A micro‐mechanical study of peak strength and critical state

We present a micro‐mechanical analysis of macroscopic peak strength, critical state, and residual strength in two‐dimensional non‐cohesive granular media. Typical continuum constitutive quantities such as frictional strength and dilation angle are explicitly related to their corresponding grain‐scale counterparts (e.g., inter‐particle contact forces, fabric, particle displacements, and velocities), providing an across‐the‐scale basis for a better understanding and modeling of granular materials. These multi‐scale relations are derived in three steps. First, explicit relations between macroscopic stress and strain rate with the corresponding grain‐scale mechanics are established. Second, these relations are used in conjunction with the non‐associative Mohr–Coulomb criterion to explicitly connect internal friction and dilation angles to the micro‐mechanics. Third, the mentioned explicit connections are applied to investigate, understand, and derive micro‐mechanical conditions for peak strength, critical state, and residual strength. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of continuous and discontinuous constitutive models to simulate concrete behaviour under mixed‐mode failure conditions

The paper presents detailed FE simulation results of concrete elements under mixed‐mode failure conditions according to the so‐called shear‐tension test by Nooru‐Mohamed, characterized by curved cracks. A continuous and discontinuous numerical two‐dimensional approach was used. In order to describe the concrete's behaviour within continuum mechanics, two different constitutive models were used. First, an elasto‐plastic model with isotropic hardening and softening was assumed. In a compression regime, a Drucker–Prager criterion with a non‐associated flow rule was used. In turn, in a tensile regime, a Rankine criterion with an associated flow rule was adopted. Second, an isotropic damage constitutive model was applied with a single scalar damage parameter and different definitions of the equivalent strain. Both constitutive laws were enriched by a characteristic length of micro‐structure to capture properly strain localization. As an alternative approach, the extended finite element method was used. Our results were compared with the experimental ones and with results of other FE simulations reported in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Kinematic models for non‐coaxial granular materials. Part I: theory

The purpose of this paper is to present a physically based plasticity model for non‐coaxial granular materials. The model, which we shall call the double slip and rotation rate model (DSR² model), is a pair of kinematic equations governing the velocity field. The model is based on a discrete micro‐analysis of the kinematics of particles in contact, and is formulated by introducing a quantity called the averaged micro‐pure rotation rate (APR) into the unified plasticity model which was proposed by one of the authors. Our macro–micro mechanical analysis shows that the APR is a non‐linear function of, among other quantities, the macro‐rotation rate of the major principal axis of stress taken in the opposite sense. The requirement of energy dissipation used in the double‐sliding free‐rotating model appears to be unduly restrictive as a constitutive assumption in continuum models. In the DSR² model the APR tensor and the spin tensor are directly linked with non‐coaxiality of the stress and deformation rate tensors. We also propose a simplified plasticity model based on the DSR² model for a class of dilatant materials, and analyse its material stability. Copyright © 2005 John Wiley & Sons, Ltd.     

ACMEG‐TS: A constitutive model for unsaturated soils under non‐isothermal conditions

This paper introduces an unconventional constitutive model for soils, which deals with a unified thermo‐mechanical modelling for unsaturated soils. The relevant temperature and suction effects are studied in light of elasto‐plasticity. A generalized effective stress framework is adopted, which includes a number of intrinsic thermo‐hydro‐mechanical connections, to represent the stress state in the soil. Two coupled constitutive aspects are used to fully describe the non‐isothermal behaviour. The mechanical constitutive part is built on the concepts of bounding surface theory and multi‐mechanism plasticity, whereas water retention characteristics are described using elasto‐plasticity to reproduce the hysteretic response and the effect of temperature and dry density on retention properties. The theoretical formulation is supported by comparisons with experimental results on two compacted clays. Copyright © 2008 John Wiley & Sons, Ltd.     

Integration and calibration of a plasticity model for granular materials

A new macroscopic constitutive model for non‐cohesive granular materials, with the focus on coarse‐sized materials (railway ballast), is presented. The model is based on the concepts of rate‐independent isotropic plasticity. The Backward Euler rule is used for integrating the pertinent evolution equations. The resulting incremental relations are solved in the strain space that is extended with the internal (hardening) variables. The model is calibrated using data from Conventional Triaxial Compression (CTC) tests, carried out at the University of Colorado at Boulder. A function evaluation method is used for the optimization, whereby a ‘multi‐vector’ strategy for choosing the appropriate start vector is proposed. Copyright © 2002 John Wiley & Sons, Ltd.     

