INVESTIGATION OF THE SCATTERING OF ANTI-PLANE SHEAR WAVES BY TWO GRIFFITH CRACKS USING THE NON-LOCAL THEORY

In this paper, the scattering of harmonic antiplane shear waves bytwo finite cracks is studied using the non-local theory. The Fouriertransform is applied and a mixed boundary value prob- lem isformulated. Then a set of triple integral equations is solved using anew method, namely Schimidt's method. This method is more exact andmore reasonable than Erigen's for solving this Kind of problem. Theresult of the stress near the crack tip was obtained. Contrary to theclassical elas- Ticity solution, it is found that no stresssingularity is present at the crack tip, which can explain theProblem of macroscopic and microscopic mechanics.

ELECTROELASTIC FIELD FOR AN IMPERMEABLE ANTI-PLANE SHEAR CRACK IN A PIEZOELECTRIC CERAMICS PLATE

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four Cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate we obtained, The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h --> infinity.

Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials

The present paper is exposed theoretically to the influence on the dynamic stress intensity factor （DSIF） in the piezoelectric bi-materials model with two symmet- rically permeable interracial cracks near the edges of a circular cavity, subjected to the dynamic incident anti-plane shearing wave （SH-wave）. An available theoretical method to dynamic analysis in the related research field is provided. The formulations are based on Green＇s function method. The DSIFs at the inner and outer tips of the left crack are obtained by solving the boundary value problems with the conjunction and crack- simulation technique. The numerical results are obtained by the FORTRAN language program and plotted to show the influence of the variations of the physical parameters, the structural geometry, and the wave frequencies of incident wave on the dimensionless DSIFs. Comparisons with previous work and between the inner and outer tips are con- cluded.

DYNAMIC ANTI-PLANE PROBLEMS OF PIEZOCERAMICS AND APPLICATIONS IN ULTRASONICS—A REVIEW

We review theoretical results on anti-plane motions of polarized ceramics based on the linear theory of piezoelectricity. Solutions to dynamic problems of the propagation of bulk acoustic waves （BAW） and surface acoustic waves （SAW）, vibrations of finite bodies, and applications to various piezoelectric devices including piezoelectric waveguides, resonators, mass sensors, fluid sensors, actuators, nondestructive evaluation, power harvesters （generators）, transformers, and power transmission through an elastic wall by acoustic waves are discussed. Complications due to material inhomogeneity, initial stress, electromagnetic coupling, electric field gradient and semiconduction are also discussed. The paper cites 82 references.

Dynamic response of cylindrical cavity to anti-plane impact load by using analytical approach

The transient response of an unlimited cylindrical cavity buried in the infinite elastic soil subjected to an anti-plane impact load along the cavern axis direction was studied.Using Laplace transform combining with contour integral of the Laplace inverse transform specifically,the general analytical expressions of the soil displacement and stress are obtained in the time domain,respectively.And the numerical solutions of the problem computed by analytical expressions are presented.In the time domain,the dynamic responses of the infinite elastic soil are analyzed,and the calculation results are compared with those from numerical inversion proposed by Durbin and the static results.One observes good agreement between analytical and numerical inversion results,lending the further support to the method presented.Finally,some valuable shear wave propagation laws are gained: the displacement of the soil remains zero before the wave arrival,and after the shear wave arrival,the stress and the displacement at this point increase abruptly,then reduce and tend to the static value gradually at last.The wave attenuates along the radial,therefore the farther the wave is from the source,the smaller the stress and the displacement are,and the stress and the displacement are just functions of the radial distance from the axis.

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.

COMPLETE EIGEN-SOLUTIONS FOR ANTI-PLANE NOTCHES WITH MULTI-MATERIALS BY SUPER-INVERSE ITERATION

Ln this paper, the super-inverse iterative method is proposed to compute the accurate and complete eigen-solutions for anti-plane cracks/notches with multi-materials, arbitrary opening angles and various surface conditions. Taking the advantage of the knowledge of the variation forms of the eigen-functions, a series of numerical techniques are proposed to simplify the computation and speed up the convergence rare of the inverse iteration. A number of numerical examples are given to demonstrate the excellent accuracy, efficiency and reliability of the proposed approach.

