Statics of loose triangular embankment under Nadai’s sand hill analogy

In structural mechanics, Nadai’s sand hill analogy is the interpretation of an ultimate torque applied to a given structural member with a magnitude that is analogously twice the volume of stable sand heap which can be accommodated on a transverse cross-section basis. Nadai’s analogy is accompanied by his observation of a loose triangular embankment, based on the fact that gravitating loose earth is stable if inclined just under the angle of repose. However, Nadai’s analysis of stress distribution in a planar sand heap was found to be inaccurate because the total pressure obtained from Nadai’s solution is greater than the self-weight calculated from the heap geometry. This raises a question about the validity of his observation in relation to the analogy. To confirm his criterion, this article presents and corrects the error found in Nadai’s solution by analyzing a radially symmetric stress field for a wedge-shaped sand heap with the purpose of satisfying both force balance and Nadai's closure. The fundamental equation was obtained by letting the friction state vary as a function of angular position and deduce it under the constraint that the principal stress orientation obeys Nadai's closure. The theoretical solution sufficiently agreed with the past experimental measurements.     

On comminution and yield in brittle granular mixtures

The mechanics of granular mixtures are pivotal in many industrial applications. Unravelling the relation between yielding and comminution, the action of mechanically induced grain size reduction, in confined mixture systems is a common and open challenge. This paper attacks this problem by adopting the breakage mechanics theory, which was originally proposed for single mineral materials. We present an extension to the theory that allows predicting: (1) the yielding pressure in granular mixtures, (2) the yield pressure increase/hardening with increasing breakage, and (3) the evolution of the grain size distributions of the separate species—all of these novel capabilities are tested and validated with experiments. Of particular appeal is the finding that the average yielding pressure is a simple generalized mean with an exponent −3/2 of the yielding pressures of the homogeneous components.     

Assessing continuum postulates in simulations of granular flow

Continuum mechanics relies on the fundamental notion of a mesoscopic volume “element” in which properties averaged over discrete particles obey deterministic relationships. Recent work on granular materials suggests that a continuum law may be inapplicable, revealing inhomogeneities at the particle level, such as force chains and slow cage breaking. Here, we analyze large-scale three-dimensional discrete-element method (DEM) simulations of different granular flows and show that an approximate “granular element” defined at the scale of observed dynamical correlations (roughly three to five particle diameters) has a reasonable continuum interpretation. By viewing all the simulations as an ensemble of granular elements which deform and move with the flow, we can track material evolution at a local level. Our results confirm some of the hypotheses of classical plasticity theory while contradicting others and suggest a subtle physical picture of granular failure, combining liquid-like dependence on deformation rate and solid-like dependence on strain. Our computational methods and results can be used to guide the development of more realistic continuum models, based on observed local relationships between average variables.     

An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material

The article concerns the complex determination process of the material parameters governing micropolar granular material with elasto-plastic material properties. Proceeding from a gradient-based method, we split the total set of parameters and the overall identification procedure into two major categories. These are, firstly, the identification of the parameters of a standard non-polar elasto-plastic continuum, and, secondly, the determination of the remaining parameters governing the micropolar part of the constitutive model. While the first set of parameters can be obtained from homogeneous triaxial tests on, e. g., granular, cohesive-frictional materials like sand, the second set can only be determined from inhomogeneous tests, such as biaxial tests including the onset and the development of shear bands. Following this, one can obtain the first part of the identification process from a simple inverse algorithm applied to the elasto-plastic material model of non-polar solids, while the second part requires a fully inverse computation in the sense of a back analysis of the underlying boundary-value problem. In the present article, this procedure is carried out by use of the semi-discrete sensitivity analysis. Finally, the whole model is applied to the data of Hostun sand taken at the universities of Grenoble and Stuttgart. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday.     

