Anisotropic Critical State Theory Research Articles

Correlation of fabric parameters and characteristic features of granular material behaviour in DEM in constitutive modelling

AbstractThe anisotropic microstructure of granular materials has a profound effect on their macroscopic behaviour and can be characterised using a fabric tensor. To include of fabric in the critical state theory (CST), anisotropic critical state theory (ACST) was proposed by modifying the state parameter $$(\psi )$$ ( ψ ) of CST to a fabric-dependent dilatancy state parameter $$(\upzeta )$$ ( ζ ) . Noteworthy that $$\uppsi$$ ψ showed a very strong correlation with characteristic features (e.g. instability, phase transformation and characteristic state) of macroscopic behaviour and, as a result, it has been adopted in many constitutive models. While $$\upzeta$$ ζ aided the inclusion of fabric in ACST models, the correlation between $$\upzeta$$ ζ and characteristic features has not been evaluated in detail yet, although a large number of works are found on micromechanics and fabric only. In this study, a large number of discrete element method simulations for drained and undrained triaxial were conducted to evaluate the correlation between $$\upzeta$$ ζ and characteristic features. To this purpose, the correlation between stress ratio and both classic and dilatancy state parameter ($$\psi$$ ψ and $$\upzeta$$ ζ ) were studied in important characteristic features (e.g. instability, phase transformation and characteristic state). It was found that this correlation was improved using $$\upzeta$$ ζ which might be due to the inclusion of fabric in our model. This observation is new and significant for inclusion of fabric evolution in constitutive modelling.

High-cycle shakedown, ratcheting and liquefaction behavior of anisotropic granular material with fabric evolution: Experiments and constitutive modelling

Although the mechanical response of granular materials strongly depends on the interplay between their anisotropic internal structure (fabric) and loading direction, such coupling is not explicitly considered in existing high-cycle experimental datasets and models. High-cycle experiments on granular specimens specifically prepared with various fabric orientations are presented. It is found that the high-cycle strain accumulation behavior can change remarkably, from shakedown to ratcheting, when the fabric orientation deviates more from the loading direction. Inspired by the experimental observations, a fabric-dependent anisotropic high-cycle model is proposed, by proper recasting of an existing model formulated within Critical State Theory, into the framework of Anisotropic Critical State Theory. The model explicitly accounts for the fabric evolution, which is linked to plastic modulus, dilatancy and kinematic hardening rules. The model can quantitatively reproduce the high-cycle strain accumulation (i.e., shakedown and ratcheting) under drained conditions, as well as pre-liquefaction and post-liquefaction responses granular materials having widely ranged fabric anisotropy, densities and cyclic loading types using a unified set of constants. It exhibits a unique feature of simulating the distinct high-cycle strain accumulation and liquefaction of granular material with various fabric anisotropy, while the existing high-cycle models treat them equally. The successful reproduction of the anisotropic sand element response under high-cycle drained and undrained conditions makes it possible to perform whole life analysis of various foundations on granular soil subjected to high-cycle loading events.

Nonlinear site response analyses for sands: investigating the influence of fabric anisotropy