ELECTRIC FIELD GRADIENT EFFECTS IN ANTI-PLANE PROBLEMS OF A CIRCULAR CYLINDRICAL HOLE IN PIEZOELECTRIC MATERIALS OF 6 mm SYMMETRY

We study electromechanical felds in the anti-plane deformation of an infnite medium of piezoelectric materials of 6 mm symmetry with a circular cylindrical hole. The theory of electro- elastic dielectrics with electric feld gradient in the constitutive relations is used. Special attention is paid to the felds near the surface of the hole.

STRESS INTENSITY FACTORS CALCULATION IN ANTI-PLANE FRACTURE PROBLEM BY ORTHOGONAL INTEGRAL EXTRACTION METHOD BASED ON FEMOL

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions （or between the derivatives of eigenfunctions） of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors （SIFs）, so any order SIF can be extracted based on a certain known solution of displacement （an analytic result or a numerical result）. Many numerical examples based on the finite element method of lines （FEMOL） show that the present method is very powerful and efficient.

APPLICATION OF TREFFTZ BEM TO ANTI-PLANE PIEZOELECTRIC PROBLEM

Anti-plane electroelastic problems are studied by the Trefftz boundary element method （BEM） in this paper. The Trefftz BEM is based on a weighted residual formulation and indirect boundary approach. In particular the point-collocation and Galerkin techniques, in which the basic unknowns are the retained expansion coefficients in the system of complete equations, are considered. Furthermore, special Trefftz functions and auxiliary functions which satisfy exactly the specified boundary conditions along the slit boundaries are also used to derive a special purpose element with local defects. The path-independent integral is evaluated at the tip of a crack to determine the energy release rate for a mode Ⅲ fracture problem. In final, the accuracy and efficiency of the Trefftz boundary element method are illustrated by an example and the comparison is made with other methods.

State-of-the-art on the anchorage performance of rock bolts subjected to shear load

Rock bolts are extensively utilized in underground engineering as a means of offering support and stability to rock masses in tunnels,mines,and other underground structures.In environments of high ground stress,faults or weak zones can frequently arise in rock formations,presenting a significant challenge for engineering and potentially leading to underground engineering collapse.Rock bolts serve as a crucial structural element for the transmission of tensile stress and are capable of withstanding shear loads to prevent sliding of weak zones within rock mass.Therefore,a complete understanding of the behavior of rock bolts subjected to shear loads is essential.This paper presents a state-of-the-art review of the research progress of rock bolts subjected to shear load in three categories:experiment,numerical simulation,and analytical model.The review focuses on the research studies and developments in this area since the 1970s,providing a comprehensive overview of numerous factors that influence the anchorage performance of rock bolts.These factors include the diameter and angle of the rock bolt installation,rock strength,grouting material,bolt material,borehole diameter,rock bolt preload,normal stress,joint surface roughness and joint expansion angle.The paper reviews the improvement of mechanical parameter setting in numerical simulation of rock bolt shear.Furthermore,it delves into the optimization of the analytical model concerning rock bolt shear theory,approached from the perspectives of both Elastic foundation beam theory coupled with Elastoplasticity theory and Structural mechanic methods.The significance of this review lies in its ability to provide insights into the mechanical behavior of rock bolts.The paper also highlights the limitations of current research and guidelines for further research of rock bolts.

Mechanical behaviors of backfill-rock composites: Physical shear test and back-analysis

The shear behavior of backfill-rock composites is crucial for mine safety and the management of surface subsidence.For exposing the shear failure mechanism of backfill-rock composites,we conducted shear tests on backfill-rock composites under three constant normal loads,compared with the unfilled rock.To investigate the macro-and meso-failure characteristics of the samples in the shear tests,the cracking behavior of samples was recorded by a high-speed camera and acoustic emission monitoring.In parallel with the experimental test,the numerical models of backfill-rock composites and unfilled rock were established using the discrete element method to analyze the continuous-discontinuous shearing process.Based on the damage mechanics and statistics,a novel shear constitutive model was proposed to describe mechanical behavior.The results show that backfill-rock composites had a special bimodal phenomenon of shearing load-deformation curve,i.e.the first shearing peak corresponded to rock break and the second shearing peak induced by the broken of aeolian sand-cement/fly ash paste backfill.Moreover,the shearing characteristic curves of the backfill-rock composites could be roughly divided into four stages,i.e.the shear failure of the specimens experienced:stage I:stress concentration;stage II:crack propagation;stage III:crack coalescence;stage IV:shearing friction.The numerical simulation shows that the existence of aeolian sand-cement/fly ash paste backfill inevitably altered the coalescence type and failure mode of the specimens and had a strengthening effect on the shear strength of backfillrock composites.Based on damage mechanics and statistics,a shear constitutive model was proposed to describe the shear fracture characteristics of specimens,especially the bimodal phenomenon.Finally,the micro-and meso-mechanisms of shear failure were discussed by combining the micro-test and numerical results.The research can advance the better understanding of the shear behavior of backfill-rock composites and contribute to the safety of mining engineering.

