Advances in Multi-Physics and Multi-Scale Couplings in Geo-Environmental Mechanics

By Francois Nicot

Advances in Multi-Physics and Multi-Scale Couplings in Geo-Environmental Mechanics reunites some of the most recent work from the French research group MeGe GDR (National Research Group on Multiscale and Multiphysics Couplings in Geo-Environmental Mechanics) on the theme of multi-scale and multi-physics modeling of geomaterials, with a special focus on micromechanical aspects.

Its offers readers a glimpse into the current state of scientific knowledge in the field, together with the most up-to-date tools and methods of analysis available.

Each chapter represents a study with a different viewpoint, alternating between phenomenological/micro-mechanically enriched and purely micromechanical approaches. Throughout the book, contributing authors will highlight advances in geomaterials modeling, while also pointing out practical implications for engineers.

Topics discussed include multi-scale modeling of cohesive-less geomaterials, including multi-physical processes, but also the effects of particle breakage, large deformations on the response of the material at the specimen scale and concrete materials, together with clays as cohesive geomaterials.

The book concludes by looking at some engineering problems involving larger scales.

• Identifies contributions in the field of geomechanics
• Focuses on multi-scale linkages at small scales
• Presents numerical simulations by discrete elements and tools of homogenization or change of scale

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 20, 2017
• ISBN: 9780081025963

Book preview

Advances in Multi-Physics and Multi-Scale Couplings in Geo-Environmental Mechanics

Edited by

François Nicot

Olivier Millet

Series Editor

Félix Darve

Table of Contents

Cover

Title page

Copyright

Foreword

Introduction

1: Multi-Scale and Multi-Physics Modeling of the Contact Interface Using DEM and Coupled DEM-FEM Approach

Abstract

1.1 Introduction

1.2 Modeling of mutlicontact systems using DEM for electrical transfer applications

1.3 DEM for modeling continuous media

1.4 DEM–FEM-based approach for multi-scale and multi-physics modeling

1.5 Conclusion

1.6 Acknowledgments

2: Adsorption-induced Instantaneous Deformation in Double Porosity Media: Modeling and Experimental Validations

Abstract

2.1 Introduction

2.2 An incremental poromechanical framework with varying porosity for single porosity media

2.3 Extension of the poromechanical framework to double porosity media

2.4 A new experimental set-up allowing the simultaneous in situ measurements of both adsorption and swelling

2.5 Validation of the extended poromechanical model by experimental comparisons on a double porosity synthetic activated carbon

2.6 Concluding remarks and perspectives

2.7 Acknowledgments

3: Granular Materials: Mesoscale Structures and Modeling

Abstract

3.1 Introduction

3.2 Mesoscale as a basis for upscaling the mechanical behavior of granular materials

3.3 Multi-scale constitutive model including mesostructures

3.4 Conclusion

4: Behavior of Granular Materials Affected by Grain Breakage

Abstract

4.1 Introduction

4.2 Size effects in rockfill materials

4.3 Challenges in modeling rockfill behavior

4.4 Detrimental effect of humid conditions on grain breakage

4.5 Conclusion

4.6 Acknowledgements

5: Multi-Scale Modeling of the Mechanical Behaviour of Clays

Abstract

5.1 Introduction

5.2 Experimental investigation on clayey microstructure

5.3 Development of micromechanics-based model for clay

5.4 Mobilized three-dimensional strength criteria

5.5 Induced anisotropy effects obtained by the micromechanical model

5.6 Inherent anisotropy in the micromechanical model

5.7 Cyclic loading effect by the micromechanical model

5.8 Comments on possible model extension

5.9 Conclusion

5.10 Acknowledgements

6: Modeling of Complex Microcracking in Quasi-Brittle Materials: Numerical Methods and Experimental Validations

Abstract

6.1 Introduction

6.2 Experimental procedures

6.3 Numerical simulation methods

6.4 Validations of crack propagation

6.5 Conclusion

7: Multi-Scale Methods for the Analysis of Creep-Damage Coupling in Concrete

Abstract

7.1 Introduction

7.2 Experimental methods for the identification of creep-damage coupling in concrete

7.3 Numerical methods for the analysis of damage during creep

7.4 Conclusion

8: Effect of Variability of Porous Media Properties on Drying Kinetics: Application to Cement-based Materials