Nonlinear effective stress site response analyses (SRAs) are commonly used to estimate dynamic soil behaviour, seismic wave propagation through the soil medium, and resulting ground motions [1]. These analyses can be used to identify potential hazards (e.g., landslides, settlements, liquefaction) and to estimate dynamic loads on superstructures in areas that are prone to natural or induced earthquakes, which can help with disaster planning and risk mitigation efforts. In this study, the influence of fabric anisotropy, which is induced during the soil formation process, on the response of sand deposits has been assessed through one-dimensional site response and response spectrum analyses (RSAs). First, a novel anisotropic critical state theory (ACST) based semi-micromechanical constitutive model accounting for the effect of fabric anisotropy has been incorporated into a fully coupled dynamic code employing the u-p formulation. Then, the initial fabric anisotropy has quantitatively (both with respect to intensity and orientation ) been changed to imitate different anisotropic formations observed in natural deposits. The proposed numerical procedure shows that fabric effects stemming from the anisotropic nature of sands can significantly influence the dynamic behaviour of sand deposits, leading to significant variations in ground motions and therefore resulting in diverse spectral accelerations at the ground surface.
 The loading direction dependent behaviour of sands, which can be associated with their anisotropic nature originating from the arrangement of the soil inner microstructure, is generally described/idealized using a second order fabric tensor by ACST based models. Similarly, in this study, a contact normal based second order fabric tensor together with a plastic strain driven fabric evolution formulation has been employed to link the influence of the changing inner microstructure to the relevant constitutive formulations. Further details on the fabric formulations and their multilaminate specific extension can be found in ref. [2] and [3]. Although numerous experimental studies have been conducted to investigate the influence of fabric on the undrained response of sands and advanced constitutive models have been developed to account for it, the majority of research efforts involving anisotropy have concentrated on the element test level, while practical boundary value problem (BVP) simulations are usually omitted. In order to ameliorate that trend, the practical aspects of the fabric effects in BVPs will be investigated in the next section.
 To investigate the repercussions of incorporating fabric effects, two identical SRAs with different initial fabric configurations, i.e., initially isotropic and anisotropic, have been carried out and the resultant response spectrums are presented in Figure 1. These SRAs were performed for a one-dimensional column of 10 m height with a water table located at m depth, subjected to a seismic load taken from the 1987 Superstition Hills earthquake. The initial field stresses were determined assuming . Even though the different initial anisotropic configurations produce similar RSA trends over the range of the period , a significant difference in peak values is observed at sec. Whereas the initially isotropic sand returns a peak value of , its anisotropic counterpart yields (i.e., higher), in which and are the spectral and peak ground accelerations, respectively. Figure 1 illustrates two different types of structure, a one storey building and a four storey building, with approximate natural periods of and , respectively. Since most of the variation in the RSA is observed in the range , it is expected that structures residing in that period range will be affected most by the fabric anisotropy.

Fabric anisotropy effects on static liquefaction under constant shear drained loading

Several case history failures of slope systems have highlighted that the instability onset in loose materials can be triggered under prevailed drained conditions and stress paths that can be represented by constant shear drained (CSD) loading. This study uses the anisotropic critical state theory (ACST) to assess the effect of fabric anisotropy and loading characteristics (e.g., Lode angle and principal stress direction) on the instability onset under CSD stress paths, comparing our numerical-based observations with available experimental information. Towards this end, the ACST-based SANISAND-F model’s performance under CSD stress paths is also assessed. In addition, multiaxial conditions are incorporated through the estimation of instability surfaces. The numerical simulations are useful in explaining that the instability onset under CSD loading is dictated by a trade-off of volumetric strain components. Moreover, the results show an important effect of fabric anisotropy on the instability stress ratio (ηf). For conditions representative of common experimental setups, ηf decreases with the increase of the Lode angle and the major principal stress inclination, and ηf increases with the increase of initial fabric intensity, consistent with available experimental evidence. However, these trends can change based on the interaction between the Lode angle and loading/fabric directions; hence, departing from typical experimental observations. Finally, we discuss the potential of a simplified approach to estimate ηf analytically, including fabric effects.

A state-dependent multilaminate constitutive model for anisotropic sands

Experimental studies show that initial fabric and its evolution under different stress paths greatly influences soil behaviour. Even though different sample preparation methods create different inherent anisotropies and cause different material responses, the same initial fabric structure under different stress paths also results in different material behaviours. In this paper, a simple state-dependent, bounding surface-based elastoplastic constitutive model, which can simulate the anisotropic nature of sands including the effect of principal stress rotation, is described. The model is developed based on a semi-micromechanical concept within the multilaminate framework and, to include the inherent anisotropy of sand, a deviatoric fabric tensor describing the initial microstructure is introduced. In addition, a fabric evolution rule compatible with anisotropic critical state theory is employed to describe the evolving fabric structure and induced anisotropy towards the critical state. In contrast to the classical strain-driven formulation for fabric evolution, a micro-level evolution rule is proposed. This paper presents concise theoretical aspects of the multilaminate framework and the anisotropic elastoplastic constitutive formulation. The model's capability under drained and undrained monotonic loading conditions at different stress states, relative densities and principal stress orientations is demonstrated by simulating experimental data for Toyoura sand.

