Gravity flow of bulk solids

There is hardly an industry in existence which does not use solid materials in bulk form. Where the volume of the solids is substantial, gravity is usually relied upon to cause the solids to flow. Such materials as ores, coal, cement, flour, cocoa, soil, to which the general term of bulk solids is applied, flow by gravity or are expected to flow by gravity in thousands of installations and by the billions of tons annually. Mining relies on gravity flow in block-caving and in ore passes; subsidence is a case of gravity flow of solids. Agriculture relies on gravity flow of its products in storage silos, in feed plants and on the farms. Every type of processing industry depends on gravity flow of some solid, often of several solids. Although vast quantities of bulk solids have been handled for many years, the author believes that this is the first comprehensive study of the subject. The fact that this work appears at this time is not accidental, but stems from the progress achieved during the past fifteen years in the mathematical theory of plasticity and in the techniques of numerical calculation. On the basis of recently developed and refined principles of plasticity, the problem of flow of bulk solids has been set up in mathematical terms. A few years ago, this would not have been possible; just as a few years ago the mathematically formulated problem would have been practically insoluble because there were no computers to carry out the necessary calculations. The careful reader of the author's previous reports and papers on the subject of flow of bulk solids will notice substantial modifications in the design formulae. No apology is offered for these seeming inconsistencies; the author has always approached the subject from the standpoint of the engineer who has had to provide definite recommendations on the basis of information at hand, at the time. Hence, as the volume of experience increased, the theory was developed, and the numerical data were computed, the design methods improved and changed - at times, radically* The work is presented in six parts. In Part I, the yield function applicable to bulk solids is described, and the flow properties of bulk solids are defined. The solids are assumed to be rigid-plastic, isotropic, frictional, and cohesive. During incipient failure, the solids expand (dilate), during steady state flow, they may expand or contract. The yield function is consistent with the principle of normality [7] which is specifically applied in incipient failure. Part II contains the theory of steady state gravity flow of solids in converging and vertical channels. The equations are first derived in a general form, applicable to problems of extrusion as well as gravity flow, in plane strain and in axial symmetry. Some of the derivations are more general than they need to be for this work. They will be referred to in other publications which are now in preparation [22, 23]. It is shown that, provided the slopes of the walls of a converging channel are sufficiently steep and mathematically continuous, the stress pattern in the neighborhood of the vertex of the channel is, primarily, a function of the slope and of the frictional conditions of the walls at the vertex, with the influence of the top boundary of the channel vanishing at the vertex . The particular stress field which develops at the vertex is called the radial stress field, because it is the field which can lead to a radial velocity field. Since the radial stress field is closely approached in the vicinity of the vertex, that field represents the stresses at the outlet of a channel. The region of the outlet of a channel is most important because it is there, that obstructions to flow originate. The radial stress field thus provides a basis for a general solution of flow in this important region of the channels. In Part III, the conditions leading to incipient failure are considered. General equations of stress are derived in plane strain and in axial symmetry. The conditions following incipient failure are discussed, and it is suggested that the velocity fields usually computed for conditions of failure are meaningless and that only initial acceleration fields can be computed. Two cases of incipient failure are analyzed: doming across a flow channel, and piping (which refers to a state of stress around a vertical, empty hole of circular cross-section). Part IV describes the flow criteria. The material developed in the previous three parts is brought together to relate the slopes of channels and the size of the outlets necessary to maintain the flow of a solid of given flowability on walls of given frictional properties. Part V describes the testing apparatus and the method which has been developed to measure the flowability of solids, their density, and the angle of friction between a solid and a wall. Finally, Part VI contains the application of the theory to the design of storage installations and flow channels, and discusses flow promoting devices, feeders, segregation, blending, structural problems, the flow of ore, as well as aspects of block-caving and miscellaneous items related to the gravity flow of solids. All these topics are approached from the standpoint of flow: their effect on flow and vice-versa. The reader will soon realize that many of the bins now in operation have been designed to fill out an available space at a minimum cost of the structure rather than to satisfy the conditions of flow. The result has been a booming business for manufacturers of flow promoting devices. While there are, and always will be, solids which are not suitable for gravity flow, the vast majority of them will flow if the bins and feeders are designed correctly. However, a correct bin will usually be taller and more expensive. It is up to the engineer to decide whether the additional cost of the correct bin will be balanced by savings in operation. This part is made as self-contained as possible to facilitate its reading to the engineer who has neither time nor inclination to study the theoretical parts. The reader versed in soil mechanics should note that the magnitude of the stresses discussed here is 100 to 1000 times smaller than that encountered in soil mechanics. Hence, some phenomena which may not even be observable in soil mechanics assume critical importance in the gravity flow of solids. For instance, the curvature of the yield loci (Mohr envelopes) in the ( cf, t ) coordinates is seldom detectable in soil mechanics, but in gravity flow the curvature assumes an important role in the determination of the flowability of a solid. By the terminology of soil mechanics, solids possessing a cohesion of 50 pounds per square foot are cohesionless: standard soil mechanics tests do not measure such low values. But a solid with that value of cohesion, an angle of internal friction of 30°, and a weight of 100 pounds per cubic foot can form a stable dome across a 3-foot-diameter channel and prevent flow from starting. In gravity flow, it is of interest to be able to predict whether or not flow will take place through a 6-inch-diameter orifice. This involves values of cohesion down to 8 pounds per square foot and even less for lighter solids.

........... BULLETIN OF THE UNIVERSITY OF UTAH Volume 52 October 1961 Bulletin No. 108 of the UTAH ENGINEERING EXPERIMENT STATION G R A V I T Y F L O W O F B U L K S O L I D S by A. W. Jenike Salt Lake City, Utah / No. 29 PREFACE There is hardly an industry in existence which does not use solid materials in bulk form. Where the volume of the solids is substantial, gravity is usually relied upon to cause the solids to flow. Such materials as ores, coal, cement, flour, cocoa, soil, to which the general term of bulk solids is applied, flow by gravity or are expected to flow by gravity in thousands of installations and by the billions of tons annually. Mining relies on gravity flow in block-caving and in ore passes; subsidence is a case of gravity flow of solids. Agriculture relies on gravity flow of its products in storage silos, in feed plants and on the farms. Every type of processing industry depends on gravity flow of some solid, often of several solids. Although vast quantities of bulk solids have been handled for many years, the author believes that this is the first compre­hensive study of the subject. The fact that this work appears at this time is not accidental, but stems from the progress achieved during the past fifteen years in the mathematical theory of plas­ticity and in the techniques of numerical calculation. On the basis of recently developed and refined principles of plasticity, the problem of flow of bulk solids has been set up in mathematical terms. A few years ago, this would not have been possible; just as a few years ago the mathematically formulated problem would have been practically insoluble because there were no computers to carry out the necessary calculations. The careful reader of the author's previous reports and papers on the subject of flow of bulk solids will notice substantial modifica­tions in the design formulae. No apology is offered for these seeming inconsistencies; the author has always approached the subject from the standpoint of the engineer who has had to provide definite recommenda­tions on the basis of information at hand, at the time. Hence, as the volume of experience increased, the theory was developed, and the numerical data were computed, the design methods improved and changed - at times, radically* The work is presented in six parts. In Part I, the yield function applicable to bulk solids is described, and the flow properties of bulk solids are defined. The solids are assumed to be rigid-plastic, isotropic, frictional, and cohesive. During incipient failure, the solids expand (dilate), during steady state flow, they may expand or contract. The yield function is consistent with the principle of normality [7] which is specifically applied in incipient failure. Part II contains the theory of steady state gravity flow of solids in converging and vertical channels. The equations are first derived in a general form, applicable to problems of extrusion as well as gravity flow, in plane strain and in axial symmetry. Some of the derivations are more general than they need to be for this work. They will be referred to in other publications which are now in pre­paration [22, 23]. It is shown that, provided the slopes of the walls of a converging channel are sufficiently steep and mathematically continuous, the stress pattern in the neighborhood of the vertex of the channel is, primarily, a function of the slope and of the fric­tional conditions of the walls at the vertex, with the influence of the top boundary of the channel vanishing at the vertex . The parti­cular stress field which develops at the vertex is called the radial stress field, because it is the field which can lead to a radial velocity field. Since the radial stress field is closely approached in the vicinity of the vertex, that field represents the stresses at the outlet of a channel. The region of the outlet of a channel is most important because it is there, that obstructions to flow originate. The radial stress field thus provides a basis for a general solution of flow in this important region of the channels. In Part III, the conditions leading to incipient failure are considered. General equations of stress are derived in plane strain and in axial symmetry. The conditions following incipient failure are discussed, and it is suggested that the velocity fields usually computed for conditions of failure are meaningless and that only initial acceleration fields can be computed. Two cases of incipient failure are analyzed: doming across a flow channel, and piping (which refers to a state of stress around a vertical, empty hole of circular cross-section). Part IV describes the flow criteria. The material developed in the previous three parts is brought together to relate the slopes of channels and the size of the outlets necessary to maintain the flow of a solid of given flowability on walls of given frictional properties. Part V describes the testing apparatus and the method which has been developed to measure the flowability of solids, their density, and the angle of friction between a solid and a wall. Finally, Part VI contains the application of the theory to the design of storage installations and flow channels, and discusses flow promoting devices, feeders, segregation, blending, structural problems, the flow of ore, as well as aspects of block-caving and miscellaneous items related to the gravity flow of solids. All these topics are approached from the standpoint of flow: their effect on flow and vice-versa. The reader will soon realize that many of the bins now in opera­tion have been designed to fill out an available space at a minimum cost of the structure rather than to satisfy the conditions of flow. The result has been a booming business for manufacturers of flow promoting devices. While there are, and always will be, solids which are not suitable for gravity flow, the vast majority of them will flow if the bins and feeders are designed correctly. However, a correct bin will usually be taller and more expensive. It is up to the engineer to decide whether the additional cost of the correct bin will be balanced by savings in operation. This part is made as self-contained as possible to facilitate its reading to the engineer who has neither time nor inclination to study the theoretical parts. The reader versed in soil mechanics should note that the magnitude iv of the stresses discussed here is 100 to 1000 times smaller than that encountered in soil mechanics. Hence, some phenomena which may not even be observable in soil mechanics assume critical importance in the gravity flow of solids. For instance, the curvature of the yield loci (Mohr envelopes) in the ( cf, t ) coordinates is seldom detectable in soil mechanics, but in gravity flow the curvature assumes an import­ant role in the determination of the flowability of a solid. By the terminology of soil mechanics, solids possessing a cohesion of 50 pounds per square foot are cohesionless: standard soil mechanics tests do not measure such low values. But a solid with that value of cohesion, an angle of internal friction of 30°, and a weight of 100 pounds per cubic foot can form a stable dome across a 3-foot-diameter channel and prevent flow from starting. In gravity flow, it is of interest to be able to predict whether or not flow will take place through a 6-inch-diameter orifice. This involves values of cohesion down to 8 pounds per square foot