Definition Of Effective Stress Research Articles

Thermodynamically consistent effective stress formulation for unsaturated soils across a wide range of soil saturation

We outline an extension of Biot’s theory of dynamic wave propagation in fluid-saturated media, which can be used to model dynamic soil-structure interaction in frictionless conditions across a wide range of soil saturation levels. In this regard, we present a comprehensive analysis of experimental evidence, the thermodynamic, and the theoretical basis of using the degree of saturation as Bishop’s parameter in unsaturated soils. The analysis highlights the limitations of using this parameter to accurately model unsaturated soil behaviour, particularly as the soil approaches dryness. Based on the analysis, a new definition of effective stress is proposed, and the associated work-conjugate pairs are identified. Recommendations are made for constitutive modelling using the new definition of effective stress. Finally, we introduce a fully coupled finite element contact model that utilises the new effective stress definition. Through numerical examples, we demonstrate the model’s capability to control the vanishing capillary effect on soil-structure interaction as the soil dries.

Coal permeability behaviors and non-uniform deformations under various boundary conditions: Part 1 – Experimental observations

Permeability is a key parameter to evaluate the ability of coal reservoir for transmitting coalbed methane (CBM). Deformations of internal components of coal caused by the variations of physical fields control the evolution of coal permeability. And there is obvious non-uniform in the deformations of coal bulk and fracture due to the heterogeneity of coal. However, at present, the evolution of non-uniform deformation and its relationship with coal permeability are unclear. In this study, first of a two-part series, we completed a series of experiments to measure the permeability, axial strain, radial strain, overall strain and fracture strain of a coal sample under different boundary conditions (constant confining pressure, constant effective stress and uniaxial strain) at different injection pressures of carbon dioxide (0.2 MPa, 1.0 MPa, 2.0 MPa, 3.0 MPa, 4.0 MPa and 5.0 MPa). And we defined the comprehensive non-uniform deformation index (CNDI) as the ratio of the fracture strain to the overall strain to quantify the degree of the non-uniform deformation between coal bulk and fracture. The experimental results show that the deformation of the coal sample in the longitudinal joint direction is smaller than that in the transverse joint direction, which indicates that the mechanical properties of the coal sample is anisotropic in different directions. And there are obvious differences between the deformations of coal bulk and fractures. When the gas pressure is in the range of 0.2 ∼ 5.0 MPa, the non-uniform deformations are vaguely related to the change of gas pressure, but closely related to the type of boundary conditions. The influence of gas pressure under displacement-controlled boundary condition (uniaxial strain) on the CNDI is greater than that under stress-controlled boundary conditions (constant confining pressure, constant effective stress). In addition, we also observed the phenomenon by the experimental results of the constant effective stress group: the permeability decreases gradually with the gas pressure even after removing the influence of gas slip effect, which is inconsistent with the traditional theoretical solution under the constant effective stress condition. Based on the fact that coal sample is a heterogeneous body, we tentatively believe that one of the reasons for this inconsistency may be that the traditional definition of effective stress does not include the effect of gas adsorption.

Poroelastic solution of a wellbore in a swelling rock with non-hydrostatic stress field

The stress distribution around a circular borehole has been studied extensively. The existing analytical poroelastic solutions, however, often neglect the complex interactions between the solid and fluid that can significantly affect the stress solution. This is important in unconventional gas reservoirs such as coalbeds and shale formations. In order to address this limitation, this paper presents the development of a poroelastic solution that takes into account the effect of gas sorption-induced strain. The solution considers drainage of the reservoir fluid through a vertical wellbore in an isotropic, homogenous, poroelastic rock with non-hydrostatic in situ stress field. The sorption-induced shrinkage of coal is modelled using a Langmuir-type curve which relates the volumetric change to the gas pressure. The redistributed stress field around the wellbore after depletion is found by applying Biot's definition of effective stress and Airey's stress functions, which leads to a solution of the inhomogeneous Biharmonic equation. Two sets of boundary conditions were considered in order to simulate the unsupported cavity (open-hole) and supported cavity (lined-hole) cases. The implementation was verified against a numerical solution for both open-hole and lined-hole cases. A comparative study was then conducted to show the significance of the sorption-induced shrinkage in the stress distribution. Finally, a parametric study analysed the sensitivity of the solution to different poroelastic parameters. The results demonstrate that the developed solution is a useful tool that can be employed alongside complex flow models in order to conduct efficient, field-scale coupled hydro-mechanical simulations, especially when the stress-dependent permeability of the reservoir is of concern.

