Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

Experimental study on 3D internal penny-shaped crack propagation in brittle materials under uniaxial compression

Fractures are widely present in geomaterials of civil engineering and deep underground engineering.Given that geomaterials are usually brittle,the fractures can significantly affect the evaluation of underground engineering construction safety and the early warning of rock failure.However,the crack initiation and propagation in brittle materials under composite loading remain unknown so far.In this study,a three-dimensional internal laser-engraved cracking technique was applied to produce internal cracks without causing damage to the surfaces.The uniaxial compression tests were performed on a brittle material with internal cracks to investigate the propagation of these internal cracks at different dip angles under compression and shear.The test results show that the wing crack propagation mainly occurs in the specimen with an inclined internal crack,which is a mixed-ModeⅠ–Ⅱ–Ⅲfracture;in contrast,ModeⅠfracture is present in the specimen with a vertical internal crack.The fractography characteristics of ModeⅢfracture display a lance-like pattern.The fracture mechanism in the brittle material under compression is that the internal wing cracks propagate to the ends of the whole sample and cause the final failure.The initial deflection angle of the wing crack is determined by the participation ratio of stress intensity factors KII to KI at the tip of the internal crack.

A PENNY-SHAPED CRACK IN A TRANSVERSELY ISOTROPIC PIEZOELECTRIC SOLID:MODES Ⅱ AND Ⅲ PROBLEMS

An exact analysis of the modes Ⅱ and Ⅲ problems of a penny- shaped crack in a transversely isotropic piezoelectric medium is performed in this paper.The potential theory method is employed based on the general solution of three-dimensional piezoelasticity and the four harmonics involved are represented by one complex potential.Previous results in potential theory are then utilized to obtain the exact solution that is expressed in terms of elementary functions.Comparison is made between the current results with those published and good agreement is obtained.

A PENNY-SHAPED CRACK IN A MAGNETOELECTROELASTIC LAYER UNDER RADIAL SHEAR IMPACT LOADING

This paper analyzes the dynamic magnetoelectroelastic behavior induced by a pennyshaped crack in a magnetoelectroelastic layer. The crack surfaces are subjected to only radial shear impact loading. The Laplace and Hankel transform techniques are employed to reduce the problem to solving a Fredholm integral equation. The dynamic stress intensity factor is obtained and numerically calculated for different layer heights. And the corresponding static solution is given by simple analysis. It is seen that the dynamic stress intensity factor for cracks in a magnetoelectroelastic layer has the same expression as that in a purely elastic material. And the influences of layer height on both the dynamic and static stress intensity factors are insignificant as h/a 〉 2.

PENNY-SHAPED CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC MATERIALS

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the dis- placement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordi- nate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r^(-1/2)) singularity.

Interactions of penny-shaped cracks in three-dimensional solids

The interaction of arbitrarily distributed penny-shaped cracks in three-dimensional solids is analyzed in this paper. Using oblate spheroidal coordinates and displacement functions, an analytic method is devel- oped in which the opening and the sliding displacements on each crack surface are taken as the basic unknown functions. The basic unknown functions can be expanded in series of Legendre polynomials with unknown coefficients. Based on superposition technique, a set of governing equations for the unknown coefficients are formulated from the traction free conditions on each crack surface. The boundary collocation procedure and the average method for crack-surface tractions are used for solving the governing equations. The solution can be obtained for quite closely located cracks. Numerical examples are given for several crack problems. By comparing the present results with other existing results, one can conclude that the present method provides a direct and efficient approach to deal with three-dimensional solids containing multiple cracks.

TORSIONAL IMPACT RESPONSE OF A PENNY-SHAPED CRACK IN A FUNCTIONAL GRADED STRIP

The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived.And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and (strip's) highness on the dynamic fracture behavior. Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip's highness.

THE APPROXIMATE SOLUTION OF PENNY-SHAPED CRACKS PERIODICALLY DISTRIBUTED IN INFINITE ELASTIC BODY

The penny-shaped cracks periodically distributed in infinite elastic body are studied. The prob- lem is approximately simplified to that of a single crack embedded in finite length cylinder and the stress intensity factor is obtained by solving a Fredholm integral equation. Numerical results are given and the effects of crack interaction on the stress intensity factor are discussed.

