Critical State Line Research Articles

Modelling the evolving critical state of crushable soils and parameter optimization with an improved genetic algorithm (AIS-RCGA)

The critical state line (CSL) plays an essential role in the constitutive modelling of granular soils. It serves as a reference line for the measurement of the state parameter. The critical state of crushable soils cannot accurately be determined from experimental data because of the ever-changing soil properties due to particle breakage. In this study, two new parameters eB and eb are introduced, which account for the final position and the evolution of CSL of crushable soils during shearing, respectively. To identify the optimal CSL-related parameters from experimental data, a hybrid genetic algorithm combining artificial immune system (AIS) and real-coded genetic algorithm (RCGA), namely AIS-RCGA is adopted. The fitness is defined to minimize the prediction error of both void ratio (or, pore water pressure) and deviatoric stress. We have refined the tuning methods for several hyperparameters of AIS-RCGA and proposed a novel method to assess the similarity of individuals within AIS-RCGA. Results show that the new method is more efficient in finding the global optimal for our problem. With optimized model parameters, the new constitutive model can accurately predict the response of crushable soil, outperforming other constitutive model reported in the literature.

Shear strength characteristics of unsaturated compacted GMZ bentonite considering anisotropy

Anisotropic microstructure would be generated in the bentonite block during unidirectional compactions. Working as buffer materials, the compacted bentonite will inevitably experience shearing processes during the long-term operation of geological repositories. In this paper, high-pressure triaxial tests were conducted on unsaturated compacted GMZ bentonite specimens with different water contents, dry densities and confining stresses, with the compaction surface of the specimens oriented in both horizontally (H-type) and vertically (V-type) configurations. Results demonstrate that an increase in water content leads to a reduction in both peak strength and residual strength, while higher confining stresses enhance these strength parameters. Normally consolidated and lightly over-consolidated bentonite specimens display substantial shear deformation, whereas heavily over-consolidated specimens tend to experience brittle failure. Water content plays a significant role in shaping both the critical state line (CSL) and the Hvorslev surface (HS), with increasing water content resulting in decreased slope parameters for both and an increased intercept parameter for the HS. Generally, V-type specimens demonstrate a steeper CSL and an outwardly extended HS in contrast to that of H-type specimens. The critical state ratio for V-type specimens is about 10 % higher, and the friction angle is 2.8° greater, than that of the H-type ones. Moreover, this difference appears to increase with increasing water content. The difference of the HS slope parameter between the two specimens is minor, while the intercept parameter is higher for the V-type specimens.

Hypoplastic constitutive model of inherently anisotropic sand

The modeling of anisotropic sand has always been a challenging and prominent issue in geotechnical constitutive research. This paper presents a hypoplastic constitutive model for inherently anisotropic sand. An anisotropic state variable A, which is defined by the joint invariants of the stress and fabric tensor, is introduced to account for the fabric anisotropy. This form of A can be referred to as a re-arrangement that allows for easier calibration. And this modification leads to the anisotropic contraction of the failure surface. The softening parameter in the critical state line is revised from a fixed to a state-dependent type, for achieving unified simulations of monotonic sand mechanical behaviour under different drainage conditions and within a wide range of stress levels and densities. Simulations of various monotonic element tests show that the salient features of inherently anisotropic sand, including barotropy, pyknotropy, strain softening, and dilatancy, can be well captured. Particularly, we make the first attempt to simulate pure stress-controlled true triaxial tests using hypoplastic constitutive under both drained and undrained conditions. Moreover, we provide a method to further optimize the flow direction by introducing an exponential formula.