Abstract

8.1 Introduction

8.2 Heterogeneous and variable nature of porous building materials

8.3 Hygrothermal transfer properties in porous media: the case of cement based materials

8.4 Most commonly used models for hygrothermal transfers

8.5 Variability effect of the moisture diffusion coefficient and the saturation water content on drying kinetics

8.6 Assessment and incidence of the spatial variability of the porous medium properties on the hygrothermal transfer at a wall scale

8.7 Conclusion

9: Internal Erosion

Abstract

9.1 Introduction

9.2 Experimental findings on interface erosion

9.3 Experimental findings on suffusion

9.4 A description of internal erosion based on flow power

9.5 Numerical approaches to describing internal erosion effects in soils

9.6 General conclusion

10: Mechanical Stability of River Banks Submitted to Fluctuations of the Water Level

Abstract

10.1 Introduction

10.2 Background and general methods of analysis

10.3 A built-in model for bank stability analysis

10.4 Application to the Lower Mekong Basin

10.5 Conclusions and perspectives

List of Authors

Index

Copyright

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2018

The rights of François Nicot and Olivier Millet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-278-6

Printed and bound in the UK and US

Foreword

In the 1970s, civil engineering benefited from a deep computational revolution with the newly introduced numerical methods for structure and engineering design. The most popular method at that time was certainly the finite element method (FEM), which is essentially based on matter continuity – even if, with some ad hoc numerical tricks, it is possible to take into consideration a few material discontinuities. However, the materials in geomechanics (the so-called geomaterials) are basically constituted by elements of a discrete nature: rock blocks, sand grains and clay aggregates are all in fact distinct elements. So geomechanics, to envisage the possibility of considering each such elemental component individually, could be considered, if desired, as an unexpected new way to investigate discrete matter, and particularly geomaterials.

So, when the discrete element method (DEM) was introduced, able to simulate the most complex mechanical properties of geomaterials – by taking into account each block/grain/aggregate in a boundary value problem – a new chapter on geomaterials mechanics could be written. It was the first topic that GDR MeGe (the National Research Group on Multi-physics and Multi-scale Couplings in Geo-Environmental Mechanics) undertook. Indeed, about half of this book is devoted to illustrating these new advances in geomaterial micromechanics. Research is still in progress on the realistic modeling of boundary value problems: on the one hand, by a relevant parallelization of DEM software – at present, it is possible to consider 10 billion interacting grains – and, on the other hand, by a spatial DEM/FEM coupling.

Besides, geomaterials are rarely dry (this is typical for soils), some chemical reactions can develop (as for concrete) and temperature effects are sometimes not negligible (the behavior of clay is strongly dependent on temperature). All these environmental effects are today commonly gathered together under the generic name of multi-physics coupling. The second half of this book is devoted to these couplings.

MeGe GDR was focused precisely on both of these most interesting and efficient (with respect to practical applications) topics – multi-scale and multi-physics couplings – in modern geomechanics. Thus, it should not be astonishing if this book tackles some of the most recent questions in geomaterial behavior with applications of tremendous socioeconomical importance, like natural hazards prevention, oil petroleum and natural gas extraction, geothermal energy, CO2 sequestration, controlled reinforcement of aging concrete dams and nuclear power plants, soil erosion, etc.

In brief, with this book, the readers will find a relevant state of the art of the most advanced studies in France in the field of geomechanics.

Félix Darve, Emeritus Professor, Grenoble Institut Polytechnique

Introduction

During the 8 years from 2008 to 2015, the MeGe GDR (National Research Group on Multi-scale and Multi-physics Couplings in Geo-Environmental Mechanics) networked the main French laboratories involved in the broad field of geomechanics, with a particular focus on environmental applications. Taking advantage of the different collaborations and connections that the partners had developed with foreign universities, important progress has been made in various fields such as failure and instability modeling, the description of aging processes, and multi-physical couplings within geomaterials. By promoting a school of thinking that bridges scales from particle to engineering issues, recurrent efforts have been made to confront micromechanical approaches with real and experimental evidence.

The objective of this book is to gather a number of up-to-date contributions issued from MeGe GDR on the theme of multi-scale and multi-physics modeling of geomaterials with particular reference to micromechanical aspects. Without pretending to be exhaustive on such a widely debated theme, the book aims at presenting a glimpse of the current state of the scientific knowledge together with the available tools and methods of analysis. The contributions will have viewpoints that alternate between phenomenological/micromechanically enriched and purely micromechanical approaches. Throughout the chapters, the authors highlight advances in geomaterial modeling, while also pointing out the practical implications to the engineer.