THE EFFECTS OF THREE-DIMENSIONAL PENNY-SHAPED CRACKS ON ZONAL DISINTEGRATION OF THE SURROUNDING ROCK MASSES AROUND A DEEP CIRCULAR TUNNEL

In this study, it was assumed that three-dimensional penny-shaped cracks existed in deep rock masses. A new non-Euclidean model was established, in which the effects of penny- shaped cracks and axial in-situ stress on zonal disintegration of deep rock masses were taken into account. Based on the non-Euclidean model, the stress intensity factors at tips of the penny- shaped cracks were determined. The strain energy density factor was applied to investigate the occurrence of fractured zones. It was observed from the numerical results that the magnitude and location of fractured zones were sensitive to micro- and macro-mechanical parameters, as well as the value of in-situ stress. The numerical results were in good agreement with the experimental data.

Penny-shaped interfacial crack between dissimilar magnetoelectroelastic layers

A penny-shaped interfacial crack between dissimilar magnetoelectroelastic layers subjected to magnetoelectromechanical loads is investigated,where the magnetoelectrically impermeable crack surface condition is adopted. By using Hankel transform technique,the mixed boundary value problem is firstly reduced to a system of singular integral equations,which are further reduced to a system of algebraic equations. The field intensity factors and energy release rate are finally derived. Numerical results elucidate the eects of crack configuration,electric and/or magnetic loads,and material parameters of the magnetoelectroelastic layers on crack propagation and growth. This work should be useful for the design of magnetoelectroelastic composite structures.

CRACK TIP PLASTICITY OF A THERMALLY LOADED PENNY-SHAPED CRACK IN AN INFINITE SPACE OF 1D QC

The present work is concerned with a penny-shaped Dugdale crack embedded in an infinite space of one-dimensional （1D） hexagonal quasicrystals and subjected to two identical axisymmetric temperature loadings on the upper and lower crack surfaces. Applying Dugdale hypothesis to thermo-elastic results, the extent of the plastic zone at the crack tip is determined. The normal stress outside the plastic zone and crack surface displacement are derived in terms of special functions. For a uniform loading case, the corresponding results are presented by simpli- fying the preceding results. Numerical calculations are carried out to show the influence of some parameters.

Recent research progress in the mechanism and suppression of fusion welding-induced liquation cracking of nickel based superalloys

Nickel-based superalloys are extensively used in the crucial hot-section components of industrial gas turbines,aeronautics,and astronautics because of their excellent mechanical properties and corrosion resistance at high temperatures.Fusion welding serves as an effective means for joining and repairing these alloys;however,fusion welding-induced liquation cracking has been a challenging issue.This paper comprehensively reviewed recent liquation cracking,discussing the formation mechanisms,cracking criteria,and remedies.In recent investigations,regulating material composition,changing the preweld heat treatment of the base metal,optimizing the welding process parameters,and applying auxiliary control methods are effective strategies for mitigating cracks.To promote the application of nickel-based superalloys,further research on the combination impact of multiple elements on cracking prevention and specific quantitative criteria for liquation cracking is necessary.

Characterization of alpine meadow surface crack and its correlation with root-soil properties

Quantifying surface cracks in alpine meadows is a prerequisite and a key aspect in the study of grassland crack development.Crack characterization indices are crucial for the quantitative characterization of complex cracks,serving as vital factors in assessing the degree of cracking and the development morphology.So far,research on evaluating the degree of grassland degradation through crack characterization indices is rare,especially the quantitative analysis of the development of surface cracks in alpine meadows is relatively scarce.Therefore,based on the phenomenon of surface cracking during the degradation of alpine meadows in some regions of the Qinghai-Tibet Plateau,we selected the alpine meadow in the Huangcheng Mongolian Township,Menyuan Hui Autonomous County,Qinghai Province,China as the study area,used unmanned aerial vehicle(UAV)sensing technology to acquire low-altitude images of alpine meadow surface cracks at different degrees of degradation(light,medium,and heavy degradation),and analyzed the representative metrics characterizing the degree of crack development by interpreting the crack length,length density,branch angle,and burrow(rat hole)distribution density and combining them with in situ crack width and depth measurements.Finally,the correlations between the crack characterization indices and the soil and root parameters of sample plots at different degrees of degradation in the study area were analyzed using the grey relation analysis.The results revealed that with the increase of degradation,the physical and chemical properties of soil and the mechanical properties of root-soil composite changed significantly,the vegetation coverage reduced,and the root system aggregated in the surface layer of alpine meadow.As the degree of degradation increased,the fracture morphology developed from"linear"to"dendritic",and eventually to a complex and irregular"polygonal"pattern.The crack length,width,depth,and length density were identified as the crack characterization indices via analysis of variance.The results of grey relation analysis also revealed that the crack length,width,depth,and length density were all highly correlated with root length density,and as the degradation of alpine meadows intensified,the underground biomass increased dramatically,forming a dense layer of grass felt,which has a significant impact on the formation and expansion of cracks.

