Elastic solutions for a transversely isotropic half-space subjected to a point load

We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the effect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite different for the displacement and stress calculated from isotropic solutions. © 1998 John Wiley & Sons, Ltd.     

Elastic solutions for a transversely isotropic half-space subjected to buried asymmetric-loads

Elastic closed-form solutions for the displacements and stresses in a transversely isotropic half-space subjected to various buried loading types are presented. The loading types include finite line loads and asymmetric loads (such as uniform and linearly varying rectangular loads, or trapezoidal loads). The planes of transverse isotropy are assumed to be parallel to its horizontal surface. These solutions are directly obtained from integrating the point load solutions in a transversely isotropic half-space, which were derived using the principle of superposition, Fourier and Hankel transformation techniques. The solutions for the displacements and stresses in transversely isotropic half-spaces subjected to linearly variable loads on a rectangular region are never mentioned in literature. These exact solutions indicate that the displacements and stresses are influenced by several factors, such as the buried depth, the loading types, and the degree and type of rock anisotropy. Two illustrative examples, a vertical uniform and a vertical linearly varying rectangular load acting on the surface of transversely isotropic rock masses, are presented to show the effect of various parameters on the vertical surface displacement and vertical stress. The results indicate that the displacement and stress distributions accounted for rock anisotropy are quite different for those calculated from isotropic solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Elastic solutions of displacements for a transversely isotropic full space with inclined planes of symmetry subjected to a point load

This work presents analytical solutions for displacements caused by three‐dimensional point loads in a transversely isotropic full space, in which transversely isotropic planes are inclined with respect to the horizontal loading surface. In the derivation, the triple Fourier transforms are employed toyield integral expressions of Green's displacement; then, the triple inverse Fourier transforms and residue calculus are performed to integrate the contours. The solutions herein indicate that the displacements are governed by (1) the rotation of the transversely isotropic planes (?), (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′), (3) the geometric position (r, φ, ξ) and (4) the types of loading (P_x, P_y, P_z). The solutions are identical to those of Liao and Wang (Int. J. Numer. Anal. Methods Geomechanics 1998; 22 (6):425–447) if the full space is homogeneous and linearly elastic and the transversely isotropic planes are parallel to the horizontal surface. Additionally, a series of parametric study is conducted to demonstrate the presented solutions, and to elucidate the effect of the aforementioned factors on the displacements. The results demonstrate that the displacements in the infinite isotropic/transversely isotropic rocks, subjected to three‐dimensional point loads could be easily determined using the proposed solutions. Also, these solutions could realistically imitate the actual stratum of loading situations in numerous areas of engineering. Copyright © 2007 John Wiley & Sons, Ltd.     

Surface load dynamic solution of saturated transversely isotropic multilayer half-space

This paper treats the dynamic response of a multilayered transversely isotropic fluid saturated poroelastic half-space under surface time-harmonic traction. The governing system of partial differential equations is uncoupled with the use of a set of physically meaningful and complete potential functions that decompose different body waves in a saturated poroelastic transversely isotropic medium. After expressing the equations in the Hankel-Fourier domain, a proper algebraic factorization is applied to generate reflection and transmission matrices for decomposed waves. All responses including displacements, stresses, and pore fluid pressure for both general patch load and point load are presented in the form of semi-infinite line integrals. The verification of the method is confirmed with the degeneration of the solutions presented here to the existing solutions for dried both homogeneous and multilayered elastic half-spaces as well as poroelastic half-space. Selected numerical results are depicted to investigate the effects of layering and pore pressure on responses of a transversely isotropic poroelastic medium. The load distribution effects are studied by comparison of the patch and point load responses. Also, resonance notion and effective parameters on this phenomenon such as layering system and anisotropy contrast are discussed. Significant influence of materials and layering configuration on number and amplitude of resonances depicted through the numerical evaluation.     