and even less for lighter solids. v ACKNOWLEDGEMENTS The work described in this report has been carried out over a period of some nine years, and during that time the author has become indebted to a number of persons who have contributed of their time and skills, and to a number of institutions which for the past five years have given financial support to the project. The author is particularly grateful to Dr. P. J. Elsey of the Utah Engineering Experiment Station for his constant and sympathetic interest in the project, and to Dr. Elsey and Professor R. H. Woolley for their assistance in setting up the Bulk Solids Flow Laboratory at the University of Utah; to Professor R. T. Shield of Brown Univer­sity for the many long discussions of the topics of plasticity and for his critical revues of the work at various stages of advancement. The author is very much in debt to his students: Joseph L. Taylor, who has contributed of his mathematical skill, and, especially, Jerry R. Johanson, whose constant assistance in every facet of the work has been most useful. Mr. Johanson, a Ph.D. candidate, also carried out all of the numerical calculations which this work required. The cost of this project has been substantial and the author wishes to acknowledge the initial support which he received from the American Institute of Mining Mineralogical and Petroleum Engineers, whose Mineral Beneficiation and Research Committees promptly recommended the author's application for AIME sponsorship. This was followed by a grant of money from Engineering Foundation and by the further support from research funds of the Utah Engineering Experiment Station. The AIME and the Engineering Foundation have remained sponsors of the project. Sincere thanks are due to Dr. Carl J. Christensen, director of the Utah Engineering Experiment Station, for his help in keeping the project alive through times of financial difficulty. The main support for the applied part of the project, entitled "Bulk Solids Flow", has come from the American Iron and Steel Institute to whom the author is most grateful. The mathematical concepts described in this report, as well as other work which is appearing separately, have been developed under a 1959 grant from the National Science Foundation to a project entitled "Flow of rigid-plastic solids in converging channels under the action of body forces". Andrew W. Jenike October, 1961 vii CONTENTS PART I - THE YIELD FUNCTION 1 Introduction 1 The coordinate system 5 Effective yield locus 9 Stresses and density during flow 10 Yield locus 15 Time yield locus 22 Stresses during failure 22 Flow-function 24 Flowfactor 26 Wall yield locus 28 PART II - STEADY STATE FLOW 35 General equations 35 Stress field 35 Velocity field 37 Superposition 40 Physical conditions 40 Grids, special lines and regions 42 Converging channels 57 Equations of stress 57 Radial stress field 59 Derivation 59 Solutions of the radial stress field 63 Resultant vertical force 68 Stresses at the walls 84 Influence of compressibility 84 General stress field 107 Proof of convergence to a radial stress field at the vertex 107 Boundaries 114 viii Radial velocity field 119 Vertical channels 124 Stress field 128 Velocity field 132 PART III - INCIPIENT FAILURE 135 General equations 135 Stress field 137 Initial acceleration field 138 Superposition 143 Physical conditions 143 Grids and special lines 143 Doming 145 Piping 148 PART IV - FLOW CRITERIA 156 Introduction 156 No-doming 156 Plane and axial symmetry 157 Plane asymmetry 158 Flowfactor plots 160 Influence of compressibility 160 No-piping 176 PART V - TESTING THE FLOW PROPERTIES OF BULK SOLIDS 182 Apparatus 182 Testing 186 Continuous flow 186 (a) Representative specimen 186 (b) Uniform specimen 188 (c) Flow 190 (d) Shear 195 Example 198 ix . Time effect 202 Density 204 Plots of flow properties 204 Angle of friction ' 206 PART VI - DESIGN 208 Introduction 208 Flow properties of bulk solids 209 Limitations of the analysis 217 Types of flow 218 Mass flow 219 Hopper& with one vertical wall 228 P uig f low 230 Calculations of the dimensions of the outlet 231 (a) Doming 231 ' (d) Piping . 234 (c) Particle interlocking ' 236 Influence of dynamic over-pressures 236 - Flow ptomor.ing devices 242 Examples of design for, flow 248 Feeders 268 Feeder Loads 272 Belt feeder 274 Side-discharge reciprocating feeder 278 Segregation and blending in flow 282 Flooding 286 Heat transfer 288 Gas counterflow 288 Structural problems 292 Stresses acting on hopper walls 292 Bin failures 292 Ore 294 . Broken rock 294 x Coarse ore 301 Block-caving 304 REFERENCES 307 xi PART I THE YIELD FUNCTION ' Introduction In gravity flow of solids, as in soil mechanics, it is convenient to assume pressures and compressive strain rates as positive, and tensions and expansive strain rates as negative. This convention is adopted throughout the work. The solids which are considered in this work are rigid-plastic. In the plastic regions, the solids are assumed to be isotropic, fric­tional, cohesive and compressible. During incipient failure an element of a solid expands, while during steady state flow, the element either expands or contracts as does the pressure along the streamline. While many problems of continuous plastic flow have been solved for isotropic, non-work-hardening solids with a zero angle of friction [e.g., 1,2,3,4,5], attempts to work out solutions of continuous flow of solids, which exhibit an angle of friction greater than zero, have not been successful. The cause of the difficulty has lain in the yield function ascribed to these solids. The yield function was a generalization of the criterium of either Tresca or von Mises into a function dependent on the hydrostatic stress. In the principal stress space such a generalization transformed the prism of Tresca and the cylinder of von Mises into, respectively, a pyramid and a cone, -1- which were assumed to be of constant size and to extend without a bound in the direction of the hydrostatic pressure. As a result, the princi­ple of plastic potential [6 ], or normality [7], required the solid to dilate continuously during flow while at the same time retaining its strength properties. Continuous dilation is not supported by physical observations. Dilation implies a reduction in density which in turn causes a loss of strength and a shrinking of the yield surface. There is ample evidence obtained from shear and triaxial tests to the effect that a solid may flow without a change of density as well as with an increase of density, and that during flow the yield surface of an element of the solid at a generic point is remarkably independent of the history of stress and strain [8 ]. In this work a yield surface recently proposed by Jenike and Shield [9] is used, This surface is shown in Fig. 1, in principal stress space. The abscissa a£v/2 = is in the cr^, c^-p lane and bisects the angle between the a^c^-axes. This surface is the Shield's pyramid [10] with three modifications: the pyramid is bounded on the pressure side (after Drucker [11])by a flat hexagonal base perpendicular to the octahedral axis; the size of the pyramid is a function of the density, the time interval of consolidation at rest, the temperature, and the moisture content of the solid; and the vertex of the pyramid is rounded off. During flow, the time interval of consolidation is zero, while the temperature and moisture content are assumed constant; density is variable and is assumed a function of the major pressure at a generic -2- Fig. 1 Yield surface -3- point. In consequence, the size of the yield surface during flow is a function of the major pressure only, while the change in the size of the yield surface (and in density) of an element becomes a function of the gradient of the major pressure along the path of that element. A change of density is measured by the normal component of the strain rate vector, which thus must be free to assume a positive or a negative direction depending on the sign of the pressure gradient, and independently of the state of stress at the generic point. The adopted yield surface allows this freedom to the strain rate vector because, during flow, the vector is located at a corner between the side walls of the pyramid and its flat, hexagonal base, as shown in Fig. 1. In the plane strain flow of an incompressible solid, normality locates the stresses on a straight side of the hexagonal base off the yield surface. In the plane strain flow of a compressible solid and in axi-symmetric flow, which involve three dimensional deformations, normality locates the stresses at a corner of the hexagonal base. It will thus be observed that in axial symmetry the principles of isotropy and plastic potential enforce the Haar and von Karman hypothesis [12] for the adopted yield function, except possibly when the principal stresses in the meridian plane are either both major or both minor. However, the latter conditions exclude all fields with body force*, hence are useless in this work. The Haar and * Assume the meridian pressures to be both minor, then equations (14) - (16) and (20) - (22) become a = a = ct0 = a(l - sin 6), t =0, cr = a, = a( 1 + sin 6) x y 2 xy (X 1 -4- von Karman hypothesis states that in axial symmetry the circumferential stress is equal to either the major of the minor stress of the meridian plane. The relationship between the size of the yield surface and the major pressure during flow is described by the effective yield locus [9]. The remarkable feature of this yield function is that not only does it not complicate the analysis of the stress fields but for steady state flow it leads to a pseudo-static system without pseudo­cohesion even though the solid may be cohesive. In the analysis of incipient failure, it will be necessary to assume a constant yield surface throughout the plastic region. This is not a serious limitation because the considered plastic regions are of small size. The stresses of incipient failure are located on the side of the yield pyramid, not on the base, and dilation accompanies failure. The coordinate systems In order to handle problems of plane strain and of axial symmetry with one set of equations, combined coordinates are introduced with a and the solution of the equations of equilibrium (48) and (49), with m = 1 , is of the form 2 sin S „ _ / , \1-sin 6 T = 0 , a = a0 (y/y0) . This requires the absence of body forces. Similar functions are obtained for two major meridian pressures, and for the conditions of incipient failure. -5- coefficient m to distinguish between the two systems. Coefficient m - 0 applies to plane strain, and m = 1 applies to axial symmetry. Two systems of coordinates will be found useful: a plane-Cartesian/ polar-cylindrical system x, y, OC, and ,a polar/spherical system r, 6, Ct, as shown in Fig. 2. Axis x is vertical and when the problems have symmetry they are symmetric with respect to this axis. The circumfer­ential coordinate OC appears only in the problems of axial symmetry and by virtue of that symmetry all the derivatives with respect of OC are zero. The positive directions of the stresses are shown in the Figure 2. It will be noted that pressures are assumed positive. The direction of the major pressure a^ with respect to the axis x is measured by angle go. Evidently co = e + t, ' (3) where \|r is the angle between the directions of and of the ray r. Two kinds of stresses are recognized: Consolidating stresses which occur during steady state flow and are denoted by letters a and t, and yield stresses which occur during incipient failure and are denoted by letters a and x. In both cases the principal pressures and and a act in the meridian plane (x,y) or (r,9), while the principal pressure a' (cL) is the circumfer- Ct c* ential pressure. The principal pressures are ordered as follows (2) (1) -6 - Fig. 2 Coordinate systems in plane strain and in the meridian plane of axial symmetry (4) The assumed yield function is of the following general form (5 ) where T is the bulk density of the solid, t the time interval of consolidation at rest, T its temperature, and H its surface moisture content„ It should be noted that the method by which a solid is consolidat­ed to the given density T may affect the yield function. For instance, a solid may be consolidated by vibration, or pounding; it may be consolidated by the application of a hydrostatic pressure, as well as by the application of pressures which are different in magnitude but whose deviator components are insufficient to cause shear. Then, and this is of main interest in our study, a solid may be consolidated under a set of pressures which cause a continuous deformation of the solid: this is the condition of flow. Finally, flow may be stopped for an interval of time t with the consolidating pressures remaining practically unchanged and with the solid undergoing additional con­solidation at rest. The bulk density of a solid is assumed to be a function of the majore consolidating pressure o^, as well as of the time t, the temp­erature T, and the moisture content H, thus T = T(J15t,T,H). (6) Effective yield locus (EYL) [9] During steady state flow, within the regions of non-zero velocity, the solid deforms continuously without abrupt changes in bulk density, and the plastic region is uniformly at yield with yield planes passing through every point of the region. In these regions, the time interval of consolidation at rest is zero, while the temperature and moisture content can usually be assumed constant, t = 0, T = constant, H = constant. (7) Density then becomes a single-valued function of the major con­solidating pressure while the yield function, eq. (5), reduces to f (a15a2) = FCap . (8) Experimental data show that the ratio between the major and the minor consolidating pressures during flow approaches a constant value*, « 1 + sin 5 . ct2 1 - sin 5 w This function is called the effective yield locus (EYL), and 6 is referred to as the effective angle of friction. In general, 5 is a function of the temperature T and the moisture content H of the solid, 5 = S (T, H) , (10) but, under conditions of flow and with relations (7) in force, 6 is constant. The equation of the effective yield locus (9) can also be expressed by means of the stress components as follows * See also reference [13], Fig. 1.2.2. -9- sin 5 = v ' * J -------(11) or 1 xQ i x 2 V + S' <i 1 + a y J p r - v2 + 4t 2 r 9 li Q + (12) In principal stress space, function (9) is represented by the side OAB of Shield's pyramid [10] with its vertex at the origin, Fig. 3. This pyramid is also of hexagonal cross-section but, unlike the yield function, Fig. 1, extends into the direction of hydrostatic pressure without a base. In the (cj,t) coordinates this function is represented by two straight lines, EYL passing through the origin and inclined at the angle S to the cr-axis, Fig. 4. These lines are envelopes of Mohr stress circles determining the consolidating pressures and 1 ■ Stresses and density during flow ' It is convenient to introduce a mean pressure " a .