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

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

Deterministic and statistical size effect during shearing of granular layer within a micro‐polar hypoplasticity

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during quasi‐static shearing of an infinite layer under plane strain conditions, free dilatancy and constant pressure. For a simulation of the mechanical behaviour of a cohesionless granular material during a monotonous deformation path, a micro‐polar hypoplastic constitutive relation was used which takes into account particle rotations, curvatures, non‐symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. In the paper, a deterministic and statistical size effect is analysed. The deterministic calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2000 mm. To investigate the statistical size effect, the Monte Carlo method was applied. The random distribution of the initial void ratio was assumed to be spatially correlated. Truncated Gaussian random fields were generated in a granular layer using an original conditional rejection method. The sufficient number of samples was determined by analysing the convergence of the outcomes. In order to reduce the number of realizations without losing the accuracy of the calculations, stratified and Latin hypercube methods were applied. A parametric analysis of these methods was also presented. Some general conclusions were formulated. Copyright © 2007 John Wiley & Sons, Ltd.     

A visco‐elasto‐plastic model for granular materials under simple shear conditions

The numerical simulation of rapid landslides is quite complex mainly because constitutive models capable of simulating the mechanical behaviour of granular materials in the pre‐collapse and post‐collapse regimes are still missing. The goal of this paper is to introduce a constitutive model capable of capturing the response of dry granular flows from quasi‐static to dynamic conditions, in particular when the material experiences a sort of solid‐to‐fluid phase transition. An ideal assembly of identical spheres under simple shear conditions is considered. In the constitutive model, void ratio and granular temperature have been chosen as state variables, and both shear and normal stresses are computed as the sum of two contributions: the quasi‐static one and the collisional one. The former is determined by using a perfect elasto‐plastic model including the critical state concept, while the latter is derived from the kinetic theory of granular gases. The evolution of the granular temperature, fundamentally governing the material phase transition, is obtained by imposing the kinetic fluctuating energy balance. The constitutive relationship has been integrated, under both constant pressure and constant volume conditions, and the influence of shear strain rate, initial void ratio and normal pressure on the mechanical response has been investigated. Copyright © 2015 John Wiley & Sons, Ltd.     

Coupled modelling of hydro‐mechanical behaviour of unsaturated compacted expansive soils

The Barcelona basic model cannot predict the mechanical behaviour of unsaturated expansive soils, whereas the Barcelona expansive model (BExM) can only predict the stress–strain behaviour of unsaturated expansive soils without the water‐retention behaviour being incorporated. Moreover, the micro‐parameters and the coupling function between micro‐structural and macro‐structural strains in the BExM are difficult to determine. Experimental data show that the compression curves for non‐expansive soils under constant suctions are shifted towards higher void ratios with increasing suction, whereas the opposite is true for expansive soils. According to the observed water‐retention behaviour of unsaturated expansive soils, the air‐entry value increases with density, and the relationship between the degree of saturation and void ratio is linear at constant suction. According to the above observation, an elastoplastic constitutive model is developed for predicting the hydraulic and mechanical behaviour of unsaturated expansive soils, based on the existing hydro‐mechanical model for non‐expansive unsaturated soil. The model takes into consideration the effect of degree of saturation on the mechanical behaviour and that of void ratio on the water‐retention behaviour. The concept of equivalent void ratio curve is introduced to distinguish the plastic potential curve from the yield curve. The model predictions are compared with the test results of an unsaturated expansive soil, including swelling tests under constant net stress, isotropic compression tests and triaxial shear tests under constant suction. The comparison indicates that the model offers great potential for quantitatively predicting the hydraulic and mechanical behaviour of unsaturated expansive soils. Copyright © 2011 John Wiley & Sons, Ltd.     

Load‐displacement and bearing capacity of foundations on granular soils using a multi‐surface kinematic constitutive soil model

A finite element approach based on an advanced multi‐surface kinematic constitutive model is used to evaluate the bearing capacity of footings resting on granular soils. Unlike simple elastic‐perfectly plastic models, often applied to granular foundation problems, the present model realistically accounts for stress dependency of the friction angle, strain softening–hardening and non‐associativity. After the model and its implementation into a finite element code are briefly discussed, the numerical difficulty due to the singularity at the footing edge is addressed. The bearing capacity factor N_γ is then calculated for different granular materials. The effect of footing size, shape, relative density and roughness on the ultimate bearing capacity are studied and the computed results compare very favourably with the general experimental trends. In addition, it is shown that the finite element solution can clearly represent counteracting mechanisms of progressive failure which have an important effect on the bearing capacity of granular foundations. Copyright © 2006 John Wiley & Sons, Ltd.     

