On the modeling of confined buckling of force chains

This paper is dedicated to Professor Robert P. Behringer on the occasion of his 60th birfthday.
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Abstract

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.

Introduction

Today, 300 years on after the birth of Euler, the mechanism of buckling not only remains at the forefront of mechanics, but its relevance has spread beyond the realm of traditional structural mechanics—perhaps none more notable than to the burgeoning areas of micromechanics and nanomechanics (e.g. Chaudhuri et al., 2007, Majmudar and Behringer, 2005). As technological advances provide new and more powerful tools to probe materials at ever-diminishing length scales, whole new worlds of structures are continuing to be discovered. Structures in naturally occurring and man-made materials abound from the mesoscale to the nanoscale, and a common mode of failure of many of these structures is that of buckling; examples can be found in biological materials, foams, polymers, micellar systems, DNA and carbon nanotubes (e.g. Ji et al., 2004, Gioia et al., 2001, Gibson and Ashby, 1988, Goodman et al., 2005, Falvo et al., 1997).

The focus of this paper is on buckling mechanisms occurring in deforming dense granular materials. Considered as the ultimate paradigm of a complex system, granular materials exhibit behavior that has eluded scientists for centuries. To date, this class of materials, despite its ubiquity in everyday life, possesses no constitutive model of the same level of reliability as the Navier–Stokes equation for fluids (Duran, 2000). Consequently, systems and processes involving granular materials rarely reach 60% of their design performance—a far cry from fluid processing which operates on average at 96% (Knowlton et al., 1994).

Numerous experimental studies, from those by Oda et al. (2004) and references cited therein, to the more recent developments in soil mechanics and physics (Rechenmacher, 2006, Corwin et al., 2005, Majmudar and Behringer, 2005), suggest that a missing piece of the puzzle of constitutive theory for granular materials lies in the manner of force transmission and associated kinematics. Specifically, in a deforming granular material, instabilities emerge at multiple length scales, and a prevalent source of instability is known to be the buckling of so-called force chains (e.g. Tordesillas et al., 2009; Rechenmacher, 2006, Oda and Kazama, 1998). Physical experiments have shown that force chains—quasi-linear, chain-like particle groups through which above average contact forces are transmitted—form to resist deformation, and that these “columnar structures” of particles preferentially align in the direction of maximum compressive stress in the system. Under continued axial compression, force chain columns ultimately buckle. Mounting evidence suggests that this event is the underlying mechanism for shear banding and slip in slip-stick phenomena (e.g. Oda and Kazama, 1998, Rechenmacher, 2006, Thornton and Zhang, 2006, Aharonov and Sparks, 2004, Alonso-Marroquin et al., 2006).

Although extensive experimental and theoretical effort has been devoted to the study of force chains (e.g. Rechenmacher, 2006, Majmudar and Behringer, 2005, Oda et al., 2004, Aharonov and Sparks, 2004, Cates et al., 1998, Radjai et al., 1998), a proper analysis of the structural stability and associated kinematics of the process of force chain failure by buckling is still lacking. This is surprising given the fundamental role of force chains in force transmission, energy storage and macroscopic strength. More recently, the role of force chain failure as an underpinning mechanism for energy dissipation and macroscopic failure, has been assessed in the context of constitutive theory for dense, cohesionless granular systems, in the absence of particle damage (Tordesillas, 2007a, Tordesillas, 2007b; Walsh et al., 2007, Tordesillas and Walsh, 2005). For these systems, the two dissipative mechanisms that generally operate under quasi-static loading conditions are frictional slip and buckling of force chains. Constitutive models which account only for plastic slip over-predict stability, and fail to capture the defining behavior of these materials: i.e. strain-softening under dilatation (see Tordesillas, 2007a, Walsh et al., 2007, Chang and Hicher, 2004 and references cited therein). Specifically, although these models predict loss of contacts in the direction of extension in a biaxial test, the normal contact force continues to grow in the direction of most compressive principal stress. In other words, the amount of energy dissipated through slip at the contacts is much less than the stored energy that is accumulated from the steady growth of the normal contact forces as loading proceeds. Consequently, the stress ratio increases monotonically, even in the presence of plastic slip and loss of contacts in the direction of extension. Viewed from the standpoint of the force chain network, this result signifies that the force chains, which initially align themselves with the major principal stress axis, continue to sustain a steady increase in load even under continuing loss of lateral supporting contacts. What essential physics then is missing in these models? The answer is twofold. First, there is nothing in the formalism of plastic slip that limits the buildup of the normal forces at the contacts. Second, slip only limits the tangential force at the contacts but in the absence of softening. To resolve this, one might be tempted to adopt a more sophisticated contact law that allows for plastic softening in the normal contact force. However, this would not necessarily resolve the problem, unless the softening is tied to dilatation—in which case, one is then confronted with a dissipative mechanism that is beyond the particle scale. The mesoscopic mechanism of force chain buckling is one event in which plastic softening seems inextricably linked to dilatation. The recent constitutive formulation in Tordesillas and Muthuswamy (2008) and Walsh et al. (2007) presents one approach to account for this mechanism via the kinematics—specifically, from the standpoint of nonaffine deformation. A rigorous analysis of both the statics and kinematics of this event is thus key to the advancement of constitutive theory for granular materials—and to bridging the gap between fundamental advances in the physics of force chains and the development of constitutive models that are of core importance to a broad range of applied settings: geomechanics, in which such models are used in mining, exploration and construction; chemical engineering for control of handling and processing of powders, pharmaceuticals and food products; and agriculture for design of cutting tools, off-road vehicles and machines—to name a few examples (Duran, 2000).