+ a „ a + a a +an , 1 2 x y r Q a - . (13) The component stresses can now be expressed in the plane-Cartesian/ polar-cylindrical coordinates by a = a(l + sin 8 cos 2oo) , (14) a = a(l - sin 5 cos 2a>) , (15) T = o sin & sin 2a>: (16) xy ' v ' -10- Fig. 3 Effective yield surface -11- and in the polar/spherical coordinates by cr^ = a(l + sin 6 cos , (17) Oq = a(l - sin 6 cos 2v|/) , (18) T Q = o sin 8 sin . (19) rt7 The principal pressures are = a(1 + sin 5), (20) a2 = o(l - sin 8), (21) Oq, = a( 1 + k sin 8), (22) where k = + 1, (23) for converging flow, locates the stresses on the edge OA of the pyramid, Fig. 3, while k = -1, (24) for diverging flow, locates the stresses on the edge OB of the pyramid. On the strength of the relations (6), (7) and (20), the bulk density during flow becomes of the form r = r(a) . (25) This relation has been found experimentally to be well represented by the equation T = To(l + a)? (26) where To and (3 are constant under conditions of flow. Tests show that for a measured in pounds per square foot, (3 does not exceed,10. The method of measuring g is described in reference [14] and the results of -12- Fig. 4 Effective yield locus Fig. 5 Yield loci -13- tests for several solids for a range of major pressure cr^ from 150 to 2,500 pounds per square foot are shown in Table 1. Table 1 * £ • - ........ i .... . Solid P Adipic acid .026 Soybean oil meal (dry) .010 Gocoa powder .096 Light soda ash .017 Foundry sand .009 Iron ore (6% H^O) .076 Taconite concentrate .029 Taconite concentrate .036 Copper concentrate (dry) .009 Copper concentrate (57o H?0) .055 Feed granules .017 Feed granules .020 Eq. (26) leads to awkward mathematical expressions and in parts of the analysis will be replaced by ........... T " TocP, (27) This is justified by the fact that, in many interesting parts of a field, a is large compared to unity, and eq. (27) is practically equivalent to eq. (26). It should be noted that with the assumption of incompressibility, (3 - 0, and both equations (26) and (27) yield T = To- -14- Yield locus (YL) The yield function defined by conditions (7) and eq. (8) is repre­sented by a family of yield loci in the ( ct, t ) coordinates. The major consolidating pressure is the parameter of the family. In Fig. 5 two yield loci denoted YL* and YL" are shown; these yield loci were generated by the major consolidating pressures and a^. The properties of a yield locus, Fig. 6 , will now be discussed in some detail. 1 The stresses acting in a cross-section of a solid are described by a stress vector whose component are: the normal pressure a and the shear stress t. The yield locus is the locus of the values of (ct,t) at which permanent deformation, or yield, occurs. In plast­icity, the equations of equilibrium are assumed satisfied, therefore, the stresses described by the yield locus cannot be exceeded. This implies that the yield locus is the envelope of the Mohr stress circles at yield. For any stress condition represented by a Mohr circle A, not touching the yield locus, the solid is rigid (or elastic). When the stress condition changes so that the corresponding Mohr circle A' comes in contact with the yield locus, yield stresses, described by the points B, develop in the two planes of the solid inclined at angles -j- _ -(#,/4 - i/2) to the direction of the major pressure G^, and the solid deforms. These two planes are called the slipplanes, and are represent­ed by two slip lines in the principal, physical plane x-y, Fig. 7. Angle i is the angle of friction of the.solid. The strain rate which accompanies a yield stress is described by -15- a e Yield locus -16- Fig. 7 Sliplines Unconfined yield pressure, f -17- the strain rate vector e , whose components are: the normal strain rate e and the shear strain rate y„ If the coordinates (e,Y) are superimposed over the coordinates (o,t), Figures 5 and 6 , then by the principle of normality [?], the strain rate vector e is normal to the yield locus at the point of contact with the Mohr stress circle. It is evident from Fig. 6 that any point of contact, B, enforces a direction of the strain rate vector which contains a negative, hence expansive, normal compo­nent of strain rate and, therefore, implies dilation of the solid. The only exception is point E, the terminus of the yield locus, at which normality only restricts the direction of the strain rate vector to within a sector <t>, - ff/2, shown in Fig. 5. When the Mohr circle is tangential to the yield locus at point E, normality allows the solid either to dilate, or to contract, or to deform without change of density. This is the condition which occurs during steady flow. It is observed that the angle, of friction <£> is not constant along the yield locus but varies from a minimum at points E to Jt/2 at the intercept with the cr-axis. The shape of the yield locus at low values of a is important in this work because it affects the value of the major pressure f which causes failure at a traction free surface. f is ,»< ....... c c defined thus °2 ~ O ' CT1 = fc" f is called the unconfined yield pressure and is obtained by inscrib­ing a Mohr yield circle through the origin 0, Fig. 8 . The curvature of the yield locus at low values of a is not generally recognized and lacking complete experimental verification, the following -18- Fig. 9 Unlikely shape of the yield locus Fig. 10 Tensile (brittle) failure -19- arguments are offered in support of this concept: (a) Direct shear tests at low values of pressure show a downward curving of the yield locus. (b) If the yield locus were to intersect the a-axis at an angle other than »/2, as shown in Fig. 9, the solid would be stable under a hydrostatic tension but would fail if the tensions were reduced to those given by the Mohr circle. This appears unreasonable. (c) The yield locus shown in Fig. 10 allows for both, tensile (brittle) failure, and shearing failure of the solid. Namely, there exists a limiting circle of a radius equal to the radius of curvature of the yield locus at point (ao ,0) such that all stress conditions represented by circles within the limiting circle approach the yield locus at point (ao ,0), where the shear stress is zero, causing failure in tension; all other stress conditions are represented by Mohr circles which approach the yield locus at non-zero values of shear, causing failure in shear. (d) The failure of a dome over a cavity, Fig. 11, often proceeds in successive stages which can be observed. The domes are usually smooth and rounded off at the top. At the two abutments of a dome, failure occurs in shear along slipines belonging to a different family at each abutment. From observations it appears that these sliplines merge at the top of the dome. Sliplines of different families are inclined to each other at an angle of jt/2 - tfS. In order for these sliplines to merge, the angle of friction at the point of mergence must equal a/2. It seems that, at the top of the dome, failure does -20- . Fig. 11 Failure of a dome -21- occur in tension, and ^ = *t/2 . It is evident that an accurate determination of the value of the yield locus at point C, Fig. 8 , is necessary to obtain a reliable value of f£. A linear extrapolation of the yield locus from test values obtained at pressures considerably higher than C would give erroneous results. Time yield locus (TYL) .• j If flow is stopped for an interval of time t, the consolidating pressures remain practically unchanged and the solid undergoes further consolidation at rest. This may cause an expansion of the yield loci throughout the solid. The new yield loci are called time yield loci. A typical time yield locus (TYL), together with a yield locus (YL), is shown in Fig. 12. It should be remembered that both, temperature and moisture content, are parameters in the yield function (5 ) and, if either of them should change during the time of consolidation, the position of the time yield locus may be affected. ■ - ' I Stresses during failure In order to express the stresses during failure in a tractable form, the yield locus of Fig. 6 is linearized as shown in Fig. 13. In linearization, the value of f and the size and position of the consolidating stress circle are left unchanged. The stresses can now be expressed in the plane-Cartesian/cylindrical-polar coordinates -22- Fig. 12 Time yield locus -23- by a = a(l + sin ^ cos 2cu) - f ^ s'i'- ^ (29) x c 2 sin p - . / n \ „ 1 - s in ^ a = o(l - sin <b cos 2d>) - f - - - - 7 , y c 2 s m p (30) T = cr sin & sin 2co; (31) xy ' and in the polar/spherical coordinates, by a = a(l + sin i cos 2\|/) - f -r^~p, (32) r c 2 s m p *= a(l - sin & cos 2t) ~ f - sin-7^ (33) y c /. s m p t ■- cr sin ^ sin 2ilr. (34) r0 The principal pressures are an = a(l + sin $>) - f - - Sin^~, (35) 1 c 2 sin 0 a = a(l - sin ^) - f -- (36) Z c Z a in p cr = afl + k sin - f - -- (37) O! ' c 2 s m p In the above equations - J 1 °2 . 1 - sin <t> . . 0 = -- - 2 - + f -----7- 1 , (38) c 2 s m p and k = +1 for converging failure, and k = -1 for diverging failure, the same as for flow. Flow-function The concept of the flow-function is introduced as a measure of the flowability of solids. This concept is obtained by substituting relation (6) for y in eq. (5), and by placing the minor yield pressure -24- Fig. 13 Linearized yield locus -25- a2 ~ 0. The corresponding value of the major pressure is the unconfined yield pressure f , eq. (28). Eq„ (5) then assumed the form fc = G(cr1 ,t,T,H), (39) and this relation is called the flow-function of a solid. It is usually plotted as *C - W - <40) Fig. 14, with t., T and H as parameters. A flow-function measured without consolidation at rest (t = 0) is referred to as the instantaneous flow-function, while a flow-function measured with consolidation at rest (t i- 0) is referred to as the time flow-function. Low values of f indicate a high flowability of the solid, and vice versa. In particular, a perfectly free flowing solid is one whose f is zero for all values of the major consolidating pressure cr^. In Fig. 14 such a flow-function line coincides with the cr^-axis. Flowfactor ff The concept of the flowfactor is introduced as a measure of the flowability of channels. The flowfactor ff is defined as the ratio a! ff = (41) c and represented by a straight line in the (a^,fc) coordinates, Fig. 15. In the design for no-piping an instantaneous flowfactor and a time flowfactor are used. The former applies to solids which are not -26- Fig. 14 Flow-function Fig. 15 Flowfactor -27- affected by consolidation at rest, while the latter applies to those whose time flow-function exceeds the instantaneous flow-function by, say, 20%. Wall yield locus (WYL) A side boundary between a region in a plastic state of stress and a rigid (or elastic) stationary region is called a wall. In gen­eral, there is a velocity discontinuity along a wall, the wall frictional strength is fully mobilized, and the stresses acting on the wall lie on a wall yield locus, which is represented by a line WYL in the (ct,t) coordinates, Fig. 16. Since the solid is in a plastic state, the stresses at the wall lie at one of the points of intersection W of the wall yield locus with a Mohr stress circle tangential to the yield locus of the solid,Y'L. During flow, the circle is tangential to the yield locus at the points E and is also tangential to the effective yield locus (not shown in Fig. 16). The position of the wall yield locus depends on the frictional conditions at the wall. These conditions may range from perfectly smooth (in concept, at least) to the full strength of the flowing solid. In the former case, the wall yield locus is represented by the positive part of the cr-axis, the stresses at the wall are defined by one of the points M, and the wall can transfer no shear stress. In the latter case, the wall yield locus merges with the yield locus of the solid and the stresses at the walls are given by one of the points E (or points B in incipient failure)„ Such a wall will be referred to as a I i T Fig, 16 Wall yield locus -29- "rough wall". A rough wall is a slip line. The wall yield locus shown by line WYL in Fig. 16 denotes a degree of weakness of the wall as compared to a rough wall and such a wall will be referred to as a "weak wall". In this work, the stresses at the walls assume values which lie on the arc E'ME", hence the stresses at a weak wall are represented either by point W' or W"„ Observations of flow patterns in models and measurements of wall yield loci indicate that the introduction of a wall made of an extran­eous material, even a coarse material, causes a significant drop in the cohesive and frictional forces at the wall. There seems to be no practical way of gradually decreasing the weakness of a wall by, say, increasing its coarseness until the wall yield locus merges with the yield locus of the solid. Experiments indicate that for weak walls the points W locate within the arc T'MT" of the Mohr stress circle, and that the wall yield locus can be linearized without a significant loss of accuracy. Since linearization reduces the amount of testing necessary to define the wall yield locus, and greatly simplifies the analysis, it will be adopted in this work. The position of a wall yield locus becomes thus fully determined by the magnitude of the angle of friction £$' between a solid and a wall. In plane strain the channel may be asymmetric and the frictional conditions at each wall may be different, as shown in Fig. 17. In this case, the relations have to be developed separately for each wall. The values relating to a point of a wall inclined at angle 9' to the -30- Fig. 17 Wall conditions -31- x-axis are denoted by primes. The corresponding part of the Mohr stress circle and the yield loci are shown in Fig. 18 (a), the wall yield locus is determined by the angle of friction The relation between the stresses at the wall are ^ 7 = tan <i>' . (42) From the geometry of the Mohr circle it follows that sin[2(\|r" + 0' - 0 ') - - Jt] = , 8 in o and the significant solution for \|r' is + 0 ' _ Q« = £ + h p + Arc sin - - " ) , (43) 2. Z s in o For rough walls, point W' merges with point. E' and it is evident from Fig. 18 (a) that \|r1 + 9 ' ~ 0 ' = "Tit + (44) or, noting that it is the period of angle \|r' , eq. (44) can also be written a|t 1 + 6 1 - 0' = - it/4 + i/Z. Similarly, the values relating to the wall inclined at angle -0" to the x-axis are denoted by double-primes, and the corresponding part of the Mohr stress circle and the yield loci are shown in Fig. 18 (b). The wall yield locus is located by the angle of friction between the solid and the wall. Relations (42) to (44) now become •^77 = tan <6" (45) + 0" . 