On a multilaminate model for soil incorporating small strain stiffness

In this paper a constitutive model for soils incorporating small strain stiffness formulated in the multilaminate framework is presented. In the multilaminate framework, the stress–strain behaviour of a material is obtained by integrating the mechanical response of an infinite number of randomly oriented planes passing through a material point. Such a procedure leads to a number of advantages in describing soil behaviour, the most significant being capture of initial and induced anisotropy due to plastic flow in a physically meaningful manner. In the past, many soil models of varying degree of refinement in the multilaminate framework have been presented by various authors. However, the issue of high initial soil stiffness in the range of very small strains and its degradation with straining, commonly referred to as ‘small strain stiffness’, has not been addressed within the multilaminate framework. In this paper, we adopt a micromechanics‐based approach to derive small strain elastic stiffness of the soil mass. Comparison of laboratory test data with results obtained from numerical simulations based on the proposed constitutive model incorporating small strain stiffness is performed to demonstrate its predictive capabilities. The model is implemented in a finite element code and numerical simulations of a deep excavation are presented with and without incorporation of small strain stiffness to demonstrate its importance in predicting profiles of deformation. Copyright © 2008 John Wiley & Sons, Ltd.     

Lateral stress caused by horizontal and vertical surcharge strip loads on a cross‐anisotropic backfill

This study derives analytical solutions for estimating the lateral stress caused by horizontal and vertical surcharge strip loads resting on a cross‐anisotropic backfill. The following loading types are employed in this work: point load, line load, uniform strip load, upward linear‐varying strip load, upward nonlinear‐varying strip load, downward linear‐varying strip load and downward nonlinear‐varying strip load. The cross‐anisotropic planes are assumed to be parallel to the horizontal surface of the backfill. The solutions proposed herein have never been mentioned in previous literature, but can be derived by integrating the point load solution in a Cartesian co‐ordinate system for a cross‐anisotropic medium. The calculations by the presented solutions are quick and accurate since they are concise and systematized. Additionally, the proposed calculations demonstrate that the type and degree of material anisotropy and the horizontal/vertical loading types decisively influence the lateral stress. This investigation presents examples of the proposed horizontal and vertical strip loads acting on the surface of the isotropic and cross‐anisotropic backfills to elucidate their effects on the stress. The analytical results reveal that the stress distributions accounting for soil anisotropy and loading types are quite different from those computed from the available isotropic solutions. Restated, the derived solutions, as well as realistically simulating the actual surcharge loading circumstances, provide a good reference for the design of retaining structures for the backfill materials are cross‐anisotropic. Copyright © 2005 John Wiley & Sons, Ltd.     

一个考虑土的各向异性的孔压公式及其应用

为了研究粘土的各向异性对孔压生成的影响,在修正剑桥模型的基础上,采用关口-太田应力比概念、结合邢义川-郑颖人方程,构造了一个各向异性旋转屈服面,建立了一个各向异性弹塑性本构模型;据此推导出一个各向异性孔压表达式,该式可以考虑Lode角以及主应力轴的平面旋转对孔压的影响;并采用新的孔压表达式着重分析了三轴应力状态下Lode角旋转以及主应力轴旋转所造成的孔压发展,并对条形荷载下的平面应变各向异性粘土地基中的孔压发展进行了计算。计算结果表明,地基中由于Lode角旋转造成的孔压不明显,但在荷载边角位置主应力轴旋转造成的孔压不容忽视,尤其是在埋深相对较浅的位置。     

Generalized solution to the anisotropic Mandel's problem

Existing solutions to Mandel's problem focus on isotropic, transversely isotropic, and orthotropic materials, the last two of which have one of the material symmetry axes coincide with the vertical loading direction. The classical plane strain condition holds for all these cases. In this work, analytical solution to Mandel's problem with the most general matrix anisotropy is presented. This newly derived analytical solution for fully anisotropic materials has all the three nonzero shear strains. Warping occurs in the cross sections, and a generalized plane strain condition is fulfilled. This solution can be applied to transversely isotropic and orthotropic materials whose material symmetry axes are not aligned with the vertical loading direction. It is the first analytical poroelastic solution considering mechanical general anisotropy of elasticity. The solution captures the effects of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction on the responses of pore pressure, stress, strain, and displacement. It can be used to match, calibrate, and simulate experimental results to estimate anisotropic poromechanical parameters. This generalized solution is capable of reproducing the existing solutions as special cases. As an application, the solution is used to study the responses of transversely isotropic and orthotropic materials whose symmetry axes are not aligned with the vertical loading direction. Examples on anisotropic shale rocks show that the effects of material anisotropy are significant. Mandel-Cryer's effects are highly impacted by the degree of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction.     