Constitutive modelling of fabric effect on sand liquefaction

Sand liquefaction under static and dynamic loading can cause failure of embankments, slopes, bridges and other important infrastructure. Sand liquefaction in the seabed can also cause submarine landslides and tsunamis. Fabric anisotropy related to the internal soil structure such as particle orientation, force network and void space is found to have profound influence on sand liquefaction. A constitutive model accounting for the effect of anisotropy on sand liquefaction is proposed. Evolution of fabric anisotropy during loading is considered according to the anisotropic critical state theory for sand. The model has been validated by extensive test results on Toyoura sand with different initial densities and stress states. The effect of sample preparation method on sand liquefaction is qualitatively analysed. The model has been used to investigate the response of a sand ground under earthquake loading. It is shown that sand with horizontal bedding plane has the highest resistance to liquefaction when the sand deposit is anisotropic, which is consistent with the centrifuge test results. The initial degree of fabric anisotropy has a more significant influence on the liquefaction resistance. Sand with more anisotropic fabric that can be caused by previous loading history or compaction methods has lower liquefaction resistance.

Modeling footing resting on anisotropic sand using material point method

Sand typically exhibits anisotropic internal structure which may significantly influence its mechanical behavior. The material point method (MPM) can eliminate mesh distortion and thus is suitable for investigating geotechnical problems with large deformation. In this study, an advanced anisotropic critical state theory (ACST)-based soil model is implemented in MPM to study the response of strip footing resting on anisotropic sand. The capability of the model is verified by simulating several element tests and strip footing tests with different soil densities and fabric bedding plane orientations. For the footing problem with a vertical load, as the fabric bedding plane orientation increases, the bearing capacity decreases and its corresponding settlement increases. The failure pattern becomes asymmetrical when the bedding plane orientation or the loading direction is inclined. A comparison between the simulation results predicted by the anisotropic and isotropic models is made, which demonstrates that neglecting the fabric anisotropy may lead to the overestimation of the bearing capacity.

Effect of fabric anisotropy on the cyclic liquefaction of sands: Insight from DEM simulations

To interpret the effects of fabric anisotropy on the cyclic liquefaction of sands, undrained cyclic triaxial tests on the vertical and horizontal specimens were simulated using the discrete element method with ellipsoidal clumps. The simulation captured the anisotropic cyclic responses of sands observed experimentally: (a) the vertical specimen is more prone to liquefaction than the horizontal one under symmetrical cyclic loading conditions; (b) the axial strain is accumulated at the extension side for the vertical specimen and at the compression side for the horizontal specimen. At the particle scale, the mechanical coordination number, redundancy index, contact-normal and particle-orientation based fabrics were investigated to provide insight into the macro response. It is found that at the onset of liquefaction, the redundancy index is equal to 1, the mechanical coordination number is equal to 3.3, and the contact-normal based fabric anisotropic variable is always negative. Furthermore, the anisotropic critical state theory was applied to investigate the relationship between the macro-mechanical behavior and the evolution of contact-normal based fabric, which facilitates the development of constitutive models for simulating cyclic liquefaction behavior of anisotropic sands.

Bounding surface rotational hardening model for overconsolidated clay accounting for evolving fabric anisotropy

AbstractSoil is inherently anisotropic owing to the sedimentation process under gravity and is generally under anisotropic stress state in the field. To reflect the influence of anisotropic consolidation, rotational hardening (RH) is favorably used in constitutive models, mimicking in‐situ stress conditions. However, most of the existing models do not account for the inherent anisotropy of clays under isotropic stress state as well as the interplay between the fabric and loading direction, and the corresponding initial yield surface is assumed to be oriented along the hydrostatic axis. Furthermore, as the rotational angle is usually deemed as the clayed soil's anisotropy measure, a critical yield surface without rotation appears to contradict the highly anisotropic fabric at critical state. In this study, an anisotropic bounding surface model that incorporates RH and fabric anisotropy is formulated to simulate the behavior of overconsolidated (OC) clay subject to different consolidation histories, while addressing the aforementioned inconsistency issue. During the loading process, both fabric anisotropy and rotational angle evolve continuously based on their respective evolution rules. Fabric anisotropy evolves to the critical state when the rotational angle returns to zero, satisfying the anisotropic critical state theory (ACST). In addition, the model employs an anisotropic elasticity to consider the influence of the initial fabric at a low‐stress level. The predictive capacity of the model can be demonstrated by simulating several compression and extension tests on clay over a wide range of overconsolidation ratios (OCRs) and anisotropic consolidation stress ratios under both drained and undrained conditions.

