An approach to model the mechanical behavior of transversely isotropic materials

The present work proposes an approach to adapt existing isotropic models to transversely isotropic materials. The main idea is to introduce equivalence relations between the real material and a fictitious isotropic one on which one can take all the advantages of the well‐established isotropic theory. Two applications of this approach are presented here: a failure criterion and a damage model that takes into account the load‐induced anisotropy. In both cases, theoretical predictions are in agreement with the experimental data. In the present paper, the developed approach is applied to sedimentary rock materials; nevertheless, it can be generalized to any material that exhibits transverse isotropy. Copyright © 2015 John Wiley & Sons, Ltd.     

On the strength of transversely isotropic rocks

Accurate prediction of strength in rocks with distinct bedding planes requires knowledge of the bedding plane orientation relative to the load direction. Thermal softening adds complexity to the problem since it is known to have significant influence on the strength and strain localization properties of rocks. In this paper, we use a recently proposed thermoplastic constitutive model appropriate for rocks exhibiting transverse isotropy in both the elastic and plastic responses to predict their strength and strain localization properties. Recognizing that laboratory‐derived strengths can be influenced by material and geometric inhomogeneities of the rock samples, we consider both stress‐point and boundary‐value problem simulations of rock strength behavior. Both plane strain and 3D loading conditions are considered. Results of the simulations of the strength of a natural Tournemire shale and a synthetic transversely isotropic rock suggest that the mechanical model can reproduce the general U‐shaped variation of rock strength with bedding plane orientation quite well. We show that this variation could depend on many factors, including the stress loading condition (plane strain versus 3D), degree of anisotropy, temperature, shear‐induced dilation versus shear‐induced compaction, specimen imperfections, and boundary restraints.     

Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity

Geomaterials such as soils and rocks are inherently anisotropic and sensitive to temperature changes caused by various internal and external processes. They are also susceptible to strain localization in the form of shear bands when subjected to critical loads. We present a thermoplastic framework for modeling coupled thermomechanical response and for predicting the inception of a shear band in a transversely isotropic material using the general framework of critical state plasticity and the specific framework of an anisotropic modified Cam–Clay model. The formulation incorporates anisotropy in both elastic and plastic responses under the assumption of infinitesimal deformation. The model is first calibrated using experimental data from triaxial tests to demonstrate its capability in capturing anisotropy in the mechanical response. Subsequently, stress‐point simulations of strain localization are carried out under two different conditions, namely, isothermal localization and adiabatic localization. The adiabatic formulation investigates the effect of temperature on localization via thermomechanical coupling. Numerical simulations are presented to demonstrate the important role of anisotropy, hardening, and thermal softening on strain localization inception and orientation. Copyright © 2016 John Wiley & Sons, Ltd.     

A transversely isotropic thermo-poroelastic model for claystone: parameter identification and application to a 3D underground structure

A transversely isotropic thermo-poroelastic constitutive law is developed and implemented in the finite element code Code_Aster (EDF, France). It is then validated using an analytic solution for an inclined borehole in a transversely isotropic medium. A strategy for identifying the parameters of the transversely isotropic thermo-poroelastic model based on an inverse method is proposed on the basis of different laboratory tests. To demonstrate the efficiency and applicability of the model, it is then applied in a three-dimensional numerical model of an underground structure in a parameter sensitivity study. The results of the modelling highlight the importance of accounting for anisotropic phenomena when determining the dimensions of underground facilities. The whole approach is presented in the paper, from model development to application to 3D numerical modelling to an engineering case study.     

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.     

列车荷载下上覆弹性层横观各向同性饱和地基的动力响应

基于Biot多孔弹性介质的波动理论,构造了轨道、道碴、枕木及弹性层的横观各向同性饱和地基在列车荷载下的动力计算模型。利用Fourier变换,得到了列车荷载作用下横观各向同性软土地基上弹性层动力响应的解析结果。利用离散Fourier逆变换得到数值计算结果,分析了荷载速度、地基的各向异性参数、弹性层刚度系数及厚度对位移和孔压响应的影响。分析结果表明：弹性层对控制地基振动作用显著,地表振动幅值随荷载速度的增加而增大,软土的横向弹性模量对地表振动及土中孔压有较大影响。     