Estimating shear strength of high-level pillars supported with cemented backfilling using the HoekeBrown strength criterion

Deep metal mines are often mined using the high-level pillars with subsequent cementation backfilling(HLSCB)mining method.At the design stage,it is therefore important to have a reasonable method for determining the shear strength of the high-level pillars(i.e.cohesion and internal friction angle)when they are supported by cemented backfilling.In this study,a formula was derived for the upper limit of the confining pressure σ3max on a high-level pillar supported by cemented backfilling in a deep metal mine.A new method of estimating the shear strength of such pillars was then proposed based on the Hoek eBrown failure criterion.Our analysis indicates that the horizontal stress σhh acting on the cemented backfill pillar can be simplified by expressing it as a constant value.A reasonable and effective value for σ3max can then be determined.The value of s3max predicted using the proposed method is generally less than 3 MPa.Within this range,the shear strength of the high-level pillar is accurately calculated using the equivalent MohreCoulomb theory.The proposed method can effectively avoid the calculation of inaccurate shear strength values for the high-level pillars when the original HoekeBrown criterion is used in the presence of large confining pressures,i.e.the situation in which the cohesion value is too large and the friction angle is too small can effectively be avoided.The proposed method is applied to a deep metal mine in China that is being excavated using the HLSCB method.The shear strength parameters of the high-level pillars obtained using the proposed method were input in the numerical simulations.The numerical results show that the recommended level heights and sizes of the high-level pillars and rooms in the mine are rational.

A multifunctional shear apparatus for rocks subjected to true triaxial stress and high temperature in real-time

Deep engineering disasters,such as rockbursts and collapses,are more related to the shear slip of rock joints.A novel multifunctional device was developed to study the shear failure mechanism in rocks.Using this device,the complete shearedeformation process and long-term shear creep tests could be performed on rocks under constant normal stiffness(CNS)or constant normal loading(CNL)conditions in real-time at high temperature and true-triaxial stress.During the research and development process,five key technologies were successfully broken through:(1)the ability to perform true-triaxial compressioneshear loading tests on rock samples with high stiffness;(2)a shear box with ultra-low friction throughout the entire stress space of the rock sample during loading;(3)a control system capable of maintaining high stress for a long time and responding rapidly to the brittle fracture of a rock sample as well;(4)a refined ability to measure the volumetric deformation of rock samples subjected to true triaxial shearing;and(5)a heating system capable of maintaining uniform heating of the rock sample over a long time.By developing these technologies,loading under high true triaxial stress conditions was realized.The apparatus has a maximum normal stiffness of 1000 GPa/m and a maximum operating temperature of 300C.The differences in the surface temperature of the sample are constant to within5C.Five types of true triaxial shear tests were conducted on homogeneous sandstone to verify that the apparatus has good performance and reliability.The results show that temperature,lateral stress,normal stress and time influence the shear deformation,failure mode and strength of the sandstone.The novel apparatus can be reliably used to conduct true-triaxial shear tests on rocks subjected to high temperatures and stress.

Cyclic shear behavior of en-echelon joints under constant normal stiffness conditions

To reveal the mechanism of shear failure of en-echelon joints under cyclic loading,such as during earthquakes,we conducted a series of cyclic shear tests of en-echelon joints under constant normal stiffness(CNS)conditions.We analyzed the evolution of shear stress,normal stress,stress path,dilatancy characteristics,and friction coefficient and revealed the failure mechanisms of en-echelon joints at different angles.The results show that the cyclic shear behavior of the en-echelon joints is closely related to the joint angle,with the shear strength at a positive angle exceeding that at a negative angle during shear cycles.As the number of cycles increases,the shear strength decreases rapidly,and the difference between the varying angles gradually decreases.Dilation occurs in the early shear cycles(1 and 2),while contraction is the main feature in later cycles(310).The friction coefficient decreases with the number of cycles and exhibits a more significant sensitivity to joint angles than shear cycles.The joint angle determines the asperities on the rupture surfaces and the block size,and thus determines the subsequent shear failure mode(block crushing and asperity degradation).At positive angles,block size is more greater and asperities on the rupture surface are smaller than at nonpositive angles.Therefore,the cyclic shear behavior is controlled by block crushing at positive angles and asperity degradation at negative angles.

