Principal Strain Increment Research Articles

Influence of Intermediate Principal Stress on Undrained Behavior of Intact Clay under Pure Principal Stress Rotation

This study presents the accumulations of the excess pore water pressure and the deformation as well as the noncoaxial behavior of intact soft clay subjected to pure principal stress rotation. Series of tests were carried out by using a dynamic hollow cylinder apparatus to highlight the influence of intermediate principal stress parameterb. It was found that the rate of PWP evolution was greatly influenced byb, but the influence was not monotonous. Specimens under the conditionb= 0.75 had the highest accumulation of pore water pressure while under the conditionb= 0 had the strongest resistance to the pore pressure generation. PWP accumulated mainly in the first cycle. The failure of specimens under principal stress rotation was controlled by the strain other than the pore pressure. The shear stiffness decreased more quickly with higherbvalue. The direction of the principal strain increment was strongly dependent on the principal stress increment orientation and less influenced by thebvalue and the number of cycles.

Noncoaxial Behavior of Sand under Various Stress Paths

In this paper, the results of three series of drained tests carried out on sands using hollow cylinder apparatus are presented. The noncoaxiality, defined as the difference between the major principal stress direction and the corresponding principal strain increment direction, is investigated. In the first series of tests, the sand was isotropically consolidated before being sheared with the fixed principal stress direction. In the other two series of tests, the sand specimens were isotropically consolidated and then sheared by rotating the major principal stress axes while the deviator stress level was either fixed (pure stress rotation) or increased continuously (combined shear loading). The experimental results provide clear evidence for material noncoaxiality when the rotation of principal stress direction is involved. The results from these series of tests show that the degree of noncoaxiality depends on the level of deviatoric stress and the stress increment direction. It tends to decrease when the specimen is approaching a failure state. It was also observed that the effect of soil density on noncoaxiality is more significant at lower shear stress levels. Test results on two different materials, Portaway sand and Leighton Buzzard sand, were also compared with the study of the influence of material anisotropy associated with sand particle characteristics. Numerical simulations of granular materials using the discrete element method (DEM) were conducted to understand the effect of the initial material anisotropy produced during sample preparation. The DEM results were consistent with those obtained from the hollow cylinder tests.

The Hencky equivalent strain and its inapplicability to the interpretation of torsion testing experiments

Experiments were carried out on a 0.06%C plain C steel to determine the shear stress–shear strain behaviour. The results obtained at 1200°C and a shear strain rate of 0.866 s−1 are described. The experimental data are converted into equivalent stress–equivalent strain form using the von Mises and Hencky formalisms. It is shown that when the equivalent stress is of interest, as in the calculation of rolling load, only the von Mises formulation can be used without the introduction of large errors. This is required so that the incremental work done/unit volume of material can remain constant in both the shear and equivalent uniaxial tension formulations. The problem with the Hencky formalism with regard to simple shear is that the orientations of the Hencky principal strains and principal strain increments do not coincide and also vary continuously. As a consequence, the absolute values of the strain increments cannot simply be summed and the description of the equivalent strain applicable to large strain torsion proposed by Onaka [Philos. Mag. Lett. 90 (2010) p.633] cannot be used to predict mechanical properties along other strain or loading paths. The problems associated with the definitions of the Hencky equivalent stress and Hencky work increment are also discussed.

A comparison of the von Mises and Hencky equivalent strains for use in simple shear experiments

The von Mises equivalent strain increment is derived for the case of large strain simple shear (torsion testing). This is used, in conjunction with the von Mises yield surface, to define the von Mises equivalent stress as well as the incremental work per unit volume. Integration of the equivalent strain increment leads to the definition of the von Mises equivalent strain for torsion. The Hencky equivalent strain increment is derived from the Hencky strain defined as the logarithm of the semi major and minor axes of the strain ellipse. This is then used, via the incremental work, to derive the ‘Hencky equivalent stress’. In the Onaka approach, the numerical values of the principal strain increments were integrated without taking into account the continuous rotation of the strain ellipse. This invalid operation leads to an expression for the equivalent strain increment that cannot be applied to the large strains considered by Onaka. Using the correct increments of the Hencky strain, it is shown that the shear strain increments turn negative and consequently, the incremental work becomes negative when the shear is large.

A non‐coaxial and unbalanced plastic flow rule for geomaterials and its application to the shear band orientation prediction

AbstractIn this paper, the non‐coaxial relation between the principal plastic strain increments and the principal stresses, which results from the internal friction in geomaterials, is analyzed, and the phenomenon of the unbalanced development of plastic flow in two conjugate directions is discussed. A non‐coaxial, unbalanced plastic flow model for Coulomb frictional materials is developed and used to determine the orientation of shear band in geomaterials. It is shown that the unbalanced index r of plastic flow has important effect on the orientation of the shear band, and the orientation determined by the conventional plastic flow theory is only a special case of the proposed model when r=0. This result soundly explains the reason that the geomaterials with the same internal friction angle and dilatancy angle can have very different shear band orientations. In addition, the difference between the intrinsic and apparent dilatancy angles is analyzed, and it is emphasized that the dilatancy angle commonly used in practice is indeed the apparent dilatancy angle. Copyright © 2010 John Wiley & Sons, Ltd.

