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.     

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     

A Transversely Isotropic Damage Model for Boom Clay

Boom clay can be considered as a transversely isotropic geomaterial. However, due to lack of experimental evidence and data base, it is still difficult to describe the transversely isotropic plastic behavior of this argillaceous rock. In this paper, we present first, by means of an experimental approach, the main features of the mechanical properties of Boom clay. Then, combining the transversely isotropic elastic model and the modified Mohr–Coulomb criterion, a suitable constitutive model is introduced so as to fully describe the mechanical behavior of the studied material, in which, an elastic damage law which takes into consideration, the transversely isotropic effect, a plastic hardening law and a plastic damage law were introduced to describe the nonlinear elastic, hardening and softening behavior of Boom clay. As a preliminary step, the evolution law of both elastic moduli and Poisson’s ratio during the elastic stage was obtained by direct analysis of the test data. The synchronism of the elastic damage in both transversal and axial directions was proved by this method. Some of the parameters of the model in the elastic stage were also determined by direct analysis method and further verified by back analysis. Other unknown parameters in the model were determined by back analysis.     

Effects of the micro-structure and micro-parameters on the mechanical behaviour of transversely isotropic rock in Brazilian tests

Based on the Brazilian test results of 23 kinds of transversely isotropic rocks, five trends are obtained for the variation of normalized failure strength (NFS) as a function of the weak plane-loading angles. For each angle, three kinds of fracture patterns are obtained. Furthermore, a new numerical approach based on the particle discrete element method is put forward to systematically investigate the influence of the micro-structure of rock matrix and strength of weak plane on NFS and fracture patterns. The results reveal that the trend of NFS and fracture patterns are slightly influenced by coordination number of rock particles and tensile strength of weak plane, but greatly influenced by percentage of pre-existing cracks and shear strength of weak plane. Micro-parameters of the numerical approach are calibrated to reproduce behaviours of transversely isotropic rocks with different trends, and the simulation results are well matched with experimental results in terms of NFS and fracture patterns. Finally, the numerical approach is applied to study the failure process of layered surrounding rock after tunnel excavation. The simulation results also agree well with observation results of engineering projects.     

Boundary element analysis of transversely isotropic bi-material halfspaces with inclined planes of isotropy and interfaces

In this paper, a single-region boundary element method (BEM) is presented for the analysis of transversely isotropic bi-material halfspaces with arbitrarily inclined planes of isotropy and material interfaces. The proposed BEM uses the fundamental solution of a transversely isotropic bi-material fullspace and five boundary element techniques. Infinite boundary elements are introduced to consider the far-fields of a transversely isotropic bi-material halfspace. The effective integration methods are proposed for dealing with various integrals in the discretized boundary integral equation. The stresses at internal points are obtained using the coordinate transformation of kernel functions, and the stresses on the boundary surface are calculated using an improved traction recovered method. Numerical verifications of displacements and stresses for a benchmark problem are conducted, and excellent agreement with previously published results is obtained. Numerical examples are presented to illustrate the influence of non-horizontal or horizontal planes of isotropy in bi-material halfspaces on the displacements and stresses induced by the tractions on the horizontal boundary surface. Results reveal that the elastic fields vary clearly with the dip angle of the isotropic plane and the stresses across the bi-material interface are closely related to the ratios of the elastic parameters of the bi-material.     

平行层状岩体的自重应力场

将平行层状复合岩体视为等效的模观各向同性体，推导出自重荷载下地应力的理论解，并分析了两个水平主应力受岩层倾角和岩体弹性参数的影响，结果表明，与各向同性岩体相比，水平主应力的变化范围增大了，并且两上水平主应力可能不再相等，而岩层倾角对水平主应力之比影响显著，当倾角从0－90度变化时，两个水平主应力能从相等到相差内倍，这也说明在地应力研究中，考察岩体的各向异性是十分必要的，所导出的自重应力场解答对地下工程问题具有一定的实用价值。     

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.     

Experimental and Numerical Studies on Determination of Indirect Tensile Strength of Rocks

Indirect tension tests using Brisbane tuff Brazilian disc specimens under standard Brazilian jaws and various loading arcs were performed. The standard Brazilian indirect tensile tests caused catastrophic, crushing failure of the disc specimens, rather than the expected tensile splitting failure initiated by a central crack. This led to an investigation of the fracturing of Brazilian disc specimens and the existing indirect tensile Brazilian test using steel loading arcs with different angles. The results showed that the ultimate failure load increased with increasing loading arc angles. With no international standard for determining indirect tensile strength of rocks under diametral load, numerical modelling and analytical solutions were undertaken. Numerical simulations using RFPA^2D software were conducted with a heterogeneous material model. The results predicted tensile stress in the discs and visually reproduced the progressive fracture process. It was concluded that standard Brazilian jaws cause catastrophic, crushing failure of the disc specimens instead of producing a central splitting crack. The experimental and numerical results showed that 20° and 30° loading arcs result in diametral splitting fractures starting at the disc centre. Moreover, intrinsic material properties (e.g. fracture toughness) may be used to propose the best loading configuration to determine the indirect tensile strength of rocks. Here, by using numerical outcomes and empirical relationships between fracture toughness and tensile strength, the best loading geometry to obtain the most accurate indirect tensile strength of rocks was the 2α?=?30° loading arc.     