Instability of binary mixtures subjected to constant shear drained stress path: Insight from macro and micro perspective

AbstractLoose granular materials may also exhibit instability behaviors similar to liquefaction under drained conditions, commonly referred to as diffuse instability, which can be studied through constant shear drained (CSD) tests. So far, the research on CSD in binary mixtures is still insufficient. Therefore, a series of numerical tests using the discrete element method (DEM) were conducted on binary mixtures under CSD path. The possible model of instability is categorized into type I and type II, type I instability occurs prior to reaching the critical state line (CSL), whereas type II instability occurs after exceeding the CSL. The study analyzes the macroscopic instability behavior and the impact of fine content (FC) on macroscopic instability behavior. The numerical results show that as FC increases, the slope of the instability line (IL) increases initially and then falls in the p‐q plane. In the e‐p plane, the IL decreases initially and then ascends. The instability type of the binary mixtures is influenced not only by relative density but also by FC. The stability index increased first and then decreased with the increase of FC. The microscopic origin of binary mixtures instability is explored by investigating the fabric‐stress relationship. The collapse of the weak contact sub‐network triggers the specimen instability, while the strong contact sub‐network dictates the difficulty of achieving instability. FC influences the evolution of fabric anisotropy of the strong and weak contact networks, thereby controlling the macroscopic instability behavior of binary mixtures.

Thermo-mechanical response of sand-silt mixtures

The thermo-mechanical response under undrained conditions is crucial in evaluating the mechanical behaviors of sand-silt mixtures. A series of consolidated undrained triaxial shear tests were conducted to investigate the mechanical response of sand-silt mixtures with different fines contents, temperatures, and initial mean effective stresses. The results showed that the volume shrinkage was observed in all specimens during heating. The amplitudes of volume shrinkage were dependent on the fines content and initial mean effective stress. Moreover, the fines content and temperature significantly influence the peak strength, flow behavior, stress ratio at the undrained instability state, and collapsibility index. Furthermore, a unified critical state line has been established in the compression plane for both clean sand and binary mixture under different temperatures and fines contents. It is reliable and suitable to use the equivalent skeleton void ratio to illustrate the thermo-mechanical response of sand-silt mixtures under undrained conditions.

On the mechanical behaviour of a coral silt from the South China Sea

During land reclamation on the reef islands, large amounts of silt-sized coral soils were created by segregation and degradation. The accumulation of silt-sized coral soils is fairy uncommon while the research on the geotechnical properties of the coral silt is very limited. In this study, a systematic experimental investigation on the mechanical behaviour of a coral silt obtained from a reclaimed reef island in the South China Sea has been performed, with comparisons to the coral sand collected from the same area. Similar to the coral sand, the coral silt particles also exhibit irregular particle shape and intra-particle pore due to their nature origin. According to the limiting water contents, the coral silt is classified as a low-plasticity clayey silt. Under one-dimensional compression, the coral silt exhibits a much quicker convergence compared to coral sand. Prior to convergence, the compressibility of coral silt is higher. After yielding, the compressibility of coral sand becomes higher due to significant particle breakage. The loose coral silt subjected to undrained shearing at low confining pressures exhibits obvious strain softening behaviour, indicating static liquefaction or flow failure. The peak and critical state friction angles of coral silt are lower than those of coral sand but much higher than those of the other terrigenous clayey silts. A curved critical state line with well-defined horizontal asymptote in the deviatoric stress – mean effective stress plane is identified for coral silt, again indicating higher potential to static liquefaction.

Stress–Dilatancy Behavior of Alluvial Sands

Based on the Frictional State Concept (FSC), the stress–dilatancy behavior of alluvial sands in drained triaxial compression was investigated. The dilatant failure state is equivalent to the minimum plastic dilatancy state for sands. The dilatant failure state slightly precedes the failure state. The straight line approximating dilatant failure states in the stress ratio–plastic dilatancy plane defines the slope of the critical frictional state line in the q-p’ plane, i.e., the critical frictional state angle. The stress ratio–plastic dilatancy relationship, obtained from the FSC, defines the sand shear angle as a function of the critical frictional state angle and plastic dilatancy. The shear angle of the tested sand is a maximum of 2° greater than that obtained from Bolton’s formula. According to the authors, these differences are affected by the transverse anisotropy of sand in the tested samples and the difference between the dilatant failure state and the failure state.

