Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

Poroelastodynamic responses and elastic moduli of a transversely isotropic porous cylinder under forced deformation test

Existing transversely isotropic poroelastodynamics solutions are limited to infinite domains and without experimental validation. Furthermore, there is a lack of analytical simulations for the elastic moduli dispersion of fluid-saturated porous cylinders. To address these three limitations and investigate the mechanisms of moduli dispersion, we present the analytical solutions of the poromechanical responses and the elastic moduli dispersion of a transversely isotropic, fluid-saturated, finite porous cylinder subjected to a forced deformation test. Through an example, we demonstrate the effects of loading frequency, boundary conditions, and material's anisotropy, dimension, and permeability on the responses of pore pressure,force, displacement, and dynamic elastic moduli of the cylinder. The specimen's responses are significantly influenced by the frequency of the applied load, resulting in a drained state at low frequencies and an undrained state at high frequencies. At high frequencies, the sample behaves identically for an open or a closed lateral boundary, and permeability has insignificant effects. The dynamic elastic moduli are mainly controlled by the loading frequency and the ratio of the sample's radius to its height. Lastly,we show excellent matches between the newly derived analytical solution and laboratory measurements on one clay and two shale samples from Mont Terri.

Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme

To deal with the numerical dispersion problem, by combining the staggeredgrid technology with the compact finite difference scheme, we derive a compact staggered- grid finite difference scheme from the first-order velocity-stress wave equations for the transversely isotropic media. Comparing the principal truncation error terms of the compact staggered-grid finite difference scheme, the staggered-grid finite difference scheme, and the compact finite difference scheme, we analyze the approximation accuracy of these three schemes using Fourier analysis. Finally, seismic wave numerical simulation in transversely isotropic （VTI） media is performed using the three schemes. The results indicate that the compact staggered-grid finite difference scheme has the smallest truncation error, the highest accuracy, and the weakest numerical dispersion among the three schemes. In summary, the numerical modeling shows the validity of the compact staggered-grid finite difference scheme.

Torsional vibration of a pipe pile in transversely isotropic saturated soil

This study considers the torsional vibration of a pipe pile in a transversely isotropic saturated soil layer. Based on Biot＇s poroelastic theory and the constitutive relations of the transversely isotropic medium, the dynamic governing equations of the outer and inner transversely isotropic saturated soil layers are derived. The Laplace transform is used to solve the governing equations of the outer and inner soil layers. The dynamic torsional response of the pipe pile in the frequency domain is derived utilizing 1D elastic theory and the continuous conditions at the interfaces between the pipe pile and the soils. The time domain solution is obtained by Fourier inverse transform. A parametric study is conducted to demonstrate the influence of the anisotropies of the outer and inner soil on the torsional dynamic response of the pipe pile.

THE NON-AXISYMMETRICAL DYNAMIC RESPONSE OF TRANSVERSELY ISOTROPIC SATURATED POROELASTIC MEDIA

The Blot's wave equations of transversely isotropic saturated poroelastic media excited hy non-axisymmetrical harmonic source were solved by means of Fourier expansion and Hankel transform. Then the components of total stress in porous media are expressed with the solutions of Biot's wave equations. The method of research on non-axisymmetrical dynamic response of saturated porous media is discussed, and a numerical result is presented.

Dynamic analysis of slab track on multi-layered transversely isotropic saturated soils subjected to train loads

The dynamic responses of a slab track on transversely isotropic saturated soils subjected to moving train loads are investigated by a semi-analytical approach. The track model is described as an upper Euler beam to simulate the rails and a lower Euler beam to model the slab. Rail pads between the rails and slab are represented by a continuous layer of springs and dashpots. A series of point loads are formulated to describe the moving train loads. The governing equations of track-ground systems are solved using the double Fourier transform, and the dynamic responses in the time domain are obtained by the inverse Fourier transform. The results show that a train load with high velocity will generate a larger response in transversely isotropic saturated soil than the lower velocity load, and special attention should be paid on the pore pressure in the vicinity of the ground surface. The anisotropic parameters of a surface soil layer will have greater influence on the displacement and excess pore water pressure than those of the subsoil layer. The traditional design method taking ground soil as homogeneous isotropic soil is unsafe for the case of RE 〈 1 and RG 〈 1, so a transversely isotropic foundation model is of great significance to the design for high train velocities.

Diffraction of SH-waves by topographic features in a layered transversely isotropic half-space

Abstract： The scattering of plane SH-waves by topographic features in a layered transversely isotropic （TI） half-space is investigated by using an indirect boundary element method （IBEM）. Firstly, the anti-plane dynamic stiffness matrix of the layered TI half-space is established and the free fields are solved by using the direct stiffness method. Then, Green＇s functions are derived for uniformly distributed loads acting on an inclined line in a layered TI half-space and the scattered fields are constructed with the deduced Green＇s functions. Finally, the free fields are added to the scattered ones to obtain the global dynamic responses. The method is verified by comparing results with the published isotropic ones. Both the steady-state and transient dynamic responses are evaluated and discussed. Numerical results in the frequency domain show that surface motions for the TI media can be significantly different from those for the isotropic case, which are strongly dependent on the anisotropy property, incident angle and incident frequency. Results in the time domain show that the material anisotropy has important effects on the maximum duration and maximum amplitudes of the time histories.