Yielding Characteristic and Non-Coaxiality of Toyoura Sand on p′-Constant Shear Stress Plane

ABSTRACT In order to investigate the deformation characteristics under a wide selection of stress history, ten series of stress probing tests on p′-constant shear stress plane on dense Toyoura sand are conducted by using hollow cylinder apparatus. This paper presents yielding behavior and non-coaxiality of Toyoura sand obtained from the tests. The stress probing tests consist of ten series of shearing tests each of which starts from systematically chosen individual initial stress point. Each initial stress point is subjected to load-unload stress history. The yielding characteristic is interpreted by means of a concept of multiple yield surfaces model which has three yield surfaces representing linearly elastic behavior limit, commencement point of rapid development in plastic strain and elasto-plastic range. As a result, an experimental evidence of the isotropic hardening, i.e., isotropic expansion of yield surface, induced by anisotropic loadings is obtained. This is what was tacitly assumed in the application of plastic flow rules. However, misalignment of directions between principal stress and principal plastic strain increment is observed. Moreover, plastic strain increment direction is found to be influenced by the given stress increment direction. These facts suggest the non-coaxiality between stress and plastic strain increment due to shear stress increment, even during monotonic loading with isotropic hardening. The plastic strain increment direction diverges from the normal direction to a surface which is a circle with its center nearly at the origin on p′-constant plane and which passes through the current stress point.

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.

Non‐coaxiality and stress–dilatancy rule in granular materials: FE investigation within micro‐polar hypoplasticity

AbstractThis paper deals with FE investigations of shear localization in dilatant granular bodies. The calculations were carried out with a hypoplastic constitutive law enhanced by micro‐polar terms to properly model the shear zone evolution. The behaviour of an initially medium dense sand specimen with very smooth and very rough horizontal boundaries was analyzed during a plane strain compression test. A stochastic distribution of the initial void ratio was assumed to be spatially correlated. Attention was focused on the non‐coaxiality of the directions of the principal strain increments and principal stresses in the shear zone and on the stress–dilatancy rule. Copyright © 2008 John Wiley & Sons, Ltd.

Experimental observations on the response of loose sand under simultaneous increase in stress ratio and rotation of principal stresses

The drained response of loose sand (relative density of 30%) under simultaneous increase in principal stress ratio (R = σ'1/σ'3) and the inclination of major principal stress to the vertical (ασ) is examined using data from hollow cylinder torsional shear testing. The study specifically pertains to the behaviour of loose sand subject to monotonic linear stress path loadings in the R–ασspace, within the domain of R ≤ 2, ασ < 45°, while intermediate principal stress parameter (b) and effective mean normal stress (σ'm) are held constant. The relationship between horizontal shear stress (τzt) and horizontal shear strain (γzt) of loose sand under such loadings is shown to be unique and stress-path independent. At any stress state, the horizontal shear stiffness (dτzt/dγzt) for a given σ'mdepends only on the current value of τzt, and not on the value of individual components of normal effective stress, or their increments. When R and ασare increased simultaneously in a linear manner, loose sand initially exhibits linear strain paths, suggesting no significant changes to the inherent anisotropy during the early stages of such loading. The directions of principal stress increment (αΔσ) and principal strain increment (αΔε) are found coincident, when αΔσ < 45°. An approach to predict the response of loose sand under simultaneous increase in R and ασwith constant b and σ'mhas been developed based on these findings.

Interpretation of strains in torsion shear tests

The normal and shear strains obtained in torsion shear tests may be interpreted in two different ways to gain insight into (1) the type of plastic potential to be employed in hardening plasticity stress–strain models, and (2) the coincidence in physical space of the plastic strain increment direction with the stress direction during principal stress rotation. Thirty-four drained torsion shear tests were performed on hollow cylinder specimens of Santa Monica Beach sand deposited by dry pluviation. Twenty-six tests were performed on tall specimens with height of 40 cm, and eight tests were performed on short specimens with height of 25 cm to investigate the effect of the specimen height on the soil behavior in hollow cylinder specimens. Each test was conducted with the same, constant inside and outside confining pressure, σ r , thus tying the value of b = ( σ 2– σ 3)/( σ 1– σ 3) to the inclination, β, of the major principal stress. The directions of strain increment vectors at failure are compared with the directions corresponding to associated and non-associated flow. The relation between the directions of major principal strain increment and major principal stress during rotation of principal stress axes in physical space are investigated.

