Radiographic texture of the trabecular bone of the proximal phalanx in horses with metacarpophalangeal osteoarthritis.

Osteoarthritis (OA) is a prevalent condition in horses, leading to changes in trabecular bone structure and radiographic texture. Although fractal dimension (FD) and lacunarity have been applied to quantify these changes in humans, their application in horses remains nascent. This study evaluated the use of FD, bone area fraction (BA/TA), and lacunarity in quantifying trabecular bone differences in the proximal phalanx (P1) in 50 radiographic examinations of equine metacarpophalangeal joints with varying OA degrees. In the dorsopalmar view, regions of interest were defined in the trabecular bone of the proximal epiphysis, medial and lateral to the sagittal groove of P1. Lower BA/TA values were observed medially in horses with severe OA (P=0.003). No significant differences in FD and lacunarity were found across OA degrees (P>0.1). FD, BA/TA, and lacunarity were not effective in identifying radiographic texture changes in the P1 trabecular bone in horses with different metacarpophalangeal OA degrees.

Magnetization Dynamics in Quasiperiodic Magnonic Crystals.

Quasiperiodic magnonic crystals, in contrast to their periodic counterparts, lack strict periodicity which gives rise to complex and localised spin wave spectra characterized by numerous band gaps and fractal features. Despite their intrinsic structural complexity, quasiperiodic nature of these magnonic crystals enables better tunability of spin wave spectra over their periodic counterparts and therefore holds promise for the applications in reprogrammable magnonic devices. In this article, we provide an overview of magnetization reversal and precessional magnetization dynamics studied so far in various quasiperiodic magnonic crystals, illustrating how their quasiperiodic nature gives rise to tailored band structure, enabling unparalleled control over spin waves. The review is concluded by highlighting the possible potential applications of these quasiperiodic magnonic crystals, exploring potential avenues for future exploration followed by a brief summary.

Unique algorithm for the evaluation of embryo photon emission and viability.

Living cells have spontaneous ultraweak photon emission derived from metabolic reactions associated with physiological conditions. The ORCA-Quest CMOS camera (Hamamatsu Photonics, Japan) is a highly sensitive and essential tool for photon detection; its use with a microscope incubator (Olympus) enables the detection of photons emitted by embryos with the exclusion of harmful visible light. With the application of the second law of thermodynamics, the low-entropy energy absorbed and used by embryos can be distinguished from the higher-entropy energy released and detectable in their environment. To evaluate higher-entropy energy data from embryos, we developed a unique algorithm for the calculation of the entropy-weighted spectral fractal dimension, which demonstrates the self-similar structure of the energy (photons) released by embryos. Analyses based on this structure enabled the distinction of living and degenerated mouse embryos, and of frozen and fresh embryos and the background. This novel detection of ultra-weak photon emission from mouse embryos can provide the basis for the development of a photon emission embryo control system. The ultraweak photon emission fingerprints of embryos may be used for the selection of viable specimens in an ideal dark environment.

Experimental, theoretical and numerical simulation-based investigations on the fabricated Cu2ZnSn thin-film-based Schottky diodes with enhanced electron transport for solar cell.

Copper-zinc-tin Cu2ZnSn (CZT) thin films are promising materials for solar cell applications. This thin film was deposited on a fluorine-doped tin oxide (FTO) using an electrochemical deposition hierarchy. X-ray diffraction of thin-film studies confirms the variation in the structural orientation of CZT on the FTO surface. As the pH of the solution is increased, the nature of the CZT thin-film aggregate changes from a fern-like leaf CZT dendrite crystal to a disk pattern. The FE-SEM surface micrograph shows the dendrite fern leaf and sharp edge disks. The 2-D diffusion limitation aggregation under slippery conditions for ternary thin films was performed for the first time. The simulation showed that by changing the diffusing species, the sticking probability was responsible for the pH-dependent morphological change. Convincingly, diffusion-limited aggregation (DLA) simulations confirm that the initial structure of copper is responsible for the final structure of the CZT thin films. An experimental simulation with pH as a controlled parameter revealed phase transition in CZT thin films. The top and back contact of Ag-CZT thin films based on Schottky behavior give a better electronic mechanism in superstrate and substrate solar cells.