Elastic solutions for stresses in a transversely isotropic half‐space subjected to three‐dimensional buried parabolic rectangular loads

In many areas of engineering practice, applied loads are not uniformly distributed but often concentrated towards the centre of a foundation. Thus, loads are more realistically depicted as distributed as linearly varying or as parabola of revolution. Solutions for stresses in a transversely isotropic half‐space caused by concave and convex parabolic loads that act on a rectangle have not been derived. This work proposes analytical solutions for stresses in a transversely isotropic half‐space, induced by three‐dimensional, buried, linearly varying/uniform/parabolic rectangular loads. Load types include an upwardly and a downwardly linearly varying load, a uniform load, a concave and a convex parabolic load, all distributed over a rectangular area. These solutions are obtained by integrating the point load solutions in a Cartesian co‐ordinate system for a transversely isotropic half‐space. The buried depth, the dimensions of the loaded area, the type and degree of material anisotropy and the loading type for transversely isotropic half‐spaces influence the proposed solutions. An illustrative example is presented to elucidate the effect of the dimensions of the loaded area, the type and degree of rock anisotropy, and the type of loading on the vertical stress in the isotropic/transversely isotropic rocks subjected to a linearly varying/uniform/parabolic rectangular load. Copyright © 2002 John Wiley & Sons, Ltd.     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。     

Analytical layer element solutions for deformations of transversely isotropic multilayered elastic media under nonaxisymmetric loading

This paper presents the analytical layer element solutions for deformations of transversely isotropic elastic media subjected to nonaxisymmetric loading at an arbitrary depth. The state vectors for the nonaxisymmetric problem are deduced through the substitution of the Hu Hai‐chang solutions into the basic equations for the transversely isotropic elastic media. From the state vectors, the analytical layer element of a single layer is obtained in the Hankel transformed domain. The analytical layer element is an exact and symmetric stiffness matrix whose elements are without positive exponential functions, which can not only simplify the calculation but also improve the stability of computation. On the basis of the continuity conditions between adjacent layers, the global stiffness matrix is obtained by assembling the interrelated layer elements. The solutions for the multilayered elastic media in the transformed domain are obtained by solving the algebraic equation of the global stiffness matrix, which satisfies the boundary conditions. The actual solutions in the physical domain are further obtained by inverting the Hankel transform. Finally, some cases are analyzed to verify the solutions and evaluate the influences of the transversely isotropic character and stratified character of the media on the load–displacement responses. The numerical results show that the variations of the elastic properties between layers have a great effect on the displacements of the multilayered media. Copyright © 2014 John Wiley & Sons, Ltd.     

Displacements under linearly distributed pressures by extended Mindlin’s equations

Analytical elasticity solutions provide an efficient means of performing a first approximate analysis in foundation engineering. One of the well-known solutions is Mindlin’s solution to the stress and displacement induced by a point load at an embedment depth in a half-space. This solution is more superior but less widely used than Boussinesq’s solution. To promote this situation, Mindlin’s displacement equations are integrated to obtain a complete set of explicit formulae for calculating the displacements at an arbitrary point. The displacements are induced by uniformly and triangularly distributed horizontal or vertical pressures, which are exerted over a horizontal or vertical rectangular area in the interior of a homogeneous, isotropic, elastic half-space. These formulae facilitate the future development of computer programs for the analysis of related practical problems in foundation engineering.     

横观各向同性饱和地基三维瞬态Lamb问题

研究了横观各向同性饱和土地基在地表动力荷载作用下的三维瞬态响应。基于饱和多孔介质的三维Biot波动理论,利用Laplace变换,建立圆柱坐标系下横观各向同性饱和土的波动方程;解耦波动方程后,根据算子理论,并借助Fourier展开和Hankel变换技术,得到瞬态荷载作用下,饱和土介质的土骨架位移和应力、孔隙水相对位移和孔隙水压力的一般解;利用一般解,给出横观各向同性饱和地基在地表集中荷载激励下的瞬态Lamb问题的解答。数值算例结果表明,采用各向同性饱和介质的动力学模型,不能准确描述具有明显各向异性特性的饱和土地基的瞬态动力特性。     

Deformations resulting from the movements of a shear or tensile fault in an anisotropic half space

Earlier solutions (Bull. Seismol. Soc. Amer. 1985; 75 :1135–1154; Bull. Seismol. Soc. Amer. 1992; 82 :1018–1040) of deformations caused by the movements of a shear or tensile fault in an isotropic half-space for finite rectangular sources of strain nucleus have been extended for a transversely isotropic half-space. Results of integrating previous solutions (Int. J. Numer. Anal. Meth. Geomech. 2001; 25 (10): 1175–1193) of deformations due to a shear or tensile fault in a transversely isotropic half-space for point sources of strain nucleus over the fault plane are presented. In addition, a boundary element (BEM) model (POLY3D:A three-dimensional, polygonal element, displacement discontinuity boundary element computer program with applications to fractures, faults, and cavities in the Earth's crust. M.S. Thesis, Stanford University, Department of Geology, 1993; 62) is given. Different from similar researches (e.g. Thomas), the Akaike's view on Bayesian statistics (Akaike Information Criterion Statistics. D. Reidel Publication: Dordrecht, 1986) is applied for inverting deformations due to a fault to obtain displacement discontinuities on the fault plane. An example is given for checking displacements predicted by proposed analytical expressions. Another example is generated for the use of proposed BEM model. It demonstrates the effectiveness of this model in exploring displacement behaviours of a fault. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic Green's functions for an anisotropic poroelastic half-space

