Critical State Line Research Articles

Mechanical behaviors of coral sand and relationship between particle breakage and plastic work

As a special geotechnical material, coral calcareous sand (CCS), also widely known as coral sand or calcareous sand, surely has similarities and differences in mechanical behaviors with terrigenous sand. As the foundation material of some important structures such as ports, aircraft runways, and marine lighthouses in the South China Sea (SCS), it is necessary and has engineering significance to study its basic mechanical properties. Taking the CCS sampled from a reclaimed land on the top of a natural coral reef in the SCS as the typical case, a series of drained and undrained triaxial shear tests were performed under conventional and high confining pressure in this study. It is found that CCS has all the characteristics that all granular materials should have. However, since the irregular particle morphology and high roughness particle surface, CCS presents a great initial shear stiffness which is significantly different from that of quartz sand. Only some CCS specimens can strictly reach the critical state, and the feature lines in the p’-q space, i.e., critical state line, and phase transformation line are not affected by the factors such as the consolidation confining pressure, initial dry density, and drainage condition. However, the critical state line in the e-lnp’ space is not unique, but dependent on the initial dry density. Based on the particle breakage measurement, a new description for the relationship between particle crushing and plastic work is proposed. It could provide an experimental basis for the development of a constitutive model considering the particle breakage for CCS.

Breakage-Dependent Fractional Plasticity Model for Sands

Particle breakage could significantly influence the shear strength, volumetric deformation, stress–dilatancy relationship, and plastic-flow rule of crushable sands. A breakage-dependent fractional-plasticity (BDFP) model was proposed in this paper to capture the evolution of the stress–strain relationship for crushable sands by using a general plastic-flow rule, which is obtained from the Riemann–Liouville fractional derivative on the modified Cam-Clay yield surface and the incorporation of the breakage-dependent void-ratio–pressure state parameter in the framework of a kinetic breakage-dependent critical state line. The BDFP model can well predict the state- and breakage-dependent strength and deformation of crushable sands. Specifically under higher confining pressures, the prediction for the evolution of the volumetric strain of crushable sands by the BDFP model is much better than that by the model not considering particle breakage.

Strength, Deformation, and Particle Breakage Behavior of Calcareous Sand: Role of Anisotropic Consolidation

The effect of an anisotropic stress state on the monotonic behavior of calcareous sand has been rarely investigated, although calcareous sands in the field mainly experience anisotropic consolidation rather than isotropic consolidation. This study provides a laboratory study of the undrained and drained behaviors of dense calcareous sand consolidated anisotropically under various initial mean effective stresses (p0′) and anisotropic stress ratios (Kc), where significant effects of p0′ and Kc on the strength-deformation-degradation characteristics of calcareous sand can be observed. For undrained tests, higher deviatoric stress can be reached under a lower p0′. This could be due to the buildup of negative excess pore-water pressure (u) induced by sample dilatancy. As Kc increased, the negative u accumulated more distinctly, whereas the maximum and minimum u decreased. For drained tests, the maximum deviatoric stress decreased as Kc increased, even though more dilatancy occurred. Undrained loading generally resulted in a higher extent of particle breakage due to the buildup of negative u that increased the mean effective stress. A unique critical-state line was observed in both isotropic and anisotropic samples regardless of the loading condition. However, unlike quartz sand, a pseudosteady state that defined the transition from a dilatancy-dominant nonflow response to breakage-dominant stress degradation was observed in calcareous sand before reaching the critical state. Results also reveal the negligible effect of Kc on the peak-state stress ratio at undrained conditions. However, Kc had significant effect on stress ratio at peak or phase-transformation state in experiments under drained conditions, which can be well represented by using the concept of state dependence. Moreover, Kc had an adverse effect on the initial secant modulus.

Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils

The non-unique critical state represents the distance between the critical state line (CSL) and the isotropic consolidation line (ICL) that significantly varies with stress paths and particle size distribution of soils. A structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was implemented using an explicit algorithm. However, the explicit algorithm did not well capture the non-unique critical state of soils with a large spacing ratio r, which prevented the soil mechanics research on non-unique critical state via finite element analysis. To overcome the limitation, the implicit algorithm of the SBSP-R model is formulated, and it mainly includes elastic prediction and plastic correction. The plastic correction is realized using the Newton–Simpson scheme with a controlling equation set related to consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. Case studies indicate that the implicit algorithm of the SBSP-R model is right and stable in predicting non-unique critical states. Comparisons between predicted and tested results indicate that the implicit algorithm of the SBSP-R model not only captures the critical state, stress-strain, and stress paths of various soils but also shows higher computational accuracy and efficiency compared with the previous explicit algorithm. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm.

Shear behaviour of peat at different stress levels

Consolidated undrained triaxial compression tests were conducted from 1.65 m depth subjected to stress levels of 10.4–40.5 kPa. The results showed that most test data failed to reach the tension cut-off line (q/p′ = 3). For peat, the membrane correction effect on peat shear resistance was strain dependent – generally small within 10% shear strain, but significant above 10% shear strain. A critical state line for peat was determined based on the maximum curvature approach. Of the data recorded for peat, 78% fell within the range 30–60°, increasing to 90.4% when ignoring points lower than 10 kPa. Organic content (OC) also had an effect on the shear behaviour. For OC > 75%, the deviator stress behaved like an organic soil; otherwise, the peat behaved more like a mineral soil. In peat samples with OC > 75%, the direct shear box test gives higher estimates of shear strength than the triaxial shear test, but which are not necessarily accurate – the mechanism of direct shear acts only at the centre of a specimen, while triaxial shear can shear throughout the specimen.

Experiment Study on the Influence of Density and Confining Pressure on Triaxial Shear Properties of Calcareous Sand.

Calcareous sand is one of the main building materials in the construction of islands and reefs, and its shear property is very important for predicting their strength and deformation. However, the correlation research on the shear properties of calcareous sand is limited. In this paper, a series of the triaxial consolidation drainage shear tests of calcareous sand with relative densities (Dr) of 70% and 90% under confining pressures of 100, 200, 400 and 800 kPa were carried out by a triaxial testing apparatus, and the effects of relative density and confining pressure on the deformation and strength characteristics of calcareous sand were analyzed. The results show that the stress-strain curves of calcareous sand show a strain softening characteristic, and both peak deviatoric stress and failure strain increase with confining pressure, but the increase in failure strain is restrained when the confining pressure is larger than 400 kPa. The initial shear modulus of calcareous sand is positively correlated with confining pressure. Additionally, the molar circular envelope of calcareous sand is linear in the range of 100~400 kPa, but it deviates from linear when confining pressure exceeds 400 kPa. The critical state line (CSL) of calcareous sand is nonlinear, with almost the same exponent for calcareous sand with different relative densities. The research results have important reference value for the foundation construction of islands and reefs.

Exploring packing density, critical state, and liquefaction resistance of sand-fines mixture using DEM

A series of monotonic and cyclic triaxial shearing tests are simulated by discrete element method (DEM) to explore packing density, critical state, and liquefaction resistance of sand-fines binary mixtures. It is found that there is a threshold fines content (FCth) that makes void ratio reach the minimum value and it is affected by the relative density of the mixture. A unique critical state line (CSL) is obtained in equivalent skeleton void ratio-mean effective stress (e*-p') plane regardless of the fines content. The critical stress ratio of the mixture slightly increases as fines content increases, and its average value is around 1.20. The e* and ψe* (state parameter based on e*) can unifiedly characterize the cyclic liquefaction resistance of the mixtures with different fines contents. The active particles of binary mixtures that participate in the force chains are explored from a microscope perspective. It reveals that there is a unique micro CSL in the mechanical coordination number-mean effective stress (MCN-p') plane despite of fines contents, and the micro state parameter (ψMCN) can also successfully and unifiedly characterize the cyclic liquefaction resistance of sand-fines mixtures.

