Geometrically exact bifurcation and post-buckling analysis of the granular elastica

Highlights

• This paper presents exact bifurcation diagrams for the post-buckling branches of the granular elastica (granular column).

• The in-plane buckling and post-buckling analysis in a geometrically exact framework is numerically investigated from a nonlinear difference eigenvalue problem.

• The granular column asymptotically behaves as an Engesser–Timoshenko column for a sufficiently large number of grains.

• An exact analytical solution of the buckling load of this discrete shear granular system is obtained from the linearization of the granular elastica problem.

• Exact analytical solutions of the post-buckling branches are also available for the granular problem with few numbers of grains.

• An asymptotic method is used to approximate the primary post-bifurcation branch.

• Bifurcation diagrams of the granular elastica problem composed of few grains are numerically obtained with the simplex algorithm.

• The post-buckling of this granular column reveals very rich behaviour in the bifurcation diagram.

Abstract

The nonlinear behaviour of a simply supported granular column loaded by an axial force is studied in this paper. The granular column is composed of rigid grains (discs) elastically connected by some shear and rotational springs. The in-plane buckling and post-buckling analysis in a geometrically exact framework is numerically investigated from a nonlinear difference eigenvalue problem. The granular column asymptotically behaves as an Engesser–Timoshenko column for a sufficiently large number of grains. An exact analytical solution of the buckling load of this discrete shear granular system is obtained from the linearization of the granular elastica problem. An asymptotic expansion is applied to the difference eigenvalue problem, to efficiently approximate the equation of the primary post-bifurcation branch of the discrete problem. Exact analytical solutions of the post-buckling branches are also available for the granular problem with few numbers of grains. Bifurcation diagrams of the granular elastica problem composed of few grains are numerically obtained with the simplex algorithm (for an exhaustive capture of all post-bifurcation branches). It is shown that the post-buckling of this granular column reveals complex behaviour similarly to the post-buckling of a generalized shear Hencky column (also called discrete Engesser elastica). Complex higher-order branches are exhibited, a phenomenon very similar to the discrete elastica problem. These branches reveal the specific nature of the discrete granular problem, as opposed to its continuum limit valid for an infinite number of grains

Introduction

In this paper, the buckling and post-buckling behaviour of an axially loaded granular column is studied from analytical and numerical perspectives. The instability of such granular structure is studied in view of a better understanding of shear band formation in geomaterials, including granular media. It has been experimentally and numerically observed that curved columns of particles or grains are formed during the shear band with a key role of rotation of particles, and chain of forces. These phenomena are essential in the localization process of granular materials [1], [2], [3], [4]; or [5]. The present granular discrete model explored in its full geometrically nonlinear range may contribute to a better understanding of granular instabilities in geomaterials. The granular system is composed of a finite number of uniform rigid grains with independent translational and rotational degrees-of-freedom. Shear and rotational interactions are taken into account at the interface of each grain. This nonlinear problem will be investigated in a geometrically exact framework. The nonlinear behaviour of such discrete structural systems is ruled by some nonlinear difference equations, as opposed to nonlinear differential or partial differential equations valid for continuous systems.

The idea to compute the macroscopic behaviour of granular systems by modelling each grain separately, thus leading to a large scale discrete system is quite old. This idea is also related to the possibility to investigate continuum elasticity problems based on molecular elastic interactions, as suggested by Boscovich [6] and the French mechanicians at the beginning of the XIXth century [7], [8], [9]. This question was of fundamental interest during all the XIXth century (see for instance the paper of [10] on the history of molecular elasticity). Investigating the macroscopic behaviour of continuous media from their fundamental discrete interactions is still a stimulating research nowadays, with the development of more complex interaction laws or more sophisticated computational possibilities [11], [12]. Even if the basic ideas behind discrete granular media based on elementary interaction laws were probably mature before the end of the XXth century, the possibility to compute effectively the nonlinear behaviour of such discrete systems only dates from the last 70’s, with the so-called Distinct Element Method, due to the availability of computational capabilities. The Distinct Element Method (DEM) has been initiated by Cundall [13], Serrano and Rodriguez-Ortiz [14] and Cundall and Strack [15]. This method assumes that the grains are rigid and interact with translational and rotational elastic and inelastic connections. This method has been widely and successfully applied to a large variety of engineering cases. The response of the granular particles is computed using numerical codes initially developed in the 70’s (program ESTIB of [14]; program BALL of [15], which gives birth to PFC — Particle Flow Code). The research in this field is still active, especially for bridging discrete granular systems with continuous media (see the monographs of [2], [3], [4]; or [5]. There are still some debates about the dissipative nature of the interaction law at the elementary level (see the discussion in McNamara et al. [16]); or more recently Nicot et al. [17] or Turco et al. [18].

