Some recent advances in 3D crack and contact analysis of elastic solids with transverse isotropy and multifield coupling

Significant progress has been made in mixed boundary-value problems associated with three-dimensional（3D） crack and contact analyses of advanced materials featuring more complexities compared to the conventional isotropic elastic materials.These include material anisotropy and multifield coupling,two typical characteristics of most current multifunctional materials.In this paper we try to present a state-of-the-art description of 3D exact/analytical solutions derived for crack and contact problems of elastic solids with both transverse isotropy and multifield coupling in the latest decade by the potential theory method in the spirit of V.I.Fabrikant.whose ingenious breakthrough brings new vigor and vitality to the old research subject of classical potential theory.We are particularly interested in crack and contact problems with certain nonlinear features.Emphasis is also placed on the coupling between the temperature field（or the like） and other physical fields（e.g.,elastic,electric,and magnetic fields）.We further highlight the practical significance of 3D contact solutions,in particular in applications related to modern scanning probe microscopes.

Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy

The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.

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.

Anisotropy measurements and characterization of the Qingshankou shale

Qingshankou shale(Gulong area,China)exhibits strong acoustic anisotropy characteristics,posing significant challenges to its exploration and development.In this study,the five full elastic constants and multipole response law of the Qingshankou shale were studied using experimental measurements.Analyses show that the anisotropy parametersϵandγin the study region are greater than 0.4,whereas the anisotropy parameterδis smaller,generally 0.1.Numerical simulations show that the longitudinal and transverse wave velocities of these strong anisotropic rocks vary significantly with inclination angle,and significant differences in group velocity and phase velocity are also present.Acoustic logging measures the group velocity in dipped boreholes;this differs from the phase velocity to some extent.As the dip angle increases,the longitudinal and SH wave velocities increase accordingly,while the qSV-wave velocity initially increases and then decreases,reaching its maximum value at a dip of approximately 40°.These results provide an effective guide for the correction and modeling of acoustic logging time differences in the region.

Characteristics of dispersion curves for Love channel waves in transversely isotropic media

Coal seams have a pronounced bedding structure with developed cracks and exhibit signifi cant anisotropy.However,few studies have examined the frequency dispersion properties of channel waves in anisotropic coal seams.In this study,numerical solutions are calculated using the generalized reflection–transmission coefficient method for the dispersion curves of Love channel waves in vertical transversely isotropic(VTI)and horizontal transversely isotropic(HTI)medium models.Moreover,the frequency dispersion characteristics of Love channel waves in several typical transversely isotropic models are analyzed.We fi nd that the dispersion curves for isotropic and VTI media diff er signifi cantly.In addition,the phase and Airy-phase velocities in VTI media are higher than those in isotropic media.Thus,neglecting this difference in practical channel wave detection will result in large detection errors.The dispersion curves for the isotropic and HTI media do not differ signifi cantly,and the Airy-phase velocities of various modes are similar.The group-velocity curve for a coal seam model containing a dirt band is found to be extremely irregular.The fundamental-mode Airy phase is not pronounced,but the fi rst-mode Airy phase can be clearly observed.Hence,fi rst-mode channel waves are suitable for detecting dirt bands.

ON THE BENDING, VIBRATION AND STABILITY OF LAMINATED RECTANGULAR PLATES WITH TRANSVERSELY ISOTROPIC LAYERS

A method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges. By introducing two displacement functions and two stress functions, two independent state equations were constructed based on the three-dimensional elasticity equations for transverse isotropy. The original differential equations are thus decoupled with the order reduced that will facilitate obtaining solutions of various problems. For the simply supported rectangular plate, two relations between the state variables at the tap and bottom surfaces were established. In particular, for the free vibration ( stability) problem, it is found that there exist two independent classes: One corresponds to the pure in-plane vibration (stability) and the other to the general bending vibration (stability). Numerical examples are finally presented and the effects of same parameters are discussed.

Line-integral representations for extended displacements, stresses,and interaction energy of arbitrary dislocation loops in transversely isotropic magneto-electro-elastic bimaterials

In addition to the hexagonal crystals of class 6 mm, many piezoelectric materials （e.g., BaTiO3）, piezomagnetic materials （e.g., CoFe2O4）, and multiferroic com-posite materials （e.g., BaTiO3-CoFe2O4 composites） also exhibit symmetry of transverse isotropy after poling, with the isotropic plane perpendicular to the poling direction. In this paper, simple and elegant line-integral expressions are derived for extended displace-ments, extended stresses, self-energy, and interaction energy of arbitrarily shaped, three-dimensional （3D） dislocation loops with a constant extended Burgers vector in trans-versely isotropic magneto-electro-elastic （MEE） bimaterials （i.e., joined half-spaces）. The derived solutions can also be simply reduced to those expressions for piezoelectric, piezo-magnetic, or purely elastic materials. Several numerical examples are given to show both the multi-field coupling effect and the interface/surface effect in transversely isotropic MEE materials.