The plan of the book is organized as follows. The first chapters will be dedicated to the multi-scale modeling of cohesion-less geomaterials, including multi-physical processes. Then, the effects of particle breakage and large deformations on the response of the material at the specimen scale are examined. Finally, concrete materials are discussed together with clays as cohesive geomaterials, before closing by looking at some engineering problems involving larger scales.

The contents, main lines and ideas as conveyed in each chapter are detailed as follows to guide the reader.

Chapter 1 deals with the description of physical phenomena in the context of the modeling of the contact interface behavior based on the discrete element method (DEM) and the DEM-finite element method (DEM-FEM) coupled approach. The purpose, related to several engineering applications where the contact interface plays a significant role, is to understand the coupled multi-physical mechanisms that are involved, and how these mechanisms influence the system from a mechanical, thermal or electrical point of view. It is obviously impossible to list all the applications involving contact interfaces; nevertheless, those reported in this chapter highlight the relevance of a multi-scale and multi-physics approach including a microscale description. Starting from physical information, micromechanically-based models have been developed, resulting in numerical modeling that reproduces fairly the experimental observations.

Chapter 2 discusses natural and synthetic porous media, which generally encompass different and distinct porosities: a microporosity where the fluid is trapped as an adsorbed phase and a meso- or a macroporosity required to ensure the transport of fluids to and from the smaller pores. Zeolites, activated carbon, tight rocks, coal rocks, source rocks, cement paste or construction materials are among the materials for which this is relevant.

In nanometric pores, the molecules of fluid are confined. This effect, denoted as molecular packing, includes fluid-fluid and fluid-solid interactions and has significant consequences at the macroscale, such as instantaneous swelling, which are not predicted by classical poromechanics. If adsorption in nanopores induces instantaneous deformations at a higher scale, the matrix swelling may close the transport porosity, reducing the global permeability of the porous system or annihilating the functionality of synthesized materials. This is important for applications in petroleum and oil recovery, gas storage, separation and catalysis and drug delivery. This chapter aims at characterizing the influence of an adsorbed phase on the instantaneous deformation of micro-to-macro porous media.

In Chapter 3, a straightforward and reliable modeling of granular materials remains an open issue due to the difficulty in addressing the evolution of their internal structure when loaded. This chapter proposes a new upscaling approach including a meso-scale.

Using two-dimensional (2D) DEM simulations of compression and extension tests, we highlight the discriminant role played by mesoloops of particles, constituting different phases, in the mechanical behavior of granular materials. The development of a constitutive model for such materials should benefit from an approach including the mesoscale.

Then, by simplifying the complex shapes and the variety of orientations of the mesoloops, the main characteristics are identified to achieve a complete constitutive model from an uspcaling approach including these mesoloops. Finally, the H-directional model [NIC 11] is presented as a constitutive relation for 2D granular materials.

Chapter 4 deals with the effect of grain breakage on the behavior of crushable granular materials as well as its impact on the behavior of rockfill structures. Since the experimental identification of the strength envelope for coarse soils is far from being an obvious task, the authors validate a predictive methodology originally developed by Frossard et al.[FRO 12] that takes into account scale effects in the resistance of the grains themselves. This theoretical methodology is supported by an extensive experimental program of triaxial tests on different specimen sizes and homothetic grain size distributions of the same material. Then a strain-hardening elastoplastic constitutive model is developed, in which the evolution of the grain size distribution due to grain breakage can be described either in an implicit way (related to the plastic work) or in an explicit way (confined comminution model). The latter couples the probability of failure of the particles, subjected to size effects, and the distribution of normal forces in a granular packing, obtained from DEM computations. Finally, moisture is known to be a detrimental factor in the onset of grain breakage. Through oedometer tests conducted in different moisture conditions, the authors investigate the relationship between moisture and the amount of grain breakage. In addition, it is demonstrated that a toughness parameter could be determined from these simple tests.

In Chapter 5, clay is treated as a collection of aggregates formed by groups of platy clay particles. At the scale of aggregate sizes, long-range forces are negligible and the aggregates interact with each other mainly through mechanical contact forces. Thus, clayey material can be modeled by analogy to granular material. Based on investigations of clayey microstructure, a clay aggregate is considered as a deformable grain. The granular mechanics approach is extended to derive the elastoplastic stress-strain relationship for clay. Then, the proposed model is used to analyze different features of clay from the interaggregate contact level to that of global phenomenon. Various numerical simulations are presented such as true triaxial tests for analyzing the mobilized strength criteria in three dimension; triaxial tests with different stress paths for analyzing the induced anisotropy of the stressstrain response and the rotational hardening of the yield surface; triaxial tests on vertical and horizontal samples and hollow cylinder tests with different angles of the principal stress axis to the sample vertical axis for analyzing the inherent anisotropy; and drained and undrained cyclic triaxial tests for analyzing the influence of cyclic loading. Furthermore, some explanations for possible model extensions are presented and the model extension for sensitive clays is presented as an example.

