Stress-dilatancy Equation Research Articles

Testing and modeling of the state-dependent behaviors of rockfill material

Dense Tacheng rockfill material (TRM) exhibits strain softening and dilation during drained triaxial tests, and therefore, an adapted Rowe’s stress–dilatancy equation was proposed for TRM. This equation incorporates an internal state index related to the density and pressure, as well as the coefficient of particle breakage and rotation. The adapted Rowe’s stress–dilatancy equation indicates that the relationship between stress and dilatancy is not constant, but varies with density and pressure. This result is in agreement with TRM test data. A state-dependent model was established for TRM using generalized plasticity theory combined with the adapted Rowe’s stress–dilatancy equation. The model includes twelve constants calibrated using TRM test data from Group A, and this model was used to predict the strain softening and dilatancy behaviors of dense TRM. Furthermore, the model predictions were validated using test data from Group B. In summary, the model accurately represented the stress–strain and dilatancy behaviors of TRM over a wide range of densities and pressures.

Modeling Cyclic Behavior of Rockfill Materials in a Framework of Generalized Plasticity

Typical triaxial compression experiments were revisited to investigate the essential mechanical behavior of rockfill materials to be reflected in constitutive modeling, such as the nonlinear dependence of the strength and the dilation on the confining pressure and the accumulation of permanent strains during cyclic loading. The mathematical descriptions of the axial stress-strain behavior during initial loading, unloading, and reloading were formulated, respectively, which enables us to represent the hysteresis loops directly without recourse to complex concepts and parameters. The axial stress-strain model was then incorporated into the constitutive framework of generalized plasticity for the modeling of cyclic behavior of rockfill materials. This task was fulfilled by defining the elastic modulus, the plastic flow direction tensor, the loading direction tensor, and the plastic modulus for different loading conditions. In particular, the plastic flow direction tensor was derived based on a stress-dilatancy equation considering the influence of loading direction, and the representation of the plastic modulus was established in terms of the tangential modulus and the elastic modulus by using the special constitutive equations under axisymmetric stress states. The cyclic model proposed in this study has three distinct features. First, the hysteresis behavior and the accumulation of permanent strains were unified and described under the framework of generalized plasticity. Second, all the loading phases were treated as elastoplastic processes so that no purely elastic regions exist in the principal stress space. Third, the introduction of two aging functions for the consideration of the hardening effect facilitates the controlling of the magnitudes of permanent strains. There are in total 13 parameters in the model, all of which can be determined easily from (cyclic) triaxial compression experiments. To check the capabilities of the model in reproducing the monotonic and cyclic behavior, typical triaxial compression experiments were simulated with the constitutive equation. Satisfactory agreement between the experimental results and the corresponding model predictions lent sufficient creditability to the effectiveness of the proposed model, which further motivates us to extend the model for more complex stress paths and apply the model in practical engineering in the future.

A 3D bounding surface model for rockfill materials

A 3D bounding surface model is established for rockfill materials, which can be applied to appropriately predict the deformation and the stabilization of rockfill dams. Firstly, an associated plastic flow rule for rockfill materials is investigated based on the elaborate data from the large-style triaxial compression tests and the true triaxial tests. Secondly, the constitutive equations of the 3D bounding surface model are established by several steps. These steps include the bounding surface incorporating the general nonlinear strength criterion, stress-dilatancy equations, the evolution of the bounding surface and the bounding surface plasticity. Finally, the 3D bounding surface model is used to predict the mechanical behaviors of rockfill materials from the large-style triaxial compression tests and the true triaxial tests. Consequently, the proposed 3D bounding surface model can well capture such behaviors of rockfill materials as the strain hardening, the post-peak strain softening, and the volumetric strain contraction and expansion in both two- and three-dimensional stress spaces.