Pore pressure coefficient in frozen soils

The Skempton pore pressure coefficient, B, is defined as the variation in pore pressure with the unit change in confining pressure under undrained conditions. The B parameter is an essential parameter to consider the coupled effects of solid–fluid compressibility and skeleton compressibility in a porous system. It is a key factor in exploring a possible definition of effective stress in frozen soil. However, limited experimental and theoretical research is available in the literature to give insight to the problem. Therefore, a series of B tests on frozen clay was conducted in this study. Results from these tests, along with tests on Ottawa sand, as available in the literature, are analysed considering the effect of the ice crystallisation mode on the skeleton stiffness. The measured B values were lower than expected compared with B values using models that consider single grain bulk stiffness. However, when the difference in bulk stiffness of ice and that of soil grains is considered, even an increase in pore volume, for an increase in fluid pressure, at constant Terzaghi effective stress is possible. The ‘pore stiffness’, which is different from the solid phase stiffness, can take a negative value and can be used to explain the low measured B values.

Surface‐Wave Dispersion in Partially Saturated Soils: The Role of Capillary Forces

AbstractImproving our understanding of the relation between the water content and the seismic signatures of unconsolidated superficial soils is an important objective in the overall field of hydrogeophysics. Current approaches to constrain the water content in the vadose zone from seismic data are based on computing the ratio between compressional and shear wave velocities Vp/Vs. While this allows for the detection of pronounced changes in saturation, such as the groundwater table, it is essentially insensitive to variations in the saturation‐depth profile. Conversely, evidence shows that surface waves are sensitive to both the location of the water table and the saturation‐depth profile. Classic rock physics models are unable to explain the corresponding observations. We propose to estimate surface‐wave signatures accounting for capillary suction effects. We extend the Hertz‐Mindlin model using Bishop's effective stress definition, thus accounting for stiffness changes associated with capillary stresses acting on the soil's frame. We then compute the elastic properties of the partially saturated medium using the Biot‐Gassmann‐Wood model. Considering a 1D unconsolidated porous medium under steady‐state saturation conditions, as given by Richards' equation, we simulate body‐wave travel times and surface‐wave dispersion characteristics for different water table depths and overlying soil textures. Our results illustrate that surface‐wave phase velocity dispersion curves are remarkably sensitive to capillary effects in partially saturated soils, exhibiting velocity changes of up to 20% in the 10–100 Hz frequency range. These effects, which are particularly important in medium‐to fine‐grained soils, are virtually nonexistent in the corresponding Vp/Vs profiles.

Generalized effective stress concept for saturated active clays

Experimental evidence shows that changes in pore-water chemistry can significantly affect the mechanical behavior of saturated active clays. Despite this evidence, how the chemical composition of the pore water can be considered in effective stress definition is questionable. This paper develops the concept of generalized effective stress for active clays. To this end, physicochemical studies on water–clay mineral interactions are used to clearly define the different types of ions and water present in an active clay. In particular, the presence of both movable and non-movable ions within the liquid water is highlighted. Taking this into account, thermodynamic and geochemistry principles are applied to the representative elementary volume scale for determining the pore-water pressure and redefine the effective stress accordingly. The theoretical development results in the dependence of the effective stress on the pore-water chemistry through the effective solute suction variable. Equations for the determination of this chemical variable are developed. The implications of the use of the proposed effective stress concept are investigated using experimental results taken from the literature. The results show advantages both in the interpretation of shear strength and volumetric data, and all support the theoretical explanation underlying the proposed concept.

From Basic Particle Gradation Parameters to Water Retention Curves and Tensile Strength of Unsaturated Granular Soils

Abstract The long-debated effective stress definition of unsaturated soils is believed to be correlated to suction, the degree of saturation, and the less mentioned interfacial areas. The tensile s...