Hydro-mechanical response of volcanic ash on removal of fines: Shear stiffness to critical state mechanics

Natural volcanic soils containing pumice particles are commonly found in Hokkaido, Japan, and this type of soil is prone to landslides, internal erosion, and liquefaction. Therefore, the purpose of this paper is to summarise the hydro-mechanical response of volcanic ash soil subjected to internal erosion using the modified erosion triaxial apparatus, based on the literature and additional investigations. The results of the study show that the rate of erosion and shear strain during the erosion process are influenced by initial density, stress state, and hydraulic gradient. Notably, anisotropic consolidation is experienced by specimens under seepage flow. Additionally, the removal of fines leads to a slight decrease in the grading state index. Moreover, suffosion increases the maximum shear modulus and Poisson’s ratio of the soil, while increasing seepage time stabilises the peak shear strength of eroded specimens. Furthermore, the critical state line does not change much with internal erosion. To sum up, this study offers valuable insights into the behaviour of volcanic ash soil subjected to internal erosion and provides an integrated interpretation of hydro-mechanical response of volcanic ash on removal of fines.

Critical state behaviour of an unsaturated kaolin mixture

In this paper, a series of suction-controlled triaxial tests were conducted to investigate the hydromechanical behaviour of an unsaturated kaolin mixture. The studied variables included the mean net stress, suction, and initial dry density. The suction was controlled over a wide range – in the span of 0–390 MPa using the axis translation method and the vapour equilibrium technique, aiming to holistically study the mechanical response of unsaturated soils. The discussion focused on the critical state behaviour and dilatancy characteristics of the tested soil. The experimental results indicate that the ductility of the soil decreases and shifts towards brittleness with increasing suctions. The critical state stress ratio (Ma) of the soil exhibits a significant increase with suction, and this increase becomes more pronounced with higher initial dry density. Conversely, the impact of suction on the critical state line in the specific volume-mean net stress plane (v-lnpnet plane) diminishes with higher initial dry density. Furthermore, suction notably influences the stress-dilatancy relationship of kaolin mixtures, while the effect of confining pressure appears to be marginal. The influence of suction on the stress-dilatancy relationship primarily stems from its impact on the critical stress ratio (Ma), which can be normalized by dividing the stress ratio by the critical state stress ratio. Lastly, the stress-dilatancy equation of the modified Cam-clay model is extended to capture the effect of suction on the dilatancy characteristics of unsaturated soils.

New critical state constitutive model considering particle breakage implicitly for uncemented carbonate sands

One of the main problems of carbonate sands is the fragile nature of particles and their susceptibility to breakage. Carbonate sands are affected by volumetric strain even at low stress levels, which is not the case with silicate sands. By defining a simple breakage model, the current study develops an elastoplastic critical state constitutive model that considers the impact of particle breakage on the mechanical behavior of carbonate sands. The particle breakage model depends on mean effective stress and critical breakage stress, which is assumed to correspond with the precompression pressure of soil in the oedometer test. In the proposed model, critical state line movement with the breakage parameter (α) considers the particle breakage effect. Based on the unified clay and sand model (CASM), a novel dynamic yield surface with a shape parameter affected by particle breaking has been created. Certain modifications are made to the modified Cam-Clay stress dilatancy to predict the behavior of carbonate sand. The current model has only ten parameters that simulate the carbonate sands’ behavior even at high-stress levels without any breakage test. Experimental data with different soil densities, loading stress paths, and stress levels were compared with the model, and the results demonstrated satisfactory conformance.