0" = £ _ + Arc sin 4 S-^) , (46) l I s m o -32- T Linearized wall yield loci -33- PART II STEADY STATE FLOW General equations In this section the differential equations required for the solution of the stress and velocity fields in steady state flow are presented. The adopted yield function allows the stress equations to be uncoupled from the velocity equations and to be solved first. The solution of the stress field produces the direction of the major pressure to(x,y) and the value of density T(x,y) throughout the field, and suitable velocity fields can then be computed. The boundary conditions have to be satisfied in both, the stress and velocity fields and certain physical conditions imposed on the stress and velocity fields have to be. met. All the differential equations are hyperbolic and each field requires the solution of a set of two partial differential equations of first order. The equations are presented in a form suitable for numberical calculations by the method of characteristics. Stress field„ In the plane-Cartesian/cylindrical-polar coordinates x, y, 01, Fig. 2, the equations of equilibrium are -35- These two equations together with the equation of the effective yield locus (11) and the empirical relation for density (26) can be solved for the four dependent variables (a , cr , t ,T) in plane strain. r ■ x y xy In axial symmetry, the fifth dependent variable $ is taken care of by the additional equation (22). The equations of equilibrium are first expressed in terms of a and 03 by means of equations (14) - (16) and (22), thus fj n n "T ('y | ^ (1 + sin 5 cos 2cd)~ + sin 5 sin 2cu - 2 ® sin 8 sin 2o> + x y x - 8 cr + 7.0 sin 8 cos 2ca = y0(l + cr)- - m - sin 8 sin 2cd, sin 8 sin 2cd + (1 sin 8 cos 2to) + 2 0 sin 8 cos 2o) + dx • ay ox ■ + 2o sin 8 sin 2cd = m ~ sin 8 (k + cos 2ao) . Now the following abbreviation is introduced [15] S = In (50) ^ Oo where a0 is an arbitrary constant. The differential equations then reduce to the following form where /-i , , S\ / , 5N , , , it , 8. To (1 + o) sm ( a ) - + -) cos (a ) + - - -) + k cos (a ) - A = - ' ' - r--1- m --- ------ - ' 2 a s i n 5 cos(oc> + - -) 2y cos(cd + - - -) To (1 + cr)^sin(o) + -r - ~ ) cos(co - y + tj) + k cos(cd + 7 - § ) B ------------------ V - V - m ------- -- ------ 7 5 - L • <52> 2o s m 8 cos(co - ^ + -) 2y cos(a) - - + -) In the first characteristic direction, dx = tan(" + 4 " 2^' ^53) the left hand side of eq.(a) is a total derivative = A. (54) d (S + co) dx Similarly, in the second characteristic direction, S = t3n(" " I + 2} 5 (55) the left hand side of eq.(b) is a total derivative d<s : m> - B. (56) CIX It will be observed that the two stress characteristics inter­sect at an angle it/2 - 8 , and form angles + (^/4 - 8 /2) with the direction of the major pressure. Velocity field. The velocity field is computed with the assumption of continuity and isotropy. The equation, of continuity in steady flow can be -37- written as follows I-(r u ym) + |^(r v ym) - 0, (57) where u and v are the components of the velocity vector in the direc­tions of the coordinate axes x and y, respectively, Fig. 19. Density T is eliminated by means of eq. (26), yielding [ ( 1 + cr)^uym ] + [ ( 1 + a)^vym ] = o, which expands into where v , f3 ,dcr . da e = m ---r T~r~<.>n t -5- y 1+ a o x ay + r g - g Z u + v) . (59) The principle of isotropy states that the directions of the principal strain rates coincide with the directions of the principal stresses. The normal, compressive strain rates and e , and the shear strain rate T a r e expressed in terms of the velocity com­ponents as follows C>U c *v cHl S v ,, Ex ■ ' X " ey ' - 3 F- Txy = ■ Ty ' S ' (60) The equation of isotropy then can be written du _j_ dv tan 20) = |2- (61) cSx By In order to find the characteristic directions and the relations -38- which hold along the characteristics, equations (58) and (61), together with the equations of the total derivatives dx + ^ dy = du, ^ dx + ^ dy = dv, are solved for du/dx to yield 3u dy + + e(fe - tan 2a,) . (c) a x " ' dx [ £ - ta„(a, + f>] [ £ . t - O - f ) ] The characteristic directions are = tan(d) + -|) . (62) Hence, the velocity characteristics are orthogonal and do not coincide with the stress characteristics. In the directions of the characteristics, the numerator of eq.(c) is zero and, with the substitution of the appropriate expression (62) for dy/dx, reduces to = (63) dy dx cos 2co In both equations (62) and (63) the top sign applies along the first characteristic and the bottom sign along the second characteristic. Sometimes it is more convenient to have the velocity vector expressed in terms of its projections v^ and v2, Fig. 19, on the directions of the characteristics. A substitution for u = - v^ sin(co - it/4) + v^ sin(oo + fl/4), (64) v = v^ cos(o) - rt/4) - v^ cos (od + it/4) in eq.(63) leads to the following relations: -39- along the 1st. characteristic: = tan (co + -^) , ' . (65) dv., dcu m dy ft da ft da + V2 + 2? <V1 + V2 + TTCf^I W x + 2W) ‘ ° ' (66> In the above equations, as well as in the two equations below, the derivatives are taken in the direction of the 1st or the 2nd character­istic as indicated by the subscript. Along the 2nd characteristic: " tan (co - -■) , (67 2 dv d.co m dy 6 v.. da ft v ~a- " v i ~a- ' + T~ v i 1- + v 0 ) - ~/i. a ~- + TTTwT ~ r ~ ~ ° * (68 dy^ 1 dy^ 2y 1 dx^ 2 2^1+u) dx^ 2(1+a) dy^ Superposition. Since both., the equation of continuity (58) and of isotropy (61), are linear and homogeneous, a linear combination of two (or more) solutions of a velocity field is also a solution. This property of the velocity field is very convenient in the development of physical solutions„ Physical conditions The following conditions are imposed on the stress and velocity fields on physical grounds: - A. Stresses are positive (tension not allowed) and bounded. B. Along a line of infinite shear strain rate, frictional and cohesive forces are fully mobilized. This implies that stresses along such a line lie either on the yield locus or on the wall yield locus, hence, such a line, is either a slip line or a weak wall. -40- ----- i** y, v X , u Fig. 19 Projections of the velocity vector on the characteristic directions -41- C. The velocity V of an element of a solid is bounded. D„ The acceleration of an element, dV dV - . v2 - .... ' d£ S + T " <69) S along the path of its travel, Fig. 20, is bounded. This implies that dV/dt is bounded everywhere, and 1/p is bounded everywhere with the * S . exception of points at which V = 0. E. Singularities in density are inadmissible. Grids, special lines and regions The solution of a stress field defines the function cd = co(x,y) , (70) where tan co = dy/dx is the direction of the major pressure Eq.(70) thus is the differentia 1 equation of a grid of lines of action of pressure o^, Fig. 21. While the solution of flow does not require the determination of this grid, eq.(70) is used to locate the following grids, special lines and regions. 1. Stress characteristics. The slope of the stress characteristics, Fig. 22, is given by cd t (jt/4 - 8/2) . (71) The fields shown in Figures 22 to 24 assume that the walls are rough. 2. Velocity characteristics. The slope of these lines, Fig. 23, is cut */4. * (72) It will be ovserved that the velocity characteristics form two bunches whose stalks are located at the vertex of the channel. The stalks are -42- Fig. 20 Velocity along a streamline in (s,n) coordinates -43- within the walls of the channel because the walls are rough. The stresses at the walls are given b y the points E, Fig. 18. If the walls were weak in such a degree that the stresses at the walls were given by the points T (T = W ) , Fig. 18, then the wall on each side of the channel would aline with a velocity characteristic. The stalks would be at the walls. If the walls were weaker yet, so that the stresses at the walls were given b y the points W, as shown in Fig. 18, the stalks of the velocity chara­cteristics would be cut off by the walls. 3. Lines of maximum shear strain rate. These lines are inclined at angles "t rt/4 to the lines of the principal strain rates and, with the assumption of isotropy, also to the lines of the principal stresses. Therefore, the slope, of these lines is 0) "t it/4 and they coincide with the velocity characteristics, Fig. 23. The lines of maximum shear strain rate are important because some of them can be observed through a transparent wall of a model and thus provide an experimental check of the analysis. 4. Slip lines. Under conditions of flow, the slip lines, Fig. 24, do not coincide with either the stress or the velocity characteristics. However, slip lines are significant because by the physical condition B a line of infinite shear strain rate across a solid can occur only along a slipline. A slipline has the slope of either of the two angles co t (it/4 - 412). (73) From Fig. 24, it is evident that the sliplines, like the velocity charac­teristics, form two bunches with stalks at the vertex of the channel. Since the walls are rough, they aline with the sliplines and, therefore, -44- Fig. 2.1 Lines of action of the major pressure an -45- the stalks of the sliplines are at the walls. Any weakness at the walls would cut off the stalks of the sliplines* , During flow, angle ^ is measured at the points E of the Mohr stress circle, Fig. 5, At these points angle ^ is always smaller than 5; this follows from the condition of convexity of the yield locus [7]. Angle <i> is also greater than zero;; this can be demonstrated by means of models in which the lines of maximum shear strain rate can be observed through a transparent front wall. If 6 were zero, then e q , (73) would be identical with eq,(72) and, in a channel wi t h tough walls, the walls would coincide with the lines of maximum shear strain rate. The stalks of the latter lines would lie at the walls while, in fact, these stalks are observed in the positions shown in Fig. 23, Another illustration of <i> > 0 is provided in the section "Flow in vertical channels", 5, Streamlines, In steady state flow the paths of flowing elements of the solid are independent of time and are called streamlines. Streamlines have the following two properties" they cannot have cusps, (except at points where V = 0), and they cannot intersect each other. The former follows from the physical condition D which requires that 1/ pg be bounded. The latter is demonstrated from the consideration of the equation of continuity in the orthogonal, curvilinear coordinates (s,n), Fig, 20, This equation is of the form S(r V pn ym ) • ^ = 0, and it follows that T V p ym = f (n) , . n It is easy to show that at the intersection of two streamlines p 0, -46- Fig. 22 Stress Characteristics -47- hence V-?*°°, which is not permitted by the physical condition C „ 6. Coincidence of a velocity characteristic with a streamline. Suppose the 1st velocity characteristic coincides with a streamline, then Vj = V and v^ - 0 in e q . (66), which integrates into V 2 ym (1 + a)P = c. (74) Since the constant c may be set equal to zero, velocity may be zero along such a line. In plane strain, velocity is either constant or, allowing for compressibility, almost constant. . 7. Velocity discontinuities. Any line along which the shear strain rate, is infinite will be referred to as a velocity discontinuity. This term thus covers both, jumps in the magnitude of velocity and infinitely large velocity gradients. Two conditions need to be satisfied along a line of discontinuity. First, a velocity discontinuity has to fallow either a slipline or a weak wall:' this follows from the physical condition B. Second, a velocity discontinuity can occur only along a streamline. The latter is demonstrated as follows: Along a line of infinite shear strain rate, T*xy» ecl*(60), is infinite. In the (s,n) system of coordinates, Fig. 20, the strain rates are given by &V V dv . V . , -v e = - XT, e = - , r = " x : + (75‘) s n P sn °n. p . n s and the relevant relation between these expressions and the strain rates in the (x,y), coordinates, equations (60), is j 2 2 2 (e - e ) ' + r = (e - e ) + r s n sn x y xy -48- Fig. 23 Velocity characteristics -49- In this equation dV _ SV dn dV SV _ dt dn dt dt 3s _ds " v ' dt since ds/dt = V and dn/dt a 0. All the terms on the left hand side of eq.(d) with the exception of dV/cki are bounded by the physical conditions C and D. Hence, along a line of velocity discontinuity, it is necessary that S V / S n » » 5 which means that a velocity discontinuity follows a streamline. A velocity discontinuity, though it is a line of infinite shear strain rate, does not, in general, coincide with a line of maximum shear strain rate; the latter strain rate being bounded. A velocity discontinuity separates two regions which may be both plastic, both rigid (or elastic), or one plastic and the other rigid (or elastic). A velocity discontinuity often originates at the top boundary of a channel. In channels with rough walls all the slip lines enter the stalks at the vertex, and discontinuities can extend from the top to the bottom of the channel. In all practical channels wi t h w e a k walls the walls cut off the stalks of the slip lines. A velocity discontinuity cannot continue to an intersection with a wall, since that would involve an intersection of the streamline, which coincides with the discontinuity, with the walljwhich itself is a streamline. A velocity discontinuity Fig. 24 Slip lines -51- cannot transfer from a slipline of one family to another slipline of the other family because that would imply a cusp in the streamline at the transfer point. In channels with weak walls a velocity discontinuity which originates at the top boundary dampens out within the field. 8. A straight velocity characteristic in incompressible plane Strain. Suppose that in plane strain a first1velocity characteristic is straight and the solid is incompressible, it then follows from eq. (66) that v^ is constant along that characteristic. . 