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Statistics and kinematics of granular materials
103
Book Reviews Physics of Granular Media Centre de Physique, Les Houches Series, edited by Daniel Bideau and John Dodds, published Nova Science, 1991, 434 pp., $89, Commack, New York, ISBN l-56072-034-4.
This book is a collection of papers taken from the winter school of the same name held in les Houches, France in 1990. The book is organised into separate sections consisting mainly of review papers. There is a section called, ‘geometry and structure of packings’; much of the discussion is about the random packing of spheres (e.g. local coordination numbers and Voronoi polyhedra). There are articles in this section by Rivier, Sadoc, Finney, Dodds, Oger and Jernot. An article by Meakin and Julien in the section on particle flow takes these geometrical aspects a step further to consider deposition and segregation of binary mixtures under a gravitational field in a simulation model. Moving on from purely geometrical aspects of random packing, there is a section on ‘mechanical properties’ in which frictional forces are considered. Articles were by Guyon, Georges, Desrues, Briscoe and Roux. Contact friction itself and the impact of friction and contact areas on localisation of deformation (i.e. relative sliding of semi-rigid volumes of material along failure boundaries) are the main themes. Local geometry was not really considered in anything more than in a simple sense (e.g. at the coordination number level) so as yet there does not appear to be much use made of the geometrical properties determined in the first section. Percolation theory and ‘fractals’ are used to provide a framework for a generic description of mechanical response of granular media. There was a particularly interesting article on shear bands by Desrues. Next followed a section on ‘transport properties in the pore space’ in which fluid flow through granular materials (treated mainly at the continuum level) was discussed. The geometry of the paths and the relative permeability of mixed fluids were two topics discussed. The authors included Dullien, Koplic and Hulin and co-workers. Another section on granular media in motion, called ‘particle flow, suspensions and fluidisation’ included articles by Ghadiri (on fluidised beds), Fauve (on the dynamics of avalanches in a rotating cylinder), Blanc (sedimentation), Couch and Hinch (aggregation during sedimentation using a fractal formalism) Meakin and Julien, and Savage (molecular dynamics simulations of couette flow of granular materials). There was an interesting article by Adams bringing contact mechanics
ideas to the flow of dense suspensions. The section was completed by articles on critical state model of compaction by Stanley-Wood, sintering by Jernot and liquid-phase sintering by Chaix. This is a very useful book, which provides an entrypoint to many areas of the granular materials literature, and I highly recommend it. D. M. Heyes
Statics and Kinematics of Granular Materials By R. M. Nedderman, published by Cambridge University Press,‘Cambridge, 1992, 352 pp., fi0, ISBN o-521-40435.
Statics and Kinematics of Granular Materials is first and foremost a comprehensive and chronological record of the research carried out by the author and his successive research students at Cambridge University from the late sixties to the present. To this extent, as one of the previous students whose work is included in the book, it is very pleasing for me to see the book in print and I am sure I will cherish my copy for many years as a ‘nostalgic’ reminder of my beginnings in the field. The book shows Nedderman as an experienced and extensive researcher in many aspects of the mechanics of granular materials ranging from bulk frictional characterization of materials to prediction of bulk stress distributions, velocity fields and discharge rates in different flow regimes set up in bulk storage vessels. He is consequently able to offer a much more complete and comprehensive theoretical treatment of the subject area than for example the two recently published textbooks by P. A. Shamlou, ‘Handling of Bulk Solids: Theory and Practice’, Butterworths, 1988, which deals with a much wider list of topics at a more qualitative level and by A. Drescher, ‘Analytical Methods in BinLoad Analysis’, Elsevier, 1991, which concentrates mainly on bulk stress analyses rather than the prediction of flow fields and discharge rates. Yet, despite its comprehensive approach, and authoritative language, the book cannot be the ‘definitive’ textbook in the field which I am sure it aspires to be. The fundamental reason lies in the failure to date of the continuum mechanics approach to produce a conclusive and unifying theory of the bulk stress and flow fields observed in practice. Using the continuum approach, the stress distributions in granular materials are calculated by incorporating
Elsevier
Sequoia
’ f!ld veIOC$ e s are generated in mass-flow and funnel-