Lateral force and centroid location caused by horizontal and vertical surcharge strip loads on a cross‐anisotropic backfill

This work presents analytical solutions for determining lateral force (force per unit length) and centroid location caused by horizontal and vertical surcharge surface loads acting on a cross‐anisotropic backfill. The surcharge loading types are point load, line load, uniform strip load, upward linear‐varying strip load, upward nonlinear‐varying strip load, downward linear‐varying strip load, and downward nonlinear‐varying strip load. The planes of cross‐anisotropy are assumed parallel to the backfill ground surface. The proposed solutions, derived by integrating the lateral stress solutions (Int. J. Numer. Anal. Meth. Geomech. 2005; 29 :1341–1361), do not exist in literature. Clearly, the type and degree of material anisotropy, loading distance from the retaining wall, and loading types markedly impact the proposed solutions. Two examples are utilized to illustrate the type and degree of soil anisotropy, and the loading types on the lateral force and centroid location in the isotropic/cross‐anisotropic backfills generated by the horizontal and vertical uniform, upward linear‐varying and upward nonlinear‐varying strip loads. The parametric study results demonstrate that the lateral force and centroid location accounting for soil anisotropy, loading distance from the retaining wall, dimension of the loading strip, and loading directions and types differ significantly from those estimated using existing isotropic solutions. The derived solutions can be added to other lateral pressures, such as earth pressure or water pressure, required for stability and structural analysis of a retaining wall. Additionally, they can simulate realistically actual surcharge loading problems in geotechnical engineering when backfill materials are cross‐anisotropic. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi-directional modeling for prediction of fabric anisotropy in sand liquefaction

Anisotropy in the onset of liquefaction and post liquefaction under cyclic loading condition causes the change in response behavior of the soil. The proposed model operates within the integration of sliding/opening/closing framework of 17 predefined planes as local deformation. This leads to the use of better stress/strain multilaminate histories with many directional effects on soil behavior specifically internal mechanism during pre and post-liquefaction. The ability of multilaminate model for fabric anisotropy has been proven by comparison with the experimental results under drain and undrained conditions and monotonic and cyclic loading. The effects of induced anisotropy was also investigated.     

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.     

A generalised D matrix for anisotropic elastic granular media

A matrix relating stress and elastic strain tensors for anisotropic particulate materials has been derived. The magnitude of the matrix depends on the state of the material anisotropy. Anisotropy in granular materials depends on strain because normal and tangential particle contact forces, as well as the spatial distribution of the contacts, vary with stress and strain. However, the rotation tensor and the strain tensor cannot be independent; they must satisfy certain constraints to meet the requirement for macroscopic stress tensor symmetry. These conditions and constraints lead to the derivation of the matrix presented in this article. The principal directions of the stress tensor and strain tensor are generally not coincident, and the values of deformation parameters, Young's modulus and Poisson's ratio, are direction dependent; these two aspects are also discussed in this paper. Whereas this matrix can be used in static numerical analyses for elastic problems, we note that this relationship can also be used as a basis upon which to derive a fully incremental stress–strain relationship for anisotropic granular materials in the plastic state, where the anisotropy is evolving with strain.     

Vertical stress distributions around batter piles driven in cross‐anisotropic media

This work presents analytical solutions to compute the vertical stresses for a cross‐anisotropic half‐space due to various loading types by batter piles. The loading types are an embedded point load for an end‐bearing pile, uniform skin friction, and linear variation of skin friction for a friction pile. The cross‐anisotropic planes are parallel to the horizontal ground surface. The proposed solutions can be obtained by utilizing Wang and Liao's solutions for a horizontal and vertical point load acting in the interior of a cross‐anisotropic medium. The derived cross‐anisotropic solutions using a limiting approach are in perfect agreement with the isotropic solutions of Ramiah and Chickanagappa with the consideration of pile inclination. Additionally, the present solutions are identical to the cross‐anisotropic solutions by Wang for the batter angle equals to 0. The influential factors in yielded solutions include the type and degree of geomaterial anisotropy, pile inclination, and distinct loading types. An example is illustrated to clarify the effect of aforementioned factors on the vertical stresses. The parametric results reveal that the stresses considering the geomaterial anisotropy and pile batter differ from those of previous isotropic and cross‐anisotropic solutions. Hence, it is imperative to take the pile inclination into account when piles are required to transmit both the axial and lateral loads in the cross‐anisotropic media. Copyright © 2008 John Wiley & Sons, Ltd.     

Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading

A new numerical approach is proposed in this study to model the mechanical behaviors of inherently anisotropic rocks in which the rock matrix is represented as bonded particle model, and the intrinsic anisotropy is imposed by replacing any parallel bonds dipping within a certain angle range with smooth‐joint contacts. A series of numerical models with β = 0°, 15°, 30°, 45°, 60°, 75°, and 90° are constructed and tested (β is defined as the angle between the normal of weak layers and the maximum principal stress direction). The effect of smooth‐joint parameters on the uniaxial compression strength and Young's modulus is investigated systematically. The simulation results reveal that the normal strength of smooth‐joint mainly affects the behaviors at high anisotropy angles (β > 45°), while the shear strength plays an important role at medium anisotropy angles (30°–75°). The normal stiffness controls the mechanical behaviors at low anisotropy angles. The angle range of parallel bonds being replaced plays an important role on defining the degree of anisotropy. Step‐by‐step procedures for the calibration of micro parameters are recommended. The numerical model is calibrated to reproduce the behaviors of different anisotropic rocks. Detailed analyses are conducted to investigate the brittle failure process by looking at stress‐strain behaviors, increment of micro cracks, initiation and propagation of fractures. Most of these responses agree well with previous experimental findings and can provide new insights into the micro mechanisms related to the anisotropic deformation and failure behaviors. The numerical approach is then applied to simulate the stress‐induced borehole breakouts in anisotropic rock formations at reduced scale. The effect of rock anisotropy and stress anisotropy can be captured. Copyright © 2015 John Wiley & Sons, Ltd.     

The effect of particle shape and grain‐scale properties of shale: A micromechanics approach

Traditional approaches for modeling the anisotropic elasticity response of the highly heterogeneous clay fabric in shale have mainly resorted to geometric factors such as definitions of particles shapes and orientations. However, predictive models based on these approaches have been mostly validated using macroscopic elasticity data. The recent implementation of instrumented indentation aimed at probing nano‐scale mechanical behaviors has provided a new context for characterizing and modeling the anisotropy of the porous clay in shale. Nanoindentation experimental data revealed the significant contribution of the intrinsic anisotropy of the solid clay to the measured elastic response. In this investigation, we evaluate both the effects of geometric factors and of the intrinsic anisotropic elasticity of the solid clay phase on the observed anisotropy of shale at multiple length scales through the development of a comprehensive theoretical micromechanics approach. It was found that among various combinations of these sources of anisotropy, the elastic response of the clay fabric represented as a granular ensemble of aligned effective clay particles with spherical morphology and anisotropic elasticity compares satisfactorily to nanoindentation and ultrasonic pulse velocity measurements at nano‐ and macroscopic length scales, respectively. Other combinations of sources of anisotropy could yield comparable predictions, particularly at macroscopic scales, at the expense of requiring additional experimental data to characterize the morphology and orientations of particles. Copyright © 2009 John Wiley & Sons, Ltd.     

Surface displacements due to batter piles driven in cross‐anisotropic media

This article derives the closed‐form solutions for estimating the vertical surface displacements of cross‐anisotropic media due to various loading types of batter piles. The loading types include an embedded point load for an end‐bearing pile, uniform skin friction, and linear variation of skin friction for a friction pile. The planes of cross‐anisotropy are assumed to be parallel to the horizontal ground surface. The proposed solutions are never mentioned in literature and can be developed from Wang and Liao's solutions for a horizontal and vertical point load embedded in the cross‐anisotropic half‐space. The present solutions are identical with Wang's solutions when batter angle equals to 0^°. In addition, the solutions indicate that the surface displacements in cross‐anisotropic media are influenced by the type and degree of material anisotropy, angle of inclination, and loading types. An illustrative example is given at the end of this article to investigate the effect of the type and degree of soil anisotropy (E/E′, G′/E′, and ν/ν′), pile inclination (α), and different loading types (a point load, a uniform skin friction, and a linear variation of skin friction) on vertical surface displacements. Results show that the displacements accounted for pile batter are quite different from those estimated from plumb piles, both driven in cross‐anisotropic media. Copyright © 2007 John Wiley & Sons, Ltd.     