Application of a transversely isotropic brittle rock mass model in roof support design

Accurate modelling of the potential failure modes in the rock mass is an essential task towards a robust design of roof support systems in coal mines.The use of generalised rock mass properties based on averaged properties(e.g.Hoek-Brown model) has been found to limit the capability to reproduce the actual rock mass behaviour which may include a wide range of interacting and complex failure mechanisms such as shear and tension fracturing of the intact rock and shear and separation of pre-existing discontinuities,including re-activation.Recent studies have also shown that traditional models,such as the Mohr-Coulomb,may not accurately describe the behaviour of the intact rock,particularly for stress induced failures where spalling and slabbing are observed.This is mainly due to the cohesion and friction components of the shear strength of the intact rock not being mobilised at the same rate with strain-softening of the former component playing an essential role in the post peak behaviour.In addition,coal measure rocks are often transversely isotropic,both by way of the preferred orientation of clay particles within the finer grained lithology and by bedding textures and bedding partings,and this is often ignored in computer simulations.A newly developed transversely isotropic brittle rock mass model is applied in the simulation of a hypothetical and simple roadway development.A Cohesion Weakening-Friction Strengthening(CWFS) approach is adopted to describe the intact rock where the mobilisation and strain-softening of the two shear strength components are linked to plastic deformation.The impacts of anisotropy and brittle rock on the development of the excavation disturbed zone or height of softening,as often referred to,are investigated and their implication in the roof support design discussed.

Analysis of stresses at the center of transversely isotropic Brazilian disk

This article presents the stresses at the center of a Brazilian disk(BD)for transversely isotropic rocks.It is shown that the solution of stresses at the center of an anisotropic disk is a function of the disk radius and the magnitude of applied load,as well as the material orientation with respect to the load axis and two dimensionless ratios with specific physical meanings and limitations.These two dimensionless parameters are the ratios of Young’s modulus and apparent shear modulus,although the ratio of apparent shear modulus will be eliminated if the Saint-Venant assumption is considered.Considerable finite element simulations are carried out to find the stresses at the disk center concerning the material orientation and the two dimensionless parameters.Also,an approximate formula obtained from analytical results,previously proposed in the literature for solving the tensile and compressive stresses at the disk center,is re-written and simplified based on these new definitions.The results of the approximate formula fitted to the analytical results are compared to those obtained from numerical solutions,suggesting a good agreement between the numerical and analytical methods.An approximate equation for the shear stress at the disk center is also formulated based on the numerical results.Finally,the influence of the assumptions for simplification of the proposed formula for the tensile,compressive,and shear stresses at the disk center is discussed,and simple and practical equations are proposed as estimations for the stresses at the center of the BD specimen for low to moderate anisotropic rocks.For highly anisotropic rocks,the reference plots can be used for more accuracy.

Propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space

The purpose of this research is to study the effect of voids on the surface wave propagation in a layer of a transversely isotropic thermoelastic material with voids lying over an isotropic elastic half-space. The frequency equation is derived after developing a mathematical model for welded and smooth contact boundary conditions. The dispersion curves giving the phase velocity and attenuation coefficient via wave number are plotted graphically to depict the effects of voids and anisotropy for welded contact boundary conditions. The specific loss and amplitudes of the volume fraction field, the normal stress, and the temperature change for welded contact are obtained and shown graphically for a particular model to depict the voids and anisotropy effects. Some special cases are also deduced from the present investigation.

Theoretical dispersion curves for borehole real-valued wave modes in vertically transverse isotropic formations

The dispersion curves of real-valued modes in a fluid-filled borehole are widely used in acoustic well logging.The accurate dispersion curves are the precondition of theoretical analysis and inversion process.Generally,these curves can be obtained by solving the conventional dispersion equation for isotropic formations and most vertically transverse isotropy(VTI)formations.However,if the real-valued solutions exist when the radial wavenumbers for the formation quasi-P and quasi-S equals to each other,the existed methods based on the conventional dispersion equation could lead to incorrect results for some VTI formations.Few studies have focused on the influence of these real-valued solutions on dispersion curve extraction.To remove these real-valued solutions,we have proposed a modified dispersion equation and its corresponding solving process.When solving the dispersion equation,the Scholte wave velocity of VTI formation at high frequency is used as the initial guess.The two synthetic examples including fast and slow VTI formations validate that these real-valued solutions do not contribute to the wavefield,and the new dispersion curve extraction method is suitable for all kinds of VTI formations.Consequently,the method can provide reliable dispersion curves for both theoretical analysis and anisotropic parameters inversion in VTI formations.

THE 3-D STRESS INTENSITY FACTOR FOR A HALF PLANE CRACK IN A TRANSVERSELY ISOTROPIC SOLID DUE TO THE MOTION OF LOADS ON THE CRACK FACES

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of normal point loads moving in a direction perpendicular to the crack edge on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Some features of the solution are discussed through numerical results.