Mechanics of an Iron Ore Tailings Exhibiting Transitional Behaviour

Laboratory critical state line (CSL) testing used in engineering practice to analyse the stability and liquefaction susceptibility of tailings explicitly assumes a unique CSL for a particular tailings. However, there is evidence of non-unique CSLs in some soils, a behaviour commonly referred to as “transitional behaviour”. There is currently limited consensus or understanding of its applicability, what factors contribute to its occurrence, and whether there is a fundamental divergence from critical state theory or an artefact of the strain limitations of laboratory element tests. In this research, normally consolidated samples of an iron ore tailings prepared using the moist tamping and slurry deposition methods were sheared to critical state in triaxial compression tests. The results of these tests showed that a different CSL was apparent for the two specimen preparation methods used. This further emphasises the need to re-evaluate the effects of fabric on the uniqueness of the CSL, which may be a consequence of different sample preparation methods typically used to prepare laboratory element tests.

Calibration of the PM4Sand model for hard-rock mine tailings based on laboratory and field testing results

A comprehensive campaign of laboratory and field tests has been conducted to characterize the behaviour of tailings from a hard-rock mine. Cyclic triaxial, direct simple shear, and triaxial simple shear tests have shown that the contractive tailings are susceptible to liquefaction and that their behaviour is similar to loose sand despite their higher fines’ content. To model the behaviour of tailings’ impoundments in dynamic simulations, the PM4Sand constitutive model is calibrated based on the dataset presented here. Some of the model parameters are defined based on consolidation and compression tests, while the critical state line (CSL) is defined based on the results of monotonic triaxial and direct simple shear tests. The CSL is lower for tailings than for most natural sands, which is consistent with previous studies. The calibrated model can reproduce reasonably well the monotonic behaviour and provides an excellent fit of the cyclic strength curves measured in the lab at different confining pressures and density index. The results presented in this paper indicate that the PM4Sand model can be used to simulate the cyclic behaviour of low-plasticity, hard-rock tailings. The proposed procedure also provides general guidelines regarding model calibration for other types of tailings.

The behaviour of a low- to medium-density chalk under a wide range of pressure conditions

Experiments are described which provided the basis for advanced numerical modelling of large-scale axial and lateral pile tests undertaken chalk to assist the design of offshore wind and other projects in northern Europe. The research explored the mechanical behaviour of chalk from a UK research site under effective cell pressures up to 12.8 MPa. When sheared from low confining pressures the chalk’s interparticle bonds contribute a large proportion of the peak deviator stresses available to specimens that crack, bifurcate and dilate markedly after failing at relatively small strains. Progressively more ductile behaviour is seen as pressures are raised, with failures being delayed until increasingly large strains and stable critical states are attained. Loading invokes very stiff responses within the chalk’s (Y1) linear elastic limits and behaviour remains stiff, although non-linear, up to large-scale (Y3) yield points. Near-elliptical Y1 and Y3 yield loci can be defined in q-p′ stress space and a critical state v-p′ curve is identified. The chalk’s initially bonded, high porosity, structure is explored by normalising the shearing and compression state paths with reference to both critical state and intrinsic compression lines. The results have important implications for pile test analysis and practical design in this challenging geomaterial.