Stresses due to vertical subsurface loading for an inhomogeneous cross‐anisotropic half‐space

In this article, we present the solutions for the stresses induced by four different loads associated with an axially loaded pile in a continuously inhomogeneous cross‐anisotropic half‐space. The planes of cross‐anisotropy are parallel to the horizontal surface of the half‐space, and the Young's and shear moduli are assumed to vary exponentially with depth. The four loading types are: an embedded point load for an end‐bearing pile, uniform skin friction, linear variation of skin friction, and non‐linear parabolic variation of skin friction for a friction pile. The solutions for the stresses due to the pile load are expressed in terms of the Hankel integral and are obtained from the point load solutions of the same inhomogeneous cross‐anisotropic half‐space which were derived recently by the authors (Int. J. Rock Mech. Min. Sci. 2003; 40 (5):667–685). A numerical procedure is proposed to carry out the integral. For the special case of homogeneous isotropic and cross‐anisotropic half‐space, the stresses predicted by the numerical procedure agree well with the solutions of Geddes and Wang (Geotechnique 1966; 16 (3):231–255; Soils Found. 2003; 43 (5):41–52). An illustrative example is also given to investigate the effect of soil inhomogeneity, the type and degree of soil anisotropy, and the four different loading types on the vertical normal stress. The presented solutions are more realistic in simulating the actual stratum of loading problem in many areas of engineering practice. Copyright © 2004 John Wiley & Sons, Ltd.     

Borehole failure analysis in a sandstone under anisotropic stresses

Borehole failure under anisotropic stresses in a sandstone is analyze numerically for various borehole sizes using a nonlinear elastic–plastic constitutive model for a Cosserat continuum. Borehole failure is identified as macroscopic failure of the borehole through the development of shear bands and breakouts. The results compare well both qualitatively and quantitatively with experimental results from polyaxial tests on Red Wildmoor sandstone. They show that the hole size effect of the borehole failure strength is independent of the far‐field stress anisotropy and follows a ? power law of the hole size. A similar scale effect equation with a ? power law is proposed for the scale effect of the maximum plastic shear strain at failure. This equation can be useful for better predicting hole‐size‐dependent failure with standard codes based on classical continua. The effect of stress anisotropy on the borehole failure stress is found to be independent of the hole size. The failure stress decreases linearly to 40% as the stress anisotropy increases. However, the maximum plastic shear strain at failure is stress anisotropy independent and therefore the critical plastic shear strain for failure is only hole‐size dependent. Copyright © 2009 John Wiley & Sons, Ltd.     

Evolution of elastic properties in finite poroplasticity and finite element analysis

The formulation of the poroelastoplastic constitutive equations at large strains of a fully saturated material is performed focusing on the usually ignored influence of large strain plasticity on the poroelastic properties. A micromechanics approach allows to take into account the evolution of the microstructure geometry which in turn induces a coupling between elasticity and plasticity. Such a coupling results in an additional term in the macroscopic Cauchy stress rate equation derived from inclusion‐based estimates that leads to a modified Jaumann derivative. The pressure rate equation is also analysed. The finite element discretization of finite poroplasticity is then presented taking into account the elasticity–plasticity coupling. Application to the consolidation situation shows that the coupling may lead to non‐negligible effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Hyperelastic modelling of small‐strain stiffness anisotropy of cyclically loaded sand

Experimental evidence shows that soil stiffness at very small strains is strongly anisotropic and depends on the stress level and void ratio. In particular, stiffness anisotropy varies considerably in sand when subjected to cyclic loading, following the stress cycles applied. To model this behaviour, an innovative hyperelastic formulation based on the elastoplastic coupling is incorporated in a new kinematic hardening elastoplastic model. The proposed hyperelastic–plastic model is the first to be capable of correctly simulating all aspects of the small‐strain behaviour of granular materials subjected to monotonic and cyclic loads. This hyperelastic formulation is generally applicable to any elastoplastic model. Copyright © 2009 John Wiley & Sons, Ltd.     