as ‘Mohr-Coulomb’ into the equations of ‘static equilibrium’ and defining the limiting ‘stress states’ before bulk failure, i.e. motion starts. This approach is largely based on the adaptation of quasi-static mechanics such as the plasticity theories of soil mechanics and appropriate modifications are proposed to deal with incompressible bulk failure and dynamic flow conditions which are encountered frequently during discharge of granular materials from storage vessels. Chapters 2 and 3 of the book define the terminology and the relationships to be used in the extensive mathematical formulations of the stress fields presented in chapters 4, 5 and 7. Whilst the approximate analytical solutions presented in chapter 5 are of considerable use in our appreciation of current silo design procedures, the ‘method of wedges’ solutions presented in chapter 4 and the exact numerical solutions presented in chapter 7 are primarily of educational benefit with limited consequence in most practical applications. The basic reason is that while the approximate theories appear to work reasonably well in axially-symmetric geometry, the exact stress field solutions which are often based on plane-strain (i.e. 2D) failure criteria require extensive numerical manipulation and are often found to produce discontinuities and numerical instabilities. Given these difficulties, the more persevering reader may feel somewhat cheated in having to master quite complicated mathematics which appear to take up more than a third of the text only to discover rather limited scope in practical application. I feel the book has much more to offer to the practising engineer in chapters 6, 8 and 10, the first of which discusses material characterization, the second the flow patterns and velocity fields in silos, and the third the prediction of discharge rates in silos. The coverage of bulk material and particle characterization techniques in chapter 6 is dealt with rather briefly and not enough reference is provided to the existing literature on the influence of particle properties on bulk frictional behaviour. However, the worked examples and the sample problems work very well in this chapter to illustrate the basic ideas to the potential student. The same commendable approach is taken in chapter 10 when presenting discharge rate calculations for both gravity-controlled and interstitial fluid-controlled flow conditions. It is also worth mentioning here that the only equations that appear to predict mass discharge rates reasonably well are the various modifications to the Beverloo et al. (1961) correlation; the purely analytical attempts to date still fail to provide accurate quantitative predictions. Perhaps the most significant shortcoming of the continuum approach is exposed in chapter 8 when
hoppers. In funnel-flow, a stress-independent model, known as the kinematic model, is adopted to predict the stagnant zone boundary and the prevailing velocity distributions in flat bottom and wide-angled hoppers. However, when dealing with mass flow hoppers, the kinematic model fails to predict the radial velocity field observed commonly in narrow-angled cones. In this case, the radial stress field solution is coupled with a radial velocity field by defining an arbitrary ‘flow rule’ such as the principle of co-axiality of the major principal stress and strain-rate directions. This approach is found to over-restrict grossly the angle of the mass flow boundary observed in conical hoppers. In an attempt to put things right, the Mohr-Coulomb failure criterion used extensively everywhere else in the book is replaced in chapter 9 with a conical yield criterion known as the ‘Extended Von Mises Failure Criterion’. Also in chapter 9, a new flow rule is introduced called the ‘Levy’s Flow Rule’ which takes into account volumetric strains as opposed to the planestrain flow rule used earlier. Predictions of the angle of the mass-flow boundary are somewhat improved with all these changes; however, in wide-angled cones, the shape of the stagnant boundary remains curved and this cannot be predicted by using the assumption of the radial stress field. On the other hand, the radial field is the only analytical solution possible as it is the only particular integral to the general stress field solution. Otherwise, a numerical method is required such as ‘the method of characteristics’ discussed extensively in chapters 7 and 8. To make matters even more difficult, the Von Mises criterion is found to result in equations which are not hyperbolic in form and are therefore not soluble using the method of characteristics. Upon reading this book, which represents a definitive mathematical treatment of the subject area, researchers like myself will I am sure feel more justified in exploring alternative approaches to flow modelling in hoppers based on discrete particle techniques such as contact mechanical and micro-structural analyses which are now fast catching up in popularity with the more established continuum approach. flow
Ugur Thin
Mixing in the Process Industries, Second Edition By N. Hamby, M. F. Edwards and A. W. Nienow, published by Butterworth - Heinemann, Oxford, 1992, 414 pp., $50, ISBN O-7506-1103.