An Investigation of Instability on Constant Shear Drained (CSD) Path under the CSSM Framework: A DEM Study

Soil liquefaction or instability, one of the most catastrophic phenomena, has attracted significant research attention in recent years. The main cause of soil liquefaction or instability is the reduction in the effective stress in the soil due to the build-up of pore water pressure. Such a phenomenon has often been thought to be related to the undrained shearing of saturated or nearly saturated sandy soils. Notwithstanding, many researchers also reported soil instability under a drained condition due to the reduction in lateral stress. This condition is often referred to as the constant shear drained (CSD) condition, and it is not uncommon in nature, especially in a soil slope. Even though several catastrophic dam failures have been attributed to CSD failure, the failure mechanisms in CSD conditions are not well understood, e.g., how the volumetric strain or effective stress changes at the triggering of flow deformation. Researchers often consider the soil fabric to be one of the contributors to soil behaviour and use this parameter to explain the failure mechanism of soil. However, the soil fabric is difficult to measure in conventional laboratory tests. Due to that reason, a numerical approach capable of capturing the soil fabric, the discrete element method (DEM), is used to investigate the CSD shearing mechanism. A series of simulations on 3D assemblies of ellipsoid particles was conducted. The DEM specimens exhibited instability behaviour when the effective stress paths nearly reached the critical state line. It can be clearly observed that the axial and volumetric strains changed suddenly when the stress states were close to the critical state line. Alongside these micromechanical observations, the study also presents deeper insights into soil behaviour by relating the macro-observations to the micromechanical aspect of the soil.

DEM simulation of coexistence type methane hydrate bearing sediments

The hydrate morphology in sandy methane hydrate (MH)-bearing sediment (MHBS) is commonly classified as pore-filling, cementation, grain-coating and load-bearing. This paper presents a numerical investigation into the mechanical behaviour of coexistence-type MHBS using the discrete-element method (DEM). The coexistence type is represented as a combination of pore-filling and cementation types. For MHBS with a given MH saturation (S MH), the relative content of pore-filling and cementation is characterised by the cement ratio λ (λ = 0 means pure pore-filling and λ = 1 means pure cementation). A series of drained triaxial compression tests is simulated, and micromechanical variables such as bond breakage are examined. The results show that (a) the strength and dilatancy of the coexistence-type MHBS are higher when the cement ratio is larger. The normalised strength and secant modulus increase linearly with the cement ratio. (b) When the cement ratio increases, the critical-state line on the mineral void ratio e cr–ln p′ projection shifts towards a higher position and the bond breakage ratio is larger. (c) The shear stress of the coexistence-type MHBS with cementation is affected by the MH generation process due to the MH bond breakages.

An elasto-plastic numerical analysis of THM responses of floating energy pile foundations subjected to asymmetrical thermal cycles

Although end-bearing energy pile foundations subjected to symmetrical thermal cycles have been extensively studied in the laboratory and in the field, the mechanisms underlying the thermo-hydro-mechanical (THM) interactions in floating energy pile groups and rafts, especially when subjected to asymmetrical thermal loads, are not well understood. In this study, an advanced thermo-mechanical bounding surface model was implemented in finite-element (FE) code to investigate the THM interactions of a two-by-two floating energy pile group and pile raft, focusing on asymmetrical thermal cycles. Computed results are compared with published centrifuge model test results in soft clay. It is revealed that the irreversible volumetric contraction of the soil adjacent to the energy piles accumulates with each thermal cycle, resulting in a decrease in the horizontal stress and hence shaft resistance of the floating piles. During thermal cycles, the stress states of the soil around the energy pile shaft and the soil beneath the pile toe approach the critical state line along different paths. The induced temperature in the soil adjacent to the non-energy pile (NEP) is 5°C lower than that in the soil at the energy pile EP1, which is flanked by the other two energy piles EP2 and EP3. Consequently, the induced excess pore pressure in the soil at the NEP is approximately 20% smaller than that in the soil at EP1. The irreversible volumetric soil contraction at the NEP is about half that at EP1, resulting in approximately 45% less toe settlement. The thermally induced ratcheting settlements of the head and toe of the NEP are less than those of the energy piles, resulting in unacceptable ratcheting tilting of the floating energy pile group. However, the excessive tilting can be reduced by the use of a pile raft.