In Chapter 6, recent modeling methodologies are presented for predicting crack propagation in quasi-brittle materials with complex heterogeneous microstructures which can be applied to rocks or civil engineering materials. First, experimental methodologies combining three-dimensional (3D) XR-μCT (X-Ray Micro Computed Tomography) imaging techniques, in situ testing and 3D image processing are described to characterize microcracking in cement-based materials at the microscale. Then, advanced simulation tools based on the phase field method are presented, developed specifically for modeling quasi-brittle damage in voxel-based models of realistic microstructures. Finally, direct validations of the simulation tools with the experiments are provided, involving full comparisons of 3D microcracking networks within realistic microstructures.

In Chapter 7, the determination of the delayed behavior of concrete structures is posited to have a major role in structure design because creep deformations could be at the origin of cracking. In order to quantify the part of damage occurring in concrete under creep, the acoustic emission (AE) technique is used. Quantitative analysis of the AE data is performed and characteristics of micromechanisms are evaluated and associated with two clusters for basic creep and three clusters for desiccation creep. According to several studies, the viscoelastic behavior of concrete is strongly linked to the main hydrates of the cement paste that could deform under a constant load and are restrained by other components. A mesoscopic model has been developed in order to describe the fracture process and the corresponding effect of concrete heterogeneities under creep. It is observed that under tensile creep, damage induced at the moment of load application increases due to strain incompatibilities between mortar and aggregates and causes a decrease in strength. Similar observations were obtained with the AE technique.

Chapter 8 discusses the variability effect of the transfer and storage properties in porous media on their drying kinetics. After some introductory definitions on the measurements, errors, uncertainty and variability, it exposes the properties that almost influence the hygrothermal behavior of porous media such as porosity, sorption isotherm, vapor and gas permeability. Afterward, it presents in the incidence of (1) the pseudo-random variability and (2) the spatial variability of the transfer and storage properties on the drying kinetics in porous media with an application to cementitious materials. These two parts are based on different probabilistic approaches detailed in this chapter. Case study results highlight the limits of deterministic approaches in the prediction of the hygrothermal behavior of some heterogeneous porous media, presenting variability in their properties.

Chapter 9 deals with the mechanisms of internal erosion, which are deeply complex and entail many parameters, a number of which are coupled. In earth structures and within their foundations, two types of internal erosion can be distinguished: suffusion and interface erosion. This chapter presents the results of experimental and numerical studies on interface erosion and suffusion. The main findings concern, on the one hand, the main soil characteristics controlling susceptibility to erosion and, on the other hand, the impact of suffusion on the mechanical properties of soils.

In Chapter 10, a simplified built-in numerical program is developed for the practical evaluation of the risk of mass slides of river banks and is applied to the conditions of the Lower Mekong River. Three problems are addressed: transient water flow within the bank, bank surface erosion and bank mass slide. Pore water pressures are determined from the hypothesis of unidirectional groundwater flow solved by a finite difference discretization with respect to time and space. Bank mass stability is analyzed by a limit equilibrium approach (method of slices). Surface erosion is determined from the critical shear stress concept. The results show that the critical situations for mass slides arise after a rapid drop of the river water level. Moreover, the river hydrograph and the bank permeability are key parameters that must be known accurately.

1

Multi-Scale and Multi-Physics Modeling of the Contact Interface Using DEM and Coupled DEM-FEM Approach

Mohamed Guessasma; Valery Bourny; Hamza Haddad; Charles Machado; Eddy Chevallier; Aymen Tekaya; Willy Leclerc; Robert Bouzerar; Khaled Bourbatache; Christine Pélegris; Emmanuel Bellenger; Jérôme Fortin

Abstract

This chapter presents the description of physical phenomena in the context of the modeling of the contact interface behavior based on discrete element method (DEM) and coupled DEM–finite element method (DEM–FEM) approach. Our interest in this area, which is related to several engineering applications where the contact interface plays a significant role, is prompted by the desire to understand the mechanisms that involved multi-physics coupling and how these influence the studied system from mechanical, thermal or electrical behavior point of view. Obviously, it is not easy to summarize all applications involving the contact interfaces, nevertheless, through those given as examples in the following sections, we have tried to highlight the relevance of a multi-scale and multi-physics approach with the aim of a modeling as close as possible to the microscale.

