accurate semianalytical solution for flow in a single vertical fracture that fully penetrates a homogeneous infinite-acting reservoir. This would be the first time that fractal theory is used to study a problem that is not related to naturally fractured reservoirs or reservoir heterogeneity. In addition, as part of the development process, we revisit the fundamentals of fractals in reservoir engineering and show that the underlying FDE possesses some interesting qualities that have not yet been comprehensively addressed in the literature.
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Shariff, Asma A., and M. Hadi Hafezi. "A Review on Fractals and Fracture, Part I: Calculating Fractal Dimensions by CAD Model." Applied Mechanics and Materials 148-149 (December 2011): 818–21. http://dx.doi.org/10.4028/www.scientific.net/amm.148-149.818.
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The objective of this paper is to consider the use of fractal geometry as a tool for the study of non-smooth and discontinuous objects for which Euclidean coordinate is not able to fully describe their shapes. We categorized the methods for computing fractal dimension with a discussion into that. We guide readers up to the point they can dig into the literature, but with more advanced methods that researchers are developing. Considerations show that is necessary to understand the numerous theoretical and experimental results concerning searching of the conformality before evaluating the fractal dimension to our own objects. We suggested examining a cloud of points of growth of fracture surface at laboratory using CATIA - Digitized Shape Editor software in order to reconstruct the surface (CAD model). Then, the author carried out measurement/calculation of more accurate fractal dimension which are introduced by [1] in the other paper as Part II.
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Ren, L., L. Z. Xie, C. B. Li, and J. Wang. "Compressive Fracture of Brittle Geomaterial: Fractal Features of Compression-Induced Fracture Surfaces and Failure Mechanism." Advances in Materials Science and Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/814504.
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Compressive fracture is one of the most common failure patterns in geotechnical engineering. For better understanding of the local failure mechanism of compressive fractures of brittle geomaterials, three compressive fracture tests were conducted on sandstone. Edge cracked semicircular bend specimens were used and, consequently, fresh and unfilled compressive fracture surfaces were obtained. A laser profilometer was employed to measure the topography of each rough fracture surface, followed by fractal analysis of the irregularities of the obtained compression-induced fracture surfaces using the cubic cover method. To carry out a contrastive analysis with the results of compressive fracture tests, three tension mode fracture tests were also conducted and the fractal features of the obtained fracture surfaces were determined. The obtained average result of the fractal dimensions of the compression-induced surfaces was 2.070, whereas the average result was 2.067 for the tension-induced fracture surfaces. No remarkable differences between the fractal dimensions of the compression-induced and tension-induced fracture surfaces may indicate that compressive fracture may occur, at least on the investigative scale of this work, in a similar manner to tension fracture.
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Wang, Xiaolin, Liyuan Yu, and Hanqing Yang. "Correlations between Geometric Properties and Permeability of 2D Fracture Networks." Advances in Civil Engineering 2021 (January 25, 2021): 1–7. http://dx.doi.org/10.1155/2021/6645238.
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The equivalent permeability of fractured rock masses plays an important role in understanding the fluid flow and solute transport properties in underground engineering, yet the effective predictive models have not been proposed. This study established mathematical expressions to link permeability of 2D fracture networks to the geometric properties of fractured rock masses, including number density of fracture lines, total length of fractures per square meter, and fractal dimensions of fracture network structures and intersections. The results show that the equivalent permeability has power law relationships with the geometric properties of fracture networks. The fractal dimensions that can be easily obtained from an engineering site can be used to predict the permeability of a rock fracture network. When the fractal dimensions of fracture network structures and intersections exceed the critical values, the effect of randomness of fracture locations is negligible. The equivalent permeability of a fracture network increases with the increment of fracture density and/or fractal dimensions proportionally.
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Qiang, Tao, and De Mei Yu. "Correlation between Fractal Dimension and Impact Strength for Wood Plastic Composites." Advanced Materials Research 411 (November 2011): 548–51. http://dx.doi.org/10.4028/www.scientific.net/amr.411.548.