Influence of grading and fabric arising from the initial compaction on the geomechanical characterisation of compacted copper tailings

Tailings dams constructed using the upstream method are generally less stable, because of its operational and constructive approach; this presents increased risks, which is the reason why such structures are being closed in Brazil. A feasible option to ensure the safety of closing operations in such dams consists of extracting tailings from behind the dam (reservoir), reducing moisture content through dewatering and compacting it into stable stacks. This demands knowledge of the compacted material's response. Thus, the present research assesses the mechanical response of compacted copper ore tailings extracted from two different sampling locations within the dam reservoir (i.e. about to be de-characterised) from northern Brazil. For this, drained and undrained triaxial compression tests were carried out on compacted dewatered dense, medium and loose specimens, which were assembled using the tailings retrieved from upper and lower beaches to evaluate the influence of grading, as well as the influence of the fabric arising from the initial compaction on the critical state lines (CSLs) of these saturated non-plastic silty sand tailings. The results indicated that the critical state parameters, in the υ–ln p′ space, are dependent on the fabric arising from the initial compaction. Distinct CSLs, which are all curvilinear and parallel to each other, were found in the research.

Stress analysis of powder beds using constant-volume and constant-load shear tests: Characterization by repeated shearing and application to quasi-steady load shearing

• Constant-volume and constant-load shear tests were conducted. • Normal and shear stresses acting on a shear plane were accurately measured. • Results obtained using the two test methods presented reasonable concurrence. • Constant-volume shear tester reproduced constant-load shear test results. • A quasi-steady load shear test method was proposed. Two types of high-performance shear test methods were employed to analyze the mechanical properties of powders. Both tests accurately measured the normal and shear stresses acting on the shear plane. Alumina powders with mass median diameters of 0.67, 4.0, 9.9, and 49 µm were used as samples. The effects of repeated shearing on the void fraction, stress relaxation ratio, stress transmission ratio, powder yield locus (PYL), consolidation yield locus (CYL), and critical state line (CSL) were determined using a constant-volume shear tester. The PYL and CSL were also measured using a constant-load shear tester, and the results obtained were compared. This comparison demonstrated a reasonable concurrence between the results obtained with the two testers. Furthermore, quasi-steady load shear tests were conducted using a constant-volume shear tester with feedback control, which demonstrated that the PYL could be obtained in a single shear test within a few minutes.

Effect of fiber-reinforcement on the mechanical behavior of sand approaching the critical state

Several types of ground improvement methods that employ fiber-reinforcement have been developed in recent years. A series of consolidated drained triaxial compression tests has been conducted here to examine the effect of short fibers on the mechanical properties of Toyoura sand. Sand with 0%, 0.2%, 0.4%, and 1% fiber contents, prepared to yield random distribution, was sheared under several confining pressures and controlled via their initial relative densities. The test results showed that the maximum and residual deviatoric stresses increased, whereas the volumetric expansion decreased with an increase in fiber content. Although the stress ratio η ( =q / p ′) and specific volume changed depending on the fiber content and confining pressure with shear progression, they each reached the same values for a definite fiber content at the end of shearing, independent of initial relative density. In other words, the unique critical state line can be found for a definite fiber content. Moreover, the greater the fiber content, the larger the slope of the critical state line at the end of shearing. Additionally, as the length of fibers shortened with the same percentage of fiber inclusions in sand, the deviatoric stress and the stress ratio decreased, approaching the shear-strain-volumetric response of unreinforced sand.

Triaxial test on glass beads simulating coarse-grained soil

It is feasible to study the mechanical characteristics of coarse-grained soil by simulated granular materials such as glass beads. In this paper, 3 mm diameter glass beads are used to conduct drained and undrained triaxial tests under different confining pressures to explore their strength, deformation and critical state characteristics. Specifically, the influence of drainage and confining pressure on the stick-slip phenomenon of glass beads is reported. The experimental findings from triaxial tests show that the stress-strain relationship of glass beads softens when the confining pressure is high. Under the undrained condition, the initial modulus increases with the increase in the confining pressure. In contrast, it is not significantly affected by the confining pressure in the drained condition. It is quite evident that the glass beads hardly contract during the shearing process, and their stress path is approximately a segmented straight line. The slope of the critical state line under the undrained condition is greater than that under the drained condition, and the friction angle of the glass beads under the undrained and drained conditions is calculated to be 28.1° and 29.5°, respectively. The phenomenon of stick-slip has been depicted for the different test conditions, and the stick-slip amplitude linearly increases with the confining pressure, especially in the undrained condition. It is also found that the maximum energy released from the phenomenon of stick-slip increases linearly with the confining pressure.