Non-coaxial version of Rowe’s stress-dilatancy relation

Non-coaxiality occurs when the directions of the principal plastic strain increments and the principal stresses deviate. Extensive experimental data have now conclusively shown that plastic flow in granular soils is non-coaxial particularly during loadings involving rotation of the principal stress directions. One way to integrate the effects of non-coaxiality is by modifying the expressions for energy dissipation and stress-dilatancy used in modeling plastic deformation of granular soils. In this regard, the paper’s main objective is to derive a non-coaxial version of Rowe’s stress-dilatancy relation, thereby making it more general and applicable to loadings involving principal stress rotation. The paper also applies Rowe’s non-coaxial stress-dilatancy equation in the determination of the effects of principal stress rotation in granular soils during simple shear loading conditions. Previous experimental data from simple shear tests on sand are used to validate the proposed non-coaxial version of Rowe’s stress-dilatancy relation.

A critical state subloading surface model of sands with shear hardening

Based on the framework of critical state soil mechanics, a subloading surface plastic model for sand, being applicable to cyclic loading, was proposed. The model can be used to describe strain softening behaviour of sand under monotonic loading when the similarity-ratio equals to unity. The characteristics of the model are as follows: 1) A reverse bullet-shaped yield surface is adopted to ensure accurate prediction of the behavior of sand, instead of bullet-shaped or elliptical yield surface in Cam-Clay model. 2) No unique relationship between void ratio and the mean normal stress for sand prevents the direct coupling of yield surface size to void ratio, so incremental deviatoric strain hardening rule is used. 3) The model combines the concept of state-dependent dilatancy by incorporating state parameter in Rowe’s stress dilatancy equation, which accounts for the dependence of dilatancy on the stress state and the material internal state. A single set of model constants, which is calibrated, can simulate stress—strain response under different initial void ratios and different confine pressures. The model is validated true by comparing predicted results with experimental results under monotonic and cyclic loading conditions.

Simulating a direct shear box test by DEM

Distinct element simulation was performed for direct shear box (DSB) tests on a dense and a loose two-dimensional (2D) sample of 3259 cylinders. Special attention was devoted to the effect that the frictional force between the inside surface of the upper shear box and the sample had on the measured shear strength in the DSB test. Some ways of minimizing this interface frictional force were introduced in the paper. Given that the deformation approximates simple shear within the deforming zone across the sample centre (referred to as the shear zone), a method was proposed to evaluate the overall strains in the DSB test. The numerically simulated data were used to interpret, on a microscopic scale, the angle of internal friction and a 2D stress–dilatancy equation for the mobilized plane in granular material. It was found that the angle of internal friction in granular material is not directly related to the interparticle friction angle (ϕµ). Instead, it relates to the average interparticle contact angle ([Formula: see text]) on the mobilized plane and the ratio k/f0, representing the degree of the probability distribution of the interparticle contact forces that is biased toward the positive zone of the contact angle θ (along the shear direction), where k is the slope of the linear distribution of the average interparticle contact forces against the interparticle contact angle; and f0is the average interparticle contact force.Key words: angle of internal friction, direct shear box test, distinct element method, friction, granular material, stress–dilatancy.

Stress Dilatancy and Fabric Dependencies on Sand Behavior

A stress dilatancy model with embedded microstructural information, originally developed by the writers, is used to illustrate the pivotal importance of dilatancy and fabric on the behavior of sand. Fabric, as a second-order tensor, enters into the stress dilatancy equation obtained from a microscopic analysis of an ensemble of rigid particles. Model simulations of sand behavior are carried out in triaxial stress conditions along strain paths with varying degrees of controlled dilation ~or compaction! including isochoric deformations as a particular case. Under particular strain paths and fabric conditions, it is shown that a relatively dense sand can succumb to instability or liquefaction under other than isochoric ~undrained! conditions. This phenomenon is in accord with laboratory experiments in which dilation or compaction is controlled by modulating the amount of water flowing in or out of a sand specimen during shearing. Mixed drained-undrained loading paths are also simulated with particular reference to fabric dependence at a fixed void ratio. Model simulations capture most of the observed characteristics of sand response, such as instability and asymptotic behavior under various conditions.