Evolution of induced fabric in a strain space multiple mechanism model for granular materials

The strain space multiple mechanism model idealizes the behavior of granular materials based on a multitude of virtual simple shear mechanisms oriented in arbitrary directions. Within this modeling framework, the virtual simple shear stress is defined as a quantity that depends on the contact distribution function as well as the normal and tangential components of inter‐particle contact forces, which evolve independently during the loading process. In other terms, the virtual simple shear stress is an intermediate quantity in the upscaling process from the microscopic level (characterized by the contact distribution and inter‐particle contact forces). The stress space fabric (i.e. the orientation distribution of the virtual simple shear stress) produces macroscopic stress through the tensorial average. Thus, the stress space fabric characterizes the fundamental and higher modes of anisotropy induced in granular materials. Comparing an induced fabric associated with the biaxial shear of plane granular assemblies obtained via a simulation using Discrete Element Method to the strain space multiple mechanism model suggests that the strain space multiple mechanism model has the capability to capture the essential features in the evolution of an induced fabric in granular materials. Copyright © 2012 John Wiley & Sons, Ltd.     

An Anisotropic Geomechanical Model for Jointed Rock Masses at Taishan Nuclear Power Station, Southern China

An anisotropic geomechanical model for jointed rock mass is presented. Simultaneously with deriving the orthotropic anisotropy elastic parameters along the positive axis, the equivalent compliance matrix for the deflection axis orthotropic anisotropy was derived through a three-dimensional coordinate transformation. In addition, Singh’s analysis of the stress concentration effects of intermittent joints was adopted, based on two groups of intermittent joints and a set of cross-cutting joints in the jointed rock mass. The stress concentration effects caused by intermittent joints and the coupling effect of cross-cutting joints along the deflection-axis are also considered. The proposed anisotropic mechanics parameters method is applied to determine the deformation parameters of jointed granite at the Taishan Nuclear Power Station. Combined with the deterministic mechanical parameters of rock blocks and joints, the deformation parameters and their variability in jointed rock masses are estimated quantitatively. The computed results show that jointed granite at the Taishan Nuclear Power Station exhibits typical anisotropic mechanical characteristics; the elastic moduli in the two horizontal directions were similar, but the elastic modulus in the vertical direction was much greater. Jointed rock elastic moduli in the two horizontal and vertical directions were respectively about 24% and 37% of the core of rock, showing weakly orthotropic anisotropy; the ratio of elastic moduli in the vertical and horizontal directions was 1.53, clearly indicating the transversely isotropic rock mass mechanical characteristics. The method can be popularized to solve other rock mechanics problems in nuclear power engineering.     

有明黏土各向异性固结特性的试验研究

由于大多数黏土矿物是片状的,在沉积和随后的固结过程中黏土颗粒和组构单元趋向水平定向排列,使天然沉积黏土呈现各向异性的固结特性。通过垂直切和水平切试样的恒应变速率固结试验研究了日本有明黏土的各向异性固结特性。试验结果表明,水平切试样的先期固结压力pch与垂直切试样的先期固结压力pcv的比值在0.5～1.0之间,主要是因为天然沉积黏土各向异性的屈服轨迹以及垂直切和水平切试样所经历的应力路径不同;在正常固结阶段,水平切试样的固结系数ch与垂直切试样的固结系数cv之比约为1.43,水平切试样的渗透系数kh与垂直切试样的渗透系数kv之比约为1.40,水平切试样的体积压缩系数mh与垂直切试样的体积压缩系数mv基本相等。因此,有明黏土固结系数的各向异性主要是因为渗透系数的各向异性,而渗透系数的各向异性本质上是因为其微观结构的各向异性。     

A bounding surface elasto‐viscoplastic constitutive model for non‐isothermal cyclic analysis of asphaltic materials

An elasto‐viscoplastic constitutive model for asphaltic materials is presented within the context of bounding surface plasticity theory, taking into account the effects of the stress state, void binder degree of saturation, temperature and strain rate on the material behaviour. A stress state dependent non‐linear elasticity model is introduced to represent time‐independent recoverable portion of the deformation. The consistent visco‐plasticity framework is utilised to capture the rate‐dependent, non‐recoverable strain components. The material parameters introduced in the model are identified, and their determination from conventional laboratory tests is discussed. The capability of the model to reproduce experimentally observed response of asphaltic materials is demonstrated through numerical simulations of several laboratory test data from the literature. Copyright © 2016 John Wiley & Sons, Ltd.