A hybrid element method for dynamics of piles and pile groups in transversely isotropic media

A hybrid analytical-numerical method is proposed for the dynamic analysis of single piles and pile groups embedded in semi-infinite transversely isotropic media. In the method proposed, the soil-pile system is modeled using finite elements combined with massless rigid radiation discs representing pile-soil-pile interaction. The elasto-dynamic response of the radiation discs buried at different depths in a transversely isotropic half-space is analytically derived in a transform domain using a set of complete potential functions. A Boussinesq-type loading distribution is introduced to act on the disc region to achieve the proper mode of deformation at the cross sections of piles. Numerical results and comparisons with known analytical/numerical solutions are presented, demonstrating the application of the method.     

裂缝型HTI介质中的弹性阻抗

在回顾封闭平行硬币状裂缝模型的基础上,将裂缝填充物性质、分布密度与背景介质的横纵波速度比,引入裂缝型水平横向各向同性介质纵波、转换横波弹性阻抗公式,并进行归一化弹性阻抗响应特征模拟。分析表明：裂缝介质弹性阻抗呈现方位各向异性,且随裂缝密度加大而增强;纵波、转换横波弹性阻抗在值域分布规律上具有较大区别;在典型砂岩介质的前提下,弹性阻抗差可以作为裂缝含气、含水指示因子定性地识别裂缝填充物性质。     

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.     

Discrete element modeling of transversely isotropic rocks with non-continuous planar fabrics under Brazilian test

Acta Geotechnica - A new numerical approach based on the particle discrete element method (PDEM) is developed to investigate the mechanical behavior of transversely isotropic rocks with...     

The Brazilian Disc Test for Rock Mechanics Applications: Review and New Insights

The development of the Brazilian disc test for determining indirect tensile strength and its applications in rock mechanics are reviewed herein. Based on the history of research on the Brazilian test by analytical, experimental, and numerical approaches, three research stages can be identified. Most of the early studies focused on the tensile stress distribution in Brazilian disc specimens, while ignoring the tensile strain distribution. The observation of different crack initiation positions in the Brazilian disc has drawn a lot of research interest from the rock mechanics community. A simple extension strain criterion was put forward by Stacey (Int J Rock Mech Min Sci Geomech Abstr 18(6):469–474, 1981) to account for extension crack initiation and propagation in rocks, although this is not widely used. In the present study, a linear elastic numerical model is constructed to study crack initiation in a 50-mm-diameter Brazilian disc using FLAC^3D. The maximum tensile stress and the maximum tensile strain are both found to occur about 5 mm away from the two loading points along the compressed diameter of the disc, instead of at the center of the disc surface. Therefore, the crack initiation point of the Brazilian test for rocks may be located near the loading point when the tensile strain meets the maximum extension strain criterion, but at the surface center when the tensile stress meets the maximum tensile strength criterion.     

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。     

Measurement of Indirect Tensile Strength of Anisotropic Rocks by the Ring Test

Summary This paper presents a new approach, combined with the Boundary Element Method (BEM) analysis and the diametrical compression on a thin disc with a small central hole, referred to as the ring test, for determining the indirect tensile strength of anisotropic rocks. The stress distribution around the hole can be successfully obtained by the proposed single-domain BEM. The complex variable function method was used for conveniently computing the tractions and displacements of a two-dimensional anisotropic body. If we assume that the tensile strength is given by the maximum absolute value of stress in the direction perpendicular to the loaded diameter at the intersection of loaded diameter and the hole, then from the failure load recorded by laboratory testing of ring (disc), the indirect tensile strength of rocks could be obtained. A marble from Hualien (Taiwan) with clearly black-white foliation, which was assumed to be transversely isotropic, was selected to conduct both ring tests and Brazilian tests for evaluating the tensile strength. The variation of the marble tensile strength with the inclination angle of foliation and with the hole size was also investigated. In general, the tensile strength of anisotropic rocks determined by ring test is not a constant, but depends on the elastic properties of rocks, the angle between the planes of rock anisotropy and the loading direction, the diameter of the central hole, and the contact condition of loading.     