Fractal geometry of fingering front morphologies of water and NAPL flow in homogeneous porous media: A comparison study.

Fingering front morphologies for water and non-aqueous phase liquids (NAPLs) infiltrating into homogeneous unsaturated porous media were quantitatively described based on the fractal assumption. Correlations of fractal dimensions with physical properties of the fluids were studied. The implications of fractal dimensions for environmental systems are further discussed. Fingering front morphologies had fractal properties, and diesels which with high capillary numbers experienced high fractal dimensions than water. Fractal dimension was suggested as an indicator for dye coverage and infiltration depth, which represent pollution area and depth, respectively. The pollution areas showed positive correlations with fractal dimensions while the infiltration depths showed negative correlations. This information is useful for contaminated soil risk management and important in the effective design of recovery and remediation schemes.

Robust arterial compliance estimation with Katz's fractal dimension of photoplethysmography.

Arterial compliance (AC) plays a crucial role in vascular aging and cardiovascular disease. The ability to continuously estimate aortic AC or its surrogate, pulse pressure (PP), through wearable devices is highly desirable, given its strong association with daily activities. While the single-site photoplethysmography (PPG)-derived arterial stiffness indices show reasonable correlations with AC, they are susceptible to noise interference, limiting their practical use. To overcome this challenge, our study introduces a noise-resistant indicator of AC: Katz's fractal dimension (KFD) of PPG signals. We showed that KFD integrated the signal complexity arising from compliance changes across a cardiac cycle and vascular structural complexity, thereby decreasing its dependence on individual characteristic points. To assess its capability in measuring AC, we conducted a comprehensive evaluation using both in silico studies with 4374 virtual human data and real-world measurements. In the virtual human studies, KFD demonstrated a strong correlation with AC (r = 0.75), which only experienced a slight decrease to 0.66 at a signal-to-noise ratio of 15dB, surpassing the best PPG-morphology-derived AC measure (r = 0.41) under the same noise condition. In addition, we observed that KFD's sensitivity to AC varied based on the individual's hemodynamic status, which may further enhance the accuracy of AC estimations. These in silico findings were supported by real-world measurements encompassing diverse health conditions. In conclusion, our study suggests that PPG-derived KFD has the potential to continuously and reliably monitor arterial compliance, enabling unobtrusive and wearable assessment of cardiovascular health.

Characterization of antiseizure medications effects on the EEG neurodynamic by fractal dimension.

Objectives: An important challenge in epilepsy is to define biomarkers of response to treatment. Many electroencephalography (EEG) methods and indices have been developed mainly using linear methods, e.g., spectral power and individual alpha frequency peak (IAF). However, brain activity is complex and non-linear, hence there is a need to explore EEG neurodynamics using nonlinear approaches. Here, we use the Fractal Dimension (FD), a measure of whole brain signal complexity, to measure the response to anti-seizure therapy in patients with Focal Epilepsy (FE) and compare it with linear methods. Materials: Twenty-five drug-responder (DR) patients with focal epilepsy were studied before (t1, named DR-t1) and after (t2, named DR-t2) the introduction of the anti-seizure medications (ASMs). DR-t1 and DR-t2 EEG results were compared against 40 age-matched healthy controls (HC). Methods: EEG data were investigated from two different angles: frequency domain-spectral properties in δ, θ, α, ß, and γ bands and the IAF peak, and time-domain-FD as a signature of the nonlinear complexity of the EEG signals. Those features were compared among the three groups. Results: The δ power differed between DR patients pre and post-ASM and HC (DR-t1 vs. HC, p < 0.01 and DR-t2 vs. HC, p < 0.01). The θ power differed between DR-t1 and DR-t2 (p = 0.015) and between DR-t1 and HC (p = 0.01). The α power, similar to the δ, differed between DR patients pre and post-ASM and HC (DR-t1 vs. HC, p < 0.01 and DR-t2 vs. HC, p < 0.01). The IAF value was lower for DR-t1 than DR-t2 (p = 0.048) and HC (p = 0.042). The FD value was lower in DR-t1 than in DR-t2 (p = 0.015) and HC (p = 0.011). Finally, Bayes Factor analysis showed that FD was 195 times more likely to separate DR-t1 from DR-t2 than IAF and 231 times than θ. Discussion: FD measured in baseline EEG signals is a non-linear brain measure of complexity more sensitive than EEG power or IAF in detecting a response to ASMs. This likely reflects the non-oscillatory nature of neural activity, which FD better describes. Conclusion: Our work suggests that FD is a promising measure to monitor the response to ASMs in FE.