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Polylactide (PLA)-based wood plastic composites (WPCs) were manufactured by extrusion blending followed by injection molding. The fracture surfaces created from the impact test were recorded with SEM. Fractal analysis has been used to calculate the fractal dimension of the fracture surfaces with four different fractal analysis techniques. Then, the correlation between the fractal dimension of the fracture surfaces and its impact strength of the PLA-based WPCs was investigated by the linear regression. The results showed that there is a positive correlation between the impact strength and the fractal dimension.
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LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (December 1995): 725–36. http://dx.doi.org/10.1142/s0218348x95000643.
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We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).
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Cherny, A. Yu, E. M. Anitas, V. A. Osipov, and A. I. Kuklin. "Scattering from surface fractals in terms of composing mass fractals." Journal of Applied Crystallography 50, no. 3 (June 1, 2017): 919–31. http://dx.doi.org/10.1107/s1600576717005696.
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It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.
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Charkaluk, E., M. Bigerelle, and A. Iost. "Fractals and fracture." Engineering Fracture Mechanics 61, no. 1 (August 1998): 119–39. http://dx.doi.org/10.1016/s0013-7944(98)00035-6.
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Shi, Rui Xiang, Chi Ai, Wan Chun Zhao, and Xiao Han Feng. "The Damage Mechanics Model of the Wellbore Surrounding Rock Based on Fractal Theory." Applied Mechanics and Materials 423-426 (September 2013): 1623–26. http://dx.doi.org/10.4028/www.scientific.net/amm.423-426.1623.
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In order to describe the mechanical characteristics of CBM wellbore surrounding rock more accurately, the article establishes the CBM wellbore surrounding rock mechanics model based on the fractal theory. According to the pores of CBM surrounding rock and the characteristics of mass fractal, people find the macro fracture and micro fracture damage characteristics and build the relationship of any scale coal strength, damage variable and fractal dimension. According to the numerical calculation method of the surrounding rock stress, people find the surrounding rock stress calculation method and the damage distribution model of the pore fractal model and fracture fractal model.
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Gong, Lei, Xiaofei Fu, Shuai Gao, Peiqiang Zhao, Qingyong Luo, Lianbo Zeng, Wenting Yue, Benjian Zhang, and Bo Liu. "Characterization and Prediction of Complex Natural Fractures in the Tight Conglomerate Reservoirs: A Fractal Method." Energies 11, no. 9 (September 2, 2018): 2311. http://dx.doi.org/10.3390/en11092311.
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Using the conventional fracture parameters is difficult to characterize and predict the complex natural fractures in the tight conglomerate reservoirs. In order to quantify the fracture behaviors, a fractal method was presented in this work. Firstly, the characteristics of fractures were depicted, then the fracture fractal dimensions were calculated using the box-counting method, and finally the geological significance of the fractal method was discussed. Three types of fractures were identified, including intra-gravel fractures, gravel edge fractures and trans-gravel fractures. The calculations show that the fracture fractal dimensions distribute between 1.20 and 1.50 with correlation coefficients being above 0.98. The fracture fractal dimension has exponential correlation with the fracture areal density, porosity and permeability and can therefore be used to quantify the fracture intensity. The apertures of micro-fractures are distributed between 10 μm and 100 μm, while the apertures of macro-fractures are distributed between 50 μm and 200 μm. The areal densities of fractures are distributed between 20.0 m·m−2 and 50.0 m·m−2, with an average of 31.42 m·m−2. The cumulative frequency distribution of both fracture apertures and areal densities follow power law distribution. The fracture parameters at different scales can be predicted by extrapolating these power law distributions.
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Hilders, O. A., A. Quintero, L. Berrio, R. Caballero, L. Sáenz, N. Peña, and M. Ramos. "Effects of Aged Conditions on the Fracture Surface Fractal Dimension and Mechanical Behavior of an Austenitic Stainless Steel." Microscopy and Microanalysis 6, S2 (August 2000): 768–69. http://dx.doi.org/10.1017/s1431927600036333.