The dynamic responses of an anisotropic poroelastic half-space under an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of Fourier transform and Stroh formalism, the three-dimensional (3D) general solutions of the anisotropic Biot's coupling dynamics equations are derived in the frequency domain. Considering the two surface conditions, permeable and impermeable, the analytical solutions for displacement fields and pore pressure in half-space under a point source (point load or a fluid source) are obtained. When the material properties are isotropic, the numerical results of the poroelastic half-space are in excellent agreement with the existing analytical solutions. For anisotropic half-space cases, numerical results show the strong dependence of the dynamic Green's functions on the material properties.     

Displacement and stress distributions under a uniform inclined rectangular load on a cross‐anisotropic geomaterial

In practical engineering, an applied rectangular area load is not often horizontally or vertically distributed but is frequently inclined at a certain angle with respect to the horizontal and vertical axes. Thus, the solutions of displacements and stresses due to such a load are essential to the design of foundations. This article yields the analytical solutions of displacements and stresses subjected to a uniform rectangular load that inclines with respect to the horizontal and vertical axes, resting on the surface of a cross‐anisotropic geomaterial. The planes of cross‐anisotropy are assumed to be parallel to the horizontal ground surface. The procedures to derive the solutions can be integrated the modified point load solutions, which are represented by several displacement and stresses elementary functions. Then, upon integrations, the displacement and stress integral functions resulting from a uniform inclined rectangular load for (1) the displacements at any depth, (2) the surface displacements, (3) the average displacements in a given layer, (4) the stresses at any depth, and (5) the average stresses in a given layer are yielded. The proposed solutions are clear and concise, and they can be employed to construct a series of calculation charts. In addition, the present solutions clarify the load inclinations, the dimensions of a loaded rectangle, and the analyzed depths, and the type and degree of geomaterial anisotropy profoundly affect the displacements and stresses in a cross‐anisotropic medium. Parametric results show that the load inclination factor should be considered when an inclined rectangular load uniformly distributed on the cross‐anisotropic material. Copyright © 2008 John Wiley & Sons, Ltd.     

横观各向同性饱和土体三维粘弹性动力分析

采用针对横观各向同性饱和土体u-w形式三维粘弹性动力方程,考虑土骨架的粘弹性性质且基于粘弹性理论,通过运用Fourier 展开、Laplace 和Hankel 积分变换方法和引入中间变量,将含有粘弹性参数的六元二阶偏微分运动控制方程组,化为2组各含4个未知变量的常微分方程组,从而给出了柱坐标系下粘弹性横观各向同性饱和土体在非轴对称动力荷载作用下的瞬态反应的土骨架位移分量、孔隙流体相对于土骨架的位移分量瞬态反应一般解。在此基础上,引入初始条件和边界条件,对垂直向和水平向动力荷载作用下半空间边值问题进行了求解。根据动力时域解答的一般解,利用Laplace和Hankel 数值逆变换技术,编制了相应的数值计算程序。并进行了实例验证和弹性、粘弹性解的对比分析。结果表明,在进行横观各向同性饱和土体动力分析时,考虑土骨架的粘弹性是必要的。     