An entropy based spatial-temporal cube with its application to assess stress of overburden due to mining.

Underground coal seam mining significantly alters the stress and energy distribution within the overlying rock, leading to eventual structural degradation. Therefore, it is imperative to quantitatively identify the temporal and spatial characteristics of stress evolution of overlying rock caused by mining. This paper introduces a novel rock stress model integrating entropy and a spatial-temporal cube. Similar material model tests are used to identify the abrupt entropy changes within the mining rock, and the trend analysis is carried out to describe the spatial-temporal evolution law of stress during mining. Experimental findings indicate elevated stress levels in the unmined rock preceding and following the panel, as well as within specific rock strata above it. Definitively, dynamic stress arches within the surrounding rock of the stope predominantly bear and distribute the load and pressure from the overlying rock, and each stress mutation is accompanied by a sudden stress entropy change. Over time, z-score shows that the noticeable reduction in mining-induced overburden stress becomes increasingly pronounced, especially in the water-conducting fracture zone. The model's bifurcation set serves as the comprehensive criterion for the entropy-induced sudden changes in the rock system, signifying overall failure.

The effect of mentoplate application on the condyle.

BACKGROUND: The aim of the study was to investigate the changes occurring in the mandibular condyle by using mentoplate together with rapid maxillary expansion (MP-RME) treatment in the correction of skeletal class III relationship, using fractal analysis (FA). METHODS: The sample consisted of 30 individuals (8-11 years) diagnosed with skeletal Class III malocclusion who underwent MP-RME treatment. Archival records provided cone-beam computed tomography (CBCT) images taken at two intervals: before MP-RME treatment (T0) and after treatment (T1). The CBCT images were obtained using standardized settings to ensure consistency in image quality and resolution. The trabecular structures in the bilateral condyles at both T0 and T1 were analyzed using FA. The FA was performed on these condylar images using the Image J software. The region of interest (ROI) was carefully selected in the condyle to avoid overlapping with cortical bone, and the box-counting method was employed to calculate the fractal dimension (FD). Statistical analysis was conducted to compare the FD values between T0 and T1 and to evaluate gender differences. The statistical significance was determined using paired t-tests for intra-group comparisons and independent t-tests for inter-group comparisons, with a significance level set at p < 0.05. RESULTS: The analysis revealed no statistically significant differences in the trabecular structures of the condyles between T0 and T1 (p > 0.05). However, a significant gender difference was observed in FA values, with males exhibiting higher FA values in the left condyle compared to females at both T0 and T1 (p < 0.05). Specifically, the FA values in the left condyle increased from a mean of 1.09 ± 0.09 at T0 to 1.13 ± 0.08 at T1 in males, whereas in females, the FA values remained relatively stable with a mean of 1 ± 0.09 at T0 and 1.03 ± 0.11 at T1. CONCLUSION: The findings indicate that MP-RME therapy does not induce significant alterations in the trabecular structure of the mandibular condyle. These results suggest the treatment's safety concerning the structural integrity of the condyle, although the observed gender differences in FA values warrant further investigation.

Banana fruit bruise detection using fractal dimension based image processing.