Numerical modelling of multiple excavations in an ultra-deep foundation using an enhanced distinct lattice spring model with modified cam clay model

The control of foundation deformation has garnered significant attention. This study focuses on understanding the deformation induced by excavation activities in real foundation construction scenarios. To accurately model the complex geomechanical responses, an enhanced Distinct Lattice Spring Model (DLSM) was developed. The Modified Cam Clay (MCC) model is employed to represent the nonlinear behaviour of the soil. Integration with one-dimensional structure elements is implemented to model support structures and their interactions with the soil. The birth–death particle method is utilized to simulate the excavation process of the foundation. To improve computational efficiency, a CPU-GPU heterogeneous parallel technique is employed. The enhanced DLSM produces consistent results with the finite element method when calculating foundation-bearing capacity and demonstrates relatively close results compared to finite difference methods when analyzing the stability of foundation slopes under different parameters. Finally, a three-dimensional numerical model is established using the proposed DLSM to simulate the deformation of the foundation caused by the excavations. The entire simulation task is divided into twelve stages, involving in-situ stress, a diaphragm wall, ten supporting structures, and ten excavations. A comparative analysis between the MCC model and the ZP (Zienkiewicz-Pande) model is conducted. The findings indicate that the calculation results of the two are consistent in the early stages of excavation, while those calculated by the MCC in the later stages are relatively small. Sensitivity analysis of the MCC parameters using orthogonal analysis reveals that the porosity of the soil has the greatest impact on the deformation of the foundation pit, followed by the slope of the swelling line, while the slope of the critical state line has a minimal impact. This work provides an alternative high-performance tool for engineering-scale deformation analysis of geotechnical underground excavation.

Experimental study on the engineering properties and failure mechanism of moraine in Southeast Tibet under freeze–thaw cycles conditions

The effect of freeze–thaw cycles (FTCs) is a prevalent hydrothermal phenomenon in seasonal permafrost regions, significantly affecting the micro and engineering properties of moraine. This study investigates the engineering properties and failure mechanisms of moraine induced by FTCs, using samples with varying dry densities subjected to 0–20 FTCs. The permeability characteristics, particle breakage characteristics, elastic modulus, stress–strain curve, excess pore water pressure, peak friction angle, and critical state line characteristics of the moraine were analyzed before and after the FTCs, and the microscopic failure mechanism of moraine induced by FTCs was revealed. The test results show a degradation in the engineering properties of moraine induced by FTCs, exhibiting the greater the dry density the more obvious the damage to the engineering properties of the moraine. In addition, the particle fragmentation and fine particle aggregation caused by the FTCs changed the distribution, shape, arrangement, and particle intercontact patterns, which revealed the failure mechanism of moraine induced by FTCs. The correlation analysis between the shear strength index and microstructural parameters demonstrated that the particle arrangement has the greatest effect on the peak friction angle and particle shape has the greatest effect on the critical state line of moraine under the FTCs conditions. The findings offer experimental evidence and theoretical support for preventing and resolving geological risks and engineering project problems resulting from the deterioration of moraine shear strength during FTCs in the alpine mountain region.

An SPH framework for drained and undrained loading over large deformations

AbstractWe propose a new approach for performing drained and undrained loading of elastoplastic geomaterials over large deformations using smoothed particle hydrodynamics (SPH), a meshfree continuum particle method, combined with the modified Cam Clay (MCC) model of critical state soil mechanics. The numerical approach draws upon a novel one‐particle two‐phase penalty‐method based formulation for handling undrained loading in saturated soils, which allows tracking of the buildup of pore‐water pressures under combined shearing and compression. Large‐scale parallelized simulations are employed to accommodate a significant number of degrees of freedom in a three‐dimensional setting. After verification and benchmark testing, the SPH based formulation is used to analyze the propagation of reverse faults through fluid‐saturated clay deposits and the rupture of strike‐slip faults across earthen embankments. The computational methodology tests the robustness of the meshfree approach in situations where the soil tends to dilate on the ‘dry’ side of the critical state line and to compact on the ‘wet’ side, but cannot, because of the incompressibility constraint imposed by undrained loading. Our results extend the current understanding of fault rupture modeling and further demonstrate the potential of our framework together with the SPH method for large deformation analyses of complex problems in geotechnics.