Response of multilayered transversely isotropic medium due to axisymmetric loads

A novel procedure associated with the precise integration method (PIM) and the technique of dual vector is proposed to effectively calculate the magnitude and distribution of deformations in a homogeneous multilayered transversely isotropic medium. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the soil system. The linearly elastic medium is subjected to four types of vertically acting axisymmetric loads prescribed either at the external surface or in the interior of the soil medium. There are no limits for the thicknesses and number of soil layers to be considered. By virtue of the governing equations of motion and the constitutive equations of the transversely isotropic elastic body, and based on the Hankel integral transform and a dual vector formulation in a cylindrical coordinate system, the partial differential motion equations can be converted into first‐order ordinary differential matrix equations. Applying the approach of PIM, it is convenient to obtain the solutions of ordinary differential matrix equations for the continuously homogeneous multilayered transversely isotropic elastic soil in the transformed domain. The PIM is a highly accurate algorithm to solve the sets of first‐order ordinary differential equations, which can ensure to achieve any desired accuracy of the solutions. What is more, all calculations are based on the standard method with the corresponding algebraic operations. Computational efforts can be reduced to a great extent. Finally, numerical examples are provided to illustrate the accuracy and effectiveness of the proposed approach. Some more cases are analyzed to evaluate the influences of the elastic parameters of the transversely isotropic media on the load‐displacement responses. Copyright © 2015 John Wiley & Sons, Ltd.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.     

弹性矩形板下横观各向同性多层地基分析

利用弹性矩形板与多层地基表面的竖向位移协调条件与光滑接触条件,由横观各向同性多层地基应力与位移非耦合的传递矩阵解,推导出弹性矩形板下竖向应力和位移的解析解。在此基础上,编制了相应的程序,并进行了数值计算。计算结果表明：矩形板刚度对板底竖向位移及板中心下的竖向应力有着较为显著的影响;板底竖向位移及板中心下的竖向应力随着板刚度的增加而减小,相同荷载作用下横观各向同性地基与均匀各向同性地基模型的计算结果差异较大,实际工程中很有必要采用更符合土体性质的横观各向同性地基模型     

Elastic fields in two joined transversely isotropic media of infinite extent as a result of rectangular loading

This paper presents the closed‐form solutions for the elastic fields in two bonded rocks induced by rectangular loadings. Each of the two bonded rocks behaves as a transversely isotropic linear elastic solid of semi‐infinite extent. They are completely bonded together at a horizontal surface. The rectangular loadings are body forces along either vertical or horizontal directions and are uniformly applied on a rectangular area. The rectangular area is embedded in the two bonded rocks and is parallel to the horizontal interface. The classical integral transforms are used in the solution formulation, and the elastic solutions are expressed in the forms of elementary harmonic functions for the rectangular loadings. The stresses and displacements in the rocks induced by both the horizontal and vertical body forces are also presented. The numerical results illustrate the important effect of the anisotropic bimaterial properties on the stress and displacement fields. The solutions can be easily implemented for numerical calculations and applied to problems encountered in rock mechanics and engineering. Copyright © 2011 John Wiley & Sons, Ltd.     

半无限平面问题的相对位移

通过Flamant和Melan的解析解答、Mindlin解答的积分蜕化公式以及有限元数值分析计算结果,展示了在半无限平面问题中线荷载作用方向位移解答的不确定性。线荷载作用方向没有绝对位移,只有相对位移,但相对位移会随着与位移约束参考点距离的增大而增大,或随着线荷载在垂直于半平面方向分布长度的增大而增大,不具收敛性。这意味着,在解析和数值分析中,纯粹的半平面问题的位移解答具有多值性,因此,将岩土工程问题作为半空间问题进行分析是必要的。     

横观各向同性土体中压力隧洞的应力和位移计算

基于Biot固结理论,考虑了土体和孔隙流体压缩性,通过对控制方程的解耦,得到在横观各向同性饱和土体中圆形隧洞边界上作用随时间变化的轴对称荷载或流体压力所引起的应力、位移和孔隙水压力场在拉普拉斯变换域中的解析表达式,运用拉普拉斯数值逆变换进行算例分析,得到在时间域中的解,讨论了单级加载和循环加载对计算结果的影响,并与瞬时加载条件下的结果进行了比较。同时也分析了土体的横观各向同性性质对应力、位移和孔隙水压力场的影响。     

The study of reflection/transmission phenomena in a corrugated interface between two magnetoelastic transversely isotropic media

In this paper, an integrated approach has been carried out with an intent to study the reflection and transmission phenomena of plane SH-type wave on a corrugated interface separating two magnetoelastic transversely isotropic half-space. Closed form expressions for reflection and transmission coefficients have been derived for both plane and corrugated surface. Rayleigh’s method of approximation have been incorporated to deduce equations for the first- and second-order approximation of corrugation. Analytical solutions for both the half-spaces have been worked out. All possible cases have been apprehended specially for anisotropic and an isotropic medium. The effects of magnetoelastic coupling parameter, angle at which the wave crosses the magnetic field, corrugation amplitude, frequency factor and wave number have been explained by collaborating with graphical analysis.