Finite element modelling of rate-dependent ratcheting in granular materials

Abstract

The present paper introduces a comprehensive model that is capable of describing the behaviour, under cyclic loading, of the granular materials used in railway tracks and road pavement. Its main thrust is the introduction of the “Chicago” law in a continuum approach to account for the ratcheting effects. It also emphasizes rate-dependency as a dissipative mechanism that acts independently or jointly with the ratcheting effect as well as the non-associated plasticity. The numerical procedure is based on the return mapping algorithm, where Newton’s method is used to calculate the nonlinear consistency parameter of the flow rule and to obtain a consistent tangent modulus. The model was applied to specific numerical examples including multi-axial and cyclic loading conditions.

Introduction

Granular materials have received increasing interest from modern industry due to their complex mechanical behaviour. In particular, the ballasts used in railway tracks or similar agglomerates used in road pavement represent key materials for the transport industry. Understanding the behaviours of these materials, especially under cyclic loading, represents a challenging subject requiring considerable research effort. Different modelling approaches, based on discrete or continuum descriptions, have been used to study the responses of these materials, especially under cyclic loading. Discrete element methods such as molecular dynamics (Cundall and Strack [8], [9], Gallas et al. [13], Roux [35], Karrech et al. [20]) and contact dynamics (Azema et al. [2], Saussine et al. [36]) are well adapted to study samples consisting of a finite number of particles. Although the discrete element methods accurately describe the local mechanisms of energy dissipation caused by ratcheting effects, a loss of contacts, and granular flow, these techniques cannot treat samples made of a large number of grains. Continuum approaches represent convenient time-saving alternatives, though they are limited in predicting local inter-granular mechanisms such as contact loss (Karrech et al. [19]).

Research studies on cyclically loaded granular materials of fine granulometry, such as sands and silts, are rather well established (Eekelen and Potts [12], Kovacevic and Vaughan [22], Lehane et al. [24]). Within the VErification of Liquefaction Analysis by Centrifuge Studies (VELACS; Arulanandan and Scott [1]) research project, several experimental and numerical studies have been conducted on these materials to assess their responses under repeated loading. However, these materials are different in terms of the micro-structure and water content than ballasts, which is of interest in our study. Studies on the cyclic loading of ballasts revealed that ratcheting, rate-dependency, and non-associated flow are necessary features that should be taken into account in the formulation of constitutive models (see Karrech [18] and the references therein). The experimental studies of the same spirit showed that ratcheting can be described with a logarithmic function known as the Chicago law (Ben-Naim et al. [3], Nowak et al. [31]). This empirical law was confirmed numerically using discrete element methods by Rosato and Yacoub [34], Philippe and Bideau [32], Rémond [33]. Both the experimental and the numerical results showed that granular materials exhibit a gradual decrease of the volume of voids, resulting in a nonlinear permanent deformation with respect to the number of cycles. A similar decay was observed by both Guérin et al. [14], Bodin et al. [4] when studying the settlement of granular materials of railway platforms due to repeated loading. In addition, Saussine et al. [36] developed a discrete element model based on contact dynamics that showed the same trend. More recently, Karrech et al. [20] developed a numerical algorithm based on molecular dynamics and showed that the logarithmic law is appropriate for long-term settlement prediction in granular materials. The obtained empirical law proved to be an accurate description of the decaying phenomenon of cyclically loaded granular materials.

The present work is based on several existing computational procedures that involve the above mechanisms of energy dissipation within the framework of continuum mechanics. The purpose of this work is to combine some of these techniques and come up with a comprehensive numerical tool that encompasses the observed logarithmic decay as well as the rate-dependency and frictional aspects for cyclically loaded ballast samples or similar materials. In this context, Lu and Wright [28] suggested a numerical model to describe the behaviour of asphalt concrete under cyclic loading, where an exponential empirical law was introduced to describe the permanent deformation. However, the cyclic effect was included in the rate-dependency mechanism rather than ratcheting. Lorefice et al. [26] introduced a different model that was intended to describe the rate-dependent behaviour of quasi-brittle materials like concrete without taking into account the ratcheting effect. Recently, Cvitanic et al. [10] formulated a theoretical and numerical framework based on non-associated plasticity with isotropic hardening. The numerical integration included the return mapping technique, which consists of projecting the trial stress on the surface of yielding at every increment (see also Key [21], Dafalias [11], Chen and Han [7], Cernocky and Krempl [6], Marques and Owen [29], Loret and Prevost [27]). This technique, which was first introduced by Wilkins [39], was improved by Simo and Taylor [37], Hofstetter et al. [15], who proposed the consistent tangent modulus. Liu [25], Kang [16], Kumar and Nukala [23] proved that return mapping coupled with a consistent tangent modulus is an effective and robust tool to integrate nonlinear constitutive equations. Although the cited models covered particular combinations of rate-dependency, ratcheting, non-associated flows, and local compaction, none of them contained a comprehensive approach to address these phenomena altogether to predict the response of granular materials.

The present study is developed in light of the above mentioned contributions while taking into account the (i) non-associated flow rule, (ii) Drucker–Prager yield and plastic potential functions (instead of the frequently used Von-Mises yield function), (iii) logarithmic ratcheting effect, and (iv) robust integration method based on the return mapping algorithm. This paper is divided into three major sections. Section 2 gives a brief description of the problem under consideration, defines the constitutive continuous model, and highlights the assumptions adopted in the study. Section 3 describes the numerical procedure used to integrate the constitutive model. Section 4 is dedicated to the application of the developed model on particular examples.