Assessing static liquefaction triggering considering fabric anisotropy effects under the ACST framework

Previous experimental studies have shown that the onset of instability in sands under undrained loading is affected by the initial state (i.e., void ratio and confinement), intermediate stresses, and fabric anisotropy. These experimental results have motivated numerical studies that investigate the conditions for instability triggering; however, most efforts have been focused on triaxial conditions, mainly addressing the role of state with few notable exceptions that extend to a multiaxial setting incorporating fabric anisotropy. In this study, we use the relatively new anisotropic critical state theory (ACST) to investigate the onset of instability in sands under undrained loading considering the role of state, multi-axial loading, and fabric anisotropy. We use the ACST-based SANISAND-F constitutive model to extend a previously established stability criterion and take into account the effect of fabric anisotropy in a multiaxial setting. The analytical instability criterion is derived from the fabric-dependent constitutive equations, and predicts the plastic modulus and the flow stress ratio at the instability point. The derived criterion highlights the benefits of the ACST framework in incorporating fabric and anisotropy effects. Lastly, we show that the stress ratio at the instability onset is not significantly affected by the extent of anisotropic consolidation.

Numerical simulation of cyclic shear tests considering the fabric change and principal stress rotation effects

AbstractCyclic loadings could induce complicated soil responses, including the fabric change and principal stress rotation, both of which would reduce the effective confining pressure and accelerate the build‐up of excess pore water pressures, thus leading to sand liquefaction in undrained conditions. The principal stress rotation, even without the change of principal stress magnitudes, would further generate the accumulative plastic shear strains. However, most of the existing studies focus on their individual effect and the combined effect was seldom considered. In this paper, a numerical approach including an elastoplastic sand constitutive model is proposed to consider the combined effect of the fabric change and principal stress rotation. In this model, the fabric change is considered with the anisotropic critical state theory and the principal stress rotation is considered by splitting the plastic strain increment into the monotonic part and rotational part. By simulating a series of cyclic shear tests, results from the numerical solutions are found showing good agreements with the existing experimental results and the importance of considering the combined effect is demonstrated. The approach can be used to predict the stress‐strain response of sand and guide the design of foundations especially in undrained cyclic loading conditions, for example, earthquake loadings conditions.

Non‐coaxial hypoplastic model for sand with evolving fabric anisotropy including non‐proportional loading

AbstractNon‐coaxial response refers to the deviation between the directions of the principal stress and plastic strain increment. In this paper, an extended hypoplastic model is proposed based on the anisotropic critical state theory, to describe the non‐coaxial and anisotropic response of sand subjected to both monotonic loading and rotation of principal stress axis. A fabric tensor that characterizes the internal structure of sand is introduced into a hypoplastic model, to reflect the effect of fabric anisotropy on the dilatancy and strength of sand. A Lode‐angle‐dependent hypoplastic potential surface is employed, rendering the flow direction no longer co‐directional with the stress tensor in the deviatoric plane. The fabric tensor is further incorporated into the flow direction, enabling the model to generate a non‐coaxial response. Upon shearing, the fabric evolves toward the loading direction following a properly defined evolution law, and the model response becomes purely coaxial at the critical state when the fabric becomes co‐directional with the loading direction. The tangential loading effect is further introduced to simulate the stiffness degradation of sand during undrained rotational shearing. The model is demonstrated to be capable of simulating the prismatic yet complex anisotropic behavior of sand under both proportional and non‐proportional loading conditions.