The study used the fractal dimension (FD), browning incidence, and grayscale values using machine vision to describe the bruise magnitude and quality of mechanically damaged 'Fard' bananas bruised from 20, 40, 60 cm drop heights by 66, 98, and 110 g ball weights conditioned at different storage temperatures (5, 13, 22 °C) after 48 h. Conventional analyses like bruise area (BA), bruise volume (BV), and bruise susceptibility (BS) were also conducted. A correlation was performed to determine the relationship between image processing and conventional assessment of bruise damage in bananas. Weight, firmness, color, sugar content, and acidity were investigated. The results demonstrated that bananas bruised from the highest force and stored at 5 and 22 °C reported the lowest FD with values of 1.7162 and 1.7403, respectively. Increasing the level of damage reduced the fractal dimension and grayscale values and increased browning incidence and bruise susceptibility values after 48 h of storage. The total color change values showed a strong Pearson's correlation coefficient (r≥-0.81) with image analysis fractal dimension and grayscale values. The findings also indicated that higher bruising and temperature can induce weight loss, firmness reduction, lightness, and yellowness increment, and sugar and acidity changes. Overall, the fractal image analysis conducted in this study was highly effective in describing the bruising magnitude of bananas under different conditions.

Development, Application and Utility of a Machine Learning Approach for Melanoma and Non-Melanoma Lesion Classification Using Counting Box Fractal Dimension.

The diagnosis and identification of melanoma are not always accurate, even for experienced dermatologists. Histopathology continues to be the gold standard, assessing specific parameters such as the Breslow index. However, it remains invasive and may lack effectiveness. Therefore, leveraging mathematical modeling and informatics has been a pursuit of diagnostic methods favoring early detection. Fractality, a mathematical parameter quantifying complexity and irregularity, has proven useful in melanoma diagnosis. Nonetheless, no studies have implemented this metric to feed artificial intelligence algorithms for the automatic classification of dermatological lesions, including melanoma. Hence, this study aimed to determine the combined utility of fractal dimension and unsupervised low-computational-requirements machine learning models in classifying melanoma and non-melanoma lesions. We analyzed 39,270 dermatological lesions obtained from the International Skin Imaging Collaboration. Box-counting fractal dimensions were calculated for these lesions. Fractal values were used to implement classification methods by unsupervised machine learning based on principal component analysis and iterated K-means (100 iterations). A clear separation was observed, using only fractal dimension values, between benign or malignant lesions (sensibility 72.4% and specificity 50.1%) and melanoma or non-melanoma lesions (sensibility 72.8% and specificity 50%) and subsequently, the classification quality based on the machine learning model was ≈80% for both benign and malignant or melanoma and non-melanoma lesions. However, the grouping of metastatic melanoma versus non-metastatic melanoma was less effective, probably due to the small sample size included in MM lesions. Nevertheless, we could suggest a decision algorithm based on fractal dimension for dermatological lesion discrimination. On the other hand, it was also determined that the fractal dimension is sufficient to generate unsupervised artificial intelligence models that allow for a more efficient classification of dermatological lesions.

Effect of Thermal Activation on the Structure and Electrochemical Properties of Carbon Material Obtained from Walnut Shells.

A simple activation method has been used to obtain porous carbon material from walnut shells. The effect of the activation duration at 400 °C in an atmosphere with limited air access on the structural, morphological, and electrochemical properties of the porous carbon material obtained from walnut shells has been studied. Moreover, the structure and morphology of the original and activated carbon samples have been characterized by SAXS, low-temperature adsorption porosimetry, SEM, and Raman spectroscopy. Therefore, the results indicate that increasing the duration of activation at a constant temperature results in a reduction in the thickness values of interplanar spacing (d002) in a range of 0.38-0.36 nm and lateral dimensions of the graphite crystallite from 3.79 to 2.52 nm. It has been demonstrated that thermal activation allows for an approximate doubling of the specific SBET surface area of the original carbon material and contributes to the development of its mesoporous structure, with a relative mesopore content of approximately 75-78% and an average pore diameter of about 5 nm. The fractal dimension of the obtained carbon materials was calculated using the Frenkel-Halsey-Hill method; it shows that its values for thermally activated samples (2.52, 2.69) are significantly higher than for the original sample (2.17). Thus, the porous carbon materials obtained were used to fabricate electrodes for electrochemical capacitors. Electrochemical investigations of these cells in a 6 M KOH aqueous electrolyte were conducted by cyclic voltammetry, galvanostatic charge/discharge, and impedance spectroscopy. Consequently, it was established that the carbon material activated at 400 °C for 2 h exhibits a specific capacity of approximately 110-130 F/g at a discharge current density ranging from 4 to 100 mA/g.

