Triaxial Stress Space Research Articles

Implementation of modified cam–clay model using closest point projection method under Cartesian coordinate system

The modified Cam–Clay model is an elastoplastic soil model that holds a prominent position in civil engineering for its precise depiction of varied soil behaviors. Despite extensive examination of its numerical processes, the predominant focus on stress integration algorithms within the triaxial stress space presents challenges for individuals less acquainted with geotechnical science. In response to this, the present study derived the stress integration equations for the modified Cam–Clay model within the Cartesian coordinate system employing the closest point projection method. Subsequently, a finite element program was developed incorporating the derived integration process. This study computed and compared the consolidation processes under both drained and undrained conditions for cubic soil samples with various over-consolidation ratios using the developed program, ABAQUS, and analytical formulas. The findings demonstrate that equations integrated within the Cartesian coordinate system offer ease of understanding and programming. The feasibility, accuracy, and stability of the closest point projection method have been validated. Comparative analysis of soil samples across various over-consolidation ratios indicated a reduction in failure stress with an increase in over-consolidation ratio, indicating a higher propensity for failure under undrained conditions than under the drained conditions.

Experimental Study of Compact Sandstone Deformation Under Axisymmetric Triaxial Loading Along Specific Paths in Stress Space

The deformation of rocks well below their ultimate strengths is frequently described through the constant stiffness moduli of the linear elasticity theory. In addition, the isotropic material approximation is usually used as the basic approach. Particularly in sedimentary rocks, local inhomogeneities, anisotropy, and irreversible processes, mainly related to the gradual breakage of grain joints, can affect their deformation behavior from the beginning of the loading. Consequently, the stiffness moduli measured under triaxial conditions are effective parameters, which may depend on the loading path in the stress space. Therefore, a complex experimental study of the deformation response of compact sedimentary rock (Brenna sandstone) along various paths in triaxial stress space was prepared to understand this dependence. In addition, a detailed analysis of the rock composition and structure of this compact sandstone was carried out. This paper presents an initial experimental study that is based on a loading method using alternative stress paths that correspond to different modes of the monotonic increase in differential stress from an initial isotropic compression state. In the experiment, the dependence of stiffness moduli on the loading path was found. Differences in rock deformation for conventional triaxial compression and extension can be attributed to a slight rock anisotropy originating from the stratification. The different behavior of deformation and related stiffness moduli along so-called reduced triaxial paths and conventional paths indicates that irreversible processes must be taken into account.

Double image stress point bounding surface model for monotonic and cyclic loading on anisotropic clays

In this paper, an anisotropic non-associative constitutive model, based on the well-established critical state soil mechanics framework, is developed for the natural and reconstituted clays subjected to monotonic and cyclic loadings. A robust non-elliptical yield surface is implemented in the model which enables it to capture a wide variety of yield stress points. In addition to a realistic soil stiffness simulation, the employed flexible plastic potential surface equips the proposed model to predict a more accurate coefficient of earth pressure at rest. The model uses a combination of conventional volumetric and comprehensive rotational hardening rules to control the evolution of the yield surface caused by plastic strain increments. The adopted rotational hardening rule uses the concept of governing plastic strain increment which not only ensures the existence of the equilibrium state of anisotropy but also takes the effect of plastic strains at different constant stress ratios into account more sensibly. The proposed model is enhanced with a novel double-image stress point bounding surface plasticity type theory to capture nonlinear behaviour inside the yield surface, and also to simulate the cyclic responses. The detailed model formulation is discussed in both triaxial stress space and the e-mathrm{ln}, p plane, and the important features of the model are elaborately explored. The capabilities of the model are also demonstrated by an extensive sensitivity analysis. In the end, the model is used to carry out simulations of monotonic and cyclic element tests on different clays and the results are compared with the available experimental data.