Microscopic Interpretation on a Stress-Dilatancy Relationship of Granular Materials

Matsuoka (1974) proposed a two-dimensional stress-dilatancy equation of granular materials on the mobilized plane through the direct box shear tests on assemblies of aluminum and photoelastic rods, which was expressed as τ/σN=λ(−dεN/dγ)+μ. In this paper, the stress ratio τ/σN and the strain increment ratio −dεN/dγ are approximated to be tan x¯ and tan θ¯, respectively, on the basis of the numerically simulated results for a simple shear test by DEM. Here, x¯ and θ¯ denote the average angle of the interparticle forces to the normal of the mobilized plane and the average interparticle contact angle on the mobilized plane, respectively. It was found that the difference between x¯ and θ¯, denoted as δ, varies slightly during shear. The angle δ is related to the average interparticle contact force f0, the slope k of the straight line that characterizes the distribution of the average interparticle contact forces against the contact angle, and the average interparticle contact angles θ¯ on the mobilized plane. The intercept μ in the stress-dilatancy equation is interpreted as tan δ. The influences of interparticle surface friction, grain shape and confining pressure on the stress-dilatancy relation are examined. It was found that the angle δ is not very sensitive to the interparticle surface friction angle ϕμ, except in the lower value of ϕμ, and independent of the confining pressure. However, it is affected to some extent by grain shapes. Various results of the newly developed in-situ direct shear tests on various granular soils are presented to support the arguments in this paper.

The strength of unsaturated bentonite-enhanced sand

A modification to Rowe's stress-dilatancy equation is presented that extends its range of application to include unsaturated soil behaviour. The results of a programme of constant water content triaxial tests on unsaturated bentonite-enhanced sand (BES) are reported, together with those of a programme of saturated drained triaxial tests on the sand. It is shown that the variation in the rate of dilation at failure with the sand relative density is similar for the two materials. It is proposed that the packing and friction angle of the sand particles and the degree of saturation control the shear strength of unsaturated BES containing modest amounts of bentonite, and that the shear strength of the bentonite component can be ignored.

The strength of unsaturated bentonite-enhanced sand

A modification to Rowe's stress-dilatancy equation is presented that extends its range of application to include unsaturated soil behaviour. The results of a programme of constant water content triaxial tests on unsaturated bentonite-enhanced sand (BES) are reported, together with those of a programme of saturated drained triaxial tests on the sand. It is shown that the variation in the rate of dilation at failure with the sand relative density is similar for the two materials. It is proposed that the packing and friction angle of the sand particles and the degree of saturation control the shear strength of unsaturated BES containing modest amounts of bentonite, and that the shear strength of the bentonite component can be ignored.

Non-Coaxiality and Energy Dissipation in Granular Materials

The paper presents a theoretical and experimental study of the effects of non-coaxiality or non-coincidence of the principal stress and the principal plastic strain increment directions on the behaviour of granular materials. Experimental results from hollow cylindrical tests on sand involving principal stress rotation which support previously published results on non-coaxiality are presented. These results imply that constitutive relations cannot be sufficiently formulated in the principal stress space unless the deviations between the principal stress and plastic strain increment directions are taken into consideration. It is shown that plasticity formulations with plastic potentials that are scalar functions of the stress invariants alone implicitly assume coaxiality and cannot be used for loading involving principal stress rotation. The paper presents a comprehensive analysis of the effects of non-coaxiality on the energy dissipation of sand. The paper shows that energy dissipation calculated from the principal stresses and the principal plastic strain increments or from the stress and plastic strain increment invariants, would be erroneous and would over-estimate the amount of dissipated energy during loading in the case of non-coaxial flow. A non-coaxiality factor is introduced in order to account for the effects of non-coaxiality on the energy dissipation equation and in a stress-dilatancy relation for granular materials. Explicit expressions of the non-coaxiality factor for two-and three-dimensional loading conditions are given at the end of the paper. Experimental results are presented to show the validity of the proposed energy dissipation and stress-dilatancy equations.