A three‐dimensional micromechanically based model

Granular materials react with complicated mechanical responses when subjected to external loading paths. This leads to sophisticated constitutive formulations requiring large numbers of parameters. A powerful and straightforward way consists in developing micro‐mechanical models embedding both micro‐scale and meso‐scale. This paper proposes a 3D micro‐mechanical model taking into account an intermediate scale (meso‐scale) that makes it possible to describe a variety of constitutive features in a natural way. The comparison between experimental tests and numerical simulations reveals the predictive capability of this model. Particularly, several simulations are carried out with different confining pressures and initial void ratios, based on the fact that the critical state is quantitatively described without requiring any critical state formulations and parameter. The model mechanism is also analyzed from a microscopic view, wherein the evolution of some key microscopic parameters is investigated. Copyright © 2017 John Wiley & Sons, Ltd.     

Modelling the time‐dependent behaviour of granular material with hypoplasticity

This paper presents a constitutive model for time‐dependent behaviour of granular material. The model consists of 2 parts representing the inviscid and viscous behaviour of granular materials. The inviscid part is a rate‐independent hypoplastic constitutive model. The viscous part is represented by a rheological model, which contains a high‐order term denoting the strain acceleration. The proposed model is validated by simulating some element tests on granular soils. Our model is able to model not only the non‐isotach behaviour but also the 3 creep stages, namely, primary, secondary, and tertiary creep, in a unified way.     

Discrete element modelling of material non‐coaxiality in simple shear flows

We investigate the quasi‐static simple shear flow of a two‐dimensional assembly of cohesionless particles using discrete element method (DEM) simulations. We focus on the unsteady flow regime where the solid would experience significant evolution of stresses, mobilised shear strength and dilation. We construct the DEM model using a discretised‐wall confined granular cell where the apparent boundary is allowed to dilate or contract synchronously with the confined solid. A rather uniform simple shear field is achieved across the whole assembly, which benefits rheological studies in generalising constitutive laws for continuum methods. We examine two aspects of the simple shear behaviour: macroscopic stress and strain rate evolution, particularly the non‐coaxiality between the principal directions of the two; and micromechanics such as evolution of fabric. For an initially anisotropic specimen sheared under constant normal pressure, the direction of principal stress rotates towards that of the principal strain rate, gradually reducing the degree of non‐coaxiality from about 45° to fluctuating around 0°. The rate in approaching coaxiality is slower in samples with larger initial porosity, stress ratio and mean stress. Generally, a faster rate in approaching coaxiality in simple shear is observed in a more dilatant sample, which often shows a larger degree of mobilised fabric anisotropy, suggesting the possible important role of instantaneous internal friction angle. The evolution of principal fabric direction resembles that of the principal stress direction. © 2013 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons, Ltd.     

Effective properties of a cemented or an injected granular material

In this paper the macroscopic elastic properties of injected or cemented sands are derived from the characteristics of the constituents and the analysis of the microstructure using a multi‐scale modelling approach. Particular interest is given to the choice of the representative elementary volume, by relying on existing microstructural data. The periodic homogenization is adopted and required numerical solutions are performed by the finite element method. An assessment of the validity of the multi‐scale approach is achieved through comparison with theoretical and experimental results on cemented and injected granular media reported in the literature. The capabilities of the model are also used to investigate the influence of geometrical and mechanical microscale parameters on the macroscopic behaviour of the treated materials. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical investigations of shear localization in a micro‐polar hypoplastic material

In this paper a micro‐polar continuum approach is proposed to model the essential properties of cohesionless granular materials like sand. The model takes into account the influence of particle rotations, the mean grain size, the void ratio, the stresses and couple stresses. The constitutive equations for the stresses and couple stresses are incrementally non‐linear and based on the concept of hypoplasticity. For plane strain problems the implementation of the model in a finite element program is described. Numerical studies of the evolution of micro‐polar effects within a granular strip under plane shearing are presented. It is shown that the location and evolution of shear localization is strongly influenced by the initial state and the micro‐polar boundary conditions. For large shearing the state quantities tend towards a stationary state for which a certain coupling between the norm of the stress deviator and the norm of the couple stress tensor can be derived. Copyright © 2003 John Wiley & Sons, Ltd.     