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There have been several attempts to find a relation between the fractal morphology of the fracture surfaces and the mechanical properties of engineering materials., although the current resuls are inconclusive. If there are correlations between the fractal dimension and such properties, this parameter could be very useful to predict them and to improve the resistance to fracture. The main part of the investigations concerned with the fractal geometry and fracture behavior concentrate on the relations between roughness and fracture toughness . In the present work, the effects of thermal aging at 850°C on the fracture topography developed during the rupture in tension at room temperature of a 304 type stainless steel and their relation with the strength and ductility, were studied using the fractal geometry approach.
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TANAKA, MANABU, ATSUSHI KAYAMA, RYUICHI KATO, and YOSHIAKI ITO. "ESTIMATION OF THE FRACTAL DIMENSION OF FRACTURE SURFACE PATTERNS BY BOX-COUNTING METHOD." Fractals 07, no. 03 (September 1999): 335–40. http://dx.doi.org/10.1142/s0218348x99000335.
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In the box-counting method, positioning of images do not significantly affect the estimation of the fractal dimension of river pattern on the brittle fracture surface, and that of dimple pattern on the ductile fracture surface of materials. A reasonable estimation of the fractal dimension can be made using the box-counting method by a single measurement on the fracture surface pattern. The fractal dimension of dimple pattern in pure Zn polycrystals (about 1.50) is larger than that of river pattern in soda-lime glass (about 1.30). Personal difference in image processing does not have a large influence on the estimation of the fractal dimension of grain-boundary fracture surface profile, compared with the effects of local variation in fracture pattern concerning image size.
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Secrieru, Claudia, and Ion Dumitru. "Fractal Analysis of Fracture Surfaces of Steel Charpy Specimens." Key Engineering Materials 399 (October 2008): 43–49. http://dx.doi.org/10.4028/www.scientific.net/kem.399.43.
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The article focuses on the technical measurements which could be applied to the fracture surfaces of the steel Charpy specimens in order to apply the Fractal Analysis. One could calculate the fractal dimension not directly for a fracture, but for a profile of the fracture. Most common methods for generation of fracture profile are cross-cut techniques and profile measurements techniques [1-2]. We apply three principal methods: Profilometer, Interferometer Light Microscope and the Vertical Section for a specimen made of XC65 after the Charpy test. We compare the advantages and the limits for each technique. We use the Box Counting algorithm applied in the Image J program for determining the fractal dimension of the fracture surface in all three experimental techniques. Then we could characterize the roughness of the fracture profile at different magnifying power by the estimated fractal dimension.
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Tsai, Y. L., and J. J. Mecholsky. "Fractal fracture of single crystal silicon." Journal of Materials Research 6, no. 6 (June 1991): 1248–63. http://dx.doi.org/10.1557/jmr.1991.1248.
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The quantitative description of surfaces that are created during the fracture process is one of the fundamental issues in materials science. In this study, single crystal silicon was selected as a model material in which to study the correlation of fracture surface features as characterized by their fractal dimension for two different orientations of fracture with the fracture toughness of the material as measured using the strength-indentation and fracture surface analysis techniques. The fracture toughness on the {110} fracture plane of single crystal silicon was determined to be 1.19 MPa m1/2 for the {100} tensile surface and 1.05 MPa m1/2 for the {110} tensile surface using the indentation-strength three-point bending method. The fracture surface features of these two orientations are correspondingly different. Within our limitations of measurements (1–100 μm), the fractal dimension appeared different in different regions of the fracture surface. It has a higher value in the branching region and a lower value in the pre-branching and post-branching regions. The fractal dimensions are about the same in the pre-branching regions and post-branching region for these two orientations (D = 1.01 ± 0.01), i.e., nearly Euclidean (smooth); but the fractal dimensions are higher in the branching region for these two orientations. The fractal dimension is 1.10 ±0.4 for the {100} tensile surface and is 1.04 ±0.3 for the {110} tensile surface. If we select the highest dimension on a surface to represent the dimensionality of the surface, then a material with a higher fracture toughness has a higher fractal dimension in the branching region.