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 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.     

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.     

Moving load response of an axially loaded Timoshenko beam on a multilayered transversely isotropic half-space comprising different media

This paper investigates the dynamic response of an axially loaded Timoshenko beam coupled with a multilayered transversely isotropic (TI) half-space subjected to a moving load. An axial force induced by the thermal expansion is taken into account in the Timoshenko beam. The half-space considers the alternate distribution of an arbitrary number of TI elastic and poroelastic layers to model foundation soils with different properties and moisture conditions. To solve the governing equations, Fourier transform is adopted. The stratified foundation is formulated by extending an “adapted stiffness matrix method” to a more general scenario with an arbitrary number of layers. The beam is then coupled with the foundation to derive solutions to the system in the frequency-wavenumber domain. The final results in the time-spatial domain are recovered by the inverse Fourier transform. After confirming the accuracy of the method in this study, the influences of the pore water existence, the transverse isotropy of different parameters, and the axial force are investigated. It can be observed that the effect of pore water existence on the maximum beam deflection can reach 22% in this study. The transverse isotropy of the elastic and shear moduli influences the critical speed of the beam deflection by altering the phase velocity of the first wave propagation mode of the beam-foundation system. The vertical permeability coefficient is more important than the horizontal one in determining the excess pore pressure. The rise of the beam temperature (axial force) decreases the critical speed and magnifies the vibrations.     

Forced vertical and horizontal movements of a rectangular rigid foundation on a transversely isotropic half‐space

An analytical investigation of a half‐space containing transversely isotropic material under forced vertical and horizontal displacements applied on a rectangular rigid foundation is presented in this paper. With the goal of a rigorous solution to the shape‐ and rigidity‐ induced singular mixed boundary value problem, the formulation employs scalar potential representation, the Fourier expansion and the Hankel integral transforms method to obtain the surface arbitrary point‐load solution in cylindrical coordinate system. The obtained Green's functions are rewritten in rectangular coordinate system, allowing the response of the half‐space because of an arbitrary distributed load on a rectangular surface area be given in terms of a double integral. The numerical evaluations of stresses are done with the use of an element, which is singular at the edge and the corner of the rectangle. Upon the imposition of the rigidity displacement boundary condition for a rigid foundation and the use of a set of two‐dimensional adaptive‐gradient elements, which can capture the singular behavior in the contact stress effectively, a set of new numerical results are presented to illustrate the effect of transverse isotropy on the foundation response. Copyright © 2012 John Wiley & Sons, Ltd.     

多层横观各向同性地基轴对称固结的传递矩阵解

从横观各向同性地基轴对称Biot固结的基本方程出发,通过关于t的Laplace变换和关于r的Hankel变换,得到关于z的一阶常微分方程组。然后,对变换域内的基本未知量进行线性化处理,建立了变换域内的基本状态变量在z = 0处和任意深度处z的显式关系。利用传递矩阵法,结合层间连续性条件和边界条件,得到了多层横观各向同性地基的Biot固结轴对称问题的解答。该解答能避免随着层数增加而需要求解大型方程组的困难,明显地提高了计算效率。     

Three-dimensional discrete element simulation of indirect tensile behaviour of a transversely isotropic rock

This paper presents the development of a three-dimensional discrete element model using flat-joint and smooth-joint contact models to investigate the effect of anisotropy on the tensile behaviour of slate, a transversely isotropic rock, under Brazilian testing from both macro and microscales. The effect of anisotropy is further realised by exploring the influence of foliation orientations (β and ψ) on the tensile strength, fracture pattern, microcracking and stress distribution of the transversely isotropic rock. The variation of tensile strength with foliation orientation is presented. The cross-weak-plane fracture growth observed in laboratory is reproduced, and the criterion for which to form is also given from the aspect of foliation orientation. Furthermore, the proportional variations of microcracks well account for the effects of foliation orientation on the tensile strength and failure pattern. Finally, it is found that the existence of weak planes increases both the heterogeneity and the anisotropy of stress distributions within the transversely isotropic rock, with the degree of influence varying with the foliation orientation.     

单轴压缩下横观各向同性岩石破裂过程的数值模拟

采用基于细观损伤力学基础上开发的RFPA2D数值模拟软件,用2种不同的岩石材料来组成7个不同岩层倾角的横观各向同性的岩石试件,通过单轴加载数值模拟试验,模拟横观各向同性岩石渐进破裂的整个过程,分析了岩层与最大主应力之间的倾角和强度之间的关系,讨论了不同岩层倾角的横观各向同性岩体的不同破裂模式及其破坏准则。     

半无限横观各向同性介质中多裂纹相互作用分析

为评估含矩形裂纹的半无限横观各向同性介质的局部强度和稳定性,采用基于双横观各向同性材料基本解的对偶边界元数值方法,分析了在沿裂纹面法向和切向均布力分别作用下矩形裂纹的应力强度因子及两个共面或平行的矩形裂纹的相互作用问题。通过数值计算考察了自由面对应力强度因子值的影响,以及裂纹间距、边长比及自由面对共面或平行双裂纹相互作用效应的影响。结果表明,自由面的存在引起该类裂纹应力强度因子值大于无限域情况;裂纹形状和裂纹间距对共面或平行双裂纹相互作用效应均有较明显的影响,但自由面对共面或平行双裂纹的相互作用效应均影响较小。     