Mixing is recognised as a ‘difficult’ area. Many practical problems are specific to particular materials
Book Reviews Physics of Granular Media Centre de Physique, Les Houches Series, edited by Daniel Bideau and John Dodds, published Nova Science, 1991, 434 pp., $89, Commack, New York, ISBN l-56072-034-4.
This book is a collection of papers taken from the winter school of the same name held in les Houches, France in 1990. The book is organised into separate sections consisting mainly of review papers. There is a section called, ‘geometry and structure of packings’; much of the discussion is about the random packing of spheres (e.g. local coordination numbers and Voronoi polyhedra). There are articles in this section by Rivier, Sadoc, Finney, Dodds, Oger and Jernot. An article by Meakin and Julien in the section on particle flow takes these geometrical aspects a step further to consider deposition and segregation of binary mixtures under a gravitational field in a simulation model. Moving on from purely geometrical aspects of random packing, there is a section on ‘mechanical properties’ in which frictional forces are considered. Articles were by Guyon, Georges, Desrues, Briscoe and Roux. Contact friction itself and the impact of friction and contact areas on localisation of deformation (i.e. relative sliding of semi-rigid volumes of material along failure boundaries) are the main themes. Local geometry was not really considered in anything more than in a simple sense (e.g. at the coordination number level) so as yet there does not appear to be much use made of the geometrical properties determined in the first section. Percolation theory and ‘fractals’ are used to provide a framework for a generic description of mechanical response of granular media. There was a particularly interesting article on shear bands by Desrues. Next followed a section on ‘transport properties in the pore space’ in which fluid flow through granular materials (treated mainly at the continuum level) was discussed. The geometry of the paths and the relative permeability of mixed fluids were two topics discussed. The authors included Dullien, Koplic and Hulin and co-workers. Another section on granular media in motion, called ‘particle flow, suspensions and fluidisation’ included articles by Ghadiri (on fluidised beds), Fauve (on the dynamics of avalanches in a rotating cylinder), Blanc (sedimentation), Couch and Hinch (aggregation during sedimentation using a fractal formalism) Meakin and Julien, and Savage (molecular dynamics simulations of couette flow of granular materials). There was an interesting article by Adams bringing contact mechanics
ideas to the flow of dense suspensions. The section was completed by articles on critical state model of compaction by Stanley-Wood, sintering by Jernot and liquid-phase sintering by Chaix. This is a very useful book, which provides an entrypoint to many areas of the granular materials literature, and I highly recommend it. D. M. Heyes
Statics and Kinematics of Granular Materials By R. M. Nedderman, published by Cambridge University Press,‘Cambridge, 1992, 352 pp., fi0, ISBN o-521-40435.
Statics and Kinematics of Granular Materials is first and foremost a comprehensive and chronological record of the research carried out by the author and his successive research students at Cambridge University from the late sixties to the present. To this extent, as one of the previous students whose work is included in the book, it is very pleasing for me to see the book in print and I am sure I will cherish my copy for many years as a ‘nostalgic’ reminder of my beginnings in the field. The book shows Nedderman as an experienced and extensive researcher in many aspects of the mechanics of granular materials ranging from bulk frictional characterization of materials to prediction of bulk stress distributions, velocity fields and discharge rates in different flow regimes set up in bulk storage vessels. He is consequently able to offer a much more complete and comprehensive theoretical treatment of the subject area than for example the two recently published textbooks by P. A. Shamlou, ‘Handling of Bulk Solids: Theory and Practice’, Butterworths, 1988, which deals with a much wider list of topics at a more qualitative level and by A. Drescher, ‘Analytical Methods in BinLoad Analysis’, Elsevier, 1991, which concentrates mainly on bulk stress analyses rather than the prediction of flow fields and discharge rates. Yet, despite its comprehensive approach, and authoritative language, the book cannot be the ‘definitive’ textbook in the field which I am sure it aspires to be. The fundamental reason lies in the failure to date of the continuum mechanics approach to produce a conclusive and unifying theory of the bulk stress and flow fields observed in practice. Using the continuum approach, the stress distributions in granular materials are calculated by incorporating
Elsevier
Sequoia
’ f!ld veIOC$ e s are generated in mass-flow and funnel-