Investigation of the effect of specific interfacial area on strength of unsaturated granular materials by X-ray tomography

This paper studies the effect of interfacial areas (air–water interfaces and solid–water interfaces) on material strength of unsaturated granular materials. High-resolution X-ray computed tomography technique is employed to measure the interfacial areas in wet glass bead samples. The scanned 3D images are trinarized into three phases and meshed into representative volume elements (RVEs). An appropriate RVE size is selected to represent adequate local information. Due to the local heterogeneity of the material, the discretized RVEs of the scanned samples actually cover a very large range of degree of saturation and porosity. The data of RVEs present the relationship between the specific interfacial areas and degree of saturation and gives boundaries where the interfacial area of a whole sample should fall in. In parallel, suction-controlled direct shear tests have been carried out on glass beads and the material strength has been corroborated with two effective stress definitions related to the specific air–water interfacial areas and fraction of wetted solid surface, respectively. The comparisons show that the specific air–water interfacial area reaches the peak at about 25% of saturation and contributes significantly to the material strength (up to 60% of the total capillary strength). The wetted solid surface obtained from X-ray CT is also used to estimate Bishop’s coefficient χ based on the second type of effective stress definition, which shows a good agreement with the measured value. This work emphasizes the importance to include interface terms in effective stress formulations of unsaturated soils. It also suggests that the X-ray CT technique and RVE-based multiscale analysis are very valuable in the studies of multiphase geomaterials.

Loading-unloading shear behavior of rammed earth upon varying clay content and relative humidity conditions

This paper investigates the impact of clay and moisture contents on the shear behavior of compacted earth taking into account loading-unloading cycles. Fine sand was added to a natural soil, thereby obtaining three different soils with clay contents of 35%, 26%, and 17%, respectively. A series of triaxial tests was conducted on samples previously equilibrated at three different values of relative humidity (RH). The evolution of failure strength fc, Young's modulus E, and residual strain εres was investigated according to the clay content and the RH, the last two parameters being measured during the loading-unloading cycles. Firstly, the relative humidity at which the samples were fabricated and conditioned was seen to have a strong impact on the mechanical characteristics of the earthen material. An increase in RH led to a decrease in both failure strength fc and Young’s modulus E, and an increase in plastic strain. The tendencies were found to depend on the clay content of the samples. Secondly, with an increasing stress level, a progressive decrease in Young’s modulus and an increase in residual strain εres (after a loading-unloading cycle) appeared. Thirdly, within the range of the investigated clay contents, both failure strength fc and residual strain εres increased with an increasing clay content at constant values of RH and confining pressure, the rate of this increase being a function of the RH. Young’s modulus E was relatively insensitive to changes in the clay content, its variation being less than 20% for all cases. Finally, based on a particular definition of Bishop's effective stress, involving a specific functional form χ(s), the failure states of all the samples were observed to lie approximately on a unique failure line crossing the origin in the (p′-q) plane regardless of the matric suction and confining pressure.

An evolving effective stress approach to anisotropic distortional hardening

A new yield surface with an evolving effective stress definition is proposed for consistently and efficiently describing anisotropic distortional hardening. Specifically, a new internal state variable is introduced to capture the thermodynamic evolution between different effective stress definitions. The corresponding yield surface and evolution equations of the internal variables are derived from thermodynamic considerations enabling satisfaction of the second law. A closest point projection return mapping algorithm for the proposed model is formulated and implemented for use in finite element analyses. Select constitutive and larger scale boundary value problems are solved to explore the capabilities of the model and examine the impact of distortional hardening on constitutive and structural responses. Importantly, these simulations demonstrate the tractability of the proposed formulation in investigating large-scale problems of interest.

A microstructural effective stress definition for compacted active clays

The volumetric deformability of the microstructure of compacted active clays is linked to the water mass exchange with the macrostructural void space. Given that this exchange is determined by the difference in the chemical potential of water in these two structural levels, this work proposes to define a microstructural effective stress (understood as the magnitude controlling microstructural strains) from this difference in chemical potentials, identifying the microstructural effective stress with the thermodynamic swelling pressure. To assess the scope of this definition, an inspection exercise has been conducted, examining the important functional dependence on the microstructural void ratio for different loads, water content and environmental salinity conditions. For the compacted active clays analysed, the thermodynamic swelling pressure is a robust and consistent definition of the microstructural effective stress.