Particle shape effects on the shear behaviors of granular assemblies: irregularity and elongation

This study examines the influence of particle shape irregularity and elongation on the shear behaviors of granular materials using the 3D discrete element method (DEM). The particles are generated by combining a Fourier shape descriptor-based method with the random field theory and reconstructed in DEM using the overlapping sphere algorithm. A series of drained and undrained triaxial and true triaxial tests are performed to shear the samples with different particle irregularity and elongation to the critical state. The stress and the strain responses, the evolution of the fabric anisotropy and the coordination number are examined and analyzed within the framework of the anisotropic critical state theory. It is found that the consideration of complex shaped particles in DEM produces more realistic sand behaviors, and the shear strength and the dilation of the granular material increase with increasing irregularity and elongation. While elongation in the range of the study has insignificant influence on the location of the critical state line (CSL) on the void ratio-mean effective stress plane, irregularity is found to positively affect both the absolute slope and the intercept of the CSL. The unique fabric anisotropy norm at the critical state is also sensitive to the particle shape and is greater with larger irregularity and elongation.

Evolution of fabric anisotropy of granular soils: x-ray tomography measurements and theoretical modelling

Fabric anisotropy is a key component to understand the behaviour of granular soils. In general, experimental data on fabric anisotropy for real granular soils are very limited, especially in the critical state. In this paper, x-ray tomography measurements are used to provide experimental data on contact fabric anisotropy inside shear bands for two granular soils. The data are then used to assess the validity of Anisotropic Critical State Theory (ACST) and the accuracy of a fabric evolution law that was previously developed from the results of DEM simulations on idealised materials. Overall, the experimental results support ACST according to which unique (i.e., independent of initial conditions) values for fabric anisotropy and coordination number are observed at large strains. With increasing roundness of the material, the rate at which the critical state is approached increases. The evolution of fabric anisotropy measured from the experiments is fairly well reproduced by the proposed evolution law.

Hypoplastic modeling of anisotropic sand behavior accounting for fabric evolution under monotonic and cyclic loading

A unified hypoplastic model is formulated by incorporating the anisotropic critical state theory to describe the fabric effect in sand under both monotonic and cyclic loading conditions. An evolving deviatoric fabric tensor that characterizes the internal microstructure of sand is introduced into the hypoplastic model in conjunction with the intergranular strain concept. A scalar-valued fabric anisotropic variable indicating the interplay between the fabric and the loading direction is employed to account for the impact of fabric anisotropy on both the dilatancy and shear strength of sand. The model is demonstrated to be capable of simulating the anisotropic behavior of sand, using a single set of parameters under both monotonic and cyclic loading conditions, as evidenced by the satisfactory match with experimental results from various sources. In particular, by considering the influence of fabric evolution on the dilatancy of sand, the model adequately accounts for the fabric change effect and accurately captures the deviatoric strain accumulation, cyclic mobility, and the flow liquefaction phenomenon under cyclic loading condition.

Failure mechanisms of sand under asymmetrical cyclic loading conditions: experimental observation and constitutive modelling

In practical engineering, natural soil deposits often sustain an initial driving force prior to cyclic shear, owing to earthquakes, traffic and waves; such asymmetrical loading conditions may significantly affect the liquefaction susceptibility and failure mechanism of sand. To understand the typical cyclic liquefaction responses, comprehensive asymmetrical cyclic loading tests were conducted on sand samples subjected to either compressional or extensional static stress. The results indicated that different stress conditions can result in three distinct failure mechanisms: flow liquefaction, cyclic mobility and residual deformation accumulation. According to the experimental observations, an anisotropic sand model was developed within the framework of the anisotropic critical state theory. The model employed a fabric-dependent dilatancy, and accounted for the effects of the fabric evolution and accumulated loading index on the plastic hardening, in order to better reflect the cyclic degradation of the plastic modulus. The predictive capacity of the model was confirmed through undrained monotonic test results for samples with different densities. Comparisons between the model responses and experimental results indicated the excellent capabilities of the developed model in terms of capturing the typical deformation, strength and fabric characteristics of different cyclic failure mechanisms of sand under either symmetrical or asymmetrical loading conditions.