Shear Behaviour of Aeolian Sand with Different Density and Confining Pressure

Different from the other roadbed material, the unique mechanical properties of aeolian sand bring great difficulties to the construction and maintenance of desert highways. However, the main attention was usually paid to the engineering properties of aeolian sand, such as collapsibility, strong permeability, and poor gradation. To investigate the shear behaviour of aeolian sand under different engineering conditions, the drained and undrained tests were performed on aeolian sand with relatively large range of density and confining pressure. Under this condition, both the drained and undrained tests tend to the same critical state line, and the shear behaviour of aeolian sand is directly dependent on its density. Under the undrained condition, the q-ε1 curves and the effective stress paths in triaxial stress space exhibit four types of undrained shear behaviour, such as flow, limited flow, strain hardening, and strain softening. Meanwhile all the specimens exhibit three types of failure, such as flow slip, bulging failure, and shear bands. In the q-p’ plane, the analogous drained and undrained stress paths can be followed by aeolian sand with same initial relative density but different confining pressures, and there are two critical state lines due to the generation of shear bands for dense sand. In addition, the critical state lines in e-lnp’ plane decrease with increasing initial relative density Dr, that is, the material constant eᴦ decreases with increasing Dr, and the λ is also not constant but decreases with the increase in Dr. The results suggest that the strength behaviours of aeolian sand can be fitted by a straight line considering relative density and confining pressure and that two empirical formulas are established to describe this feature.

Creep behaviour of structured clays in triaxial stress space: theory and experimental investigation

ABSTRACT This paper investigates the time-dependent behaviour of structured clays in triaxial stress space both theoretically and experimentally. A range of existing theoretical frameworks and phenomenological equations are reviewed and discussed in their ability to interpret the creep behaviour at different states in stress space. New experimental results are presented for two structured clays subjected to complex loading conditions. Each test consists of a defined stress path with intermediate stages of creep. Considerable creep deformations were observed at all anisotropic effective stress conditions. Measured viscous deformations were observed to be related to the stress state, approaching stress path and degree of structure present in the soil. The incremental strain ratio, , measured during constant stress was found to rotate counter-clockwise as a result of drained shearing towards failure for all tests. The experiments also show that for very small stress increments, phenomenological and empirical relations incorrectly predict the development of creep strains with time.

Research on Constitutive Model of Unsaturated Sand

For unsaturated sand, due to the existence of suction, under low confining pressure, it has a large volume change during shearing and obvious dilatancy and peak characteristic points appear in the whole stress-strain behavior. Based on the above situation, the boundary surface constitutive model of unsaturated sand is established in general stress space and triaxial stress space based on phase transition state and hydraulic-mechanical coupling. The model has smooth transition among different saturation state and model parameters are easily calibrated via common triaxial tests and some other simple tests.

Effective skeleton stress and void ratio for constitutive modeling of fiber-reinforced sand

Inclusion of flexible fibers such as polypropylene and polyester is an effective method for soil improvement, as it significantly enhances the soil strength and ductility. A proper constitutive model is essential for assessing the stability and serviceability of fiber-reinforced slopes/foundations. A new method for constitutive modeling of fiber-reinforced sand (FRS) is proposed. It assumes that the strain of FRS is dependent on the deformation of the sand skeleton only, while the effective skeleton stress and effective skeleton void ratio, which should be used in describing the dilatancy, plastic hardening and elastic stiffness of FRS, are affected by fiber inclusion. The effective skeleton stress is dependent on the shear strain level, and the effective skeleton void ratio is affected by the fiber content and sample preparation method. A critical state FRS model in the triaxial stress space is proposed using the concept of effective skeleton stress and void ratio. Four parameters are introduced to characterize the effect of fiber inclusion on the mechanical behavior of sand, all of which can be easily determined based on triaxial test data on FRS, without measuring the stress–strain relationship of individual fibers. The model is validated by triaxial compression test results on four fiber-reinforced sands under loading conditions with various confining pressures, densities and stress paths. Potential improvement in the model for incorporating fiber orientation anisotropy is discussed.