Sediment Liquefaction: A Pore‐Water Pressure Gradient Viewpoint

Liquefaction of fully saturated sediments subjected to earthquake motion is often defined by the condition when effective stress reaches zero. On the other hand, the gradient of excess pore‐water pressure represents the net local surface force, which plays a crucial mechanical role for liquefaction. Herein, we consider two definitions of liquefaction, which yield different liquefaction initiation times and depths. Two hypothetical cases of liquefaction, earthquake‐induced and tsunami‐induced, are given to highlight the differences in the definitions of liquefaction. The results from the hypothetical cases show that the effective stress definition of liquefaction is a special case of the pore‐water pressure gradient definition of liquefaction.

Seismic Velocity Prediction in Shallow (<30 m) Partially Saturated, Unconsolidated Sediments Using Effective Medium Theory

Seismic velocity models of the near-surface (<30 m) better explain seismic velocities when all elements of total effective stress are considered, especially in materials with large cohesive and soil suction stress such as clays. Traditional constitutive elastic models that predict velocities in granular materials simplify the effect of total effective stress by equating it to net overburden stress, while excluding interparticle stresses and soil suction stress. A new proposed methodology calculates elastic moduli of granular matrices in near-surface environments by incorporating an updated definition of total effective stress into Hertz-Mindlin theory and calculates the elastic moduli of granular materials by extending Biot-Gassmann theory to include pressure effects induced by water saturation changes and cohesion. At shallow depths, when water saturation increases, theoretically calculated seismic velocities decrease in clay and increase in sand because interparticle stresses suppress the Biot-Gassmann effect. For standard sand and clay properties, net overburden stress becomes more influential than interparticle stresses at depths greater than 10 cm in sand and 100 m in clay. Pore pressure in the new model also incorporates the effect of layer thickness and pore size variation. Traditional calculation of pore pressure assumes a constant pore size medium, but may lead to an under- or overestimation of velocity by up to 20%. In clays, the variation of seismic velocity with water saturation is almost double the range predicted when only net overburden stress is considered to influence stress at the grain contacts. The proposed model predicts seismic velocities that compare well with measured field velocities from the literature.

Effect of water content on the resilient behavior of non standard unbound granular materials

Abstract The results of an experimental study on non-standard road granular materials are presented, including small strain precision triaxial tests under cyclic loading and wetting tests. The materials have attrition coefficients, plasticity index, and fines content above the recommended limits. The influence of each parameter and the influence of the water content were studied. This work emphasizes the role of suction in the interpretation of the results, and the possibility to use an effective stress approach to express the changes in normalized resilient modulus as a function of a single parameter representing both total stresses and suction. Among the three effective stress definitions tested, those of Terzaghi and Taibi were shown to yield satisfactory results. This approach noticeably simplifies the characterization of UGM specimens at different water contents in the perspective of a more rational design of pavement layers.

Evaluation of effective stress along the border of lateral notches

AbstractThis paper analyses the issue of defect nucleation due to fatigue loading in notched components where cracks occur in the proximity of the zone where the major stress gradient occurs. On the basis of recent experimental results where the defects do not nucleate at the notch tip, the effective stress evaluated by means of the implicit gradient approach, can explain the reasons for this apparent discrepancy.Furthermore, in order to find a kind of notch that underlines the nucleation outside the maximum stress zone, rectangular lateral notches have been investigated. Rectangular notches could provide an interesting theoretical and experimental benchmark or reference case in order to validate effective stress definitions. The linear elastic stress field at edges, corners and in the surrounding material of rectangular, sharp or rounded lateral notches has been analysed by means of the implicit gradient approach. The consequent effective values of these notches are evaluated in relation to brittle fractures or their predicted fatigue strength values.