Study on Sensing Urine Concentrations in Water Using a Microwave Sensor Based on Hilbert Structure.

In this study, a two-port network-based microwave sensor for liquid characterization is presented. The suggested sensor is built as a miniature microwave resonator using the third iteration of Hilbert's fractal architecture. The suggested structure is used with the T-resonator to raise the sensor quality factor. The suggested sensor is printed on a FR4 substrate and has a footprint of 40×60×1.6mm3. Analytically, a theoretical investigation is made to clarify how the suggested sensor might function. The suggested sensor is created and put to the test in an experiment. Later, two pans to contain the urine Sample Under Test (SUT) are printed on the sensor. Before loading the SUT, it is discovered that the suggested structure's frequency resonance is 0.46 GHz. An 18 MHz frequency shift is added to the initial resonance after the pans are printed. They monitor the S-parameters in terms of S12 regarding the change in water content in the urine samples, allowing for the sensing component to be completed. As a result, 10 different samples with varying urine percentages are added to the suggested sensor to evaluate its ability to detect the presence of urine. Finally, it is discovered that the suggested process' measurements and corresponding simulated outcomes agreed quite well.

Fractal analysis of left ventricular trabeculae in heart failure with preserved ejection fraction patients with multivessel coronary artery disease.

OBJECTIVES: Endocardial trabeculae undergo varicose changes and hyperplasia in response to hemodynamic influences and are a variable phenotype reflecting changes in disease. Fractal analysis has been used to analyze the complexity of endocardial trabeculae in a variety of cardiomyopathies. The aim of this paper was to quantify the myocardial trabecular complexity through fractal analysis and to investigate its predictive value for the diagnosis of heart failure with preserved ejection fraction (HFpEF) in patients with multivessel coronary artery disease (CAD). METHODS: The retrospective study population consisted of 97 patients with multivessel CAD, 39 of them were diagnosed with HFpEF, while 46 healthy volunteers were recruited as controls. Fractal dimension (FD) was obtained through fractal analysis of endocardial trabeculae on LV short-axis cine images. Logistic regression analyses were used to confirm the predictors and compare different prediction models. RESULTS: Mean basal FD was significantly higher in patients with HFpEF than in patients without HFpEF or in the healthy group (median: 1.289; IQR: 0.078; p < 0.05). Mean basal FD was also a significant independent predictor in univariate and multivariate logistic regression (OR: 1.107 and 1.043, p < 0.05). Furthermore, adding FD to the prediction model improved the calibration and accuracy of the model (c-index: 0.806). CONCLUSION: The left ventricular FD obtained with fractal analysis can reflect the complexity of myocardial trabeculae and has an independent predictive value for the diagnosis of HFpEF in patients with multivessel CAD. Including FD into the diagnostic model can help improve the diagnosis. CRITICAL RELEVANCE STATEMENT: Differences show in the complexity of endocardial trabeculae in multivessel coronary artery disease patients, and obtaining fractal dimensions (FD) by fractal analysis can help identify heart failure with preserved ejection fraction (HFpEF) patients. KEY POINTS: The complexity of myocardial trabeculae differs among patients with multivessel coronary artery disease. Left ventricular fractal dimensions can reflect the complexity of the myocardial trabecular. Fractal dimensions have predictive value for the diagnosis of heart failure with preserved ejection fraction.

Lineages of Fractal Genera Comprise the 88-Million-Year Steel Evolutionary Spine of the Ecosphere.