Determination of Fracture Toughness of Anisotropic Rocks by Boundary Element Method

Summary  This paper presents a systematic procedure for determining fracture toughness of an anisotropic marble using the diametral compression test (Brazilian test) with a central crack on the discs. Additionally, a novel formulation to increase the accuracy in Stress Intensity Factor (SIF) calculations using Boundary Element Method (BEM) is applied to determine the stress intensity factors and the fracture toughness of anisotropic rocks under mixed-mode loading. The numerical results show that the SIFs for both isotropic and anisotropic problems are in good agreement with those reported by previous authors. The marble with clear white-black foliation from Hualien (in eastern Taiwan), was selected for the Brazilian tests. Diametral loading was conducted on the Cracked Straight Through Brazilian Disc (CSTBD) specimens to evaluate their fracture toughness. In addition, a new fracture criterion was developed to predict pure mode I, pure mode II or mixed mode (I–II) fracture toughness of the anisotropic marble. The new fracture criterion is based on the examination of mode I, mode II and mixed mode (I–II) fracture toughness for different crack angles and anisotropic orientation. Authors’ address: Associate Professor Chao-Shi Chen, Department of Resources Engineering, National Cheng Kung University, No. 1 Dasyue Rd., East District, Tainan 701, Taiwan     

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。     

一种横观各向同性岩体蠕变模型

层状岩体由于层理面或者结构面的存在,在力学上具有横观各向同性的特点,现有的各向同性蠕变模型难以全面反映横观各向同性岩体的蠕变力学特性。为构建能够反映横观各向同性岩体的三维蠕变模型,以能反映瞬时应变、减速蠕变和稳态蠕变特征的黏弹性Burgers模型为基础,基于常泊松比假定,在三维各向同性蠕变本构方程的基础上,按照算子替换的方法,将横观各向同性柔度矩阵代替各向同性柔度矩阵,并考虑了平行和垂直层理方向岩体蠕变力学行为的差异性,推导了横观各向同性岩体的三维蠕变本构方程。根据本构方程的特点,提出了根据平行和垂直方向岩体蠕变试验结果进行三维蠕变本构模型中蠕变参数的辨识方法。将提出的模型应用于三轴蠕变试验参数辨识,从而获得了一套完整的三维蠕变参数,并与试验结果进行对比分析,从而验证了所提模型与方法的合理性和有效性。进一步,指出了传统蠕变试验设计方案的局限性,给出了横观各向同性材料蠕变试验设计建议。研究成果为研究岩体三维蠕变机制提供了新思路,可对岩体蠕变试验设计提供相应的科研支撑。     

A discrete numerical method for brittle rocks using mathematical programming

A computational formulation of discrete simulations of damage and failure in brittle rocks using mathematical programming methods is proposed. The variational formulations are developed in two and three dimensions. These formulations naturally lead to second-order cone programs and can conveniently be solved using off-the-shelf mathematical programming solvers. Pure static formulations are derived so that no artificial damping parameters are required. The rock is represented by rigid blocks, with interfaces between blocks modelled by zero-thickness springs based on the rigid-body–spring network method. A modified Mohr–Coulomb failure criterion is proposed to model the failure of the interfaces. When the interface’ strength limits are reached, a microscopic crack forms and its strength is irreversibly lost. The microscopic elastic properties of the springs are related to the observed elastic behaviour of rocks with the developed empirical equations. The program is first validated with three simple tests. Then, numerical uniaxial and biaxial compression tests and the Brazilian tests are conducted. Furthermore, the proposed approach is employed to study the rock crack propagation and coalescence using cracked Brazilian disc test. The results are in good agreements with reported experimental data, which shows its potential in modelling mechanical behaviour of brittle rocks.     

Assessments of Strength Anisotropy and Deformation Behavior of Banded Amphibolite Rocks

Assessment of strength anisotropy in transversely isotropic rocks has been one of the most challenging subjects in rock engineering. However, far too little attention has been paid to banded amphibolite rocks. This study aim to evaluate strength and deformation anisotropy behavior of banded amphibolite rocks. The dynamic mechanical tests including ultrasonic pulse test, uniaxial compressive strength, Brazilian test and deformability test were performed on drilled rock samples as a function of foliation plane angle (β = 0°, 30°, 60° and 90°). The results obtained have shown that the dynamic mechanical properties of amphibolite rocks have different values concerning banding plane. Compression and shear waves taken parallel to the foliation plane show highest values than those obtained in the other directions. Under uniaxial test, the banded amphibolite has a U-shaped anisotropy with maximum strength at β = 90° and minimum strength is obtained when β = 30°. Strength anisotropic index ranges between 0.96 and 1.47. It seems that the high range value of anisotropic index is mainly due to slight undulation of foliation planes, that being not perfectly straight. The results of elastic deformation test show that there is no clear dependence on microstructures characteristics of subtype-amphibolite rocks that controlling modulus “shape-anisotropy”. However, in this study, Young modulus values of amphibolite rocks with β follow both types of shape-anisotropy, “U-shape” and “decreased order-shaped”. Thus, this study recommended that further research be undertaken regarding the role of modulus “shape-anisotropy” within the same lithotype.