On the constitutive modeling of partially saturated interfaces

Generalization of soil–structure interface models from dry/saturated states to consider partially saturated states is studied in this paper. For this purpose, basic constitutive equations of a conventional elasto-plastic interface model are firstly presented. Then, consideration is given to the effect of partial saturation on definition of effective stress, location of the critical state line as well as the impact of interface state on plastic hardening modulus and dilatancy. For each concern, proper independent approaches together with associated constitutive equations are discussed to be included in the basic model as complementary ingredients. Among many different possibilities to combine complementary constitutive equations for effective stress, relocation of the critical state line with degree of saturation, and impact of the interface state on plastic hardening modulus and dilatancy, six essential cases are selected. Evaluations show that all six cases can realistically consider the impact of partial degree of saturation on the peak and residual shear strengths as well as the volume change behavior of unsaturated interfaces.

Coal failure during primary and enhanced coalbed methane production — Theory and approximate analyses

This paper presents a study on reservoir-scale failure in coal seams during primary and enhanced coalbed methane production. Two sets of formulations for reservoir-scale coal failure analysis are presented: one is based on the routine effective stress definition, while the other is based on an extended definition. Two application examples – the Fruitland reservoir in San Juan Basin and a reservoir in the Sydney Basin – are investigated in terms of the approximate treatment proposed, which employs the common uniaxial strain and constant vertical total stress assumptions. The Fruitland case study found that the (reservoir-scale) friction angles fitted with the routine effective stress, at a failure pressure equal to 1.9MPa (observed) and with the assumed weakest low-volatile–medium volatile coal, would be 22–24°. If the extended effective stress definition is used, the friction angle would be 12–15°. The former results are apparently closer to some core-scale laboratory results (usually above 30°) than the latter. However, as discussed in this paper, using the routine effective stress for coal may result in some theoretical anomalies that seem to be fundamentally against the concept of the ‘effective stress’ in a porous rock: i.e., the pore fluid pressure reduces the effective stress in the rock. In contrast, the extended effective stress invoked in this text is theoretically more plausible for coal.

Development of a coupled thermo-hydro-mechanical double structure model for expansive soils

In this paper, development of a thermo-hydro-mechanical model for expansive soils including double structure is described. The model is based on hypoplastic model by Masin [6], in which the hydro-mechanical coupling is considered at each of the two structural levels. The model also includes separate effective stress definitions and water retention curves for the two levels of structure. In the proposed model, an approach by Masin and Khalili [8] to include thermal effects into hypoplastic models is followed. This is combined with temperature-dependent water retention curve of macrostructure, temperature-induced deformation of microstructure and an enhanced double-structure coupling law. Good predictions of the model are demonstrated by comparing the model simulations with experimental data on MX80 bentonites taken over from literature.

Average Soil Skeleton Stress for Unsaturated Soils and Discussion on Effective Stress

AbstractThe principle of effective stress is the most fundamental and important principle in soil mechanics. It serves as the foundation of modern soil mechanics. This paper starts with a discussion of effective stress expressions for unsaturated soils. The relationship between the total stress and stress of each phase in unsaturated soils is developed based on three-phase equilibrium equations of a representative element volume (REV). The expression of average soil skeleton stress, defined as effective stress, for unsaturated soils is then derived. This expression is consistent with the effective stress equation in unsaturated soils derived based on deformation work in the energy conservation equation. The general philosophy for the definition of effective stress in unsaturated soils is pointed out, followed by the discussion of some fundamental problems of unsaturated soils, including effective stress, choice of stress state, capillary phenomenon, and so on. The limitations and application scope of the ...

Effective stress assessment at rectangular rounded lateral notches

Rectangular lateral notches are not common engineering components, thus little research attention has been directed towards the investigation of their stress field properties. Indeed, no in-depth investigations have been conducted to date to assess their effective stress distributions according to the effective stress definitions provided by more recent non-local approaches (i.e. critical distance, average values, implicit gradient values, etc.). In fact, the potential applications of this kind of investigation are not even particularly relevant. However, rectangular notches could provide an interesting theoretical and experimental benchmark or reference case in order to validate the effective stress definitions. The aim of this paper is to investigate the linear elastic stress field at edges, corners and in the surrounding material of rectangular, sharp or rounded lateral notches. The consequent effective values of these notches are evaluated in relation to brittle fracture or their predicted fatigue strength values. The main goal of this paper is to investigate the relationship between geometrical proportions and the location of critical failure points according to the definitions of effective stress proposed in the literature.