Plastic Anisotropy of A2024BE-T6 and O Aluminum Alloys

Multiaxial stress tests were carried out on thin-walled cylindrical specimens of the A2024BE-T6 alloy aged at 120°C for 1 hr and the A2024BE-O alloy fully annealed at 495°C for 1hr. The shear strength of these alloys is much lower than the tensile one, and the equi-strain surfaces in the tension-torsion stress field lie within Tresca's criterion. Planar anisotropy was determined from two types of multiaxial stress tests by means of the combined loading of axial load, internal pressure and torsion, referred to as “off-axis tension test” and “off-axis torsion test”. The degree of anisotropy is expressed by using two different measures : one is the proof stress and the other is the angle deviation between the directions of the principal stress (or principal shear stress) and principal strain increment (or principal shear strain increment). The A2024BE-T6 alloy exhibits more severe anisotropy than the A2024BE-O alloy and both alloys reveal more severe anisotropy in off-axis tension test than in off-axis torsion test. The plastic deformation behavior can be analyzed precisely by the constitutive equation derived from the anisotropic yield function, hardening law and associated flow rule.

A non‐coaxial constitutive model for sand deformation under rotation of principal stress axes

AbstractA constitutive model for the simulation of non‐coaxiality, an aspect of anisotropic behavior of sand subjected to the rotation of the principal stress axes, is presented in this paper. Experimental studies have shown that non‐coaxiality or non‐coincidence of principal plastic strain increments with principal stress axes under loadings involving the rotation of principal stress axes may be considerable. Besides, the rotation of the principal stress axes results in dramatic effects on stiffness and dilatant behavior of sand. Therefore, the consequences of principal stress axes rotation on deformational behavior, dilatancy and soil stiffness must be taken into account in theoretical and practical problems.To this aim, the following steps are taken: (1) A general relationship for flow direction with respect to possibility of non‐coaxial flow is developed. Moreover, special circumstances linking non‐coaxiality to instantaneous interaction between loading and soil fabric are proposed. (2) Proposing novel expressions for plastic modulus and dilatancy function, the model is enforced to provide realistic simulations when sand is subjected to the rotation of the principal stress axes.Finally, with numerous examples and comparisons, the model capabilities are shown under various stress paths and drainage conditions. Copyright © 2007 John Wiley & Sons, Ltd.

Discrete-continuum analysis of shear banding in the direct shear test

The direct shear test is a widely used method for determining peak and critical state strength parameters for soil and for soil-manufactured material interfaces. The objective of this research was to examine the direct shear test by way of a discrete-continuum approach. Specifically, the effects of non-coaxiality between the principal stress and the principal strain increment direction and the deviation of zero extension direction from horizontal are examined. A new, mesh-free method of evaluating internal strains is proposed, and a new flow rule is also presented. The results from discrete simulations indicate that development of shear bands proceeds from the side boundaries toward the centre with both primary and secondary bands formed. A distinct shear band along the middle plane develops around peak state in a dense sample. Simulation data using the new flow rule and laboratory test data using a conventional flow rule are in good agreement, indicating that the effect of non-coaxiality is small at peak state. Deviation of the zero linear extension direction at peak state is less than 3°. It is concluded that boundary measurements made on the direct shear apparatus are suitable for determining strength and volume change behaviour of granular soils.

Principal Deviatoric Strain Increment Ratios for Sand Having Inherent Transverse Isotropy

The behavior of sand on the π plane, including the relation between the principal deviatoric strain increment ratios, is not fully understood. The objectives are: To clarify experimentally the relation between the principal deviatoric strain increment ratios for sand in a wide range of b values and to discuss the basic shear behavior, taking particular notice of inherent transverse isotropy and noncoaxiality. From the experimental results on sand with inherent transverse isotropy, it was observed that the relation between the principal deviatoric strain increment ratios is fundamental and is related to the shape of the failure surface and the extent of strain localization. A new equation for the principal deviatoric strain increment ratios is proposed to model the influence of the incremental stress.

