Impacts of saturation-dependent anisotropy on the shrinkage behavior of clay rocks

Geomaterials such as soils and rocks can exhibit inherent anisotropy due to the preferred orientation of mineral grains and/or cracks. They can also be partially saturated with multiple types of fluids occupying the pore space. The anisotropic and unsaturated behaviors of geomaterials can be highly interdependent. Experimental studies have shown that the elastic parameters of rocks evolve with saturation. The effect of saturation has also been shown to differ between directions in transversely isotropic clay rock. This gives rise to saturation-dependent stiffness anisotropy. Similarly, permeability anisotropy can also be saturation-dependent. In this study, constitutive equations accommodating saturation-dependent stiffness and hydraulic anisotropy are presented. A linear function is used to describe the relationship between the elastic parameters and saturation, while the relative permeability–saturation relationship is characterized with a log-linear function. These equations are implemented into a hydromechanical framework to investigate the effects of saturation-dependent properties on the shrinkage behavior of clay rocks. Numerical simulations are presented to demonstrate the role of saturation-dependent stiffness and hydraulic anisotropy in shrinkage behavior. The results highlight that strain anisotropy and time evolution of pore pressures are substantially influenced by saturation-dependent stiffness and hydraulic anisotropy.

     

Elastic shear stiffness anisotropy and fabric anisotropy of natural clays

Natural clays usually show anisotropic stiffness due to their deposition process and anisotropic in situ stress state. The stiffness anisotropy depends on both of the stress anisotropy and fabric anisotropy, while the latter can be quantified by the stiffness anisotropy at isotropic stress states. This paper measures the K₀ value (i.e., stress anisotropy) and elastic shear stiffness anisotropy of natural Shanghai clay in a triaxial apparatus with horizontal and vertical bender elements. The results show that the K₀ value of Shanghai clay lies in the range of 0.40–0.66, and an empirical equation is proposed to estimate the K₀ value based on the plasticity index and initial void ratio. The fabric anisotropy of natural Shanghai clay lies in the range of 1.2–1.4 with a stronger fabric in the horizontal plane. Moreover, the experimental data of the stiffness anisotropy and fabric anisotropy of different clays in the literature are reviewed and analyzed. It reveals that the stiffness anisotropy generally increases, while the fabric anisotropy remains nearly the same during K₀ consolidation. For normally consolidated clay, the fabric anisotropy generally lies in the range of 1.1–1.7. For overconsolidated clays, the fabric anisotropy generally increases as the overconsolidation ratio increases. Empirical equations are proposed to approximately estimate the fabric anisotropy of clays based on its stress normalized elastic shear stiffness.

     

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.     

An efficient optimization method for identifying parameters of soft structured clay by an enhanced genetic algorithm and elastic–viscoplastic model

Soft structured clays usually exhibit complex behaviors, which can lead to difficulties in the determination of parameters and high testing costs. This paper aims to propose an efficient optimization method for identifying the parameters of advanced constitutive model for soft structured clays from only limited conventional triaxial tests. First, a new real-coded genetic algorithm (RCGA) is proposed by combining two new crossover and mutation operators for improving the performance of optimization. A newly developed elastic–viscoplastic model accounting for anisotropy, destructuration and creep features is enhanced with the cross-anisotropy of elasticity and is adopted for test simulations during optimization. Laboratory tests on soft Wenzhou marine clay are selected, with three of them being used as objectives for optimization and others for validation. The optimization process, using the new RCGA with a uniform sampling initialization method, is carried out to obtain the soil parameters. A classic genetic algorithm (NSGA-II)-based optimization is also conducted and compared to the RCGA for estimating the performance of the new RCGA. Finally, the optimal parameters are validated by comparing with other measurements and test simulations on the same clay. All comparisons demonstrate that a reliable solution can be obtained by the new RCGA optimization combined with the appropriate soil model, which is practically useful with a reduction in testing costs.     

Soil resistances for laterally loaded rigid piles in multi-layered elastic soil

ABSTRACT

Short stubby piles like monopiles and large diameter drilled shafts undergo rigid body translation and rotation when subjected to a lateral force and/or a moment at the head. A method of analysis for these piles embedded in multi-layered elastic soil is developed using the variational principles of mechanics. Using this analysis, the soil resistance against pile movement can be rigorously related to the soil elastic constants, and the pile head displacement and rotation can be quickly calculated. The equilibrium equations for pile and soil displacements are obtained using the principle of virtual work and solved using an iterative algorithm. Pile responses obtained from the analysis match well with those obtained from three-dimensional finite element analyses in which the same inputs of loads, geometry, and material properties are given. Based on the new analysis, fitted equations for soil resistance parameters are developed, which can be used to directly calculate the pile head displacement and rotation without the use of the iterative algorithm. Numerical examples are provided that demonstrate how the method can be used to analyse practical problems.     