Modeling the mechanical response of gas hydrate reservoirs in triaxial stress space

Exploitation of gas from deep-sea methane hydrate-bearing layers might lead to some geological disasters, including marine landslides and excessive settlement of marine ground. The first offshore gas production tests for methane hydrate-bearing sediments were carried out in eastern Nankai Trough. However, knowledge on mechanical behavior of gas hydrate reservoirs with similar gradation and minerology component to the marine sediment is still insufficient. Consequently, proper modeling of geomechanical properties of methane hydrate-bearing sediments is crucial for reservoir simulation and deep ocean ground stability analysis for long-term gas production in the future. This study conducted a series of triaxial shear tests to examine the shear response of methane hydrate-bearing sediments with a similar grading curve and minerology components to the hydrate-rich sediments in Nankai Trough. The test results demonstrated that the presence of hydrate mass between sand grains altered the stress-strain pattern from strain-hardening to postpeak strain-softening. A simple constitutive model based on several empirical relationships of granular materials is proposed to describe the stress-strain relationship of methane hydrate-bearing sediments under triaxial stress condition. This model can reproduce the enhancement of shear strength, initial stiffness, and dilation behavior of methane hydrate-bearing sediments containing different amounts of fines content with a rise in the methane hydrate saturation at a wide range of effective confining pressures. The numerical results indicate that the parameter A associated with initial stiffness of stress-strain curve and the parameter α related with dilation properties are jointly governed by the confining pressure, fines content, and hydrate saturation.

Creep behaviour of undisturbed London Clay in triaxial stress space

The evolution of the creep strain component in triaxial stress space was investigated through performing a series of multistage drained compression tests on London Clay using a specially designed locally instrumented triaxial apparatus. Experiments along specifically defined stress paths showed significant rotation of the local creep strain component as the samples were sheared towards failure. The results indicate a need for a more complex plastic potential function to correctly predict incremental creep strains at different states in triaxial stress space. Creep deformations for stress path controlled drained compression tests were also found to require a reinterpretation of the classic secondary compression behaviour. Creep strain-rates were found to fall well outside the normal power decay function. Test data and previously reported drained creep test results on London Clay have been combined to provide a complete understanding of the incremental creep component. The experiments show how creep behaviour significantly depends on the stress conditions imposed and the approaching strain rate.

A Constitutive Model for Unsaturated soils based on a Compressibility Framework dependent on Suction and Degree of Saturation

The Modified Cam Clay model is extended to account for the behaviour of unsaturated soils using Bishop’s stress. To describe the Loading – Collapse behaviour, the model incorporates a compressibility framework with suction and degree of saturation dependent compression lines. For simplicity, the present paper describes the model in the triaxial stress space with characteristic simulations of constant suction compression and triaxial tests, as well as wetting tests. The model reproduces an evolving post yield compressibility under constant suction compression, and thus, can adequately describe a maximum of collapse.

A thermo-hydro-mechanical model of unsaturated soils based on bounding surface plasticity

This paper presents a thermo-hydro-mechanical model of unsaturated soils based on effective stress concept and bounding surface plasticity. It is formulated in terms of effective saturation Se and temperature T unlike existing THM models, which use the suction and temperature to predict the behavior of unsaturated soils. An isotropic yield limit is defined in a 3D space p′-Se-T and subsequently generalized to the triaxial stress space. Regarding the hydraulic component, the non-isothermal water retention curves of deformable soils are combined with the present model. However, the part treating the flow of water and temperature is not covered. Finally, the proposed formulations are supported by comparisons with existing experimental results under isotopic and oedometric paths.

A New Model for the Cross Anisotropic Nonlinear Elastic Behavior of Unbounded Soils for Road Pavements

This study presents a new generalized small strain nonlinear anisotropic constitutive model for use in the elastic analysis of unbounded soils for road pavements. The model rests on a method to determine the cross anisotropic moduli of unbounded soils based on the classic repeated-load triaxial compression test. The model incorporates nonlinear small-strain behavior in a triaxial stress space by considering the inter-dependency of the bulk K*, shear G* and cross anisotropic modulus, J. In agreement with the laboratory test results the moduli, at a given water content, are dependent on the initial state of stress in p–q space. The model is derived by considering pure isothermal elastic behavior and by letting the net elastic work per unit volume along a closed stress path equal to zero. An approximate stress–strain curve, using solely the non linear constitutive model, is obtained using a linear analysis that simulates a typical triaxial test. Finally a non linear finite element analysis, which incorporates the non linear constitutive model, is carried out to provide a more elaborate assessment of model performance.