The elastic response of a cohesive aggregate—a discrete element model with coupled particle interaction

A model is presented for the deformation of a cohesive aggregate of elastic particles that incorporates two important effects of large-sized inter-particle junctions. A finite element model is used to derive a particle response rule, for both normal and tangential relative deformations between pairs of particles. This model agrees with the Hertzian contact theory for small junctions, and is valid for junctions as large as half the nominal particle size. Further, the aggregate model uses elastic superposition to account for the coupled force–displacement response due to the simultaneous displacement of all of the neighbors of each particle in the aggregate. A particle stiffness matrix is developed, relating the forces at each junction to the three displacement degrees of freedom at all of the neighboring-particle junctions. The particle response satisfies force and moment equilibrium, so that the model is properly posed to allow for rigid rotation of the particle without introducing rotational degrees of freedom. A computer-simulated sintering algorithm is used to generate a random particle packing, and the stiffness matrix is derived for each particle. The effective elastic response is then estimated using a mean field or affine displacement calculation, and is also found exactly by a discrete element model, solving for the equilibrium response of the aggregate to uniform-strain boundary conditions. Both the estimate and the exact solution compare favorably with experimental data for the bulk modulus of sintered alumina, whereas Hertzian contact-based models underestimate the modulus significantly. Poisson's ratio is, however, accurately determined only by the full equilibrium discrete element solution, and shown to depend significantly on whether or not rigid particle rotation is permitted in the model. Moreover, this discrete element model is sufficiently robust, so it can be applied to problems involving non-homogeneous deformations in such cohesive aggregates.     

Elasticity and strength of partially sintered ceramics

A discrete element model for the elastic and fracture behavior of partially sintered ceramics is presented. It accounts for the granular character of the material when a large amount of porosity (typically >0.2-0.4) is left after sintering. The model uses elastic force-displacement laws to represent the bond formed between particles during sintering. Bond fracture in tension and shearing is accounted for in the model. Realistic numerical microstructures are generated using a sintering model on random particle packings. In particular, packings with fugitive pore formers are used to create partially sintered microstructures with large pores. The effective elastic response and the strength of these microstructures are calculated in tension and compression. The link between important microstructural features such as bond size or coordination number and macroscopic behavior is investigated. In particular, it is shown that porosity alone is not sufficient to account for the mechanical properties of a partially sintered material.     

The formation of tabular compaction-band arrays: Theoretical and numerical analysis

The bifurcation analysis of compaction banding is extended to the formation of a tabular discrete compaction-band array. This analysis, taken together with the results of finite-difference simulations, shows that the bifurcation results in the formation of intermittent loading (elastic-plastic) and unloading (elastic) bands. The obtained analytical solution relates the spacing parameter χ (the ratio between the band thickness to the band-to-band distance) to all constitutive and stress-state parameters. Both this solution and numerical models reveal strong dependence of χ on the hardening modulus h: χ increases with h reduction. The band thickness in the numerical models is mesh dependent, but in terms of mesh-zone-size varies only from ∼2 to 4 depending on the constitutive parameters and independently on the mesh resolution. The thickness of the “elementary” compaction bands in real granular materials is equal to a few grain sizes. It follows that one grid zone in the numerical models corresponds approximately to one grain in the real material. The numerical models reproduce both discrete and continuous propagating compaction banding observed in the rock samples. These phenomena were shown to be dependent on the evolution of h and the dilatancy factor with deformation.     

On the nonlocal nature of dislocation nucleation during nanoindentation

Using static atomistic simulations, we study the full details of the mechanism by which dislocations homogeneously nucleate beneath the surface of a initially defect-free crystal during indentation. The mechanism involves the collective motion of a finite disk of atoms over two adjacent slip planes, the diameter of which depends on the indenter size. The nucleation mechanism highlights the need for nonlocal considerations in the development of a nucleation criterion. We review three nucleation criteria from the literature, each of which is based on purely local measures of the state of stress, and show that none are sufficiently general to predict nucleation in realistic atomic systems. We then propose a criterion based on an eigenmode analysis of the atomic-scale acoustic tensor. We demonstrate the accuracy of the criterion, which also works in the presence of existing topological defects like free surfaces or dislocation cores. The dependence of the size of the nucleated disk on the indenter radius leads to a self-similar nucleation process and virtually no indentation size effect (ISE), suggesting that homogeneous nucleation is only possible for very small indenters.     