9. Walls. A wall separates a plastic region from a stationary rigid (or elastic) region. Usually, there is a velocity discontinuity along a wall. The wall then coincides with a streamline, and stresses along the wall are defined by either the yield locus or the wall yield locus. W h e n the solid flows within rough walls, the walls coincide with sliplines and the stalks of the bunches of sliplines intersect the lower boundary of the channel. Therefore, any slipline which intersects the top boundary of the channel may form a wall. In consequence, the walls can shift readily in the upper part of the channel, adjusting to the top boundary conditions. In accordance with eq„(3), at the walls, Fig. 17, there is \|/' + 0' - 0)' and f " + 0" - co". (76) For rough walls t' and V are eliminated b y means of equations (44) and (47) leading to the following expressions for the slopes of the walls It will be noted that, since it is the period of angle co, eq.(77) can also be written 0' = oo1 + Jt/4 - i/2. Wh e n the solid flows within weak walls, the slopes of the walls are found from expressions (43) and (46) with appropriate substitutions of (76), thus For straight walls intersecting at the origin, 6' = O' and 6" = 0". W h e n a velocity characteristic coincides with a wa l l along which there is a velocity discontinuity, the wall is we a k to such a degree that the wall yield locus passes through the point T of the M o h r circle. For the linearized wall yield locus, Fig. 18 (a), this implies In this case, the velocity discontinuity also coincides w i t h a streamline and that enforces the restriction on the magnitude of the velocity along the wall expressed by eq.(74). W h e n there is no velocity discontinuity along a wall, the wall need not be a streamline and the stresses along the wall need not lie on a yield locus or on a wall yield locus. If such a wall coincides with a velocity characteristic then the velocity boundaries at the top and at the bottom of the channel have to be continuous at the w a l l s . In practice, this is seldom attained and these conditions lead to n o n ­steady flow. This is prevalent in axi-symmetric flow within rough walls. If such a wall does not coincide with a velocity characteristic then a (79) q" = co" . [-2 _ h i " + Arc sin S i n 'f ) ] . 2 2 s in o (80) tan i ' = sin 6. (81) -53- zero velocity region is enforced. This is discussed below. 10. Zero velocity regions within a plastic field. This concept is very useful in the development of fields in channels with sharp changes in cross-section, as shown in Fig. 25, because it allows the use of a continuous stress field. The stress characteristics of the 2nd family have to intersect the wall at an angle of 40° (in this example) i to satisfy the wall yield locus and allow a velocity discontinuity along the wall. The stress characteristics above point A and below point B do so satisfy the WYL, Between the points A and B, the stress char­acteristics bend in gradually and the stresses at the wall are below yield values, hence no velocity discontinuity can o:cur along A B . Since AB is not a velocity characteristic, zero velocity along AB enforces a zero velocity region ABC where C is the intersection of two velocity characteristics (heavy lines) through A and B, respectively. Since, further, there can be no velocity discontinuity along the line ACB (not a slip line) that line is not a streamline. Streamlines develop smoothly around ACB. Zero velocity regions can be observed in a model with a transparent wall. Indeed, it was from observations of models that this concept arose. In a bin, zero velocity regions occur at the transition from the vertical portion to the hopper. They have the effect of narrowing down the channel at the transition and explain the drop of the vertical pressure along the axis of symmetry. This drop was measured b y the author several years ago, Fig, 26, and was reported in references [16, 17]. -54- 2nd stress characteristics velocity characteristics f x Fig. 26 Drop of vertical pressure at a transition -56- 11. Stress discontinuities may occur only along streamlines. This follows from the fact that all real materials are compressible, and a discontinuity in stress implies a discontinuity in density. An element of solid flowing across a stress discontinuity would undergo a discontinuity of density and, therefore, an infinite acceleration, which is contrary to the physical condition D. Since the lines along which stress discontinuities might be expected to arise usually cross the streamline field, the existence of discontinuities is unlikely. 12. Stress singularities are inadmissible, because they would imply singularities in density which in a real solid are not acceptable by the physical condition E. Converging channels Equations of stress In this section it will be advantageous to use the polar/spherical coordinates r, 0, OC. The equations of equilibrium in these coordinates are d a , S t „ H----- nT"-- 1---[o - oa + m ( a - a r/) + m t a cot 0] + or r o0 r r 0 r OC' r 0 + T cos 0 = 0, (82) dr- + r ~ d d + 7 [m ^a0 " U0t) cct 6 + (2 + m ^Tr0J - T sin 0 = 0.(83) These equations are now transformed as follows: first, expressions (17), (18), (19) and (20) (with k = +1), and their appropriate deriva­tives are substituted for the component stresses; second, the substitution -57- o - = r r ( r , e ) s(r,0) . (84) is made; third, the derivatives ds/d© and ds/c)r are separated, leading to the two equations „ ds + s f(r,0) + g(r,0) = 0, (85) r ^ + s h(r,0) + j(r,0) = 0 , ' (86) where f(r,0) = 2 ( ^ + l)-2^" ^ sin 2^ + 2r ^ -S:Ll^--(sin 5 + cos 2f) + cos 5 cos S + - + m -^- " ( 1 + sin 8) [sin 2\(r - cot 0 ( 1 + cos 2\|/) ], (8/ ^ cos , sin 6 . .... , „, N sin 0 /r>n\ g ( r ,0) = - --- s m ^0 -t 2\|/) - ------- - , (88) cos 5 cos & h(r,g) = 1 + 2(^g + 1) ™ - ^ ~ ( c o s 2^ - sin &) - 2r ^ S ^n ^ + | cos S cos 8 I + ~ + m (1 + sin 8) (cot. 0 sin 2\|r + cos 2\|r - 1) , (89) T cos & j (r ,0) = - ™ 1-n"2_5, cos(0 + 2\(r) + (90) cos 5 cos 8 The converging channels under consideration will be assumed to possess a vertex at which the extensions of the walls intersect. It will be shown that, with some continuity conditions satisfied, stress fields in all converging channels, irrespective of their top boundary | and walls away from the vertex, approach a unique and relatively simple stress field at the vertex. That unique stress field will be called the "radial stress field',1 because it is the stress field compatible w i t h a radial velocity field. The radial stress field is fully defined by the slopes of the tangents to the walls at the vertex (S', -0") and I -58- 4 the physical parameters of the solid and the walls (6,g$',^"). While a physical channel can never be brought to a vertex, it is expected that the radial stress field is closely approached within a substantial region of the vertex, a region which will usually include the outlet of the channel. Since the knowledge of the stress field at the outlet of the channel is required in the derivation of flow criteria, the uniqueness and the simplicity of the radial stress field are of a great, advantage in this derivation. In the considerations which follow the origin of the coordin­ates will be located at the vertex of the channel. f ' Radial stress fi e l d . Der i v a t i o n . If it is assumed that t = K e ) (91) and Y = const. (92) Then the coefficients of the equations (85) and (86) assume the sim­plified form f(6) = 2(^- + 1) -- sin 2\|r + cos 5 + m (1 + sin S)[ sin 2\j/ - cot 6(1 + cos 2^r)] , (93) cos 6 sin 6 . , „,x sin 6 ,„,s g (6) = - --- 2" s i n (0 + 2^) - --- (94) cos S cos 5 -59- h(0) = 1 + 2(~^ + 1) S,:i'n 0'~ (cos 2\|r - sin &) + O.C7 cos o + m S '*‘n 2^(1 + sin 8) (cot & sin 2^ + cos 2\jr - 1) , (95) cos 6 . j(6) = - ~ - n-2- cos(0 + 2\|r) + -°-^e-. (96) cos 5 cos 5 Equations (85) and (86) now become £ + s f(0) + g (0) = 0, (a) r | + s h(0) + j(0) = 0, (b) and integrate into . -/f(0)d0 -/f(0)d0 / f(0)d0 s = c(r) e - e /g(0) e d0, s = k(0) rr" h - h ( 0 ) . Hence (c) . . -/f(0)d0 / f(0)d0 = e /g(0) e , (d) and there are two alternatives: either k(0) ~ c(r) = 0, (e) or ~/f(0)d0 k(0) = e , and c(r) = r . Consider the latter case. Evidently h = const. ^ 0. Eq. (d) is differentiated, thus g(0) h = f(0) j(0) + (f) Now differentiation of eq.(96) yields -60- In this equation and in eq. (93), the derivative d\|//d0 is eliminated by means of eq.(95) to yield These expressions and expressions (94) and (96) are now substitut­ed for the appropriate functions in eq.(f), which after transformations becomes In plane strain h = 1 and eq.(95) yields d\|r/d0 = -1. This implies an elementary field with rectilinear characteristics, which enforce a constant velocity throughout the channel. This does not provide a solution to converging flow. In axial symmetry, it follows from eq.(95) that, at the axis, for 0 = 0, \jr equals either 0 or rt/2. Any other value would enforce d\|//d0 = at the axis, and that is inadmissible since it would imply an unbounded strain rate. Consider the initial condition 0 = 0, \)r = tc/2. For these values, the numerator and denominator of e q . (g) vanish and the limit i _ 9 dj _ sin(0 + 2^) |(h - l)cos 5 - sin 5(cos 2\|i - sin 5)1 sin 0 d0 ~ 2C , ' 2* cos 6(cos 2\||r - sin 6) cos 6 m sin S(1 + sin 5) sin (0 + 2\[/) (cot 0 sin 2\|/ + cos 2\|r - 1) 2 cos 6 (cos 2\|r - sin 6) and sin 5[sin 2^ - cot 0(1 + cos 2^)] cos 2t - sin 6 o h = 1 + m £ sin S[cos 2i|f + cos 2(0 + 2\|f) ] - sin (0 + 2\|r) + t) r\ - cos 2(0 + \|/) - cos 0 j- /cos 5 [cos 2(0 + i|/) - cos 2i|f], (g) established b y twice applying l'Hospital's rule, yielding h . , , M n S 31(^ )012 * 4(^ )0+ 1 * 1 + sin 6 2(i i }o + x ' dC7 2 2 because, from eq,(95), d \Jr/d0 is bounded except, possibly, for cos 2\|r - sin6 = 0. Eq.(95) at the axis of symmetry evaluates at h - 1 - *1 -S l sn .mS 5c 1[ (d§0 )°+ 1)1. Elimination of h yields a quadratic in d\)r/d0, whose roots are both netative, A ° = _i and /itno = _ 5 ± .3 Bin_b Cde; 1 and (.d0; H + 5 sin 6* This produces two particlar solutions. These solutions have been checked out numerically and found not to lead to useful boundary conditions. The solution to radial flow is then provided by the first alterna­tive, eq.(e) which reduces eq.(c) to 5 - s(e) ■ - w r <97) Equations (a) and (b) now become + s f(e) + g(e) « o, (98) s h(e) + j(e) - o. (99) They are solved for the derivatives dl = F(6'* ' s) = = - 1 - [ m s sin 6 ( 1 + sin 6 )(cot0 sin + cos 2a)/ - 1) + cos 0 + 2 - sin 6 cos(0 + 2\|r) + s cos 6]/2 s sin S(cos 2i|r - sin 5), (h) -62- § - F<e,*,s) - s sin 2jf + sin (0 + 2\jf) + m s sin S[cot 8(1 + cos 2\|/) -sin2jj] ,. cos 2'Jr - sin 5 " 1 These equations are equivalent to the integral equations t(0) = T|r(0°) + J° F[t,i|r(t),s(t)] dt, (100) 0° s(0) = s (0°) + /e QG[t,^(t),s(t)] dt, (101) 0 which are solved for a given set of boundary conditions, \|r° = ^(0°) and s° = s(0°). a m ay then be determined from the eq,(84) which now reduces to a = r r s (0) . (102) Solutions of the radial stress fi e l d . One conclusion which is evident from the eq.(102) is that a radial stress field cannot extend upward to a traction-free top boundary. In gravity flow, the top boundary is usually traction-free and, therefore, the actual field in the upper part of a channel deviates significantly from a radial stress field. The radial stress fields are computed from the equations (100) and (101). It will be observed that these equations contain boundary conditions given along a single ray 0 = 0°. In a physical channel the boundary conditions are different, they are given by the slopes of the wall 0 1 and 0" ( 0 1 = 9 ' and 9" = 0" in a radial stress field, since the walls are straight and pass through the origin) and by the angles of friction 6' and between the solid and the walls { 6 1 = 6" = b for rough walls). It follows from Fig. 18 that two angles; \|r1 and \|/j, measured at the points W' and W|, respectively, correspond to one value -63- of . Thus, mathematically, the boundary conditions are not uniquely defined. The location of these points is shown in Fig. 27, in the (0,^) coordinates for plane strain. In axial symmetry, 9' - - 0" and \|r1 = ?t - i|f", while \Jr^ = - i)/", and the boundary points are located sym­metrically relative to either the point (0,it/2) or (0,0), Fig. 28. In the (0,\j/) coordinates, a solution is expressed by a line = \|/(0) which connects two boundary points. It is easy to show that a solution = \|/(0) cannot cross a line cos 2\|; - sin 6 = 0. This follows from the analysis of equations (h.) and (i) . Along that line, the deriva- 2 2 tive d0/d\|/ is zero, while the second derivative d^S/d^ ' 0. Hence the inverse function Q = 0(\|/) reaches an extremum when passing the line cos 2i|r -■ sin 5 = 0 and the field backtracks into the sanje physical region. Thus, only solutions connecting either the point ( 0 !,a|/') with (-0",\|r")a or the point (0*,\!/p with ( - 0 " , ^ ) are physically acceptable. Further, also on physical grounds, solutions connecting the points (0'3^|) with (~0",\|/'p are rejected because they are not. observed in practice. Thus, only solutions connecting the points ( 0 f,\|f!) and (-0",^") will be considered in this work. In axial and in plane symmetry these boundary values imply \|/ = Jt/2 at the axis, i.e. for 0 = 0. There is no direct way of finding a solution connecting two boundary points. The method adopted in this work is to compute a sufficient number of randomly spaced solutions from a boundary 9° = 0, \|/°, s° and to interpolate the required functions. To assist in the interpolation, contours of constant values of s are drawn in the (0,\|/) coordinates, -64- Fig. 27 Solutions of the radial stress field in plane strain -65- for the five values of 5: 30°, 40°, 50# , 60° and 70°. The solutions for symmetric plane strain are shown in Figures 29 to 33, for axial symmetry in Figures 34 to 38, and three cases of asymmetric plane strain at 8 = 50° are shown in Figures 39 to 41 for 4* m 20°, 30° and 40°. In axial symmetry mathematical solutions are also available with a discontinuity in \|r at the axis of symmetry. However, these solutions imply s ■ 0 at the axis. This is physically unlikely to occur and, therefore, these solutions are not computed. While no formal proof of uniqueness of solution for boundaries (0',^') and (-0",^") is submitted, the large number of numerical cal­culations which has been carried out seems to indicate that, within the range of application to the physical problems under consideration, the radial flow solutions obtained by the above method are unique. In the discussion of Boundaries it will be indicated that in gravity flow the fields are unlikely to extend outside of the j ■ 0 lines. Therefore, all the plots in plane strain are bounded by the lines ] » 0 and cos - sin 6 ■ 0. In axial symmetry the available solutions cover more restricted regions in the (0,^) coordinates and do not reach the above specificied lines. Indeed, in axial symmetry, for solutions with velocity discontinuities at the walls to exist, the walls must be i sufficiently weak. For instance, it is most unlikely that axi-symmetric converging flow can occur withih sliplines. However, flow without velocity discontinuity is possible within rough walls. In such flow, the walls are velocity characteristics, which means that i|r1 " 3rt/4 and \|/" ■ n/4. A value oft* - 90° ■ 45° cuts across the regions of solution Fig. 28 Solutions of the radial stress field in axial symmetry -67- and indicates the largest possible value of S' within rough walls. These values of Q r are listed in Table 2 as a function of 5. T ab 1 e 2 5 30° 40° 50° 60° o0 Max. S' 15° 8 „ 1° 4.4° 2.2° .5° Especially, for the larger values of 5, an axi-symmetric channel can open out but very little. In addition, this type of flow implies a continuous velocity profile at the top and at the bottom of the channel and is, in practice difficult to attain. As a result, this type of flow, if it occurs, is usually unsteady. Resultant v e r t i c a l f o r c e . It is of practical interest to know the resultant vertical force Q acting at a horizontal cross-section of a channel. A n expression for this force will now be derived. The h o r i ­zontal cross-section is of width B = 2 y ', and is elevated a distance -x0 above the wertex, Fig. 42. The vertical pressure a is found from e q .