In the present paper, we will formulate the geometrically exact nonlinear difference equations of a granular structural system composed of rigid circular grains with elastic granular interactions. This system can be also viewed as the analytical formulation of a Distinct Element Method applied to granular matter (where the normal interaction is neglected in the present study). The equations of motion will be deduced from an energy formulation for this nonlinear conservative elastic system. The granular column is assumed to be loaded by some axial loads, which may cause the granular system lose stability. A similar system has been considered by Satake [1] who also studied the buckling of a granular column with constrained shear/bending interaction laws. Satake [1] introduced this granular chain as a paradigmatic structural model which may play a key role in the shear band formation. Satake [1] only presented some linearized equations for the calculation of the buckling load. The geometrically nonlinear exact formulation of this granular column is given by Hunt et al. [19] or Tordesillas et al. [20] based on energy arguments. Hunt et al. [19] numerically solved the buckling and post-buckling behaviour of the granular column on Winkler elastic foundation by using a path-following continuation code. Tordesillas et al. [20] numerically investigated the effect of boundary conditions on the linearized buckling behaviour of the granular column. The paper of Tordesillas and Muthuswamy [21] should be also mentioned regarding the elastic and inelastic instabilities of granular systems, including longitudinal and cyclic granular chains. Challamel et al. [22] obtained some closed-form solution of the buckling load of a granular system by solving a linear difference eigenvalue problem. Challamel et al. [22] also highlighted the link between this discrete stability problem and its asymptotic continuous analogue, i.e. the Engesser–Timoshenko column [23], [24]. As shown by Pasternak and Mühlhaus [25], Challamel et al. [22], [26] or Massoumi et al. [27], the discrete granular column may behave as a discrete Bresse–Timoshenko beam element, both in statics and in dynamics. Challamel et al. [26] more recently reconsidered the buckling of a granular column under discrete Winkler and Pasternak elastic foundations. Challamel et al. [26] also developed a higher-order gradient-type continuous beam theory for capturing the length scale effects of the granular chain. Alternative discrete Bresse–Timoshenko systems have been recently proposed by Kocsis [28], Kocsis et al. [29], Kocsis and Challamel [30], Battista et al. [31] or Turco et al. [32] also in a geometrically nonlinear framework. It appears that there is a strong mathematical connection between one-dimensional granular systems and discrete beam mechanics, both in the linear and in the nonlinear ranges.

The buckling and post-buckling behaviours of the granular column are investigated in this paper, based on a geometrically exact framework. Such nonlinear problem has been numerically computed by Hunt et al. [19] or Tordesillas et al. [20] who started from an energy expression of the structural interactions. The model presented in this paper leads to exactly the same total potential energy function as used in the model of Hunt et al. [19] or Tordesillas et al. [20] without lateral supporting springs and stiff normal springs, which may be labelled as a granular elastica (or equivalently, a geometrically exact DEM granular column). The nonlinear difference eigenvalue problem is presented both from an energy approach and from a direct approach. We numerically solve the nonlinear difference boundary value problem with the simplex algorithm (for an exhaustive capture of all post-bifurcation branches). It is shown that the post-buckling of this granular column reveals complex behaviour similarly to the post-buckling of a generalized shear Hencky column (also called discrete elastica or discrete Engesser elastica in presence of shear). An exact analytical solution of the buckling load of this discrete shear granular system is obtained from the linearization of the granular elastica problem. An asymptotic expansion method is also applied to the nonlinear difference eigenvalue problem of the granular elastica to approximate analytically the primary bifurcation branches of the discrete granular problem. A similar method has been successively applied to a nonlinear elastic discrete repetitive system, i.e. the Hencky discrete beam problem formulated in a geometrically exact framework, by Challamel et al. [33]. Hencky beam is composed of rigid elements connected by rotational springs [34], which asymptotically converges towards Euler beam at the continuum limit. Even if Hencky introduced his model one century ago as a numerical method based on physical arguments to approximate the buckling loads of elastic Euler columns, the treatment of the geometrically exact Hencky problem is more recent and dates from the 80’s (with contributions of El Naschie et al., Gáspár and Domokos [35], [36] or Domokos [37]). El Naschie et al. [35] numerically computed the curvature of the primary bifurcated branches of the geometrically exact Hencky system, from an asymptotic procedure. [36] or [37] highlighted the very rich structure of the bifurcation diagram of Hencky system with a very exhaustive portrait. Challamel et al. [33] decomposed the nonlinear eigenvalue problem of the nonlinear Hencky system into a set of difference equations, and analytically confirmed the curvature values computed by El Naschie et al. [35] for Hencky system. In the present paper, we will apply the same mathematical approach for a nonlinear difference eigenvalue problem which is similar (even if not equivalent) to the geometrically exact Hencky column.

We will show that exact analytical solutions of the post-buckling branches are also available for the granular problem with few numbers of grains. The granular column asymptotically behaves as an Engesser–Timoshenko column (or continuous Cosserat beam model) for a sufficient large number of grains.