In an attempt to fill this gap in the current state of knowledge in the physics and mechanics of granular media, this study seeks to establish the first structural mechanical model of confined force chain buckling. Model predictions are to be validated against data drawn from an analysis of local buckling events from two-dimensional DEM simulations. Next, we undertake two further investigations. First, we use this model to: demonstrate the influence of various resistances to force chain buckling and their interplay on the progression of buckling; and elucidate the mesoscopic origin of strain-softening under dilatation (Rechenmacher, 2006), and other well-established trends from soil mechanics (e.g. effects of confining pressure and particle interlocking on macroscopic shear strength, as recently underlined by Kuhn and Chang, 2006). Second, we test the veracity of conclusions drawn from earlier studies on the importance of confined force chain buckling in constitutive development (Tordesillas, 2007a, Tordesillas, 2007b; Walsh et al., 2007). The first investigation is performed in this paper. The second is the subject of a companion paper (Tordesillas and Muthuswamy, 2008). Therein, we employ the buckling model to derive a thermomicromechanical constitutive law and assess this law's predictive capabilities for strain-softening under dilatation and shear banding. The findings in this companion paper will be summarized in a later section.

In preparation for the development of the buckling model, a series of preliminary studies on the evolution of force chains and its implications for constitutive modeling have been undertaken. These studies were aimed at the quantitative characterization of force chain particles and their supporting weak network neighbors, their associated kinematics and stability, and their role in unjamming transitions and shear banding (Tordesillas et al., 2008a, Tordesillas et al., 2008b, Tordesillas et al., 2004; Tordesillas, 2007a, Muthuswamy and Tordesillas, 2006, Peters et al., 2005). By unjamming transition, we refer to the transition from solid-like to fluid-like behavior, characterized by a decrease in macroscopic stress (also known as the slip phase in stick-slip phenomena, e.g. Aharonov and Sparks, 2004, Alonso-Marroquin et al., 2006). In the first of these preliminary studies, we developed an algorithm to distinguish force chain particles from the complement set termed the weak network particles (Peters et al., 2005). Using discrete element analysis, we then performed a detailed analysis of buckling events in simulations of two-dimensional, densely packed, cohesionless granular assembly subject to quasi-static, boundary driven biaxial compression (Tordesillas, 2007a). This involved the development of another algorithm, aimed at identifying parts of the force chain particle network that have undergone buckling, i.e. buckled force chain segments. A strain interval [εA,εB] is chosen, usually an unjamming transition, and a set of three filters applied, as shown in Fig. 1: (a) eliminate all particles not in force chains at εA; (b) out of those remaining, eliminate those which have not decreased in potential energy; (c) out of those remaining, identify all three-particle segments which have buckled. It was shown that these buckled force chain segments are a key mechanism responsible for nonaffine modes of deformation, dilatation and energy dissipation (Tordesillas et al., 2008a, Tordesillas et al., 2009; Tordesillas, 2007a). Moreover, results confirm the well-known hypothesis of Oda and co-workers that these buckling events are confined to and represent the governing mechanism behind the formation of shear bands. The implications of these findings for continuum modeling are significant, since these formulations are generally based on the assumption of affine deformation and thus ignore this important source of instability. Indeed, recent studies have shown that this assumption, which ignores the inherently nonaffine mode of deformation of force chain buckling, is the main reason why such models fail to reproduce the defining behavior of densely packed granular systems, e.g. strain-softening under dilatation and shear banding (Walsh et al., 2007, Agnolin et al., 2006, Tordesillas and Walsh, 2005, Chang and Hicher, 2004).

In summary, the specific problem to be addressed in this paper is the development of a two-dimensional analytical model for the elastic and plastic buckling behavior of a force chain, under an applied axial compression, with lateral support from weak network particles. Several system properties will be drawn from this model, including: buckling modes, progression of plastic contacts during buckling, evolution of the load-carrying capacity of the force chain, and evolution of particle kinematics in the cluster comprising the force chain and its supporting weak network neighbors. To achieve this, we proceed in two steps. In the first step, presented in Section 2, we conceive the physical model from a suitable choice of initial configuration, degrees of freedom (DOFs), boundary conditions, and contact models, in accordance with discrete element method (DEM) simulations. The second step, presented in Section 3, involves the development of the mathematical model for buckling, using structural stability theory for elastic buckling, and thermodynamical principles for plastic buckling. Next, the kinematics of confined force chain buckling and its implications for constitutive modeling are addressed in Section 4. Input parameters and calibration of the model with DEM simulations are discussed in Section 5. Results from model validation and application are presented in Section 6. Conclusions are drawn in Section 7.