Wave-induced flow of pore fluid in a cracked porous solid containing penny-shaped inclusions

An aim of current study is to analyze the contribution of reflected longitudinal waves to wave-induced fluid flow(WIFF) in the cracked porous solid.Initially,we investigate the time harmonic plane waves in cracked porous solid by employing the mathematical model proposed by Zhang et al.(2019).The solution is obtained in form of the Christoffel equations.The solution of the Christoffel equations indicates that there exist four(three dilatational and one shear) waves.These waves are attenuated in nature due to their complex and frequency-dependent velocities.The reflection coefficients are calculated at the sealed pore stress-free surface of cracked porous solid for the incidence of P1 and SV waves.It is found that three longitudinal waves contribute to WIFF and the contribution of these waves to the induced fluid in the cracked porous solid is analyzed using the reflection coefficients of these longitudinal waves.We analytically show that the fluid flow induced by these longitudinal waves is linked directly to their respective reflection coefficients.Finally,a specific numerical example is considered to discuss and to depict the impact of various parameters on the characteristics of propagation like phase velocity/attenuation,reflection coefficients and WIFF of longitudinal waves.

Effect of carbon material and surfactant on ink property and resulting surface cracks of fuel-cell microporous layers

Ensuring the consistency of electrode structure in proton-exchange-membrane fuel cells is highly desired yet challenging because of wide-existing and unguided cracks in the microporous layer(MPL). The first thing is to evaluate the homogeneity of MPL with cracks quantitatively. This paper proposes the homogeneity index of a full-scale MPL with an area of 50 cm~2, which is yet to be reported in the literature to our knowledge. Besides, the effects of the carbon material and surfactant on the ink and resulting MPL structure have been studied. The ink with a high network development degree produces an MPL with low crack density, but the ink with high PDI produces an MPL with low crack homogeneity. The polarity of the surfactant and the non-polarity of polytetrafluoroethylene(PTFE) are not mutually soluble,resulting in the heterogeneous PTFE distribution. The findings of this study provide guidelines for MPL fabrication.

Particle Discontinuous Deformation Analysis of Static and Dynamic Crack Propagation in Brittle Material

Crack propagation in brittle material is not only crucial for structural safety evaluation,but also has a wideranging impact on material design,damage assessment,resource extraction,and scientific research.A thorough investigation into the behavior of crack propagation contributes to a better understanding and control of the properties of brittle materials,thereby enhancing the reliability and safety of both materials and structures.As an implicit discrete elementmethod,the Discontinuous Deformation Analysis(DDA)has gained significant attention for its developments and applications in recent years.Among these developments,the particle DDA equipped with the bonded particle model is a powerful tool for predicting the whole process of material from continuity to failure.The primary objective of this research is to develop and utilize the particle DDAtomodel and understand the complex behavior of cracks in brittle materials under both static and dynamic loadings.The particle DDA is applied to several classical crack propagation problems,including the crack branching,compact tensile test,Kalthoff impact experiment,and tensile test of a rectangular plate with a hole.The evolutions of cracks under various stress or geometrical conditions are carefully investigated.The simulated results are compared with the experiments and other numerical results.It is found that the crack propagation patterns,including crack branching and the formation of secondary cracks,can be well reproduced.The results show that the particle DDA is a qualified method for crack propagation problems,providing valuable insights into the fracture mechanism of brittle materials.

Solid-state NMR reveals that oxygen tri-clusters make glass highly crackresistant

In a recent paper,advanced solid-state nuclear magnetic resonance(SSNMR)technology was employed to reveal the underlying mechanism contributing to the high hardness and exceptional resistance to fragmentation observed in certain special glasses[1].This study utilized SSNMR to analyze the atomic-scale internal structure of glass,enabling the quantification of the fraction of three-coordinated oxygen([^((3))O]).The research findings demonstrate a quantitative relationship between[^((3))O]and the resistance of glass to crack initiation.