Deformation analysis of transversely isotropic coal-rock mass with porous and cracks

Coal-rock as a typical sedimentary rock has obvious stratification,namely it has transversely isotropic feature.Meanwhile,deformation leads to coal-rock mass having the characteristics of different porous and crack structures as well as local anisotropy.Equivalent axial and circumferential strain' formulas of the pure coal-rock mass specimen with a single crack were derived through the establishment of equivalent mechanical model of standard cylindrical coal-rock specimen,and have been widely used to a variety of media combined different structures containing multiple cracks.The complete stress strain curve of a real coal-rock specimen was obtained by the CTC test.Additionally,according to the comparison with the theoretical value,the theoretical mechanical model could well explain the deformation characteristics of coal-rock mass and verify its validity.Further,following features were analyzed:strain normalized coefficient and elastic modulus(Poisson's ratio) in vertical and parallel direction to the stratification,stratification angle,porosity,pore radius,normal and tangential stiffness of crack,and the relationship of different crack width with different tangential stiffness of crack.Through the analysis above,it substantiate this claim that the theoretical model with better reliability reflects the transversely isotropic nature of the coal-rock and the local anisotropy caused by the porous and cracks.

DYNAMIC CONTACT STIFFNESS OF VIBRATING RIGID SPHERE CONTACTING SEMI-INFINITE TRANSVERSELY ISOTROPIC VISCOELASTIC SOLID

Dynamic contact stiffness at the interface between a vibrating rigid sphere and a semi-infinite transversely isotropic viscoelastic solid is investigated. An oscillating force superimposed onto a static compressive force in the vertical direction excites the vibration of a rigid sphere, which causes variable contact radius and contact pressure distribution in the contact region. The assumption of a sufficiently small oscillating force yields a dynamic contact-pressure distribution of a constant contact radius, which gives dynamic contact stiffness at the interface between the rigid sphere and the semi-infinite solid. Numerical calculations show the influence of vibration frequency of the sphere, and elastic constants of the transversely isotropic solid on dynamic contact stiffness, which benefits quantitative evaluation of elastic constants and orientation of single hexagonal grains by resonance-frequency shifts of the oscillator in resonance ultrasound microscopy.

A modified failure criterion for transversely isotropic rocks

A modified failure criterion is proposed to determine the strength of transversely isotropic rocks. Me-chanical properties of some metamorphic and sedimentary rocks including gneiss, slate, marble, schist, shale, sandstone and limestone, which show transversely isotropic behavior, were taken into consider-ation. Afterward, introduced triaxial rock strength criterion was modified for transversely isotropic rocks. Through modification process an index was obtained that can be considered as a strength reduction parameter due to rock strength anisotropy. Comparison of the parameter with previous anisotropy in-dexes in literature showed reasonable results for the studied rock samples. The modified criterion was compared to modified Hoek-Brown and Ramamurthy criteria for different transversely isotropic rocks. It can be concluded that the modified failure criterion proposed in this study can be used for predicting the strength of transversely isotropic rocks.

3-D DYNAMIC RESPONSE OF TRANSVERSELY ISOTROPIC SATURATED SOILS

A study on dynamic response of transversely isotropic saturated poroelastic media under a circular non-axisymmetrical harmonic source has been presented by Huang Yi et al. using the technique of Fourier expansion and Hankel transform. However, the method may not always be valid. The work is extended to the general case being in the rectangular coordinate. The purpose is to study the 3-d dynamic response of transversely isotropic saturated soils under a general source distributing in arbitrary rectangular zoon on the medium surface. Based on Biot＇s theory for fluid- saturated porous media, the 3-d wave motion equations in rectangular coordinate for transversely isotropic saturated poroelastic media were transformed into the two uncoupling governing differential equations of 6-order and 2-order respectively by means of the displacement functions. Then, using the technique of double Fourier transform, the governing differential equations were easily solved. Integral solutions of soil skeleton displacements and pore pressure as well as the total stresses for poroelastic media were obtained. Furthermore, a systematic study on half-space problem in saturated soils was performed. Integral solutions for surface displacements under the general harmonic source distributing on arbitrary surface zone, considering both case of drained surface and undrained surface, were presented.

In-plane dynamic Green's functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic（TI）half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf（1985）,and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.

Pure bending of simply supported circular plate of transversely isotropic functionally graded material

This paper considers the pure bending problem of simply supported transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate. First, the partial differential equation, which is satisfied by the stress functions for the axisymmetric deformation problem is derived. Then, stress functions are obtained by proper manipulation. The analytical expressions of axial force, bending moment and displacements are then deduced through integration. And then, stress functions are employed to solve problems of transversely isotropic functionally graded circular plate, with the integral constants completely determined from boundary conditions. An elasticity solution for pure bending problem, which coincides with the available solution when degenerated into the elasticity solutions for homogenous circular plate, is thus obtained. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a simply supported circular plate of transversely isotropic functionally graded material (FGM).