State-Dependent Constitutive Model for Rockfill Materials

AbstractA series of large-scale triaxial compression tests on Tacheng rockfill material (TRM) showed that the dilatancy and stress-strain behaviors were significantly influenced by density and pressure. The critical state friction angle of TRM decreased with an increase in the initial confining pressure. In addition, the critical state line of this material in the e-log⁡p′ plane ascended with an increase in the initial void ratio. On the basis of the specific critical state behaviors of TRM, a state-dependent constitutive model was slightly adapted in the triaxial stress space. The model, with only a set of model parameters, could capture the state-dependent behaviors of the dilatancy, stress-strain, and mobilized friction angle of TRM at various densities and pressures. The peak friction angle of TRM was found to be linear, with a coefficient of 0.21, in relation to the maximum dilatancy angle. The coefficient for rockfill materials was smaller than that for sands, indicating that the influence of dilata...

Explicit integration of a thermo-mechanical model for clays

This paper considers the numerical implementation of an existing constitutive model that was developed to simulate the thermo-mechanical behavior of clays. The model, which combines two plastic mechanisms, has been reformulated using generalized stress and strain variables accounting for the presence of temperature as an additional variable.Mixed control conditions are introduced to simulate standard thermo-mechanical laboratory tests. An explicit integration scheme is used with an adaptive time-stepping scheme specifically designed for the considered thermo-mechanical model. This sub-stepping procedure enables the constitutive model to be integrated accurately and efficiently by automatically adjusting the size of sub-steps to maintain the local integration error below a specified tolerance. Special attention is paid to the cases where the two yield surfaces are activated simultaneously. The correction of the drift of the stress point from the yield surface is also accounted for.This thermo-elastoplastic model has been implemented in a constitutive driver aiming at simulating thermo-mechanical tests in the triaxial stress space. The efficiency of this integration algorithm is discussed and the clear advantage of using the adaptive sub-stepping procedure is shown. The constitutive model is finally used to simulate thermo-mechanical tests. The results are compared to available experimental data and show the good capabilities of the constitutive model.

Simple Yield Surface Expressions Appropriate for Soil Plasticity

The objective of this paper is to present a number of simple, practical, and useful analytical expressions of a yield surface for geomaterials. In classical plasticity, the analytical expression of a yield surface defines the locus of points in stress space at which plastic flow initiates, and the corresponding function must depend on direct and mixed invariants of stress and tensor-valued internal variables. One single function describes a yield surface in order to avoid singularities and computational difficulties arising from the use of multiple functions representing intersecting surfaces in stress space that are often used for cap-type models in soil plasticity. The presented functions are conveniently subdivided in three main categories depending on the type of analytical expression used, and they all describe properly closed yield surfaces which are continuous and convex. The internal variables in these functions can be used in order to address classical plasticity features such as isotropic and kinematic hardening, the latter in the form of rotational hardening. The effects of parameters on the shape of yield surfaces are clearly demonstrated and illustrated for all functions in triaxial stress space. The generalization of these functions to the multiaxial stress space is presented using a consistent method such that if one applies triaxial loading conditions on the multiaxial expressions, the triaxial ones are retrieved. Finally, the appropriateness of the yield functions in regards to the soil type is discussed.

Thermomechanical model for saturated clays

This paper aims to simulate the thermomechanical behaviour of saturated clays in the triaxial stress space at states wetter than critical condition. A thermoelastic-plastic (TEP) mechanical model was developed under the framework of critical state soil mechanics and the modified Cam-Clay model. It consists of two parts. The first part involves an isotropic TEP model that is able to reproduce the thermally induced volume change of saturated clay specimens under normally consolidated and overconsolidated conditions. This part was introduced earlier and validated by the authors. The second part, the focus of this paper, involves extension of the isotropic TEP model into triaxial stress space. The model is simple enough to require merely a few constants more than those of the modified Cam-Clay model. The adequacy of the proposed model has been verified using the experimental results from three different types of clay reported in the literature.