(14) with substitutions X (3) and (102) for to and a , thus a = r y s [1 + sin 5 cos 2(\|r + 0)] „ X The total vertical force is . „ m _ 1-m rV1 m , Q = 2rt L F a y dy, (j) \J X where L is the length of the channel in plane strain. To compute Q it is necessary to express y as a function of x Q and the angles 8 and \|f. -68- Function s, 8 = 30° Plane symmetry (symmetric plane flow) -69- 0 10 20 30 40 50 60 6 ' 70 Fig. 30 Function s, 6 = 40° Plane symmetry (symmetric plane flow) -70- 70 i|r'- 90° 60 50 40 30 20 10 0 E 20 30 Fig. 31 Function s, 6 = 50° Plane symmetry (symmetric plane flow) -71- 60 6' 70 10 X 10 PER I N C H - 50 60 40 - 30 - 20 10 Fig. 32 Function s, 5 =• 60° Plane symmetry (symmetric plane flow) -72- Function s, S = 70° Plane symmetry (symmetric plane flow) 60 0 10 20 30 40 50 60 0 1 70 Fig. 34 Function s, S = 30° Axial symmetry (conical flow) -74- Fig. 35 Function s, 5 = 40° Axial symmetry (conical flow) -75- Fig. 36 Function s, 8 = 50° Axial symmetry (conical flow) -76- V 30 40 20 10 Fig 37 Function s , 8 = 60* Axial symmetry (conical flow) -77- 50 ' - 90° 0 10 20 30 40 50 60 0 1 70 Fig. 38 Function s } S = 70° Axial symmetry (conical flow) -78- Fig. 39 Function s, 5 = 50°, = 20° Plane asymmetry (Plane flow - one vertical wall) -79- t'- 90 F i g . 40 Function s, 5 = 50°, = 30° Plane asymmetry (Plane flow - one vertical wall) -80- Fig. 41 Function s, 5 = 50°, = 40° Plane asymmetry (Plane flow - one vertical wall) -81- } (k) The relations between the (x,y) and (r,0) coordinates are, (Fig. 2), x = - r cos y = - r sin The total derivatives are: dx = -cos 0 dr + r sin 0 d 0 , dy = - sin 6 dr - r cos 0 d 0 . For x 0 constant, dx = 0 and elimination of dr between the above two equations leads to , r d0 dy " ' ^ T e - (1) From conditions at the wall, Fig. 42, there is x 0 = y 1 cot 6 1. Within the channel , r = - X° ' cos or, eliminating x 0 , y'cot 0' r = cos Hence expression (1) for dy becomes dy _= y' cot o ._d0_ 2 cos 0 Elimination of r in the second of equations (k) yields y = y r cos 0' tan 0. Substitutions for a , r, y and dy in eq.(j) transform it into x' where Q = q r L 1_m B 2+m, (103) , 2+m 0 1 m q = 2itm (-°-- ■■) / s - [1 + sin S cos 2(0 + i|r) ] d 0 . (104) ° cos 0 -82- Fig. 42 Resultant vertical force -83- Lines of constant values of q are plotted in Figures 43 to 48 for 5 = 30°, 40° and 50° in symmetric plane strain and in axial symmetry. Stresses at the w a l l s . The normal and shearing stresses a' and t' which act between a flowing solid and the walls are computed from equa­tions (18) and (19): a' = Oq and t ' = - ,respectively. In these equations, a is replaced by expression (102) with B r = 2 sin 0 ' in accordance with Fig. 49, leading to ___ _______,1 - sin 5 cos 2\|f' T B r B 2 sin 0' ' U } , sin S sin 2\lr1 (106) r B r B 2 sin 0' • These equations apply in plane and axial symmetry. Lines of constant values of 0 !/Y b and t '/Y B are plotted in figures 50 to 61 for plane strain and axial symmetry in (0',o') coordinates for 6 = 30°, 40° and 50°. Influence of compressibility. The influence of compressibility on the solutions of radial flow can be estimated by using the expression (27) Y = Y o C ^ wi t h a eliminated b y means of eq.(102) to yield 1 Y = (rf3YoSP ) 1"P , (107) and the derivatives 1 d r _ P 1 ds r S Y = P n n R * Y c*0 1 ■ P s d 0' T ^ r 1 - p' U ; With these substitutions, and = \|/(0) for the radial field, the coefficients (87) to (90) of the differential equations (85) and (86) -84- k' Fig. 43 - Vertical force q, 6 = 30° Plane symmetry (symmetric plane flow) -85- Fig. 44 Vertical force q, 5 = 40° Plane symmetry (symmetric plane flow) -86- 70 - 90° 60 50 40 s 30 20 10 0 E 50 - 40 - 30 - 20 - 10 30 40 Fig. 45 Vertical force q, 5 = 50° Plane symmetry (symmetric plane flow) -87- 60 O' 70 20 30 40 50 Fig. 46 Vertical force q, 8 = 30° Axial symmetry (conical flow) -88- 1 0 X 1 0 PER I N C H 30 20 - 10 Fig. 47 Vertical force q, 5 = 40° Axial symmetry (conical flow) -89- Fig. 48 Vertical force q, 5 = 50° Axial symmetry (conical flow) -90- - 40 h- 30 - 20 - 10 F i g . 49 Stresses at the walls -91- i Fig. 50 Function cr'/V B, 5 = 30° Plane symmetry (symmetric plane flow) -92- - 1 10 20 30 40 50 Fig. 51 Function j '/T B, 5 = 40° Plane symmetry (symmetric plane flow) -93- 60 e' 70 Function ff'/V B, 5 = 50° Plane symmetry (symmetric plane flow) 0 10 20 Fig. 53 Function a'/T B, 6 » 30° Axial symmetry (conical flow) 30 40 50 Fig. 54 Function cr'/Y B, 8 = 40° Axial symmetry (conical flow) -96- Fig. 55 Function cr'/T B, 5 = 50° Axial symmetry (conical flow) -97- 20 30 40 50 Fig. 56 Function t '/t B, 8 = 30° Plane symmetry (symmetric plane flow) -98- 30 i' 20 10 0 Fig. 57 Function t'/T B, 5 = 40° Plane symmetry (symmetric plane flow) -99- 70 - 90° 60 50 40 30 20 10 20 30 40 50 Fig. 58 Function t '/t B, 5 = 50° Plane symmetry (symmetric plane flow) -100- i' 30 20 10 Fig. 59 Function t'/tB, & = 30° Axial.symmetry (conical flow) - 101- Fig. 60 Function x'/T B, S = 40° Axial symmetry (conical flow) - 102- Fig. 61 Function t '/t B, 5 = 50° Axial symmetry (conical flow) -103- become f(0) = (1 - P){2(^| + D ^ ~ sin 2* + cos 5 + m (1 + sin 5) [sin 2\Jr - cot 0(1 + cos 2\|/) ]"| , (109) * J g(0) = - (1 - sin(0 + 2t) + ^ ^ 1 , (110) cos S cos S h(0) = 'Y~7~b + 2(d0 + D (cos " sin 6)""n2" + cos 5 + m ^ (1 + sin 5) (cot 0 sin 2\|r + cos 2^ - 1) , (111) cos 5 j(0) = - cos(0 + 2\|r) + C° S20 » (112) cos 5 cos 6 and the differential equations reduce to the form (98) and (99) as for an incompressible solid. Equations (98) and (99) with the above coeffi­cients are now solved for the derivatives HJ = $(0,t,s) = * - 1 - | m s sin 8(1 + sin 8) (cot 0 sin 2i|r + cos 2\J/ - 1) + cos 0 + - sin 8 cos(0 + 2\|f) 4- s cos^S/(l -|3)|/2s sin 8(cos 2\|r - sin B)? (m) 4§ -r(0,t,s) = f s sin 2f , . \ , = | i _ (3 s m ( 0 + 2i|/) + + m s sin 5 [cot 0 (1 + cos 2\|/) - sin 2\|r j) 0 . ^--- I- r1 (n) Y Y JJ co s 2tJt - s rn 8 -104- 0 10 20 30 40 50 60 Q' 70 Fig. 62 Function s, P =* 0.10, 6 = 50° Plane symmetry (symmetric plane flow) lO X to PER I N C H 0 10 20 30 40 50 60 6' 70 Fig. 63 Function s, (3 = 0.10, 5 = 50° Axial symmetry (conical flow) -106- The above equations are equivalent to the integral equations \|r(0) = \|r(e°) + f6 ®[t,t(t) ,s(t) ]dt, (113) 0° s(0) = s(0°) + f [t,t(t),s(t) ]dt, (114) 0° from which the functions i|r and s are computed in the same way as for incompressible solids. In order to estimate the quantitative effect of compressibility, several numerical calculations were carried out for £3 = .10, which is a rather extreme value. Function s(0) is plotted in Figures 62 and 63 for symmetric plane strain and for axial symmetry with 5 = 50°. A compar­ison with the corresponding values fox- an incompressible solid, Figures 31 and 36, shows that the influence of compressibility is small. It should be noted that these equations apply at some distance from the vertex where cr[ lb per sq ft] » 1„ For (3 = 0, these equations reduce to the form obtained for incompressible solids. General stress field Proof of convergence to a radial stress field at the vertex. The significance of the radial stress field is greatly enhanced by the fact that all useful stress fields converge to radial stress fields at the vertex. This statement will now be proved under the assumption that \|/, T and their first derivatives are continuous in a sufficiently small neighborhood of the vertex. This assures the continuity of f, g, h, and j, equations (87) - (90). -107- Eq. (85) is integrated to yield -/p0f (r,t)dt -/qo f (r, t)dt /Lf(r,u)du s(r,0) = s(r,0°)e -e /0Og(r,t)e dt. (115) These integrals are bounded over a field satisfying the above conditions of continuity. Therefore, | s (r ,0) j < J s (r ,0° ) | + M2 for some constants and . Hence, if there is one angle 0° for which j s(r,0°) | < °°, then | s (r ,0) | < °° for all values of 0 within that field. Eq„ (86) is now integrated, yielding -,-r _fr M M > dt r . . / :(r,8) - s(r0,0)e r" C -e r° 1 / J % ^ e r° t *&d» r. t - dt- <U 6 > where r0 > r. The value of this function will now be analyzed in the limit as r o . It will be observed that the signs of lim h and lim j play an r ^ o r o important role in the analysis, and the following two cases are distin-gulshed: _; r h(t.e) 1. Lim h < 0, lim ] f 0. In this case lim e r° C = 0 , the r^s-o r ^ o r^-o first term of eq. (116) vanishes, while in the second term rt h(u,0) , fr i(t,e) r„ u u lim / '- ° dt = " . The second term is evaluated by using rQ t r->o L'Hospital's rule, yielding -108- ,r M u ^ l du j j (t;9) e r0 u dt lim s (r,0) = lim - t ----------- = lim ' h ^ f f (117) r+o r*o J h(t,0)^ r*o r0 u e It will be noted that the boundary value s(ro,0) does not affect lim s(r,0), r»o and that, in this case, no condition is imposed on the value at the boundary. _;r M ^ l dt x ^ 2. Lim h > 0, lim j f 0. Now lim e ° = °° and, for a r->o r^o r*o useful solution to exist, it is necessary that the boundary value satisfy the condition ft h(u,0) * ( p) \ J du s(ro,0) = lim / ^ ?- e r° U dt, (118) r»o r„ then, as in the case 1, the application of L'Hospital's rule yields lim s(r,0) - lim - r*o r-so ' The satisfaction of the condition (118) implies s(ro,0) ^ 0, i. e. no traction-free boundary, within the region of 0 in which lim h > 0. r»o If the condition (118) is not satisfied, then lim s(r,0) = <». While a mathematical solution may exist in the neightborhood of the vertex, the large values of s and, in turn, the relatively large values of pressure a in the region of the outlet of the channel would consolidate a real solid to a degree likely to cause the development of a stable dome, -109- and flow would fail to develop. Hence, it is expected that in physical channels condition (118) is satisfied. The physically acceptable solutions are further restricted by the condition that the solid does not transfer tensions, hence, s(r,0) > 0 . i (r O') This implies lim - - 2- r > 0 and requires: in case 1, lim h < 0, r*o ' r-*-o lim j > 0; in case 2, lim h > 0, lim j < 0. To conform with this condi-r* o r»o r*o tion, the change of sign of lim h and lim j must occur along the same r-yo r^o ray 0. It is easy to show that the latter is also necessary for a bounded solution to exist along that ray, t ^ M £ t0ldt r*o r*o 27 t - because for lim h = 0 , lim e is finite and non-zero, while ,fc h(u,0) , r i(t 0) ^ro""V " lim / 3- e ° dt = 00 and, hence, lim s(r,0) = °°, unless lim j = 0 r>o r° ' r-»o r-»o The significance of the direction established by lim j = 0 will be r*o further discussed in the next section on Boundaries. It has thus been shown that the useful solutions of the equations (85) and (86) approach the form (117) in the neighborhood of the vertex. It will now be shown that lim s(r,0) of eq.(117), is the same as s(0), r*o eq.(97), obtained for the radial stress field. In view of the relation (117), eq.(86) may be written where s h(r,0) + j (r,0) = e19 lim e ^ = 0 . r-^o -110- Equations (85) and (86) are now solved for d\j//d0 and ds/d0: e^cos^S- cos^S ~ 2rs ^(sin & + cos 2\|r) sin 8 (o) F( ,!/,&) 2 s sin 5 (cos 2if - sin 8) x 2rs 'nt: sin 8 + e, sin 2if - s sin 2f - ^r- ds _ _/r. N s dr 66______________ 1_________________ r or , s d0 G(e,^,s) - r - cog 2i)f _ sin 5 > (p) where F (0 ,i|/, s) and G(0,i|/,s) are given by equations (h) and (i) . It will now be shown that the remainders of the right hand sides of equa­tions (o) and fpj approach zero as r -*• 0. Since the walls of a channel at most follow a slipline and ^ < 8 , nowhere within the field does cos 2\|; - sin 8 -> 0. Therefore, all the terms of the remainders approach 1 dr zero for r -» 0, including - . The latter is shown as follows: eq. (25), with the substitution (84) for a, is differentiated with respect to 0 , yielding dr , / v M . , ST dr , ds-. 30 - r - r ' t o l r ^ a + r r 5 l , and from it ds 1. d r _______r d0 r de ~ i r ' (o) ~ r 3 1 St since ds/d0 is bounded, lim - ^r- = 0 , ' r d 0 r-»o Equations (o) and (p) can now be written df = F(0,^,s) + e2 (q) ds = G(0,t,s) + e3 (r) where lim = lim 6 = 0 . r^o r^-o -111- The equivelant integral equations are: 0 0 t(r,e) = t(r,0°) + /0OF[t,\|r(r,t) , s (r, t) ] dt + JQOe 2 (r, t) dt, (s) e e s(r,0) = s(r,0°) + /g0G[ t ,\|r (r, t) , s (r, t) ] dt + JQO e3(r,t)dt. (t) now as r+o, \|/(r,0 °) and s(r,0°) approach limits \|r(0 °) and s(0 °). Equations (100) and (101) give the radial flow solution corresponding to these boundary conditions. Equations (100) and (101) are now sub­tracted from eq.(s) and (t) yielding: \|r(r,0) - = ^(r , 0° ) - \K0°) + 0 6 + /eo^F[t,\|r,(r,t),s (r,t) ] - F[t,\|/(t), s(t)]|dt + IQo&2 (r, t) dt, s(r,0) - s(0) = s(r,0°) - s(0°) + 0 . 