ANTI-PLANE CRACK MOVING ALONG THE INTERFACE OF DISSIMILAR PIEZOELECTRIC MATERIALS

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement around the crack tip are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will be reduced to the result far the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials.

Failure transition of shear-to-dilation band of rock salt under triaxial stresses

Great potential of underground gas/energy storage in salt caverns seems to be a promising solution to support renewable energy.In the underground storage method,the operating cycle unfortunately may reach up to daily or even hourly,which generates complicated pressures on the salt cavern.Furthermore,the mechanical behavior of rock salt may change and present distinct failure characteristics under different stress states,which affects the performance of salt cavern during the time period of full service.To reproduce a similar loading condition on the cavern surrounding rock mass,the cyclic triaxial loading/unloading tests are performed on the rock salt to explore the mechanical transition behavior and failure characteristics under different confinement.Experimental results show that the rock salt samples pre-sent a diffused shear failure band with significant bulges at certain locations in low confining pressure conditions(e.g.5 MPa,10 MPa and 15 MPa),which is closely related to crystal misorientation and grain boundary sliding.Under the elevated confinement(e.g.20 MPa,30 MPa and 40 MPa),the dilation band dominates the failure mechanism,where the large-size halite crystals are crushed to be smaller size and new pores are developing.The failure transition mechanism revealed in the paper provides additional insight into the mechanical performance of salt caverns influenced by complicated stress states.

Anisotropic time-dependent behaviors of shale under direct shearing and associated empirical creep models

Understanding the anisotropic creep behaviors of shale under direct shearing is a challenging issue.In this context,we conducted shear-creep and steady-creep tests on shale with five bedding orientations (i.e.0°,30°,45°,60°,and 90°),under multiple levels of direct shearing for the first time.The results show that the anisotropic creep of shale exhibits a significant stress-dependent behavior.Under a low shear stress,the creep compliance of shale increases linearly with the logarithm of time at all bedding orientations,and the increase depends on the bedding orientation and creep time.Under high shear stress conditions,the creep compliance of shale is minimal when the bedding orientation is 0°,and the steady-creep rate of shale increases significantly with increasing bedding orientations of 30°,45°,60°,and 90°.The stress-strain values corresponding to the inception of the accelerated creep stage show an increasing and then decreasing trend with the bedding orientation.A semilogarithmic model that could reflect the stress dependence of the steady-creep rate while considering the hardening and damage process is proposed.The model minimizes the deviation of the calculated steady-state creep rate from the observed value and reveals the behavior of the bedding orientation's influence on the steady-creep rate.The applicability of the five classical empirical creep models is quantitatively evaluated.It shows that the logarithmic model can well explain the experimental creep strain and creep rate,and it can accurately predict long-term shear creep deformation.Based on an improved logarithmic model,the variations in creep parameters with shear stress and bedding orientations are discussed.With abovementioned findings,a mathematical method for constructing an anisotropic shear creep model of shale is proposed,which can characterize the nonlinear dependence of the anisotropic shear creep behavior of shale on the bedding orientation.

Anti-plane pull-out of a rigid line inclusion from an elastic medium

The transient and static anti-plane problem of a rigid line inclusion pulled out from an elastic medium is studied.The singular integral equation method is used to solve the stress field.Under the static load,the stress intensity factor(SIF)at the inclusion tips increases with the medium length.The problem becomes equivalent to an inclusion in a medium with an infinite length when the length of the medium is 3.5times longer than that of the inclusion.However,under the transient load,the maximum value of the SIF occurs when the medium length is about two times the inclusion length.Besides,the relation between the pull-out force and the anti-plane displacement is given.The conclusions are useful in guiding the design of fiber reinforced composite materials.

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode Ⅲ conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor （SIF） results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.