Fractal evolution is apparently effective in selectively preserving environmentally resilient traits for more than 80 million years in Streptotrichaceae (Bryophyta). An analysis simulated maximum destruction of ancestral traits in that large lineage. The constraints enforced were the preservation of newest ancestral traits, and all immediate descendant species obtained different new traits. Maximum character state changes in ancestral traits were 16 percent of all possible traits in any one sub-lineage, or 73 percent total of the entire lineage. Results showed, however, that only four ancestral traits were permanently eliminated in any one lineage or sub-lineage. A lineage maintains maximum biodiversity of temporally and regionally survival-effective traits at minimum expense to resilience across a geologic time of 88 million years for the group studied. Similar processes generating an extant punctuated equilibrium as bursts of about four descendants per genus and one genus per 1-2 epochs are possible in other living groups given similar emergent processes. The mechanism is considered complexity-related, the lineage being a self-organized emergent phenomenon strongly maintained in the ecosphere by natural selection on fractal genera.

Complexity synchronization in living matter: a mini review.

Fractal time series have been argued to be ubiquitous in human physiology and some of the implications of that ubiquity are quite remarkable. One consequence of the omnipresent fractality is complexity synchronization (CS) observed in the interactions among simultaneously recorded physiologic time series discussed herein. This new kind of synchronization has been revealed in the interaction triad of organ-networks (ONs) consisting of the mutually interacting time series generated by the brain (electroencephalograms, EEGs), heart (electrocardiograms, ECGs), and lungs (Respiration). The scaled time series from each member of the triad look nothing like one another and yet they bear a deeply recorded synchronization invisible to the naked eye. The theory of scaling statistics is used to explain the source of the CS observed in the information exchange among these multifractal time series. The multifractal dimension (MFD) of each time series is a measure of the time-dependent complexity of that time series, and it is the matching of the MFD time series that provides the synchronization referred to as CS. The CS is one manifestation of the hypothesis given by a "Law of Multifractal Dimension Synchronization" (LMFDS) which is supported by data. Therefore, the review aspects of this paper are chosen to make the extended range of the LMFDS hypothesis sufficiently reasonable to warrant further empirical testing.

Novel Use of Fractal Analysis for Quantifying Polymethylmethacrylate Distribution Patterns in Osteoporotic and Malignant Vertebral Compression Fractures Following Vertebroplasty.

Purpose: Fractal analysis is a mathematical tool which allows the evaluation of complex microstructural features within materials that cannot be expressed in traditional geometric terms. The purpose of this study is to quantify the differences in polymethylmethacrylate intravertebral cement spatial distribution patterns following vertebroplasty using fractal analysis through the examination of osteoporotic and malignant compression fractures. Methods: Frontal and lateral post-vertebroplasty radiographs were evaluated from 29 patients with osteoporotic and malignant compression fractures who underwent vertebroplasty. The individually treated vertebra were divided into osteoporotic (n = 35) and malignant groups (n = 41). Images underwent segmentation, thresholding, and binarization prior to fractal analysis. Fractal dimension and lacunarity values were derived from the region of interest in treated vertebrae using the "box-counting" and "gliding-box" techniques respectively using ImageJ. The mean values of both parameters were compared between the 2 groups. Results: The mean fractal dimension was significantly higher in the malignant vertebral compression fracture group (1.53 ± 0.08) compared to the osteoporotic group (1.34 ± 0.17; P < .001). Similarly, mean lacunarity values were significantly higher in the malignant fracture group (0.50 ± 0.09) compared to the osteoporotic group (0.37 ± 0.10; P < .001). Conclusions: Fractal dimension and lacunarity values of cement spatial distribution patterns obtained from the post-vertebroplasty radiographs can differentiate between benign osteoporotic and malignant vertebral compression fractures. This novel technique may be useful for evaluating cement spatial distribution patterns in spine augmentation procedures, although further research is warranted in this area.

EVALUATION OF MANDIBULAR TRABECULAR AND CORTICAL STRUCTURE BY FRACTAL ANALYSIS IN HYPERLIPIDEMIA AND HYPERTENSION PATIENTS.