J2-deformation-type soil model coupled with state-dependent dilatancy and fabric evolution: multiaxial formulation and FEM implementation

In this paper, an original J2-deformation type model coupled with state-dependent dilatancy is further extended to a multiaxial stress space. Compared with most existing models, the proposed model only requires ten parameters, which can be easily calibrated, and does not involve any abstruse concepts in elastoplasticity theory. The model is then implemented into ABAQUS through a user-defined subroutine (UMAT), and the capability of the extended model in capturing the mechanical behavior of sand is demonstrated by element tests, including conventional triaxial tests and true triaxial tests. To demonstrate the robustness of the model, two typical boundary value problems are illustrated, i.e., the biaxial compression test and the strip footing test. The model is further incorporated with anisotropic critical state theory to simulate anisotropic soil behavior. The biaxial compression test with smooth and rough boundaries is simulated with the anisotropic model. The results confirm the impact of fabric anisotropy and boundary constraints on the shear strain distributions and patterns of shear bands. The simulation results show the capability of the proposed model, which can be applied in practical analyses and geotechnical engineering designs.

An Anisotropic Critical State Plasticity Model with Stress Ratio-Dependent Fabric Tensor

This paper proposes a new constitutive model within the recent anisotropic critical state theory (ACST) to take into account the influence of evolving fabric on the stress–strain–strength response of anisotropic sands. Analytical proofs in support of stress–force–fabric correlations corroborate a clear connection between mobilization of the continuum level stress ratio and particle level fabric in granular media. A normalized stress ratio-dependent second-order fabric tensor is suggested here which fulfills the ACST. Dilatancy and plastic hardening modulus of a stress ratio-driven elastoplastic model with zero pure elastic domain are explicitly interrelated to the proposed stress ratio-dependent normalized fabric tensor. It is shown that the fabric-dependent model simulations agree well with the monotonic triaxial and hollow cylinder torsion shear tests data.

A hypoplastic model for granular soils incorporating anisotropic critical state theory

SummaryIt is well known that soil is inherently anisotropic and its mechanical behavior is significantly influenced by its fabric anisotropy. Hypoplasticity is increasingly being accepted in the constitutive modeling for soils, in which many salient features, such as nonlinear stress‐strain relations, dilatancy, and critical state failure, can be described by a single tensorial equation. However, within the framework of hypoplasticity, modeling fabric anisotropy remains challenging, as the fabric and its evolution are often vaguely assumed without a sound basis. This paper presents a hypoplastic constitutive model for granular soils based on the newly developed anisotropic critical state theory, in which the conditions of fabric anisotropy are concurrently satisfied along with the traditional conditions at the critical state. A deviatoric fabric tensor is introduced into the Gudehus‐Bauer hypoplastic model, and a scalar‐valued anisotropic state variable signifying the interplay between the fabric and the stress state is used to characterize its impact on the dilatancy and strength of the soils. In addition, fabric evolution during shearing can explicitly be addressed. Modifications have also been undertaken to improve the performance of the undrained response of the model. The anisotropic hypoplastic model can simulate experimental tests for sand under various combinations of principle stress direction, intermediate principal stress (or mode of shearing), soil densities, and confining pressures, and the associated drastic effect of different principal stress orientations in reference to the material axes of anisotropy can be well captured.

Fabric evolution and dilatancy within anisotropic critical state theory guided and validated by DEM

Fabric, expressed by means of an evolving deviatoric fabric tensor F, plays a very important role in the anisotropic mechanical response of granular materials. The Anisotropic Critical State Theory (ACST) addresses fabric anisotropy by rendering dilatancy a function of F, in addition to other state variables. In this paper, 3D DEM is used to guide the specific grain-level definition of F, the formulation of its continuum evolution equation and its effect on anisotropic dilatancy within ACST. DEM provides stress-ratio and shear strain variations as input for ACST analytical calculations of evolving fabric tensor and dilatancy, which are then favourably compared with totally independent direct measurements of these quantities by DEM. Dilatancy is shown to be strongly affected by the contact normal-based fabric tensor Fc, whose evolution is best described by a continuum equation within ACST that includes dilatancy and a quantity related to particle orientation-based fabric tensor Fp. The aforementioned favourable comparison of the results for fabric tensor and dilatancy obtained independently by ACST and DEM, confirms the validity of the core framework of ACST irrespective of any constitutive model that addresses the deviatoric stress-strain relations.