A unified thermo-mechanical bounding surface model for saturated clay and sand

To accurately predict soil thermal effects is of great importance for simulations of complex boundary value problems such as the energy foundations and nuclear waste disposal. Existing thermo-mechanical constitutive models only account for clay, and cannot simulate the sand’s behaviour under thermo-mechanical conditions. In this study, a unified thermo-mechanical bounding surface (UTMBS) model is proposed for saturated clay and sand. Based on the thermal effects on the isotropic compression line, the model proposes a new unified thermal softening relationship and a plastic modulus for clay and sand, with the thermal cyclic behaviour replicated by a memory surface. The unified model considers the thermal effects on the critical state line and the shape of the bounding surface, accounting for both drained and undrained shearing of clay and sand at different temperatures. In addition, the non-linear elasticity relationship represents the hysteresis loops of the stress–strain relationship in the mechanical cycles. The performance of the proposed model is evaluated against existing experimental results for clay and sand in terms of their thermal cyclic behaviour, drained/undrained triaxial compression, and mechanical cyclic behaviour at different temperatures. It is evident that the UTMBS model is able to simulate various thermo-mechanical behaviours of clay and sand.

A nonlinear analytical model for consolidated geotextile-encased sand columns

A nonlinear model for consolidated geotextile-encased sand columns (GESACs) was formulated in this study. The model is based on the power law and predicts the stress-strain curve of a consolidated GESAC based on the superposition rule, wherein the stress-strain curve of the unconsolidated GESAC is superposed by the stress-strain curve of the soil alone in a consolidated triaxial test. A uniaxial compression test was conducted to study the failure mechanism of the GESAC. In addition, unconsolidated and consolidated triaxial tests on loose GESACs were conducted to investigate the effect of initial stresses on the shear behavior of GESACs. To further investigate the interaction between the soil and geotextile, and to assess the GESAC model, finite element method simulations were conducted. Based on the results, internal lateral stresses developed in the GESAC due to the confining effect of the geotextile, which increases the circumferential tension force on the geotextile while the p-q path of the GESAC approaches the critical state line, and follows the line when the shear strength of the soil is mobilized. The model was verified based on data on dense consolidated GESACs found in the literature, and results have shown good agreement between the measured and predicted data.

A novel method for imitating true-triaxial stress path with conventional triaxial apparatus

The stress paths of the cylindrical specimen in the p–q stress space by controlling the ratio of the axial and the radial loading is guaranteed to be consistent with the cuboid specimen, a novel method for imitating true-triaxial stress path by conventional triaxial apparatus was presented. Under the condition that p and q were variables and b was constant, the true-triaxial stress paths were realized by conventional triaxial apparatus strictly and easily. Under the condition that b and p were invariants, the b was used to control the ratio of axial and radial loading to ensure p constant, the method can be used to measure the strength on the π plane. If the tests were conducted at the different p with the same b, the critical state line of different b could be obtained. Under the condition that p and q were constant, the proposed method of nonlinear loading with b as a parameter could be used to design the various stress paths of true-triaxial under the condition of deviatoric stress consolidation, and which could be used to determine the deformation and the plastic flow of soil in 3D space. The proposed method could be used to achieve the equivalent stress path in the p–q stress space to obtain the 3D mechanical properties, and the stress path controlled by stress, strain, and a hybrid of stress and strain. Once the software of conventional triaxial apparatus was developed by the novel method, the measuring range of stress paths could be expanded greatly.

Effects of microbially induced calcite precipitation on static liquefaction behavior of a gold tailings sand

Loose tailings are susceptible to static liquefaction during which they lose a substantial amount of their strength. This study examines a sustainable technique known as Microbially-Induced Calcite Precipitation (MICP) to improve the static liquefaction resistance of gold mine silty sand tailings. These materials were enriched with Sporosarcina pasteurii, consolidated in a direct simple shearing apparatus, and subjected to several injections of a cementation solution. Calcified tailings were then sheared under constant-volume and constant vertical stress conditions to evaluate their undrained and drained shearing behaviors. Results showed that bio-mineralization can prevent the occurrence of static liquefaction in tailings by reducing their contraction tendency. This is demonstrated by the strong strain-hardening behaviors of the treated tailings specimens compared to the strain-softening and undrained strength loss in specimens of the untreated tailings. Substantial increases in the tailings undrained and drained shear strengths (by up to 30 - 50 kPa), improvements (by up to 5 MPa) in their tangent moduli, and more than 5° rise in their friction angles are observed in the direct simple shear tests following MICP-treatment. The critical state line of tailings is also found to be steeper and shifted to denser void ratios following MICP treatment. These changes reduce liquefaction susceptibility of tailings and enhance their resistance against static liquefaction. Post-treatment acid dissolution further indicates that CaCO3 contents of about 4% to 11% precipitated in the treated specimens. This amount decreases with increasing specimens void ratio. Changes in the microstructural fabric of the cemented tailings particles are also characterized using scanning electron microscopic (SEM) images and X-ray diffraction (XRD) analyses.

