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.     

Evolution of anisotropy with saturation and its implications for the elastoplastic responses of clay rocks

Many clay rocks have distinct bedding planes. Experimental studies have shown that their mechanical properties evolve with the degree of saturation (DOS), often with higher stiffness and strength after drying. For transversely isotropic rocks, the effects of saturation can differ between the bed-normal (BN) and bed-parallel (BP) directions, which gives rise to saturation-dependent stiffness and strength anisotropy. Accurate prediction of the mechanical behavior of clay rocks under partially saturated conditions requires numerical models that can capture the evolving elastic and plastic anisotropy with DOS. In this study, we present an anisotropy framework for coupled solid deformation-fluid flow in unsaturated elastoplastic media. We incorporate saturation-dependent strength anisotropy into an anisotropic modified Cam-Clay (MCC) model and consider the evolving anisotropy in both the elastic and plastic responses. The model was calibrated using experimental data from triaxial tests to demonstrate its capability in capturing strength anisotropy at various levels of saturation. Through numerical simulations, we demonstrate the role of evolving stiffness and strength anisotropy in the mechanical behavior of clay rocks. Plane strain simulations of triaxial compression tests were also conducted to demonstrate the impacts of material anisotropy and DOS on the mechanical and fluid flow responses.     

Triaxial behaviour of transversely isotropic materials: application to sedimentary rocks

Failure and long‐term behaviour of oriented solids are studied. Transversely isotropic materials are considered and a mathematical formulation that respect the material symmetry is developed and applied to model the triaxial behaviour of sedimentary rocks. Two failure criteria and a viscoplastic constitutive model that describe, respectively, triaxial failure and triaxial creep tests are presented and discussed. The application of the developed models to describe the mechanical behaviour of Tournemire shale shows that theoretical predictions are in good 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 © 2007 John Wiley & Sons, Ltd.     

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.     

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。     

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。     

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。     

横观各向同性冻土弹性常数及强度预测

在分析冻土形成过程及其组构前提下,从冻土细观结构特征出发,将冻土视为由冰层与干硬土层黏结组成的横观各向同性固体介质。根据复合材料力学理论,求出冻土各向同性面内及面外的弹性模量和泊松比;基于Tsai-Wu 准则,预测了横观各向同性冻土的强度。根据实验数据,分析了不同组分对冻土弹性常数及强度的影响。在工程应用中可根据环境温度及土质、系统有无水分补给、温降速度与冻结速率是否同步、冻土受力方向与冻结封面方向的具体情况选用即可,从而大幅降低室内外试验次数。为冻土工程数值计算奠定了基础,为进一步了解冻土力学特性提供了理论依据     

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。     

On the effects of deformation induced anisotropy in isotropic materials

Effects of recoverable deformation induced anisotropy in the elastic stiffness of isotropic materials are described. In isotropic materials, thermodynamics predicts coupling of hydrostatic and deviatoric responses. It is shown that the coupling of the two responses is more significant than previously recognized in the literature. Properly accounting for the coupling of hydrostatic and deviatoric responses requires re‐evaluating elastic materials characterization data, allowing for the coupled response. The result is an apparent decrease in the pressure sensitivity of the elastic shear modulus. The decrease in the pressure sensitivity of the shear modulus leads to stress paths that are more tangential to the yield surface in stress space, resulting in an increase in predicted elastic strain at each step of an elastic–plastic stress update. Consequently, predicted plastic strains and, in particular, volumetric plastic strains, are smaller than if recoverable deformation induced anisotropy had been neglected. The result is an associated plasticity model, which appears to be non‐associated. Copyright © 2012 John Wiley & Sons, Ltd.     

Fundamental solutions for a fluid‐saturated,transversely isotropic,poroelastic solid

The fundamental solutions were obtained for step‐like point forces acting in three orthogonal directions and an instantaneous fluid point source in a fluid‐saturated, porous, infinite solid of transversely isotropic elasticity and permeability. After expressing the governing equations in the form of matrix in the Laplace space, we employed Kupradze's method together with the triple Fourier transforms. This method reduces the simultaneous partial differential equations with respect to three displacement components and a pore fluid pressure to a differential equation in terms of only one potential scalar function, which can be operationally solved in the transformed space. After the Laplace inversion of the potential, the residue theorem was applied to its Fourier inverse transform with respect to one of the transformation variables. The Fourier transforms with respect to two other variables were rewritten into the Hankel transforms. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

有限元和有限层元研究横向同性土对建筑物沉降的影响

从常规三轴压缩试验得到横观各向同性土层的力学参数,综合9种来自于不同国家和地区的横观各向同性土层的土样参数,研究分析了横观各向同性土层参数对建筑物沉降的影响。作为典型例子,对上海一幢剪力墙结构的箱形基础与横观各向同性土进行共同作用分析,并与各向同性土层上的分析结果进行比较,结果表明：横观各向同性土层参数对沉降的影响有些不同,尤其是不排水加载条件确定的土层参数影响的结果大不相同。     

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.     

Extended precise integration method for axisymmetric thermo‐elastic problem in transversely isotropic material

This paper presents a numerical solution for the analysis of the axisymmetric thermo‐elastic problem in transversely isotropic material due to a buried heat source by means of extended precise integral method. By virtue of the Laplace–Hankel transform applied into the basic governing equations, an ordinary differential matrix equation is achieved, which describes the relationship between the generalized stresses and displacements in transformed domain. An extended precise integration method is introduced to solve the aforementioned matrix equation, and the actual solution in the physical domain is acquired by inverting the Laplace–Hankel transform. Numerical examples are carried out to demonstrate the accuracy of the proposed method and elucidate the influence of the character of transverse isotropy, the anisotropy of linear expansion coefficient, the anisotropy of thermal diffusivity, and medium's stratification on the thermo‐elastic response. Copyright © 2015 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.     

三维横观各向同性成层地基的传递矩阵解

通过解耦变换推导出三维直角坐标系下横观各向同性地基的非耦合状态方程;利用双重Fourier变换以及Cayley-Hamilton定理得到了单层地基的传递矩阵;结合边界条件和层间连续条件进而得其传递矩阵解。编制了相应程序并进行了数值计算与分析,结果表明：数值结果与已有文献结果十分吻合,地基的横观各向同性性质与成层性质对受荷地基中竖向位移和应力的影响较为显著。