0 + (r,t), s (r, t) ] - G[t,t(t) , s(t) dt + /@ oe 3 (r,t)dt. In axial symmetry, f(0,\|r,s) is not Lipschitz in in any region about the axis of symmetry. This is due to the term containing cot 0. However, in plane strain or in axial symmetry, if the axis of symmetry is avoided, F and G are Lipschitz in both \|r and s. Then: |f [ t ,\[r (r, t) , s (r, t) ] - F[ t (t) , s (t) ] j ^ J^lr (r, t) - \|r(t) | + + L2 |s(r,t) - s(t)|, |G[t,\|r (r,t) ,s(r,t) ] - G[ t ,t (t) ,s (t) ] | ^ L |t(r,t) - \|f(t)| + + I7J s (r, t) - s (t) | , and therefore -112- |\|r (r,0) - Hr (0)1 ^ |^(r,0°) - ^ (0°) | + 0 0 + ■^0°[L1|\lf(r,t) - \Jr (t) | + L2 js(r,t) - s(t)|]|dt| + J Qt> | e2(r,t) | |dt | , |s(r,0) - s(0)| |s(r,0°) - s(0°) | + 0 0 + /0O[L3 |t(r,t) - i(r (t) | + L4 |s(r,t) - s(t) | ] |dt| + / 0 o |e3 (r, t) | jdt j „ These inequalities are now added, yielding: 0 °Kr,0) <C L /0Oar(r,t) Jdt I + |3(r,0,0u), (u) where: a(r,0) = |^(r,0) - \|r(0) | + |s(r,0) - s(0)|, P(r,0,0°) = | i|r(r,0°) - 1^(0°) | + |s(r,0°)- s(0°)| + 0 + /go[|e2(r,t)I + |e3(r,t)j ] |dt j and L = max^Lp L^, J . The method of successive approximations, used on (u) yields: a (r,0) ^ P(r,0 ,0°) e L l9 + 8 'L (v) But lim (3(r,0,0°) = 0 and, therefore, lim CC(r,0) = 0, or r->o r-^o lim i|r (r,0) = ilr (0) , (w) r-»o lim s(r,0) = s(0). (x) r-»o The significance is this: Given a converging stress field satisfying the conditions stated in this proof, and a ray 0 = 0° along which \(r(r,0o) and s(r,0°) approach limits \K0°) and s(0°) as r-»o, there is a radial stress field given by -113- i|/(8) and s(0) from eqs.(lOO) and (101), with boundary conditions i|/(0°) and s(0 such that throughout the field ty(r,0) ->i|r(0) and s(r,0)-»s(0) as r_^o. Althou the proof, as given, does not hold if 0 and 0° are separated by 0 = 0 in axial symmetry, the axi-symmetric field is completely determined if it is known on only one side of 0 = 0. Hence the axis, 0= 0 , need not be crossed. Boundaries. A general problem satisfying equations (85) and (86) is defined by the specification of adequate boundary conditions. Such conditions are ilr0 = \|/o(0) and s0 = so(0)} at the top boundary, and i|r' = \|r1 (0) , \|r" = t"(6), at the walls, Fig. 64. The stress field is then defined down to the stress characteristics passing through the end points of the walls. Hence, the stresses at these two points are inde­pendent of the bottom boundary conditions, and, as has been shown in the previous section, approach the magnitude of the appropriate radial stress field. This is important, because these two stresses are used in the derivation of the flow criteria which are, therefore, independent of the top boundary. This justifies the development of the design criteria on the basis of the radial stress fields and eliminates the need for a large number of particular solutions of general fields. Several general fields are discussed in a separate publication [15]. In gravity flow, the slopes of the walls in the neighborhood of the vertex seem to satisfy the condition lim j > 0. There is a physical r->o meaning to this bound, namely the horizontal component cs^ of the stress vector acting on the solid from a wall, Fig. 65, is given by -114- Fig. 64 Boundary conditions of a general stress field -115- a, = a cos 0 + t sin Q . h e re With sibstitutions (18) and (19), the inequality cr^ > 0 yields 1 - sin 5 cos 2jr' - sin 5 sin 2V _ * JL " o l l l U U U S i -t r\\ tan 0' < ---- g 0"; "7, (119) and is identical with the condition lim j > 0. The latter then implies a positive direction of the horizontal component of the stress vector at the wall. In a channel with a straight wall, Fig. 66, it is apparent that, if the above condition is not satisfied, the top boundary cannot be traction-free, but requires stresses which have a resultant force con­taining a positive horizontal component. This follows from the considera­tion of the equilibrium of the horizontal forces acting on the flowing mass between the wall and a vertical plane through the vertex. The rc pressure across the vertical wall cannot be balanced by H = Jo cos 6 'dr if 0 , hence the requirement for the positive horizontal component at the top boundary. While such top boundary conditions are possible, they are unlikely to occur in gravity flow, and cannot occur with a traction-free boundary. Observations of flow in physical channels confirm this conclusion: they indicate that inequality (119) is satisfied. The above argument tallies with the results of the analysis of the previous section which indicated that for case 1, lim h < 0 , lim j > 0, r>o r*-o a bounded solution lim s(r,0) = lim - (j/h) is obtained for all top r»o r^o boundary values s(ro,0); but for Case 2, lim h > 0, lim j < 0, a bounded r^-o r-f-o -116- Fig. 65 Stress vector at a wall Fig. 66 Equilibrium of a flowing mass -117- solution requires the satisfaction of the condition (118) for the top boundary stress. There thus seems to be sufficient reason to limit the considera­tion of the radial stress fields (which occur in the neighborhood of r->o) to regions in which j > 0 . The substitution of expression (44) with 9' =0' in inequality (119) yields 0' < | - </>' . (120) For a rough wall, similarly, 9' < | - )6 . These bounds for 9' apply only in plane strain. In axial symmetry, the acceptable solutions of the radial stress fields occur within regions of (9,10 which are more restricted than the bound j = 0 , cos 2i\r - sin 5 = 0. The shape of the walls, r' = r'(0) and r" = r"(9), away from the vertex, is a function of the shape of the top boundary, r0 = rQ(9), and the functions = ^ 0(9), s0 = s0(9) along the top boundary. In gravity flow, the top boundary usually varies within wide limits and,therefore, the shape of the walls should vary accordingly. How­ever, this is possible only when a solid flows within itself and forms its own rough walls. Even then, the free adjustment of the shape of the walls is inhibited by the fact that a solid develops cohesion at different rates in the plastic and in the rigid regions . In the rigid regions, the time effect comes into play and, after a while the rigid regions may, in effect, behave as if they were made of a different solid and a plane which is weak with respect to the -118- rigid solid may develop along the walls. When the channel is built of weak walls, the solid will flow along the walls, provided the walls are sufficiently steep and do not contain sharp corners. Typical shapes of channels are discussed in Chapter VI.„ Radial velocity fieltj. As was shown in the previous section, radial stress fields are approached in all channels in the vicinity of the vertex. A radial velocity field is compatible with a radial stress field. It should be emphasized that there is no unique velocity field which corresponds to a given stress field and that there are other velocity solutions compatible with a radial stress field. Indeed, radial velocity fields, which include the lines \|r = Jt/4 and \|r = 3Jt/4, are physically unacceptable because these fields require velocity to be zero along these lines and, in some cases, also require a velocity discontinuity along these lines, which is impossible since they are not sliplines. However, for wall conditions \Jr' < 3n/4 and \|r" > rt/4, the radial velocity fields closely represent the fields observed in physical channels. Since velocity fields are subject to superposition, the radial field provides one readily obtainable solution which can be combined with other fields to satisfy the required boundary conditions. It will again be convenient to use polar/spherical coordinates. The equation of continuity in steady state flow is !^[yurr(r sin 0)m ] + -^[rUgCr sin 0)™] = 0 , (121) -11.9- where u and u are the two components of the velocity vector, Fig. 67. r U The equation of isotropy is In radial flow " 2*<'-e> ■ a f ur a l <122> _r _r __0 dr r rdO ur = V, uQ = 0, (123) and the above two equations reduce to r I? + (1 + m+ y = °> || + (- r || + V) tan 2f = 0. Elimination ofcSV/dr in the last equation leads to |^ + (2 + m + |^) V tan 2\|/ = 0. (z) X* - is now eliminated in equations (y) and (z) by means of the second expression (108) yielding r i + <1 + " + r ^ p > v - ° - || + (2 + m + - ) V tan 2^ = 0. Integration of these equations produces O V » £(e) r ' < 1 + m+ r r p ) , V - g(r) e"<2 + m + 2t d8 To satisfy the differential equatiorB it is necessary that -120- Fig. 67 Velocity components in polar-spherical coordinates -121- f(0) - e'<2 + " + I?p)/tan 2* dlS and t \ - (1 + m + t%t) g(r) = r 1-p , which enforce ^ = ^(6) , as was assumed by the eq.(91) in the radial stress field. The expression for the velocity is o 1 + m + - (2 + m + T^r)/fl0 tan 2^ d0. (124) V = V° (- ) 1_P e 1_P 6 * r It will be noticed that along the rays xjr = 3it/4 and i|f = it/4, tan = °°? and V = 0. The derivative 3V/S0 along one of these rays, e. g. \|f = it/4, will now be determined. Eliminate V in eq. (z) to obtain || - F(r) tan 2* e‘(2 + " + 2* d0. It is necessary to find the limit of cOV/bO as 0 approaches the value at which \|/(0) = it/4. The function is rewritten as follows 6 Q || - F(r) tan 2* e'(2 + m + T ^ e ° tan 2* d6 e‘(2 + n + T ^ e ^ " 1 2t d® where 0^ is a constant, while 0 - 0^ is arbitrarily small. The first exponential is constant and the function can further be transformed into Q AQ I ^ Tji / \ j_ 0 . -(2 + m + T o ) [ j?o'; \ ] /. tan 2i|/ d (2\|r) ^ = c F(r) tan 2i|r e 1-P d(2\j/)J v Y , i where d0/d (2>Jr) is evaluated in the interval \|r^ - i|r. Then, through integration, n + - + .ft- ir- ^' i Sv s . ,cos 2i|r 2 2( 1-B) dilr " Se C F(r) t m ^ (cos 2*'> -122- r -j + JB 4- - ft i (- 'f _ i = c!F(r)sin 2\(/(cos 2^) 2 2(l-f3) dvjr . In the limit dV f m P , d0 i , m °°» f°r t1 + 2 + I 7 F i y ]# ' 1 < 0 ' Lim§|->0, for [1 + f + 2 ^ 1 ) ] ^ " 1 > °* For incompressible solids, (3 = 0, and the limits are tabulated below: Table 3 Limit Plane strain Axial symmetry 00 i-1 A di > 3 de 2 = de 0 - - . ^ < i de di . 3 de 2 It should be noted that the above expressions for P > 0 apply only at some distance from the vertex of the channel where a[ lb. per sq.ft.] » 1. Typical radial flow profiles are shown in Fig. 68 for the two Sv/Se limits. In both cases velocity is zero along the two rays i|r = 3n/4 and \|r = Jt/4. These two rays coincide with velocity characteristics and, of course, are streamlines. The profiles shown in the figure are physically acceptable only within the internal region, that is for channels with weak walls whose wall yield loci intersect the Mohr stress circle within the arc T'MT". -123- Lines of constant velocity in the internal regions for f3 = 0 are shown in Figures 69 and 70 for plane strain and axial symmetry for 5 = 50°. These lines show ratios V/V° along an arc r = r° and, therefore, in accordance with eq. (124) present the function 0 | o = e~(2 + m H tan de. (125) Vertical channels Vertical channels are a limiting condition of converging channels. However, the relations derived for flow in converging channels are not suitable for application to vertical channels, hence the latter will now be considered separately. The radial stress and velocity fields of the converging channels go over to fields independent of the vertical coordinate x in vertical channels. The solutions to such fields in vertical channels are obtain­ed in closed form. There is one major difference between converging and vertical channels. In the former there is no doubt as to the value of the circumferential pressure: that pressure is major. In the latter, assuming complete independence of the vertical coordinate, there is no basis for the selection of either the major or the minor value for the circumferential pressure. The physical consequence of this uncertainty appears in the erratic and unsteady flow pattern which is observed in tall vertical channels. Some conclusions can be drawn, however, on the basis of compress­ibility if even a slight gradient of pressure exists in the vertical -124- Fig. 68 Radial velocity profiles -125- - 40 50 - 30 20 - 10 Fig. 69 Sanction V/V°, 5 = 50° Plane symmetry (symmetric plane flow) -126- Fig. 70 Function V/V°, 6 = 50° Axial symmetry (conical flow) direction. If pressure increases downward then the solid consolidates as it flows and flow has to be divergent (k = -1) to keep each horizontal cross-section filled out. This condition is expected to exist in the upper part of a channel from a stress-free top surface down. In fact, this condition seems to prevail in most channels. It was shown in the analysis of the radial stress fields in axial symmetry that fields with a velocity discontinuity at the walls can be generated only within walls of a sufficient degree of weakness. This limitation carries over to the vertical axi-symmetric channel with the assumption of convergence (k = +1). Hence, it would seem that diver­gence more likely represents the actual stress conditions in a vertical channel of moderate height, while the erratic flow, when it is observed, may be explained by shifts of the circumferential pressure between minor and maj or. Stress field. The purpose of this analysis is twofold: first, to derive an expression for the mean pressure a, second, to find the ratio y"/ye, where y" is the half width of the channel, and y^ the distance from the axis of symmetry to the line of co = Jt/4. Along this line, as will be shown in the analysis of the velocity field, rapid non-steady velocity changes may occur and are, indeed, observed in full size channels. In models with transparent walls this ratio can be observed, measured, and compared with the computed values. Plane-Cartesian/cylindrical-polar coordinates will be used. Pressure -128- r is found from eq.(48) which in this case reduces to dT T + m - s - r dy y and, with the constant of integration evaluated at the axis y = 0 , t = 0 , integrates to xy r = X-3~. xy 1+m T is now eliminated by means of eq. (16), yielding ° (1+m) sin 5 sin 2(jd' (126) The ratio y"/y will now be determined. j j e -P--l--a--n---e-- ---s--t--r--a--i--n--- --f--l--o---w- is considered first. Eq.(49) now applies in the fonn dcr^/dy = 0, hence cr^ is constant. The elimination of o between eq.(126), with m = 0, and eq. (15) yields \ sin 5 sin 2cb-- y r 1 - sin 6 cos 2cd k u For cjd = tc/4, this becomes while for od" = it/4 - <4/2, it is ii _ -H sin 5 cos ^ y l r - s i n S s i n j J ' Hence, in plane strain, the ratio is y" ______ cos <j> ye 1 - sin 6 sin <{>' (128) In axially symmetric flow, m = 1, and the equation of equilibrium (49) is of the form -129- The substitution of expression (126) for o (with m = 1) in equations (15) and (22) yields a - y 1 - sin S cos 2co v - ---- - -- - y 2 sin S sin 2co ' a = r . 1 + k sin 5 y 2 sin 5 sin 2co These expressions and the appropriate derivative are now substituted in the equation of equilibrium yielding, after transformations, dy _ (cos 2co - sin S) d(2co)________ y (1 - k sin 5 - 2 sin 5 cos 2co) sin 2co' which integrates into /■., o\A,-, . ,B ,1 - k sin 5 „ ,C r>\ y = c(l + cos 2co) (1 - cos 2co) i.- y ~ T n ~5- " cos ^cu) > (129) where the exponents A? B , C are as presented below in Table 4. Table 4 Converging Diverging A 1 - sin 5 2(3 sin 5 - 1 ) - 1/2 B 1/2 1 + sin 6 2(3 sin 5 + 1 ) C 2 sin 5 - 1 2 sin 5 + 1 3 sin 5 - 1 3 sin 5 + 1 Within the field, including the boundary oj = co", the observed fields are of the type dcu/dy ^ 0. Since, at the axis of symmetry for y = 0, there is 0) = it/2, this implies 1 - k sin 8 - 2 sin 8 cos 2cd ^ 0 , or cos 2cd.