A simple gradation-state-dependent model for granular materials

This paper proposes a gradation-state-dependent constitutive model to capture the granular materials' mechanical behaviours considering the granular crushing's influence. Firstly, a gradation evolution law is developed to reflect the change in grain size distribution (GSD) with the growth of stress levels, and then, based on the evolution law, the elastic compliance matrix is modified to reflect the effects of particle breakage. Secondly, as the critical state line (CSL) exhibits gradation dependence, a novel method is developed to describe the family of CSLs under different fixed gradations. Subsequently, by combining the gradation evolution law with the CSL equation, the paper explains the steepening of the slope of the critical state line when the crushing is activated. With the improvement of CSL, the material's shear dilatancy and contraction properties are better reflected, assisting in establishing a plastic compliance matrix. Employing the derived elastic and plastic compliance matrix, the elastic–plastic model reflecting the influence of gradation and state effects is established and verified with a series of experimental and discrete element method (DEM) simulation results.

Discrete element study of the effect of particle shape on critical state characteristics and constitutive parameters of sand

Based on the discrete element particle flow software PFC3D, four irregular particles with the same aspect ratio, convexity, and sphericity as the binary pictures are constructed. To examine the influence mechanisms of particle shape on mechanical properties, microstructure, and critical characteristics, an index F characterizing the regularity of the particle shape is introduced. By examining the link between the regularity index and the constitutive parameter, a statistical model is developed to predict the peak friction angle while considering the impacts of relative density and particle shape. The findings demonstrate that decreasing the value of the regularity index always causes an increase in the stress ratio, shear dilatancy, and void ratio. In p'-q, e-logp’, and MCN-logp’ spaces, the distribution pattern of critical state lines is affected by the particle shape. At the same strain instant, when the particle becomes more irregular, the distribution direction of the maximum contact force of irregular particles tends to be more horizontal to resist the greater external loads. In comparison to spherical particles, the Δq/p' and Δφp of irregular particles increase linearly with the increase of ΔF. The measured and predicted values of the peak friction angle for the statistical model are essentially consistent. The critical parameters ecs、ηcs、φcs、MCNcs decrease linearly with the increase of the regularity index. The above conclusions serve as a guide for revising the empirical formula.

Verification of a novel stress path method by true-triaxial test

To verify the novel method of achieving a true-triaxial stress path with the pseudo-triaxial apparatus, a series of drained and undrained tests were carried out for the identical scheme with pseudo-triaxial apparatus and true-triaxial apparatus respectively. The differences between the two types of tests were quantified. The results show that the novel method effectively achieved the true-triaxial stress path by controlling the loading ratio of the pseudo-triaxial apparatus. The relationships of q − ε1 and η − εs measured by the two apparatuses had a higher similarity which decreases slightly with the b increase. When 0 ≤ b < 0.5, the slope of the critical state line measured by both apparatuses was almost identical. When 0.5 ≤ b ≤ 1, the slope of the critical state line measured by the novel method was slightly lower, but the biggest change was within 10% compared with the two Mohr–Coulomb criteria, the peak strength measured by the two apparatuses was distributed near the criteria, indicating the feasibility and rationality of the novel method. The tests show that the novel method greatly enriches the test range of pseudo-triaxial apparatus, which not only simplifies the process of soil 3D testing but also reduces the test cost.