Undrained expansion of a cylindrical cavity in clays with fabric anisotropy: theoretical solution

This paper presents a novel, exact, semi-analytical solution for the quasi-static undrained expansion of a cylindrical cavity in soft soils with fabric anisotropy. This is the first theoretical solution of the undrained expansion of a cylindrical cavity under plane strain conditions for soft soils with anisotropic behaviour of plastic nature. The solution is rigorously developed in detail, introducing a new stress invariant to deal with the soil fabric. The semi-analytical solution requires numerical evaluation of a system of six first-order ordinary differential equations. The results agree with finite element analyses and show the influence of anisotropic plastic behaviour. The effective stresses at critical state are constant, and they may be analytically related to the undrained shear strength. The initial vertical cross-anisotropy caused by soil deposition changes towards a radial cross-anisotropy after cavity expansion. The analysis of the stress paths shows that proper modelling of anisotropic plastic behaviour involves modelling not only the initial fabric anisotropy but also its evolution with plastic straining.     

Shear wave velocity anisotropy of salt- and polymer-amended kaolinite

Shear wave velocity (V_s) anisotropy of kaolinite mixed with sodium chloride (NaCl) and organic polymer (polyethylene oxide, xanthan gum, and chitosan) solutions was investigated using a custom-made floating wall consolidometer-type bender element testing system. The addition of salt and polymers influenced the microfabric anisoopy of platy kaolinite particles, thus resulted in the increment or decrement in the V_s anisotropy. The V_s of kaolinite in all three orthogonal directions increased as the NaCl concentration increased; however, the V_s anisotropy decreased. PEO and chitosan increased the V_s of kaolinite, while xanthan gum exhibited counter-effects. V_s anisotropy (V_s?hh/V_s?vh and V_s?hv/V_s?vh) of polymer amended kaolinite was found to decrease. In addition, both salt- and polymer-modified kaolinite did not show V_s cross-anisotropy.

     

Consolidation analysis of a cross-anisotropic homogeneous elastic soil using a finite layer numerical method

This paper presents a finite layer procedure for Biot's consolidation analysis of layered soils using a cross-anisotropic elastic constitutive model. The program is first verified using published results. Then, using this program, the influences of cross-anisotropy on the immediate settlement, the final settlement, and the consolidation behaviour are investigated by changing one model parameter at one time. The results obtained using the cross-anisotropic elastic model are compared with results using an isotropic elastic model. It is found that the cross-anisotropy has very large influences on the immediate settlement, the final settlement, and the consolidation behaviour. Curves or tables of the immediate settlement coefficients, the final settlement coefficients, and the average degree of consolidation are obtained and presented in the paper. These curves or tables can be easily used to estimate the immediate settlement, the final settlement, and the consolidation settlement of a cross-anisotropic soil. Copyright © 2004 John Wiley & Sons, Ltd.     

Torsional mode frequency and elastic anisotropy in alkali vermiculites

The torsional mode frequency _t in Cs_1–xRb_x-Vermiculite has been determined using an angular force constant model and a virtual crystal approximation. The sensitivity of _t has been examined with respect to the force constants between the Kagome' oxygen frame and the interlayer cations. These force constants were used to calculate the longitudinal elastic constants, C₁₁ and C₃₃ and are consistent with the observed elastic anisotropy in related layer silicates. The observed nonlinear x-dependence of the torsional mode frequency in ternary systems can be related to the polarizability of the interlayer cations (Cs⁺,Rb⁺).     

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.     

Simplified Method for the Assessment of the Stiffness Anisotropy of Rocks at Small Strains

Summary A new, practically applicable method for characterizing the stiffness anisotropy of rocks is presented. The anisotropy of geo-materials is often ignored in engineering applications, with potentially serious ramifications, because of the number of parameters required for characterization. The elastic anisotropy has often been considered to be a function of mathematical symmetry, and the restrictions due to layering, microcracking and granularity of the materials have not been considered in the assessment of the anisotropy. The practicality of the method proposed here is achieved by rationally reducing the number of independent anisotropy parameters, typically 9 for orthotropic anisotropy, to a system of 4 independent parameters through a systematic theoretical and experimental analysis of these structural restrictions. These 4 parameters are shown to be sufficient for describing the anisotropy of some rocks and sands at small strains, and parameter determination by back-analysis is demonstrated to be stable using appropriate measurement systems involving 9 elastic wave velocities even when the directions of anisotropic axes are unknown and the velocity data contains appreciable error.     