A simple constitutive model for granular soils: Modified stress-dilatancy approach

The dependencies of granular soil behaviour on void ratio and stress are modelled within plasticity theory enriched by a modified stress-dilatancy law. Simplicity has been kept in the development of the constitutive law for easy future implementation in afinite element code. By using a void ratio dependent factor which measures the deviation of the current void ratio from the critical one, Rowe's stress dilatancy equation is modified. This modification indeed corresponds to a new energy dissipation equation for a granular assembly which sustains kinematical constraints under the action of stresses. The proposed constitutive law predicts in a very consistent manner the response of sands in monotonic loading conditions for a large range of initial void ratios and confining pressures without the need to make any adjustments to the material parameters. Well known published data for the drained triaxial compression of Sacramento River sand was successfully modelled.

Stress Dilatancy of Sand at Small Stress Ratio States

The validity of the stress dilatancy equation for representing the small stress ratio state response of sand is examined. Test equipment development and procedures that were followed aimed at precise loading control and reliable monitoring of small displacements and volume changes. A variety of load controlled stress paths were applied to conventional triaxial test samples of Ottawa sand. The results obtained show that at small stress ratio states a relationship between stress state and strain increment direction as prescribed by the stress dilatancy equation does not apply. Stress increment directions were found to be related to strain increment directions for low stress ratio states. The relationship has some resemblance to the behavior of a cross anisotropic elastic material with a stiffer vertical to radial response. The range of low stress ratio states over which a stress dilatancy relationship does not apply tends to increase with density. In stress ratio regions where the stress dilatancy equation began to apply, the relationship between R and D tends toward KCv with increasing confining stress for conventional triaxial but not for constant mean normal stress paths.

Response of Granular Soil along Constant Stress Increment Ratio Path

The stress‐strain response of a granular soil along constant stress increment ratio paths is experimentally investigated for the condition of σ2=σ3. Since the strains manifested along such paths can be very small, several measures have been developed to improve the accuracy of the strain measurements. A more accurate method of isolating the elastic strain has also been developed to enable reliable determination of plastic strain increment ratio. Test data unambiguously indicate that the elastic response is anisotropic, and the degree of anisotropy increases with the principal stress ratio. The plastic strain increment ratio so deduced from test data is dependent on the stress increment ratio, but can be predicted by combining Rowe's stress dilatancy equation and the energy equation of the Cam‐Clay model. This leads to a new method of predicting the stress‐strain response along such paths. The input parameters can be determined from constant σ3 tests and constant principal‐stress ratio tests. Excellent agreement with test results is achieved.

On stress-dilatancy equations of sand subjected to cyclic loading : Pradhan, T B S; Tatsuoka, F Soils FoundV29, N1, March 1989, P65–81

On stress-dilatancy equations of sand subjected to cyclic loading : Pradhan, T B S; Tatsuoka, F Soils FoundV29, N1, March 1989, P65–81

On Stress-Dilatancy Equations of Sand Subjected to Cyclic Loading

ABSTRACT Based on the results of a series of cyclic triaxial tests and cyclic torsional simple shear tests on saturated Toyoura sand performed under drained conditions, stress-dilatancy equations under cyclic loadings were studied. It has been found from the experimental work that in each of the testing methods a unique relationship between the stress ratio and the rate of dilatancy (strain increment ratio) exists which is independent of void ratio and pressure level. It has also been found experimentally that the rate of dilatancy, as defined as the rate of volumetric strain increment to shear strain increment (positive for volume expansion), becomes negative by unloading after the reversing of loading direction, and it increases continuously with shearing without showing any discontinuous behavior at the moment when the sign of shear stress changes, and subsequently it becomes positive. Thus, at the same stress condition, two different rates of dilatancy are possible to exist depending on the direction of shear straining. Four different representative stress-dilatancy relations based on (i) the sliding block theory, (ii) the Rowe’s theory, (iii) the Roscoe’s energy dissipation theory, and (iv) the Taylor’s energy dissipation theory, were modified to apply to cyclic loading conditions with reversed loading directions. It was found that after these modifications some of these theories can well simulate the stress-dilatancy relations under drained cyclic conditions obtained by the tests. Finally, curves of equal plastic potential in cyclic drained simple shear were portraied on a stress plane. It is a double plastic potential function in the sense that two different plastic potential curves are passing through any given stress point, which are for two different directions of shear-straining.