SANISAND: Simple anisotropic sand plasticity model

AbstractSANISAND is the name used for a family of simple anisotropic sand constitutive models developed over the past few years within the framework of critical state soil mechanics and bounding surface plasticity. The existing SANISAND models use a narrow open cone‐type yield surface with apex at the origin obeying rotational hardening, which implies that only changes of the stress ratio can cause plastic deformations, while constant stress‐ratio loading induces only elastic response. In order to circumvent this limitation, the present member of the SANISAND family introduces a modified eight‐curve equation as the analytical description of a narrow but closed cone‐type yield surface that obeys rotational and isotropic hardening. This modification enables the prediction of plastic strains during any type of constant stress‐ratio loading, a feature lacking from the previous SANISAND models, without losing their well‐established predictive capability for all other loading conditions including the cyclic. In the process the plausible assumption is made that the plastic strain rate decomposes in two parts, one due to the change of stress ratio and a second due to loading under constant stress ratio, with isotropic hardening depending on the volumetric component of the latter part only. The model formulation is presented firstly in the triaxial stress space and subsequently its multiaxial generalization is developed following systematically the steps of the triaxial one. A detailed calibration procedure for the model constants is presented, while successful simulation of both drained and undrained behavior of sands under constant and variable stress‐ratio loadings at various densities and confining pressures is obtained by the model. Copyright © 2007 John Wiley & Sons, Ltd.

SANICLAY: simple anisotropic clay plasticity model

AbstractSANICLAY is a new simple anisotropic clay plasticity model that builds on a modification of an earlier model with an associated flow rule, in order to include simulations of softening response under undrained compression following Ko consolidation. Non‐associativity is introduced by adopting a yield surface different than the plastic potential surface. Besides, the isotropic hardening of the yield surface both surfaces evolve according to a combined distortional and rotational hardening rule, simulating the evolving anisotropy. Although built on the general premises of critical state soil mechanics, the model induces a critical state line in the void ratio–mean effective stress space, which is a function of anisotropy. To ease interpretation, the model formulation is presented firstly in the triaxial stress space and subsequently, its multiaxial generalization is developed systematically, in a form appropriate for implementation in numerical codes. The SANICLAY is shown to provide successful simulation of both undrained and drained rate‐independent behaviour of normally consolidated sensitive clays, and to a satisfactory degree of accuracy of overconsolidated clays. The new model requires merely three constants more than those of the modified Cam clay model, all of which are easily calibrated from well‐established laboratory tests following a meticulously presented procedure. Copyright © 2006 John Wiley & Sons, Ltd.

Analytical Model of Concrete-Filled Fiber-Reinforced Polymer Tubes based on Multiaxial Constitutive Laws

A model is proposed for predicting the compression behavior of axially loaded concrete cylinders confined by fiber reinforced polymer (FRP) shells. To capture the active confinement applied by the FRP, an orthotropic hypoelasticity-based constitutive law is used for the concrete. This law is defined the triaxial stress space. The confinement-induced concrete strength enhancements are computed from a four-parameter failure surface. The corresponding peak strain increase is computed using a newly proposed strain enhancement factor. The FRP shell behavior is modeled using a stress-strain relation for plane stress orthotropic laminated composites. The proposed model is implemented into an incremental approach for the analysis of compression tests. No iterations are needed to find the stresses corresponding to prescribed axial strains. The model is verified with correlation studies with different experimental tests on concrete-filled FRP cylinders.

State-Dependent Strength of Sands from the Perspective of Unified Modeling

This paper discusses the state-dependent strength of sands from the perspective of unified modeling in triaxial stress space. The modeling accounts for the dependence of dilatancy on the material internal state during the deformation history and thus has the capability of describing the behavior of a sand with different densities and stress levels in a unified way. Analyses are made for the Toyoura sand whose behavior has been well documented by laboratory tests and meanwhile comparisons with experimental observations on other sands are presented. It is shown that the influence of density and stress level on the strength of sands can be combined through the state-dependent dilatancy such that both the peak friction angle and maximum dilation angle are well correlated with a so-called state parameter. A unique, linear relationship is suggested between the peak friction angle and the maximum dilation angle for a wide range of densities and stress levels. The relationship, which is found to be in good agreement with recent experimental findings on a different sand, implies that the excess angle of shearing due to dilatancy in triaxial conditions is less than 40% of that in plane strain conditions. A careful identification of the deficiency of the classical Rowe's and Cam-clay's stress-dilatancy relations reveals that the unique relationship between the stress ratio and dilatancy assumed in both relations does not exist and thereby obstructs unified modeling of the sand behavior over a full range of densities and stress levels.