Experimental study on critical state behaviour of unsaturated silty sand under constant matric suctions

There are relatively few experimental studies on the hydromechanical and critical state behaviours of unsaturated silty sand. In this paper, suction-controlled, consolidated, drained triaxial shear tests were conducted on silty sand under saturated (net confining pressures of 100, 200, 300 and 400 kPa) and unsaturated (suctions of 30, 100 and 200 kPa and net confining pressures of 100, 200 and 300 kPa) conditions. For unsaturated silty sand, the effects of dry density, matric suction and net confining pressure on the hydromechanical and critical state behaviour were investigated, and the critical state parameters were obtained. The main findings for unsaturated silty sand included shear shrinkage at low dry density (1·6 g/cm3) and dilatancy at higher density (1·96 g/cm3). The peak shear strength was positively correlated with dry density, matric suction and net confining pressure. The critical state volumetric strain and critical state degree of saturation were negatively correlated with dry density and matric suction but positively correlated with net confining pressure. A new dilatancy equation that includes the state parameter for unsaturated silty sand was established. A unique critical state line passes through the origin in the effective stress [Formula: see text]–deviator [Formula: see text] plane, and its slope is the saturated critical state stress ratio [Formula: see text]. In the lnp*–v plane ([Formula: see text] is the specific volume), the critical state lines corresponding to different matric suctions are parallel.

A state parameter-dependent constitutive model for sands based on the Mohr-Coulomb failure criterion

Experimental data have demonstrated that a strong relationship exists between the state parameter and the peak strength and dilatancy characteristics of sands. This paper proposes a way of reproducing this behaviour using a modified Mohr-Coulomb failure criterion, which retains its simplicity while improving substantially its modelling capabilities. The formulated constitutive model is calibrated for Nevada sand following a well-defined procedure and used in the prediction of four centrifuge tests investigating the behaviour of axially loaded footings. It is shown that the proposed model reproduces well both the element tests and the more complex footing problems, demonstrating its usefulness for engineering practice. Moreover, a simplified version which does not require the definition of the Critical State Line is proposed for situations when this aspect of soil behaviour cannot be determined with confidence. It is shown that such simplification results in only slightly less accurate predictions than the full version of the model, while simulating aspects of soil response that cannot be reproduced using constant values for strength and dilatancy parameters.

DEM studies on the effect of particle breakage on the critical state behaviours of granular soils under undrained shear conditions

A number of discrete element analyses of undrained triaxial shear tests on crushable assemblies are performed using 3-dimensional particle flow code (PFC3D). The undrained shear of the samples is simulated by assigning different velocities at the boundaries to control the constant volume of the samples during the shear. Particle breakage is studied using the octahedral shear stress breakage criterion and the fragment spawning mode satisfying an Apollonian sphere packing. The microparameters of the soils are obtained by simulating the test results reported on a Nansha calcareous sand. The effects of particle breakage and drainage condition on the critical state behaviours of the sand are intensively examined. It is found that particle breakage and drainage condition do not affect the slope of the critical state p′−q line. In the ecs−(p′cs/pa)0.7 space, the critical state line translates downward and rotates clockwise when particle breakage is considered, whilst drainage condition does not affect the critical state line. The intrinsic macro–micro correlations underlying the critical state behaviours are explored by analysing the evolution of representative micromechanical indicators such as coordination number, redundancy ratio, and fabric and force anisotropies.