横观各向同性地基中管桩扭转振动响应解析解

考虑地基沉积过程中产生的竖向和水平向力学性质的差异,对横观各向同性地基中管桩扭转振动频域响应进行了理论研究。基于横观各向同性材料的本构关系以及桩-土耦合扭转振动,建立了桩土系统定解问题,通过Laplace变换和分离变量法求得了桩周土和桩芯土扭转振动位移形式解。通过桩-土接触面的连续条件,求得了管桩扭转频域响应解析解,并得到了桩顶复动刚度和速度导纳的表达式。将所得解退化到横观各向同性地基中实心桩解以及均匀地基中管桩解,并与已有文献进行了对比,验证了解的合理性。通过数值算例,分析了桩周土和桩芯土的横观各向同性力学参数对桩顶扭转复刚度及速度导纳的影响。     

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.     

无限域横观各向同性岩体介质中折线裂纹边界元方法

岩体材料的各向异性导致其中的光滑裂纹在扩展后会变为折线裂纹。对于折线裂纹,其折线处裂纹面外法线方向不惟一,不能用连续单元离散。为此,在光滑裂纹问题方法的基础上,引入4种新的不连续单元来离散折线处裂纹面,建立了适用于折线裂纹问题的对偶边界元方法,该方法基于横观各向同性基本解。算例验证表明,该方法具有较好的精度。最后用该方法分析了横观各向同性岩体中的折线裂纹,得到了该类裂纹的应力强度因子。当裂纹面上作用法向均布力,横观各向同性岩体介质中的矩形光滑裂纹发生弯折时,折线两侧的裂纹面在张开时存在抑制效应,从而导致折线裂纹裂尖应力强度因子小于原光滑裂纹。同时还发现,随裂纹面的逐渐弯折,其裂纹面对各向同性面的倾角发生变化,因此,其裂尖断裂特性还受到岩体各向异性的影响。     

Displacement ring load Green's functions for saturated porous transversely isotropic tri‐material full‐space

Based on the Biot's poroelastic theory and using scalar potential functions both the ring load and point load displacement Green's functions for a transversely isotropic saturated porous full‐space composed of an upper half‐space, a finite thickness middle layer and a lower half‐space is analytically presented for the first time. It is assumed that each region consists of a different transversely isotropic material. The equations of poroelastodymanics in terms of the solid displacements and the pore fluid pressure are uncoupled with the help of two scalar potential functions, so that the governing equations for the potential functions are either a second order wave equation or a repeated wave‐heat transfer equation of sixth order. With the aid of Fourier expansion with respect to circumferential direction and Hankel integral transforms with respect to the radial direction in cylindrical coordinate system, the response is determined in the form of line integrals in the real space, followed by theorem of inverse Hankel integral transforms. The solutions degenerate to a single phase elastic material, and the results are compared with previous studies, where an excellent agreement may be observed with the results provided in the literature. Some examples of displacement Green's functions are finally given to illustrate the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

The Reissner–Sagoci problem for a transversely isotropic half‐space

A transversely isotropic linear elastic half‐space, z?0, with the isotropy axis parallel to the z‐axis is considered. The purpose of the paper is to determine displacements and stresses fields in the interior of the half‐space when a rigid circular disk of radius a completely bonded to the surface of the half‐space is rotated through a constant angle θ₀. The region of the surface lying out with the circle r?a, is free from stresses. This problem is a type of Reissner–Sagoci mixed boundary value problems. Using cylindrical co‐ordinate system and applying Hankel integral transform in the radial direction, the problem may be changed to a system of dual integral equations. The solution of the dual integral equations is obtained by an approach analogous to Sneddon's (J. Appl. Phys. 1947; 18 :130–132), so that the circumferential displacement and stress fields inside the medium are obtained analytically. The same problem has already been approached by Hanson and Puja (J. Appl. Mech. 1997; 64 :692–694) by the use of integrating the point force potential functions. It is analytically proved that the present solution, although of a quite different form, is equivalent to that given by Hanson and Puja. To illustrate the solution, a few plots are provided. The displacements and the stresses in a soil deposit due to a rotationally symmetric force or boundary displacement may be obtained using the results of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

3D dynamic response of a transversely isotropic multilayered medium subjected to a moving load

The dynamic problem of a transversely isotropic multilayered medium is reducible to quasi‐static problem by introducing a moving system that travels synchronously with the load. Based on the governing equations in the moving system, the ordinary differential equations in the Fourier transformed domain are deduced. An extended precise integration method is adopted to solve the ordinary differential equations, and the solution in the physical domain is recovered by the inverse Fourier transform. Numerical examples are performed to verify the accuracy of the presented method and to analyze the influence of material properties and the load characteristic.