Keywords

Ball bearings; Composite materials; Coupled DEM-FEM; Discrete element method (DEM); Electrical transfer applications; Microscale; Ring/wire contact

This chapter presents the description of physical phenomena in the context of the modeling of the contact interface behavior based on discrete element method (DEM) and coupled DEM–finite element method (DEM–FEM) approach. Our interest in this area, which is related to several engineering applications where the contact interface plays a significant role, is prompted by the desire to understand the mechanisms that involved multi-physics coupling and how these influence the studied system from mechanical, thermal or electrical behavior point of view. Obviously, it is not easy to summarize all applications involving the contact interfaces, nevertheless, through those given as examples in the following sections, we have tried to highlight the relevance of a multi-scale and multi-physics approach with the aim of a modeling as close as possible to the microscale.

1.1 Introduction

The work in relation to the developments conducted on the multi-scale and multi-physics modeling of the contact interface is discussed in the following three sections. Section 1.2 concerns the electrical transfer where the addressed issue is related to the diagnosis of multicontact systems starting from the contact interface. Building on the experimental measurements carried out on test beds, a promising monitoring technique is proposed taking advantage of the richness of the electrical signal. Used as a new diagnosis technique for monitoring the mechanical state of bearings and preventing material wear in wind plant application, the electrical measurement provides a signature that accurately reflects the status of the contact interface. Section 1.3 addresses the issue of the modeling of continuous media with DEM-based approach. Considering granular assemblies with an isotropic contact distribution and an optimal packing density (random close packing [RCP]), the DEM is able to qualitatively describe the macroscopic behavior of a continuous material. For this purpose, a lattice model based on a beam model is introduced in order to make each contact between two particles cohesive. To achieve the macroscopic behavior of a given material, the geometrical and mechanical parameters of the beam model are required before a calibration procedure. The discrete approach is also extended to the case of composite materials. The idea behind the proposed discrete description is that it enables taking into account discontinuities, local phenomena, debonding effects, etc. In addition, the DEM is successfully implemented for modeling thermal transfer in continuous compact, homogeneous and isotropic material. Section 1.4 deals with the coupling approach combining DEM and FEM that has been used to model the thermomechanical behavior of a multicontact interface. Typically, this configuration is encountered in many engineering applications, namely in braking systems, railway transport (catenary-pantograph interface), polishing systems, etc. Such configurations might be schematically represented by two bodies kept in contact under a compressive force where one of them is animated by a sliding velocity. This schematic modeling involving the contact interface is known under the name of the tribological triplet. The two bodies in contact represent the first bodies and the contact interface the third body. This last section is then dedicated for investigating the physical phenomena taking place at the multicontact interface (third body), in terms of mechanical and thermal behaviors.

1.2 Modeling of mutlicontact systems using DEM for electrical transfer applications

The use of electrical transfer in multicontact systems is of prime importance in a wide range of the engineering applications such as electronic components [SAL 00], miniaturization of moving components [PER 00], powder metallurgy in the sintering process [EID 09], line-pantograph discontinuity contact in railway transportation systems [GON 09], etc. In recent times, the electrical transfer is also used to deduce the mechanical conditions in which the ball bearings operate [BOU 13a, BOU 13b]. With the help of the richness of the electrical signal, the information gathered over the ball bearing reveals the presence of defects and also the defect type [MAC 15a]. In addition to the case of a rolling contact, the electrical signal obtained across the contact interface in the case of a sliding contact makes it possible to track the evolution of the main features of the contact during the surface wear process [CHE 14a]. Therefore, the measure of electrical quantities (voltage, electric resistance or impedance) offers a promising method for the monitoring operations of mechanical systems, which still remain a fundamental issue for reducing maintenance costs and for prevention of industrial risk.