Section snippets

Development of the physical model

The aim of this section is to establish the physical model of force chain buckling under lateral support, accounting for the effect of particle shape irregularities. Here key results from a recent analysis of buckling events in DEM simulations and photoelastic disk experiments (Tordesillas, 2007a, Tordesillas et al., 2009), as well as past observations on force chain evolution (Taboada et al., 2005, Aharonov and Sparks, 2004, Radjai et al., 1998), are used to guide model development. In what

Mathematical model of confined force chain buckling

The aim of this section is to present a mathematical model of confined force chain buckling, by employing structural stability theory for elastic behavior, and thermodynamical principles for plastic behavior. Structural stability theory has been applied to simple rigid-link mechanical models (e.g. Bazant and Cedolin, 2003), and to N-link systems in the context of buckling of highly oriented polymer fibres in kink band formation (DeTeresa et al., 1985). However, several key differences exist

Kinematics of confined force chain buckling

Knowledge of the kinematics of key rearrangement events is a critical ingredient in resolving many open problems in the constitutive modeling of dissipative media. For dense granular systems, two challenges come to the fore. The first is the identification of the underlying origins of strain-softening under dilatation—a hallmark phenomenon exhibited by these systems on the macroscale. The second concerns the constitutive formulation, specifically, the establishment of a physical meaning for

Procedure for model calibration and validation

Before we can apply the buckling model, make predictions and assess its performance, all input parameters must first be defined, so that a comparison can be made with DEM simulations. Thus, the aim of this section is twofold: (i) to identify the method employed for the acquisition of data for the calibration and validation of the model, and (ii) to properly characterize the system under study by establishing all input parameters to the model. Recall that the effect of particle shape

Results of model validation and application

In this section, we validate model predictions in three studies. In the first (Section 6.1) we are to compare the buckling properties as predicted by the model against the DEM buckling data, while accounting for the initial imperfection and initial displacement that are inherent in the latter and in real granular assemblies. The question we wish to answer is: can the structural mechanical model for N=3 reasonably reproduce trends observed in buckling force chains in DEM simulations? In the

Conclusions

The direct modeling of self-organized emergent phenomena in complex media, especially of those in the technologically important and prevalent class of granular materials, has obvious intrinsic value. In this paper, we focussed on force chains, in particular, their structural evolution and failure—an underlying physical cause of the loss of macroscopic strength under dilatation of dense granular materials. Developed herein was a structural mechanical model for the mechanism of buckling of a

Acknowledgments

We thank our reviewers whose constructive comments and insights have helped us improve this paper. We acknowledge the support of the Australian Research Council (Discovery Grants DP0558808 and DP0772409) and the US Army Research Office (Grant W911NF-07-1-0370) to AT and the Pratt Foundation Scholarship for postgraduate support to MM.

References (57)

  • F. Alonso-Marroquin et al.

    Effect of rolling on dissipation in fault gouges

    Phys. Rev. E

    (2006)
  • J.P. Bardet et al.

    The structure of persistent shear bands in idealized granular media

  • Z.P. Bazant et al.

    Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories

    (2003)
  • F. Calvetti et al.

    Experimental micromechanical analysis of a 2d granular material: relation between structure evolution and loading path

    Mech. Cohes. Frict. Mater.

    (1997)
  • M.E. Cates et al.

    Jamming, force chains, and fragile matter

    Phys. Rev. Lett.

    (1998)
  • C.S. Chang et al.

    An elasto-plastic model for granular materials with microstructural consideration

    Int. J. Solids Struct.

    (2004)
  • O. Chaudhuri et al.

    Reversible stress softening of actin networks

    Nature

    (2007)
  • E.I. Corwin et al.

    Structural signature of jamming in granular media

    Nature

    (2005)
  • J.G.A. Croll et al.

    Elements of Structural Stability Theory

    (1972)
  • S. Deboeuf et al.

    Memory of the unjamming transition during cyclic tiltings of a granular pile

    Phys. Rev. E

    (2005)
  • J. Desrues et al.

    Strain localization in sand: an overview of the experimental results obtained in grenoble using stereophotogrammetry

    Int. J. Numer. Anal. Meth. Geomech.

    (2004)
  • S.J. DeTeresa et al.

    A model for the compressive buckling of extended chain polymers

    J. Mater. Sci.

    (1985)
  • J. Duran

    Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials

    (2000)
  • M.R. Falvo et al.

    Bending and buckling of carbon nanotubes under large strain

    Nature

    (1997)
  • L.J. Gibson et al.

    Cellular Solids: Structure and Properties

    (1988)
  • G. Gioia et al.

    The energetics of heterogeneous deformation in open-cell solid foams

    Proc. R. Soc. London A

    (2001)
  • R.P. Goodman et al.

    Rapid chiral assembly of rigid DNA building blocks for molecular nanofabrication

    Science

    (2005)
  • G. Hunt et al.

    Finite element modelling of spatially chaotic structures

    Int. J. Numer. Meth. Eng.

    (1997)
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