Stress, dilatancy and fabric in granular materials

On the basis of a simple micromechanical model, expressions are developed for the overall stress and deformation rate measures in a granular mass which carries the applied loads through contact friction. Various measures of the granular fabric are examined, and explicit connections between a fabric measure and the stress and the deformation rate tensors are obtained. Then, from the balance of energy, a general stress-dilatancy equation is deduced, which includes the effect of fabric. Several other published dilatancy equations are discussed and compared with the new equation.

A Stress-Dilatancy Model of Granular Material Under General Stress Condition

A three-dimensional isotropic stress-dilatancy model is proposed. The model is formed on the basis of the basic hypothesis that the shear deformation of granular materials results mainly from the sliding between grains. The model is also based on such a new concept relating to shear mechanism that the position of sliding contacts is closely related to the applied stresses and accordingly shifts in turn as shearing proceeds. The general stress-strain increment relations obtained from the model contain the influence of stress path in a general form. It is verified that the stress-dilatancy equations derived from the model agree well with the experimental results under the conventional and true triaxial stress conditions.

A double hardening model for sand

An elastoplastic model is established for the initial loading, unloading and reloading of sand. The plastic strains are described as the sum of two independent parts. The most important part of the plastic strains is described by means of a shear yield surface and a non-associated flow rule in which Rowe’s stress-dilatancy equation is incorporated. During continued deformation the shear yield surface changes not only in size but also in shape. Close agreement is obtained with experimental data on yield surfaces. The other part of the plastic strains is purely volumetric. This part accounts for the plastic volume strain in isotropic compression. It cannot be disregarded, particularly in the case of loose sands. Un modèle élastoplastique est établi pour le chargement initial, le déchargement et le rechargement de sable. Les déformations plastiques sont décrites comme la somme de deux parties indépendantes. La partie la plus importante des déformations plastiques est dérite à l'aide d'une surface de cisaillement et d'une régle d'ecoulement non-associé, dans laquelle l'équation dilatancecontrainte de Rowe est incorporée. Pendant la déformation continue de la surface de cisaillement change non seulement de taille mais aussi de forme. Un accord excellent est obtenu avec les données expérimentales sur les surfaces de cisaillement. L'autre partie des déformations plastiques est purement de volume. Cette partie rend compte dé la déformation plastique du volume en compression isotrope. Elle ne peut pas être negligée, particulierement dans le cas de sables lâches.

A Consideration about Rowe’s Minimum Energy Ratio Principle and a New Concept of Shear Mechanism

Rowe (1962) introduced a new concept called minimum energy ratio principle to get a stress-dilatancy relation of granular materials under axisymmetric stress condition. According to the principle, the slinding contacts are restricted to those with the preferred angle βc=(π/4)-(φμ/2). However, the physical basis of this principle is questioned by many investigators. In this paper, the author investigates this principle only on the basis of equilibrium between interparticle forces with the following results ;(1) Horne's study on Rowe's stress-dilatancy equation is incomplete to clarify the physical meaning of the minimum energy ratio principle. Accordingly, his indication such that the principle could be rigorously established as an equilibrium condition is not acceptable.(2) The minimization of the incremental energy ratio E in Rowe's theory is equivalent to maximize a driving tangential force Fd acting in the same direction as sliding direction and a resisting one Fr in the opposite direction of Fd. The physical ground is not clear.A new and simple concept of shear mechanism of granular materials can be proposed only on the basis of equilibrium between interparticle forces without using the minimum energy ratio principle. The concept is characterized by such a description that the positions of sliding contacts shift as shear stress increases. According to the concept, the development of anisotropy of grain structure during shear and the deformation behaviours under cyclic loadings can be explained well.