INTRODUCTION: This study aimed to evaluate the mandibular trabecular and cortical changes in patients with hyperlipidemia (HL) and/or hypertension (HT) using fractal dimension (FD) analysis, mandibular cortical width (MCW), panoramic mandibular index (PMI) and mandibular cortical index (MCI). MATERIALS AND METHODS: Panoramic radiographs of 100 patients were evaluated. FD measurement of three region of interest (ROI) including the angulus, corpus and interdental bone area were made. MCW, PMI and MCI were also measured and noted. RESULTS: Angulus, corpus and interdental FD values were significantly lower in three disease groups than the control group. Angulus, corpus, and interdental FD values were significantly lower in the HL+HT group than in the HL group and HT group. MCW value was significantly lower in the HL group, HT group, and HL+HT group than the control group. The cortical index C1 was more common in the control group while C2 was more common in the HT, HL and HL+HT group. CONCLUSION: The fact that FD was significantly lower in the HL+HT group compared to the HL and HT groups indicates the positive effect of their association on bone loss and quality. FD measurements on images obtained using a direct digital panoramic system can be used for treatment planning and follow-up of patients with HL and/or HT.

Research on the dynamic tensile characteristics and surface crack evolution of coal under impact loading.

The tensile properties of coal under dynamic loading are important mechanical characteristics of coal and are highly important for controlling coal rock stability under impact loading conditions, selecting blasting engineering parameters, and studying the mechanism of rockburst disasters. To investigate the dynamic tensile failure process of coal subjected to impact loading, this study used high-speed photography and digital image correlation technology to capture the dynamic tensile failure of coal under impact loading. The dynamic tensile evolution was quantitatively analyzed from the beginning of coal sample being loaded to failure. The captured images of the coal were processed, and the fractal dimension was used to quantitatively describe the evolution of the coal surface cracks under impact loading. The following conclusions were drawn from the experimental results: (1) An empirical formula was established to describe the dynamic tensile strength characteristics of coal under different loading rates. (2) Under impact loading, the maximum strain of a Brazilian disc coal sample first appeared at the contact end between the sample and the incident rod. (3) Under impact loading, a Brazilian disc coal sample cracked from the center of the sample outward, and the crack subsequently extended toward both ends. The fractal dimension of the crack exhibited a power function relationship with time, and the variation range of the fractal dimension of the crack was 1.05-1.39.

Experimental study on fractal dimension of energy dissipation and crack growth in saturated tuff at different strain rates.

In order to investigate the effects of strain rate and water saturation on the energy dissipation and crack growth of tuff, uniaxial compression tests were carried out on dry and water saturated tuff with different strain rates using an electro-hydraulic servo press and a 50 mm diameter split Hopkinson pressure rod (SHPB) device. High-speed camera and Image J image analysis software were used to obtain the crack growth process of the specimen under impact load, and fractal dimension was introduced to quantitatively study the crack growth degree. The results show that more than 90% of the energy is stored in the specimen as elastic energy when it reaches the peak stress under static load. The average total energy of water-saturated specimens is 67.55% of that of dry specimens. The average energy dissipation density of water-saturated specimens under 0.3 MPa, 0.4 MPa and 0.5 MPa air pressure is 0.79, 0.91 and 0.92 times of that of dry specimens, respectively. Water-saturated specimens will deteriorate and thus reduce their energy storage and energy absorption effects. The reflected energy, transmitted energy, absorbed energy and incident energy are linear, logarithmic and linear functions, respectively, and the energy absorptivity and specific energy absorptivity of water-saturated specimens are lower than those of dry specimens. Due to the existence of "stefan" effect, the increase of energy dissipation density of water-saturated specimen at high strain rate is greater than that of dry specimen. The mean fractal dimension of water-saturated specimens under 0.3 MPa, 0.4 MPa and 0.5 MPa is 1.09, 1.05 and 1.16 times that of dry specimens. At the same strain rate, the number and width of cracks in water-saturated specimens are larger than that in dry specimens. Water-saturated behavior reduces the energy absorption capacity of tuff, increases the fractal dimension of crack growth, and significantly reduces the resistance of water-saturated rock to external loads.