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.

EXACT SOLUTIONS FOR FREE VIBRATIONS OF TRANSVERSELY ISOTROPIC CIRCULAR PLATES

By virtue of the general solution of dynamic elasticity equations for transverse isotropy as well as the variable separation method, three-dimensional exact solutions of circular plates are obtained under two types of boundary conditions. The solutions can consider both axisymmetric and non-axisymmetric cases. Solutions based on the classical plate theory and Mindlin plate theory are also presented under the corresponding boundary conditions. Numerical results are finally presented and comparisons between the three theories are made.

EXACT SOLUTION TO THE PROBLEM OF A HALF-PLANE CRACK IN A TRANSVERSELY ISOTROPIC PIEZOELECTRIC BODY SUBJECTED TO ANTISYMMETRIC TANGENTIAL POINT FORCES

An exact and complete solution of the problem of a half-planecrack in an infinite transversely isotropic piezoelectric body ispresented. The upper and lower crack faces are assumed to be loadedantisym- metrically by a couple of tangential point forces inopposite directions. The solution is derived through a lim- itingprocedure from that of a penny-shaped crack. The expressions for theelectroelastic field are given in terms of elementary functions.Finally, the numerical results of the second and third mode stressintensity factors k_2 and k_3 of piezoelectric materials and elasticmaterials are compared in figures.

Non-axisymmetrical vibration of elastic circular plate on layered transversely isotropic saturated ground

The non-axisymmetrical vibration of elastic circular plate resting on a layered transversely isotropic saturated ground was studied. First, the 3-d dynamic equations in cylindrical coordinate for transversely isotropic saturated soils were transformed into a group of governing differential equations with 1-order by the technique of Fourier expanding with respect to azimuth, and the state equation is established by Hankel integral transform method, furthermore the transfer matrixes within layered media are derived based on the solutions of the state equation. Secondly, by the transfer matrixes, the general solutions of dynamic response for layered transversely isotropic saturated ground excited by an arbitrary harmonic force were established under the boundary conditions, drainage conditions on the surface of ground as well as the contact conditions. Thirdly, the problem was led to a pair of dual integral equations describing the mixed boundaryvalue problem which can be reduced to the Fredholm integral equations of the second kind solved by numerical procedure easily. At the end of this paper, a numerical result concerning vertical and radical displacements both the surface of saturated ground and plate is evaluated.

Nonlinear amplitude versus angle inversion for transversely isotropic media with vertical symmetry axis using new weak anisotropy approximation equations

In VTI media,the conventional inversion methods based on the existing approximation formulas are difficult to accurately estimate the anisotropic parameters of reservoirs,even more so for unconventional reservoirs with strong seismic anisotropy.Theoretically,the above problems can be solved by utilizing the exact reflection coefficients equations.However,their complicated expression increases the difficulty in calculating the Jacobian matrix when applying them to the Bayesian deterministic inversion.Therefore,the new reduced approximation equations starting from the exact equations are derived here by linearizing the slowness expressions.The relatively simple form and satisfactory calculation accuracy make the reduced equations easy to apply for inversion while ensuring the accuracy of the inversion results.In addition,the blockiness constraint,which follows the differentiable Laplace distribution,is added to the prior model to improve contrasts between layers.Then,the concept of GLI and an iterative reweighted least-squares algorithm is combined to solve the objective function.Lastly,we obtain the iterative solution expression of the elastic parameters and anisotropy parameters and achieve nonlinear AVA inversion based on the reduced equations.The test results of synthetic data and field data show that the proposed method can accurately obtain the VTI parameters from prestack AVA seismic data.

Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium

The governing equations of a transversely isotropic dissipative medium are solved analytically to obtain the speeds of plane waves. The appropriate solutions satisfy the required boundary conditions at the stress-free surface to obtain the expressions of the reflection coefficients of reflected quasi-P (qP) and quasi-SV (qSV) waves in closed form for the incidence of qP and qSV waves. A particular model is chosen for numerical computation of these reflection coefficients for a certain range of the angle of incidence. The numerical values of these reflection coefficients are shown graphically against the angle of incidence for different values of initial stress parameter. The impact of initial stress parameter on the reflection coefficients is observed significantly.

THE NON-AXISYMMETRICAL DYNAMIC RESPONSE OF TRANSVERSELY ISOTROPIC SATURATED POROELASTIC MEDIA

The Biot’s wave equations of transversely isotropic saturated poroelastic media excited by non_axisymmetrical harmonic source were solved by means of Fourier expansion and Hankel transform. Then the components of total stress in porous media are expressed with the solutions of Biot’s wave equations. The method of research on non_axisymmetrical dynamic response of saturated porous media is discussed, and a numerical result is presented.