层状横观各向同性地基平面应变问题的扩展精细积分解

利用扩展精细积分法求解横观各向同性地基的平面应变问题。扩展精细积分法具有高精度和较高计算效率的特点,是求解微分方程的有效方法,相比于解析法可以节省大量理论推导工作量。从直角坐标下弹性力学控制方程出发,推导出Fourier变换域内地基的常微分矩阵方程;之后对地基微层元进行消元合并,进一步得到荷载作用在地基内部时层状地基的扩展精细积解。与已有文献的对比验证了方法的精确性,并分析了横观各向同性参数、层状性质和荷载作用点对计算结果的影响。结果表明：土体竖向位移随着横观各向同性参数n的增大而减小,而随着横观各向同性参数m的增大而增大;荷载作用点 的变化只对作用点以上的土体有影响,而上层土体的模量对竖向位移计算结果的影响更为显著,土体成层性对沉降的影响要比对竖向应力的影响更为显著。     

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.     

横观各向同性岩体中圆形巷道反分析的惟一性

惟一性研究是位移反分析的基础工作之一。推导了横观各向同性岩体中圆形巷道的位移解析解,利用参数可辨识条件对横观各向同性岩体中圆形巷道位移反分析的惟一性进行了探讨。结果表明,无论量测多少个点的位移也不能惟一地反演出所有6个参数;必须至少已知3个参数时,才有可能惟一地反演其他参数;2个地应力分量是否相等对反分析结果有明显的影响;2个地应力分量的可辨识性最好,各向同性面上的弹性模量和泊松比次之,垂直各向同性面方向的弹性模量和泊松比最差。     

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.     

Static analysis of vertically loaded pile group in multilayered transversely isotropic soils

In this paper, a coupling approach is presented to study the static responses of vertically loaded pile group embedded in multilayered transversely isotropic soils. The individual pile in pile group is modeled by the finite element method, while the analytical layer-element method is applied to represent the soil's behavior. Then, the interaction equation of piles and soils is obtained by considering the force equilibrium and displacement compatibility conditions and solved by a FORTRAN program. The results computed by the proposed approach compare favorably with those from some existing solutions and field test. Some typical parametric analysis cases are investigated to study the effect of soil anisotropy, pile stiffness ratio, and pile spacing on the behavior of vertically loaded pile group.     

多孔结构矿物（岩石）及其环境修复材料的实用性

具有多孔结构的矿物(岩石)作为一类特殊的工业原材料，在环境修复领域的应用前景正在成为研究热点。本文从微孔的尺度、结构、形态、成分等表面固体特征出发，概括性地论述了凹凸棒石、海泡石、蒙脱石、沸石、硅藻土等十几种多孔结构矿物(岩石)在环境修复工程中用作过滤材料、吸附剂、离子交换剂和催化剂等材料时的基本特征和使用时应注意的问题。     

考虑剪应力作用时横观各向同性岩体中圆形巷道位移反分析的惟一性

推导了考虑剪应力作用时横观各向同性岩体中圆形巷道的位移解析解,利用参数可辨识条件对考虑剪应力作用时横观各向同性岩体中圆形巷道位移反分析的惟一性进行了探讨。结果表明,无论布置多少个测点也不能惟一地反分析出所有7个参数,必须至少已知3个参数,才有可能惟一地反分析出其他参数;水平与竖直地应力分量是否为 0 对反分析惟一性有一定影响;3个地应力分量的可辨识性最好,各向同性面上的弹性模量和泊松比次之,垂直于各向同性面方向的弹性模量和泊松比最差。与不考虑剪应力时位移反分析惟一性结果相比,考虑剪应力时参数的可辨识性顺序不变,但是条件惟一的比例降低,不惟一和绝对惟一的比例增加,说明考虑剪应力能够提高位移反分析的惟一性。工程实例计算表明,推导的位移解析解和反分析惟一性探讨结果能够很好地指导岩体参数反演计算。     

The effects of intermediate principle stress on the mechanical behavior of transversely isotropic rocks: Insights from DEM simulations