as ‘Mohr-Coulomb’ into the equations of ‘static equilibrium’ and defining the limiting ‘stress states’ before bulk failure, i.e. motion starts. This approach is largely based on the adaptation of quasi-static mechanics such as the plasticity theories of soil mechanics and appropriate modifications are proposed to deal with incompressible bulk failure and dynamic flow conditions which are encountered frequently during discharge of granular materials from storage vessels. Chapters 2 and 3 of the book define the terminology and the relationships to be used in the extensive mathematical formulations of the stress fields presented in chapters 4, 5 and 7. Whilst the approximate analytical solutions presented in chapter 5 are of considerable use in our appreciation of current silo design procedures, the ‘method of wedges’ solutions presented in chapter 4 and the exact numerical solutions presented in chapter 7 are primarily of educational benefit with limited consequence in most practical applications. The basic reason is that while the approximate theories appear to work reasonably well in axially-symmetric geometry, the exact stress field solutions which are often based on plane-strain (i.e. 2D) failure criteria require extensive numerical manipulation and are often found to produce discontinuities and numerical instabilities. Given these difficulties, the more persevering reader may feel somewhat cheated in having to master quite complicated mathematics which appear to take up more than a third of the text only to discover rather limited scope in practical application. I feel the book has much more to offer to the practising engineer in chapters 6, 8 and 10, the first of which discusses material characterization, the second the flow patterns and velocity fields in silos, and the third the prediction of discharge rates in silos. The coverage of bulk material and particle characterization techniques in chapter 6 is dealt with rather briefly and not enough reference is provided to the existing literature on the influence of particle properties on bulk frictional behaviour. However, the worked examples and the sample problems work very well in this chapter to illustrate the basic ideas to the potential student. The same commendable approach is taken in chapter 10 when presenting discharge rate calculations for both gravity-controlled and interstitial fluid-controlled flow conditions. It is also worth mentioning here that the only equations that appear to predict mass discharge rates reasonably well are the various modifications to the Beverloo et al. (1961) correlation; the purely analytical attempts to date still fail to provide accurate quantitative predictions. Perhaps the most significant shortcoming of the continuum approach is exposed in chapter 8 when
hoppers. In funnel-flow, a stress-independent model, known as the kinematic model, is adopted to predict the stagnant zone boundary and the prevailing velocity distributions in flat bottom and wide-angled hoppers. However, when dealing with mass flow hoppers, the kinematic model fails to predict the radial velocity field observed commonly in narrow-angled cones. In this case, the radial stress field solution is coupled with a radial velocity field by defining an arbitrary ‘flow rule’ such as the principle of co-axiality of the major principal stress and strain-rate directions. This approach is found to over-restrict grossly the angle of the mass flow boundary observed in conical hoppers. In an attempt to put things right, the Mohr-Coulomb failure criterion used extensively everywhere else in the book is replaced in chapter 9 with a conical yield criterion known as the ‘Extended Von Mises Failure Criterion’. Also in chapter 9, a new flow rule is introduced called the ‘Levy’s Flow Rule’ which takes into account volumetric strains as opposed to the planestrain flow rule used earlier. Predictions of the angle of the mass-flow boundary are somewhat improved with all these changes; however, in wide-angled cones, the shape of the stagnant boundary remains curved and this cannot be predicted by using the assumption of the radial stress field. On the other hand, the radial field is the only analytical solution possible as it is the only particular integral to the general stress field solution. Otherwise, a numerical method is required such as ‘the method of characteristics’ discussed extensively in chapters 7 and 8. To make matters even more difficult, the Von Mises criterion is found to result in equations which are not hyperbolic in form and are therefore not soluble using the method of characteristics. Upon reading this book, which represents a definitive mathematical treatment of the subject area, researchers like myself will I am sure feel more justified in exploring alternative approaches to flow modelling in hoppers based on discrete particle techniques such as contact mechanical and micro-structural analyses which are now fast catching up in popularity with the more established continuum approach. flow
Ugur Thin
Mixing in the Process Industries, Second Edition By N. Hamby, M. F. Edwards and A. W. Nienow, published by Butterworth - Heinemann, Oxford, 1992, 414 pp., $50, ISBN O-7506-1103.
Mixing is recognised as a ‘difficult’ area. Many practical problems are specific to particular materials