A SIMPLE STRESS-STRAIN RELATION BASED ON STRESS-PATH BEHAVIOR IN STRAIN-PATH CONTROLLED TRIAXIAL TESTS

ABSTRACT A new devise is developed for conducting a strain-path controlled test with a conventional triaxial test apparatus. By controlling lateral stress during axial strain controlled test, this devise can keep the principal strain increment ratio, Re , defined as a ratio between axial and lateral strain increments, constant. Toyoura sand specimens at relative densities of 55% and 80%, consolidated under four different initial confining stresses, are axially loaded with a Re -value between — 0.5 and 1.0. The specimens are then unloaded to the initial stress states and reloaded with different Re -values. The test results during compression show that the principal stress ratio, K, converges at a constant value. The converged K- value is strongly dependent on the Re -value but insensitive to the strain history and the initial stress state. There is a well-defined relationship between the principal stress ratio K and the total strain ratio Re,t where the K-value decreases with decreasing Re,t-value. The K-Re,t relation is independent of its strain path and uniquely expressed in terms of the internal friction angle and the dilatancy angle of the sand. Based on the K-Re,t relationship, a simple equivalent linear constitutive model for axisymmetric condition is developed. Two series of stress-controlled drained triaxial compression tests are performed, and the results are compared with those simulated by the model. The computed results show excellent agreements with the test results, indicating that the proposed constitutive model, although based on the equivalent linear concept, is promising for simulating a stress-strain relation including dilatancy behavior of sand in the stress-controlled triaxial test.

A stress vector-based constitutive model for cohesionless soil (II)—Application

The stress vector-based constitutive model for cohesionless soil, proposed by SHI Hong-yan et al., was applied to analyze the deformation behaviors of materials subjected to various stress paths. The result of analysis shows that the constitutive model can capture well the main deformation behavior of cohesionless soil, such as stress-strain nonlinearity, hardening property, dilatancy, stress path dependency, non-coaxiality between the principal stress and the principal strain increment directions, and the coupling of mean effective and deviatoric stress with deformation. In addition, the model can also take into account the rotation of principal stress axes and the influence of intermediate principal stress on deformation and strength of soil simultaneously. The excellent agreement between the predicted and measured behavior indicates the comprehensive applicability of the model.

予ひずみを受けた軟鋼の多軸塑性挙動と６次後続降伏関数による解析

The subsequent yield function previously proposed by one of the authors is modified by introducing the fourth rank tensor of anisotropic moduli. This yield function accounts for the effects of the third deviatoric stress invariant and anisotropy and the Bauschinger effect. Multiaxial loading tests are carried out on tubular specimens of mild steel subjected to various amounts of torsional prestrain. Two modes of loading are used to assess the theoretical predictions: one is the combined tension- torsion and the other is off-axis torsion by means of the combined loading of compression, internal pressure and torsion. The equi-strain loci and strain behavior are examined as a function of prestrain at three levels of offset strain ranging from 0.002% to 2.0%. In the tension-torsion stress field, the equi-strain locus specified by a 0.002% offset forms a nose in the prestress direction and flattens on the opposite side. The locus expands with increasing offset strain, so that both the sharpness of nose and the degree of flattening decrease. The deviation between the directions of the principal shear stress and principal shear strain increment is determined from the off-axis torsion test. It rises to a maximum value with a change in the principal shear stress direction and then decreases. The maximum deviation is larger when a smaller offset strain is used, and it increases as the prestrain becomes larger. Such behavioral characteristics can be expressed precisely by the constitutive equation derived from the proposed yield function, hardening law and associated flow rule.

Off-Axis Torsion Tests on Tubular Specimens of Steel

In order to analyze the anisotropic hardening behavior of metals, an off-axis torsion test by combined loading is developed. In this test, the maximum shear stress direction φ can be changed from 0 deg to 90 deg while the ratio of maximum and minimum principal stresses is kept at −1. With increasing angle φ, the yield stress of the torsional-prestrained steel decreases; the difference between the directions of the maximum shear stress and principal shear strain increment rises to a maximum value and then decreases. It is experimentally verified that anisotropy is more severe when a smaller offset strain is used in defining the yield stress.

Effects of Stress Rotation and Changes of B-Values on Cross-Anisotropic Behavior of Natural, K0-Consolidated Soft Clay

ABSTRACT A series of consolidated-undrained torsion shear tests was performed on large, hollow cylindrical specimens of undisturbed San Francisco Bay Mud to investigate the influence of stress rotation and changes in b-values on the stress-strain, pore pressure, and strength characteristics of a cross-anisotropic, natural clay. The specimens were first K0-consolidated and then sheared along various undrained stress paths to study the clay behavior in the full range of stress rotation from 0° to 90° relative to vertical. For this cross-anisotropic clay, the test results indicate that the major principal strain increment directions in physical space do not coincide with the directions of major principal stress at failure. The deviation between these two directions varied continuously and the maximum deviation was measured at approximately 20°. The normalized, undrained shear strength decreases systematically with inclination of the major principal stress at failure, ψ, from triaxial compression with ψ = 0° and b = 0 to triaxial extension with ψ = 90° and b = l. The effective friction angle remains constant at approx. 35° in the range of ψ from 0° to 30° after which increases gradually to almost 50° at ψ = 90°. These results are unlike those obtained for a laboratory prepared deposit of kaolin, which tended to behave more like an isotropic material despite the initial A0-consolidation. Thus, effects of aging are very important in the behavior of clays, and it may not be possible to simulate such effects with sufficient accuracy and reliability with laboratory prepared clay deposits.

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.