The discrete element method (DEM) is used to study the response of anisotropic rocks under true triaxial testing. Numerical samples of seven different bedding orientations (β = 0^o, 15^o, 30^o, 45^o, 60^o, 75^o, and 90^o) are created. Six series of test simulations (σ₃ = 0, 10, 30, 50, 70, and 100 MPa) are conducted on each sample, with five different σ₂ values, varied from σ₃ to σ₁. The effects of anisotropy and intermediate stress on the peak strength, brittle-ductile transition, and degree of anisotropy are subsequently explained through underlying micromechanics. Results show a “fan-shaped” variation of the peak strength with σ₂, displaying an ascending-then-descending trend. An increasing brittleness with σ₂ is observed at lower confining pressures for all, but medium anisotropy angles. For higher confining pressures, increasing ductility with σ₂ is seen for every anisotropy angle. A U-shaped variation of peak strength with anisotropy angles is noted that flattens under high intermediate stress. Hence, for numerical models of Posidonia shale under normalized σ₂ higher than 0.76, the anisotropy effect is found to be negligible. Micromechanical analyses reveal that the stress asymmetry, suppression of weak plane action as well as the localization and coalescence of microcracks in the intact rock matrix, due to σ₂, are the contributors towards the obtained trends. Since existing failure criteria do not weigh in these features in geotechnical assessments, this paper helps future studies by providing a deeper understanding of these effects and a comprehensive data set for the analyses of anisotropic rocks under polyaxial stress conditions.     

列车荷载下轨道系统-层状横观各向同性饱和地基动力响应

基于Biot波动理论,构建列车荷载-轨道系统-双层状横观各向同性饱和地基模型,将模型分为上覆路轨系统和地层系统。对上覆路轨系统和地层系统处理,并利用双重Fourier变换技术,在变换域中将横观各向同性饱和地基动力响应的求解简化为求解一个6阶控制方程的特征值问题,进而得到了列车荷载作用下双层横观各向同性饱和地基力响应的解析结果。利用离散Fourier逆变换得到数值计算结果,重点分析了上下土层的刚度和泊松比对位移和孔隙水压力和剪切应力响应的影响,结果表明,上、下土层刚度差异对地基动力响应有较大影响,土层各向异性参数中模量的影响较泊松比大。计算结果可为软土路基加固深度的确定提供理论依据。     

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。     

Evaluation of two multidimensional discrimination diagrams from beach and deep-sea sediments from the Gulf of Mexico and their application to Precambrian clastic sedimentary rocks

Two discriminant-function-based multidimensional major-element diagrams for the tectonic discrimination of siliciclastic sediments were recently published from a coherent statistical methodology of log_e-ratio transformation and linear discriminant analysis. These diagrams were constructed based on worldwide examples of Neogene–Quaternary siliciclastic sediments from known tectonic settings. In this work, these two tectonic discrimination diagrams were first successfully tested from Holocene (<0.0117–0 Ma) beach and deep-sea sediments from the Gulf of Mexico. These diagrams were used to decipher tectonic settings of 11 case studies of the Precambrian clastic sedimentary rocks (~512–2800 Ma) from Argentina, USA, Ghana, Spain, Norway, India, China, and Australia. The test and application results obtained from these discrimination diagrams were generally consistent with the geology of the Precambrian source areas. Therefore, the two multidimensional diagrams can be considered as a useful tool for successfully discriminating the tectonic setting of older sedimentary basins, which may consist of one or more tectonic assemblages. Comparison of results of this study with the previously published tectonic discrimination diagrams is illustrated and the probable reasons for some inconsistent inferences were also discussed.     

Modelling of the hydro-mechanical response of sedimentary rocks of southern Ontario to past glaciations

Glaciation is considered as one of the main natural processes that can have a significant impact on the long term performance of a deep geological nuclear waste repository (DGR) located in the Northern Hemisphere. The northern part of the American continent has been subjected to a series of strong glaciation and deglaciation events over the past million years. The last glacial cycle in the Northern Hemisphere started approximately 110,000 year ago. During that cycle, southern Ontario was buried under a continental ice sheet, with a maximum thickness of up to 3000 m at about 20,000 years ago. The ice cap retreated approximately 10,000 years ago. However, field data from deep boreholes in sedimentary rocks of southern Ontario show anomalous pore water pressure including underpressure and overpressure zones. In this paper, a large-scale coupled hydro-mechanical (HM) model is developed to investigate the hydro-mechanical (HM) response of the sedimentary rocks of southern Ontario to past glacial cycles. Particular emphasis has been placed on the evolution of pore water pressures and surface displacements. The HM model is verified using analytical solutions. The results of the large-scale HM modeling study shows that the past glaciation, particularly the second cycle (22,000 apb) had significant impact on the pore water pressure gradient and effective stress distribution in the sedimentary rocks of southern Ontario. Furthermore, good agreement between the large scale modeling results and anomalous pressures leads us to the conclusion that these anomalies could be glacially induced. The results of this research can provide information that will contribute to a better understanding of the impact of future glaciations on the long term performance of DRGs in sedimentary rocks.