Automatic Road Tunnel Crack Inspection Based on Crack Area Sensing and Multiscale Semantic Segmentation

The detection of crack defects on the walls of road tunnels is a crucial step in the process of ensuring travel safetyand performing routine tunnel maintenance. The automatic and accurate detection of cracks on the surface of roadtunnels is the key to improving the maintenance efficiency of road tunnels. Machine vision technology combinedwith a deep neural network model is an effective means to realize the localization and identification of crackdefects on the surface of road tunnels.We propose a complete set of automatic inspection methods for identifyingcracks on the walls of road tunnels as a solution to the problem of difficulty in identifying cracks during manualmaintenance. First, a set of equipment applied to the real-time acquisition of high-definition images of walls inroad tunnels is designed. Images of walls in road tunnels are acquired based on the designed equipment, whereimages containing crack defects are manually identified and selected. Subsequently, the training and validationsets used to construct the crack inspection model are obtained based on the acquired images, whereas the regionscontaining cracks and the pixels of the cracks are finely labeled. After that, a crack area sensing module is designedbased on the proposed you only look once version 7 model combined with coordinate attention mechanism (CAYOLOV7) network to locate the crack regions in the road tunnel surface images. Only subimages containingcracks are acquired and sent to the multiscale semantic segmentation module for extraction of the pixels to whichthe cracks belong based on the DeepLab V3+ network. The precision and recall of the crack region localizationon the surface of a road tunnel based on our proposed method are 82.4% and 93.8%, respectively. Moreover, themean intersection over union (MIoU) and pixel accuracy (PA) values for achieving pixel-level detection accuracyare 76.84% and 78.29%, respectively. The experimental results on the dataset show that our proposed two-stagedetection method outperforms other state-of-the-art models in crack region localization and detection. Based onour proposedmethod, the images captured on the surface of a road tunnel can complete crack detection at a speed often frames/second, and the detection accuracy can reach 0.25 mm, which meets the requirements for maintenanceof an actual project. The designed CA-YOLO V7 network enables precise localization of the area to which a crackbelongs in images acquired under different environmental and lighting conditions in road tunnels. The improvedDeepLab V3+ network based on lightweighting is able to extract crack morphology in a given region more quicklywhile maintaining segmentation accuracy. The established model combines defect localization and segmentationmodels for the first time, realizing pixel-level defect localization and extraction on the surface of road tunnelsin complex environments, and is capable of determining the actual size of cracks based on the physical coordinatesystemafter camera calibration. The trainedmodelhas highaccuracy andcanbe extendedandapplied to embeddedcomputing devices for the assessment and repair of damaged areas in different types of road tunnels.

Effects of orientation on the fatigue crack growth behaviors of the ZK60 magnesium alloy in air and PBS

Strong anisotropic corrosion and mechanical properties caused by specimen orientations greatly limit the applications of wrought magnesium alloys.To investigate the influences of specimen orientation,the corrosion tests and(corrosion)fatigue crack growth tests were conducted.The rolled and transverse surfaces of the materials show distinct corrosion rate differences in the stable corrosion stage,but the truth is the opposite for the initial stage of corrosion.In air,specimen orientations have a significant influence on the plastic deformation mechanisms near the crack tip,which results in different fatigue fracture surfaces and cracking paths.Compared with R-T specimens,N-T specimens show a slower fatigue crack growth(FCG)rate in air,which can be attributed to crack closure effects and deformation twinning near the crack tip.The corrosion environment will not significantly change the main plastic deformation mechanisms for the same type of specimen.However,the FCG rate in phosphate buffer saline(PBS)is one order of magnitude higher than that in air,which is caused by the combined effects of hydrogen-induced cracking and anodic dissolution.Owing to the similar corrosion rates at crack tips,the specimens with different orientations display close FCG rates in PBS.

Development and Catalytic Cracking Performance of Ultrastable Y Zeolite Rich in Secondary Pores

A novel ultra-stable zeolite, NSZ, rich in secondary pores was developed through the combination of gas-phase andmild hydrothermal methods. This zeolite was successfully tested in an industrial setting for the first time in the world. The porestructure characteristics of the NSZ zeolite prepared for industrial use were analyzed and characterized using BET. The resultsindicate a significant increase in the secondary pore volume of NSZ zeolite compared to the existing ultra-stable zeolite HSZ-5, which is produced through a conventional gas-phase method. The average secondary pore volume to total pore volume ratioin NSZ zeolite was found to be 58.96% higher. The catalytic cracking performance of NSZ zeolite was evaluated. The resultsshowed that the NSC-LTA catalyst, with NSZ as the active component, outperformed the HSC-LTA catalyst with HSZ-5 zeolitein terms of obtaining more high-value products (gasoline and liquefied petroleum gas) during the hydrogenated light cycle oilprocessing. Additionally, the NSC-LTA catalyst showed a significant improvement in coke selectivity.