By knowing the sensitivity and the richness of the local electrical signature at the contact interface, a new method for the monitoring and the diagnosis of defects has recently been introduced by means of an experimental and phenomenological approach and numerical simulations [CHE 14b, MAC 15b]. Before presenting the novel diagnosis procedure from the electrical signal analysis, it would be judicious to briefly introduce the electrical transfer issue in granular media. Retrospectively, the operation of the electrical response of a multicontact system was investigated by E. Branly [BRA 90] in the 19th Century, when he established the extreme sensitivity of the electrical conductivity of a metal powder subjected to an electromagnetic wave. This discovery led to a fundamental relation reflecting the interaction between the mechanical forces and the electrical quantities in a multicontact system. This very complex phenomenon, which is named Branly’s effect (Figure 1.1(a)), was studied many times, from the standpoints of experimental approaches and physical modeling, aiming to understand the phenomena activated at the contact interface, or to at least be able to model some of these physical aspects in accordance with certain simplifying assumptions [HOL 00, FAL 05a, CRE 07, TEK 11, BOU 12]. This phenomenon, related to Branly’s effect, has been also observed in static when the ball bearing is subjected to an electrical potential as shown in Figure 1.1(b).

Figure 1.1 Similar electrical response in terms of the evolution of the electrical potential U as a function of the current I in multicontact systems: a) granular medium and b) ball bearing. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

1.2.1 Diagnosis of defects in ball bearings

Rolling bearings are commonly encountered components in domestic and industrial rotating machinery. There is a large number of bearings for all possible applications. Statistical studies show that industrial bearings are considered critical mechanical components, which represent between 40 and 50% of the malfunctions in rotating machinery. In order to ensure the industrial system’s availability and the safety of goods and persons, the monitoring and diagnosis of bearing defects have to be considered with prime importance and the challenges in terms of productivity are non-negligible. First of all, a feasibility study was carried out on a useful and experimental device showing the relevance of the original monitoring procedure to diagnose defects in bearings (Figures 1.2(a) and (b)). The second step deals with the physical modeling and the numerical simulations by the DEM, observing that the the DEM has been used in this study on the monitoring of bearings for the first time (Figure 1.2(c)). The mechanical model of ball bearing for discrete simulations is based on smoothed formulation such as established by Cundall et al.[CUN 79], where the contact forces Fn,t in normal and tangential directions according to local frame at the contact between particles are described with a contact model depending on elastic force displacement law, Coulomb friction and viscous damping coefficients (equation [1.1]).

Figure 1.2 a) CAD drawing of an experimental device for ball bearing monitoring; b) experimental measurements of the variation in time of electrical resistance of ball bearing; c) ball bearing model using DEM. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

   [1.1]

where Kn,t are the normal and tangential stiffnesses, Cn,t is the normal and tangential damping coefficients, v is the relative speed according to local frame and μ is the Coulomb’s friction coefficient in the case of dry sliding contact. The viscous damping coefficients Cn,t are set critical to ensure a satisfactory mechanical steady state. The coefficients Kn,t are related to mechanical characteristics and dimensions of particles in contact according to nonlinear model proposed by Mindlin et al., related to the radial preload/clearance Δr and the ring radial shift δr (Figure 1.2(c)). Typically, the ball bearing is under an external radial load Fr applied on the inner ring in rotation with the shaft at an angular speed ω. Therefore, the numerical predictions of the load distribution Qψ using the DEM provide the mechanical load applied on each ball by means of the parameters previously introduced.

Generally, electrical transfer in multicontact systems depends on the intrinsic mechanical and electrical properties of materials, the number and shape of particles and the contact number [BOU 12]. Electrical response also depends on the mechanical load applied to the ball bearing. In the present context, we assume that the temperature is constant and the oxide layer on the surface of particles and the lubricant effects are not taken into account. The effect of roughness is neglected. The forces applied to the bearing are sufficiently large to validate these approximations. However, we are aware that these assumptions may be questionable since the contact interface is very intricate and several physical phenomena are involved. Even so, the contact is assumed perfect and the electrical transfer is described by Ohm’s law that corresponds to the linear phase of the experimental curve f(I) (Figure 1.1). The formulation of the electrical problem is based on the contact network of a multicontact system. The main idea is first to use Kirchhoff’s law and Ohm’s law to solve the problem. The electrical potentials at each contact point are the unknowns of the electrical problem [BOU 13a]. Therefore, the electrical resistance is related to contact surfaces Si and Sj located on the surface of a homogeneous spherical particle k as follows:

   [1.2]

where γ is the electrical conductivity of steel (γ = 5.8 × 10⁷S.m− 1), Vb is the volume of a rolling component, θ is the angle formed by the points i and j; θ is equal to π for a radial ball bearing (Figure 1.3(a)). The coupling between the mechanical and electrical