Meso‐scale particle modeling of concrete deterioration caused by alkali‐aggregate reaction

A meso‐scale particle model is presented to simulate the expansion of concrete subjected to alkali‐aggregate reaction (AAR) and to analyze the AAR‐induced degradation of the mechanical properties. It is the first attempt to evaluate the deterioration mechanism due to AAR using the discrete‐element method. A three‐phase meso‐scale model for concrete composed of aggregates, mortar and the interface is established with the combination of a pre‐processing approach and the particle flow code, PFC2D. A homogeneous aggregate expansion approach is applied to model the AAR expansion. Uniaxial compression tests are conducted for the AAR‐affected concrete to examine the effects on the mechanical properties. Two specimens with different aggregate sizes are analyzed to consider the effects of aggregate size on AAR. The results show that the meso‐scale particle model is valid to predict the expansion and the internal micro‐cracking patterns caused by AAR. The two different specimens exhibit similar behavior. The Young's modulus and compressive strength are significantly reduced with the increase of AAR expansion. The shape of the stress–strain curves obtained from the compression tests clearly reflects the influence of internal micro‐cracks: an increased nonlinearity before the peak loading and a more gradual softening for more severely affected specimens. Similar macroscopic failure patterns of the specimens under compression are observed in terms of diagonal macroscopic cracks splitting the specimen into several triangular pieces, whereas localized micro‐cracks forming in slightly affected specimens are different from branching and diffusing cracks in severely affected ones, demonstrating different failure mechanisms. Copyright © 2012 John Wiley & Sons, Ltd.     

Effects of periodic fluctuations of micro‐polar boundary conditions on shear localizations in granular soil–structure interaction

In this work, the interface behavior between an infinite extended narrow granular layer and a rough surface of rigid body is investigated numerically, using finite element method in the updated Lagrangian (UL) frame. In this regard, the elasto‐plastic micro‐polar (Cosserat) continuum approach is employed to remove the limitations caused by strain‐softening of materials in the classical continuum. The mechanical properties of cohesionless granular soil are described with Lade's model enhanced by polar terms, including Cosserat rotations, curvatures, and couple stresses. Furthermore, the mean grain diameter as the internal length is incorporated into the constitutive relations accordingly. Here, the evolution and location of shear band, within the granular layer in contact with the rigid body, are mainly focused. In this regard, particular attention is paid to the effects of homogeneous distribution and periodic fluctuation of micro‐polar boundary conditions, prescribed along the interface. Correspondingly, the effects of pressure level, mean grain diameter, and stratified soil are also considered. The finite element results demonstrate that the location and evolution of shear band in the granular soil layer are strongly affected by the non‐uniform micro‐polar boundary conditions, prescribed along the interface. It is found that the shear band is located closer to the boundary with less restriction of grain rotations. Furthermore, the predicted thickness of shear band is larger for higher rotation resistance of soil grains along the interface, larger mean grain diameter, and higher vertical pressure. Regarding the stratified soil, comprising a thin layer with slightly different initial void ratio, the shear band moves towards the layer with initially higher void ratio. Copyright © 2011 John Wiley & Sons, Ltd.     

On some advanced thermo‐mechanical models for saturated clays

This paper aims to evaluate the performance of several recent constitutive models in simulating the thermo‐mechanical behaviour of saturated clays. A classic thermo‐mechanical test on natural Boom Clay, commonly used in constitutive modelling, was first clarified. Different methods commonly used to measure volumetric strain in drained heating tests were then discussed. Model evaluation was performed in terms of thermodynamic and elasto‐plastic requirements. The capability of the models to capture the observed behaviour was assessed on the basis of the experimental evidence. It was shown that all the models provide reasonable predictions of the thermo‐mechanical behaviour of saturated clays. However, each constitutive model has its own limitations or unclear points from the theoretical point of view. Copyright © 2013 John Wiley & Sons, Ltd.     

Physical and numerical investigation of a cemented granular assembly of steel spheres

In this paper, steel spheres embedded in a cement matrix were studied using numerical and physical ISRM testing procedures. A challenge in discrete element simulations is to select appropriate micro‐mechanical models and parameters, to recover the observed macro‐mechanical behavior. An ideal experiment on cohesive granular assemblies constructed identical to numerical ones would validate these micro models for a set of measured micro‐parameters. The first part of the paper summarizes the previous studies in this area, outlines such experimental methodology and depicts the steps followed for the preparation and the testing of cemented granular assemblies together with the derivation of micro‐parameters. The second part discusses the results of numerical and physical ISRM standard tests including uniaxial and triaxial compression, Brazilian tensile and shear box tests. Physical samples were prepared using steel balls bonded with Portland cement, cured under controlled laboratory conditions and tested in compression, tension and shearing. Acoustic emissions were monitored in uniaxial tests to characterize the damage thresholds relative to volumetric strains. Numerical simulations were conducted with PFC 3D using micro‐mechanical parameters derived from physical testing. Parametric sensitivity studies were carried out to look into the dependency of macroscopic responses on the parameters. The results from both numerical and physical tests showed good correspondence in macroscopic behavior i.e. peak strength, stages of damage, mode of failures. However, the numerical simulations reflected a stiffer mechanical response than physical assemblies. Copyright © 2010 John Wiley & Sons, Ltd.