Influence of grading on the undrained behavior of granular materials

This paper aims at investigating the influence of grading on the undrained behavior of granular materials. Series of undrained triaxial tests were carried out with two different materials, glass balls and Hostun sand. For each material, samples with different gradings and similar relative densities were prepared. Experimental results show that the undrained shear strength decreases when the coefficient of uniformity C_u=d₆₀/d₁₀$C_{\mathrm{u}}=d_{60}/d_{10}$ increases from 1.1 to 20. The conditions of instability for the selected granular materials were also analyzed, based on the sign of the second-order work during undrained triaxial loading. The results demonstrate a significant influence of the grading: increasing the coefficient of uniformity heightens the potential of static liquefaction and the materials become more unstable. Furthermore, numerical tests using the three-dimensional discrete element method (DEM) were conducted on assemblies of spheres. The DEM inter-particle parameters were derived from the experimental test results on glass balls. The DEM simulations showed similar behaviors compared to experimental results and confirmed the influence of the grain size distribution on the stress–strain relationship and instability phenomena.     

Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

Crushing of particles in idealised granular assemblies

Four idealised assemblies of equally sized spherical particles are subjected to a range of macroscopic compressive principal stresses and the contact forces on individual particles are determined. For each set of contact forces the stress fields within individual particles are studied. A failure criterion for brittle materials is imposed and indicates that crushing (or rupture) occurs when the maximum contact force reaches a threshold particle strength value, irrespective of the presence and magnitude of other lesser contact forces acting on the particle and the material properties of the particle. Combining the crushing mechanism with an assembly instability mechanism enables failure surfaces to be drawn in the three-dimensional stress space. A simple spatial averaging technique has been applied to the failure surfaces to remove the effects of assembly anisotropies. Sections of the failure surfaces on π planes have similarities to those commonly used in sand modelling.     

On the definitions of failure and critical states in hypoplasticity

A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.      

The influence of particle fluctuations on the average rotation in an idealized granular material

Granular materials are a simple example of a Cosserat continuum in that the average particle rotations may differ from the rotation of the average deformation. In the absence of couple stress, this difference insures that the stress is symmetric. This has been shown in theories that assume that the displacement at particle contacts is given by the average deformation and spin. Here, we indicate how the difference between the average rotation of the particles and the average rotation of the deformation can be determined when fluctuations in particle displacements and rotations satisfy local force and moment equilibria in a random aggregate of identical spheres. The predictions based on this model are in better agreement with numerical simulation than that given by the simple average strain theory.     

On the modeling of confined buckling of force chains

Buckling of force chains, laterally confined by weak network particles, has long been held as the underpinning mechanism for key instabilities arising in dense, cohesionless granular assemblies, e.g. shear banding and slip-stick phenomena. Despite the demonstrated significance of this mechanism from numerous experimental and discrete element studies, there is as yet no model for the confined buckling of force chains. We present herein the first structural mechanical model of this mechanism. Good agreement is found between model predictions and confined force chain buckling events in discrete element simulations. A complete parametric analysis is undertaken to determine the effect of various particle-scale properties on the stability and failure of force chains. Transparency across scales is achieved, as the mechanisms on the microscopic and mesoscopic domains, which drive well-known macroscopic trends in biaxial compression tests, are elucidated.     