Anisotropy and Empirical Relations for the Estimation of Anisotropy Parameters in Niger Delta Depobelts

The difficulty in achieving well-to-seismic ties due to errors arising from wrong time-to-depth conversions has been as a result of ignoring anisotropy in seismic processing. Anisotropy plays a vital role in the processing and interpretation of seismic data. In this work, an inversion method based on the elastic stiffness tensors was adopted to estimate and quantify anisotropy in two depobelts using petrophysical well logs in Niger delta (Central Swamp and Greater Ughelli). Results show that the estimated delta (δ), epsilon (ε), gamma (γ) and eta (η) exhibit a high degree of anisotropy in the shales than in the sands. The parameters were observed to be higher in the Central Swamp than the Greater Ughelli depobelt. This behavior could be associated with the alternating sequence of massive shale and sand beds geologically observed within this depobelt. This work was also able to derive empirical relations that could be used in estimating these parameters in the depobelts once appropriate information or data for any one parameter is available.

The reflection and transmission of elastic waves at a interface between a transversely isotropi celastic soils and saturated soils

According to Biot′s wave equation of transversely isotropic saturated soil, this paper deduces the general equation of the reflection coefficients and transmission coefficients when qP 1 wave goes through from saturated soil to elastic media. The effects of anisotropies and boundary drainage condition on reflection coefficients and transmission coefficients are analyzed by numerical method. The idea of this paper can be applied to the case when qSV wave or qP 2 wave goes through from saturated soils to elastic soils.

Effects of 3D deformation and nonlinear stress–strain relationship on the Brazilian test for a transversely isotropic rock

To improve the accuracy of indirect tensile strength for a transversely isotropic rock in the Brazilian test, this study considered the three-dimensional (3D) deformation and the nonlinear stress–strain relationship. A parametric study of a numerical Brazilian test was performed for a general range of elastic constants, revealing that the 3D modeling evaluated the indirect tensile strength up to 40% higher than the plane stress modeling. For the actual Asan gneiss, the 3D model evaluated the indirect tensile strength up to 10% higher and slightly enhanced the accuracy of deformation estimation compared with the plane stress model. The nonlinearity in stress–strain curve of Asan gneiss under uniaxial compression was then considered, such that the evaluated indirect tensile strength was affected by up to 10% and its anisotropy agreed well with the physical intuition. The estimation of deformation was significantly enhanced. The further validation on the nonlinear model is expected as future research.

Vibration of an infinite inhomogeneous transversely isotropic viscoelastic medium with cylindrical hole

This paper investigates the influences of higher order viscoelasticity and the inhomogeneities of the transversely isotropic elastic parameters on the disturbances in an infinite medium, caused by the presence of a transient radial force or twist on the surface of a cylindrical hole with circular cross section. Following Voigt＇s model for higher order viscoelasticity, the nonvanishing stress components valid for a transversely isotropic and higher order viscoelastic solid medium have been deduced in terms of radial displacement component. Considering the power law variation of elastic and viscoelastic parameters, the stress equation of motion has been developed. Solving this equation under suitable boundary conditions, due to transient forces and twists, radial displacement and relevant stress components have been determined in terms of modified Bessel functions. The problem for the presence of transient radial force has been numerically analysed. Modulations of displacement and stresses due to different order of viscoelasticity and inhomogeneity have been graphically depicted. The numerical study of the disturbance caused by the presence of twist on the surface may be similarly done but is not pursued in this paper.

POINT FORCE SOLUTION FOR A TRANSVERSELY ISOTROPIC ELASTIC LAYER

By introduction of transmitting matrices'technique for layered structure,mixed equations with stresses and displacements are derived from the basic equations of transversely isotropic elasticity.Then,using Fourier transformation and the general solutions in Zhou et al.[7],the point force solution for transversely isotropic elastic layer is obtained and it can be degenerated to the corresponding solution of isotropic medium. In this paper,all equations are derived by the use of computer algebra software.

STRESS IN TRANSVERSELY ISOTROPIC HALF-SPACE WITH TYPICAL LOADS ACTING ON ITS SURFACE

Based on the results of Hu and Lekhnitskii, the united solution of additional vertical stress coefficient for both transversely isotropic and isotropic half-space was obtained. Five typical load cases, namely, vertical circular uniform load, rectangular uniform load, linearly distributed rectangular load, uniform linear and strip loads are studied in detail. The final solutions are expressed in terms of elementary functions. Numerical results show that there are anisotropic effects on the variation of additional vertical stress coefficients.

Phase velocity in 2D TTI media

We derive an expression for phase velocity in 2D tilted transverse isotropy （TTI） media. Snapshots of phase velocity in TTI and transverse isotropy （TI） model media are simulated and analyzed using the derived expression. In addition, the x-component character differences between the modeled phase velocities of the two media models are compared and analyzed.

BENDING OF FUNCTIONALLY GRADED PIEZOELECTRIC RECTANGULAR PLATES

By introducing two displacement functions as well as two stressfunctions, two independent state equations with variable coefficientsare derived from the three-dimensional theory equations of piezo-elasticity for transverse isotropy. A laminated approximation is usedto transform the state equations to those with constant coefficientsin each sub-layer. The bending problem of a functionally gradedrectangular plate is then analyzed based on the state equations.Numerical results are presented and the effect of material gradi- entindex is discussed.