Hollow Cylinder Tests on Boom Clay: Modelling of Strain Localization in the Anisotropic Excavation Damaged Zone

Boom Clay is extensively studied as a potential candidate to host underground nuclear waste disposal in Belgium. To guarantee the safety of such a disposal, the mechanical behaviour of the clay during gallery excavation must be properly predicted. In that purpose, a hollow cylinder experiment on Boom Clay has been designed to reproduce, in a small-scale test, the Excavation Damaged Zone (EDZ) as experienced during the excavation of a disposal gallery in the underground. In this article, the focus is made on the hydro-mechanical constitutive interpretation of the displacement (experimentally obtained by medium resolution X-ray tomography scanning). The coupled hydro-mechanical response of Boom Clay in this experiment is addressed through finite element computations with a constitutive model including strain hardening/softening, elastic and plastic cross-anisotropy and a regularization method for the modelling of strain localization processes. The obtained results evidence the directional dependency of the mechanical response of the clay. The softening behaviour induces transient strain localization processes, addressed through a hydro-mechanical second grade model. The shape of the obtained damaged zone is clearly affected by the anisotropy of the materials, evidencing an eye-shaped EDZ. The modelling results agree with experiments not only qualitatively (in terms of the shape of the induced damaged zone), but also quantitatively (for the obtained displacement in three particular radial directions).     

Numerical study of excavation induced fractures using an extended rigid block spring method

This paper presents a numerical study of fracturing process induced by excavation around a gallery using an extended rigid block spring method (RBSM). The surrounding rock mass is characterized by an assembly of rigid blocks based on a degraded Voronoi diagram. The macroscopic mechanical behavior of rock is related to that of interfaces between blocks. The mechanical behavior of each interface is described by its elastic stiffness and failure criterion. The failure process of interface is controlled by both normal stress and shear stress. Both tensile and shear failures are considered. The macroscopic fracturing process is described by the coalescence of cracked interfaces. The rock structural anisotropy is taken into account through a spatial variation of elastic stiffness and failure strength of interfaces. A series of sensitivity studies are performed to investigate effects of gallery orientation, failure strength of interfaces and rock structural anisotropy on gallery deformation and fracturing. Numerical results are compared with in-situ observations in terms of fracture patterns.     

岩体工程特性研究中弹性波速参数取值方法探讨

工程岩体的各向异性特征导致了弹性波速的各向异性。因此，在岩体工程特性研究、岩体质量评价等方面，应注意合理地应用岩体弹性波速资料。笔者分析了岩体波速的各向异性特征，并指出：由于波速各向异性的存在，在岩体完整性系数计算中，岩体波速应取不同方向波速的几何平均值。当利用文中所给公式进行岩体孔隙率计算时，建议波速取最小值；在进行节理、裂隙发育的优势方向判断时，则取不同方向的波速值分别进行计算；当计算弹性模量时，必须取与岩体受力方向相一致的波速值。并给出了利用波速进行有关参数计算的实例。     

Finite element modelling of thick plates on two‐parameter elastic foundation

This paper is intended to give some information about how to build a model necessary for bending analysis of rectangular and circular plates resting on a two‐parameter elastic foundation, subjected to combined loading and permitting various types of boundary conditions. The formulation of the problem takes into account the shear deformation of the plate and the surrounding interaction effect outside the plate. The numerical model based on an 18‐node zero‐thickness isoparametric interface element interacting with a thick Reissner–Mindlin plate element with three degrees of freedom at each of the nine nodes, which enforce C⁰ continuity requirements for the displacements and rotations of the midsurface, is proposed. Stiffness matrices of a special interface element are superimposed on the global stiffness matrix to represent the stiffening elastic foundation under and beyond the plate. Some numerical examples are given to illustrate the advantages of the method presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Lattice dynamics of corundum

The short-range force constants and the contribution of the potential energy to the elastic constants have been calculated from explicit expressions for the elastic constants and for the optical frequencies. The large elastic constants of corundum are attributable to the contribution of the large short-range stretching forces. The frequencies of the phonon spectrum and the contribution of the potential energy to the frequencies have also been calculated based on a rigid-ion and polarizable-ion model. The effective charge of an oxygen ion is found to be ?0.955e. The electronic polarizability of an aluminium and oxygen ion is found to be 0.05 and 1.36 ų, respectively.     

Elastic Solution for Deep Tunnels. Application to Excavation Damage Zone and Rockbolt Support

Summary  A new formulation is presented for deep circular tunnels in rock with cylindrical anisotropy. The formulation is an exact solution since it satisfies equilibrium, strain compatibility, and the anisotropic constitutive model. Complete solutions have been found for two scenarios: tunnel with excavation damage zone, and tunnel with rockbolt support. The solution is based on the assumption of a deep, circular tunnel in a medium with two homogeneous zones: an inner zone surrounding the tunnel, which is either isotropic or anisotropic, and an outer zone, for the remainder of the medium, which is isotropic. Plane strain conditions, elastic response of rock, rockbolts and support, and simultaneous excavation and support installation are also assumed. For tunnels surrounded by an excavation damage zone with reduced rock properties, the tangential stresses and the radial deformations at the tunnel wall are very sensitive to both the magnitude of stiffness reduction of the damaged rock and the size of the damaged zone. The effect of the rockbolts on the rock is approximated by treating the rockbolt-rock composite as a material with cylindrical anisotropy with stiffnesses related to the properties of the rock and rockbolts, and spacing of the rockbolts. Comparisons between the analytical solution and a numerical method show small differences and provide confidence in the approach suggested.