On criticism of the nature of objectivity in classical continuum physics

Murdoch (J. Elasticity 60, 233-242, 2000) showed that restrictions imposed upon response functions by material frame-indifference are the consequences of five distinct aspects of observer agreement (that is, of objectivity) and involve only proper orthogonal tensors. Accordingly it is unnecessary to invoke the principle of invariance under superposed rigid motions (in the sense of one observer, two motions), which imposes a restriction upon nature. Liu (Continuum Mech. Thermodyn. 16, 177-183, 2003, and Continuum Mech. Thermodyn. 17, 125-133, 2005) has challenged, misinterpreted and misrepresented the content of both Murdochs work and this work. Here all criticisms of Liu are answered, his counter-examples are used to amplify the tenets of Murdochs work, and a key modelling issue in the controversy is indicated. Further, the response function restrictions for a given observer, derived on the basis of considering other observers, are shown to be independent of possible differences in the scales of mass, length, and time employed by other observers.Received: 2 March 2004, Accepted: 20 August 2004, Published online: 24 March 2005PACS: 83.10Ff     

Micromechanical investigation of granular ratcheting using a discrete model of polygonal particles

We use a two-dimensional model of polygonal particles to investigate granular ratcheting. Ratcheting is a long-term response of granular materials under cyclic loading, where the same amount of permanent deformation is accumulated after each cycle. We report on ratcheting for low frequencies and extremely small loading amplitudes. The evolution of the sub-network of sliding contacts allows us to understand the micromechanics of ratcheting. We show that the contact network evolves almost periodically under cyclic loading as the sub-network of the sliding contacts reaches different stages of anisotropy in each cycle. Sliding contacts lead to a monotonic accumulation of permanent deformation per cycle in each particle. The distribution of these deformations appears to be correlated in form of vortices inside the granular assembly.     

Micromechanical investigation of granular ratcheting using a discrete model of polygonal particles

We use a two-dimensional model of polygonal particles to investigate granular ratcheting. Ratcheting is a long-term response of granular materials under cyclic loading, where the same amount of permanent deformation is accumulated after each cycle. We report on ratcheting for low frequencies and extremely small loading amplitudes. The evolution of the sub-network of sliding contacts allows us to understand the micromechanics of ratcheting. We show that the contact network evolves almost periodically under cyclic loading as the sub-network of the sliding contacts reaches different stages of anisotropy in each cycle. Sliding contacts lead to a monotonic accumulation of permanent deformation per cycle in each particle. The distribution of these deformations appears to be correlated in form of vortices inside the granular assembly.     

Constitutive response of idealized granular media under the principal stress axes rotation

This paper is dedicated to the understanding of the phenomena, which give rise to anisotropy and non-coaxiality in granular materials. In achieving three-dimensional numerical simulation under static condition of granular media, granular element method (GEM) is adopted in this study. The method has been incorporated into the so-called mathematical homogenization theory for quasi-static equilibrium problems, which enables us to obtain the macroscopic/phenomenological inelastic deformation response of a representative volume element (RVE). To examine the anisotropic macroscopic deformation properties of the assumed RVE, which is solved by granular element method (GEM), a series of numerical experiments involving the pure rotation of the principal stress axes are carried out, and its results are discussed in relation to induced anisotropy and non-coaxiality.     

Analysis of local behaviour in granular materials

A local scale, called the meso-scale, has recently been introduced to the multi-scale approach for 2D granular materials. This local scale is defined at the level of meso-domains enclosed by particles in contact. Stress and strain have been defined at this local scale, and their relation with the local structure has been studied. The purpose of this paper is to analyse the behaviour of granular materials at the meso-scale, i.e. the stress–strain–structure relationship at this scale. Analyses are performed on a 2D numerical granular sample subjected to a biaxial compression test and simulated with the Discrete Element Method (DEM). The sample is quite dense and it is loaded at a relatively low strain rate so that the state of the sample can be considered as being quasi-static. The size of sub-domains in the sample varies largely from 3 to 12 particles. It is shown that the evolution of the internal state of the sample corresponds, at the meso-scale, to a clear evolution of the quantity of meso-domains oriented in different directions. In addition, the behaviour of meso-domains is highly governed by their orientation rather than their density, especially for the strongly elongated meso-domains: the meso-domains oriented in the compression (resp. extension) direction behave like a dense (resp. loose) granular material.