<; 1 - k sin 8 N 2 sin 8 For diverging flow this condition is always satisfied, but for converging flow 1 , 1 - sin 8 „ - Arc cos - -- :- r- 03 ^ it 2 2 sin 8 N 1 , 1 - sin 8 - Arc cos - z-- :--r- , 2 2 sin o The bounds for 0) are Table 5 8 30° o O -4 50° ON O o O O 0)" 30.00° 36.94° 40.61° 42.75° 44.09° 0)' 150.00° 143.06° 139.39° 137.25° 135.51° It will be observed that these bounds match the values of 1 for 0-^-0 plotted for the radial fields in converging channels, Figures 34 to 38. At the wall, y = y", for a solid flowing on a rough wall, 2a>" = rt/2 - fa. From this relation it is possible to determine the maximum values of fa which will allow the formation of a vertical, converging field within rough walls fa ^ Jt/2 - 20)" This is given in Table 6 Table 6 8 30° .p­oo 50° 60° O O r^. fa 30.00° 16.12° O 00 oo 4.50° 1.82° -131- These values of jf> are small for 6 > 40°, hence it would appear that this field is unlikely to develop in most solids. The ratio y"/y& is now computed for axially symmetric flow, as follows: for co = Jt/4 C ,1 - k sin 5, ye ' C<" 2 sin 8 > > while for co" = Jt/4 - 6/2 ii /-I i i /\^ 1 . /nB /l - k s m 5 . / \ C y = c(i + sin 6) (1 - s m 6) (- ---:---- - s m 6) , I s m o and the ratio becomes V . /\A ,, . /nB 2 sin 5 s m 6 nC ■*- = (1 + s m (6) (1 - s m <6) (1 - - ----:----- :----- ) y 1 - k s m 5 e Velocity field The velocity field is also computed with the assumption of independence of the vertical coordinate. The equations of continuity and isotropy (57) and (61) reduce to m T v y = c , and du + dv tan 2<d = 0 . Since the constant of integration c evaluates at zero for y = y", at the wall, v = 0 throughout the field. From the second equation it now follows that u = V is constant, except possibly for co = Jt/4 and co = 3rt/4, where the velocity field allows velocity discontinuities in V, However, these lines are not sliplines and, therefore, steady state velocity discontinuities do not occur along these lines. Unsteady slips are observed- -132- This flow pattern is shown on the photograph, Fig. 71. Since unsteady effects cannot be. reproduced on a still photograph the steady velocity discontinuities at the walls, where oo" = 3it/4 + 6/2 and cd" = jt/4 - S/2, and the unsteady slips along the lines co = rt/4 and co = 3 it/4 appear identical. The measured ratio y"/y = 1.6. While the flow pattern of this model was almost perfectly of the plane strain type, the front and back glass walls obviously prevent a plane strain stress field from developing. The actual stress field deviates somewhat from plane strain toward the axi-symmetric. Ratio y"/ye is now computed for the plane strain stress field from eq.(128), and for the axially symmetric, diverging stress field from eq.(130). Tests of flowability of the solid indicate that 6 = 55°, i> = 50° . For plane strain 1.73 ye For axial symmetry ^ = 1.766"'5x ,234‘263x .310-'762 = 1.25 "e The observed value of 1.6 falls between these two bounds. -133- Fig. 71 Observed flow pattern in a vertical channel -134- PART III INCIPIENT FAILURE General equations The derivation of the differential equations for incipient failure follows the pattern described for steady state flow with the following differences: First, during steady flow the stresses are governed by the effective yield pyramid, and are located at the edge of the base of the pyramid. The latter implies that normality provides only a cone within which the direction of the strain rate vector has to lie. During incipient failure the stresses are govered by the yield pyramid, and are located either on a face or on a side edge of the pyramid, away from its base. Here, normality imposes either a unique direction of the strain rate vector or a plane sector within which the vector has to lie. Second, in steady state flow, the solid is flowing and a velocity field is computed. In incipient failure, velocities are zero, and an initial acceleration field is calculated. In earlier work [e.g. 8 , 19, 20, 21] attempts have been made to compute the velocity fields which follow incipient failure. In these computations it has been assumed that the yield surface remains constant as failure progresses, and that the stress field also remains essentially constant. With the concurrent assumption of normality this enforced continuous expansion of the solid. Since continuous expansion could never be verified experimentally, a serious doubt was cast on the validity of the principle of normality. This doubt appears unjustified, -135- the error seems to lie not. in normality but in the assumption of a constant yield surface. The size of the yield surface certainly is a function of density, and changes as the solid expands. In fact, as will be shown presently, these computations produced not velocity fields but initial acceleration fields. The process of failure is complicated: failure seldom takes place simultaneously throughout a stress field, usually failure occurs along some active sliplines while regions of material bounded by those slip­lines remain rigid (or elastic). The mass becomes non-homogeneous and unisotropic, the yield function varies across the mass with discontinui­ties along the active sliplines. The moment the first active slipline has developed, the incipient stress field does not apply any more and neither does the initial acceleration field. The value of the accelera­tion field associated with a stress field lies in showing that failure can start under the prescribed boundary condition. No attempt will be made in this work to determine how failure proceeds and how steady state flow is ultimately approached. The adopted yield function allows the stress equations to be solved independently of acceleration. The solution produces the function o>(x,y) throughout the field and the acceleration field can then be computed. It is shown that, at zero velocity, the direction of the strain rate vector coincides with the direction of the time derivative of that vector, so that normality and isotropy provide the necessary relations for the solution of the acceleration field. -136- Stress field The equations are derived in plane-Cartesian/cylindrical- polar coordinates. The two equations of equilibrium (48) and (49), together with the equations (29) - (31), which contain the condition of plasticity, provide a solution for the three variables (cr , a ) in v x y xy plane strain. In axial symmetry, the fourth variable Oq, is taken care of by the additional equation (37). In these considerations, the yield locus given by <b and f^, is assumed constant throughout the plastic field. The equations of equilibrium are expressed in terms of cr and CD as follows (1 + sin <£> cos 2co) + sin <t> sin 2o) - 2a sin <b sin 2o> + + 2 a sin <{> cos 2cd = T - m ~ sin sin 2cd, (131) sin sin 2co + (1 - sin <b cos 2oo)|^ + 2a sin 6 cos 2ca + + 2cr sin i sin 2co = m - sin i> (k + cos 2co) . (132) dy y ' Now the following abbrevation is introduced COt O S = - -- In •=- , (133) ^ U0 where a0 is an arbitrary constant. The differential equations then reduce to the form c)(S + cn) , d(S + ffl) . , , « v X T + v sn * tan(" Sx ay *-dUVUJ T+ 74 "- f2); = A > (*> d(S - CD) , d(S - cn) , it , v iSi + 3y tan.(ai - 4 + 2> ' B ~ (b) where -137- In the first characteristic direction, = tan (oj + - |) , (136) the left hand side of eq.(a) is a total derivative d(S + co) _ A dx Similarly, in the second characteristic direction, li ■ t m - 1 + 1 the left hand side of eq. (b) is a total derivative, (137) dx tan (co - - + -), (138) d('S ~ ^ = B. (139) dx The two stress characteristics intersect at an angle rt/2 - (6, and form angles "t (jt/4 - {5/2) with the direction of the major pressure. In incipient failure the stress characteristics coincide with the sliplines. Initital acceleration field The equations of the acceleration field are derived on the basis of isotropy, eq.(61), and the conditions imposed by normality on the direction of the strain rate vector. The three principal strain rates are ordered e. > 0 , e < 0 , m k e > 0 . J. Z '-*» Following Shield's reasoning [10], normality requires e » X,(l - sin fa) , = -(A. + M-) (1 + sin fa) , m ea = p.( 1 + k sin fa) , where ^ and |_i are positive scalars, whose elimination yields e - e2 + (e^ - e p sin + m e^(l + k sin fa) = 0. The principal strain rates are now replaced by e1+e2 - EK +£y - - S - | > £1 - e2 ■ [<£x - v 2 + 4 1% ■ [(t - ! )2 + (| + which, with the use of the equation of isotropy (61), reduces to Bu civ _ 5x dy €1 " €2 " " cos 20) ' and v £0! y leading to ^(cos 2o) + sin fa) + -^(cos 2ca - sin fa) + m ^(1 + k sin fa) cos 2oi = 0 . (c) It is now easy to show that the velocities u and v are proportional to the initial accelerations u and v and, since the equation (c) as well as the equation of isotropy are homogeneous, the velocities can be replaced by the accelerations. The total derivative of the velocity component u, du . , 5u . c>u di ■ u + X u + a^ v> is integrated and, upon application of the mean value theorem, becomes (d)'t+ ( | U + | , ) ' , -139- evidently u = a t and, similarly, v = b t, where a and b are bounded functions, therefore, the second term on the right hand side contains t and becomes insignificant as t - > 0 , hence lim u = u t, similarly, lim v = v t. t - >o t- 3.0 Since t is independent of the coordinates (x,y), the expressions for the strain rates become . cki 3u . Su _ Su t-*o t-»o } . . dv Sv ^ , . c)v dv lim = ■*:- t, lim n- ■ = -=7- t. ox ox dy dy t-»o t-»o Substitution of these expressions for the strain rates in equations (c) and (61) produces 'N • j ® • n^(cos 2cd + sin <b) + -^(cos 2cjd - sin ^) + m -^(1 + k sin ^) cos 2cd = 0, (d) Si . 5u Sv 5v ^ . . , s ■n~ tan 2cd - -=r- - ^r- - tan 2oo = 0. (e) ox oy ox oy The above two equations, together with the equations of the total deriva­tives , di = ^ dx + I ? dy' = ^ dx + % dy' are solved for Su/Sx, yielding = 4^ C (4^ + (sin <b - cos 2cd) - m - (1 + k sin <0(4^ cos 2cd - sin 2o>)J dx dx ^ dx dx y dx /[■£ " tan(cD + 4 " 2> ] [dJ " tan(" " I + 2 y] ■ (£> The directions of the characteristics are 2 -140- ---- T-** y, v V *1 jt/4 - (zS/2 jt/4 - j>/2 x, u Fig. 72 Projections of the acceleration components on the characteristic directions -141- = tan[o) + (~ - -|)] , (140) Along the characteristics, the numerator of eq.(f) is zero, With the substitution of the ^bove expressions for dy/dx in the numerator of (f) that equation becomes du , dv - , v 1 + k sin fa „ --1- ---b m - - - - ---- --- -- = 0. \141) dyJ dx Jy , or„N-i- sin 2[u) T (- - - ) ] In both equations (140) and (141) the top sign applies along the first characteristic and the bottom sign along the second characteristic. For the purpose of numerical calculation, it is often more convenient to express these equations not by the u and v components but by their projections v^ and v^ on the characteristic directions, Fig. 72. The relations are "V ^ s i n ( ( D " f + f ) + s i n ( o ) + - -j) u = --- - - - --- -- --- - - -- ------ -- - , cos fa V■ ^ cos (/a ) - - +, -fa)\ - V• - cos (/'c o +. - - -fa\) V = - - - - - - -------------7 - ■- - --- -- ------- • cos fa The transformation leads to: along the first characteristic: = tan (to + - - ~) , (142) QX ^ Z ^V 1 , V2 v do> , v(l + k sin fa) „ /n/nN -- - (v1 tan fa - - - -t) - + m ---11- - ' • = 0. (143) dyy 1 cos fa dyy „ . , . rt p, 2y sm(a) + - - |) along the second characteristic: ^ = tan(o> - jr + ^) , (144) dv v . , 2 / 1 . . da), v(l+ksin«$) _ \ -- - (---- t - v tan fa) - + m - -- -- --- 'y- = 0 (145) dy cos fa 2 dy . it fa \ ' 2y sm(co - - + j) -142- It will be noted that the acceleration characteristics, as well as the stress characteristics, coincide with the slip lines. Superposition. Since both, the equation of continuity (d) and isotropy (e), are linear and homogeneous, a linear combination of two (or more) solutions of an initial acceleration field is also a solution. Physical conditions. The physical conditions which need to be applied to incipient failure are less restrictive than they were for steady state flow. A. Some, small, tensile (negative) stresses are permissible. B. A line of infinite shear strain rate may occur along a slip line or a weak wall. C. The velocity V of an element of a solid is zero. D. The acceleration of an element, dV dV - , V2 - dt dt S p n ' Ms along the path of its travel, Fig. 18, is bounded. This implies that dV/dt is bounded but 1/n need not be bounded since V = 0. ^s E. Singularities in density are inadmissible. Grids and special lines. There is a great deal of similarity between the grids and special lines of steady state flow and incipient failure. The differences will be particularly emphasized here. As before, the solution of the stress field contains the direction of the major pressure a in the differential form oo = co(x,y) . 1. Sliplines, stress and acceleration characteristics. These three grids coincide in incipient failure. They form lines inclined at angles to "t (rt/4 - 6/2) . 2. Lines of maximum shear strain rate. These lines are, of course, inclined at co "t jt/4 but they are of little significance in incipient failure since they are not observed. 3. Trajectories. Failure is a time dependent process and, therefore, paths of travel of the elements will be referred to as trajectories. The trajectories may have cusps at the initial points, since velocity starts from zero at those points. This is important, because the stress field may be bounded by a stress envelope and incipient failure may occur along a slipline merging into the envelope. This slipline may have a cusp at the envelope yet become a trajectory. 4. Acceleration discontinuities. A line of acceleration discontinuitj develops into a line of velocity discontinuity. Such a line has to be a trajectory (or a streamline) and has to follow a slipline or a weak wall. It separates two regions which may be both plastic, both rigid (or elastic), or one plastic and the other rigid (or elastic). 5. A straight acceleration characteristic in plane strain enforces a constant projection of the acceleration vector on that characteristic. This is evident from, say, eq. (143). 6. Walls. A wall separates a plastic region from a stationary rigid (or elastic) region. Usually, there is a velocity discontinuity -144- along a wall: the wall then is a streamline and stresses along it are defined either by the yield locus, for a rough wall, or by the wall yield locus, for a weak wall. When the solid fails along a rough wall, the wall is a slipline and a stress and acceleration characteristic. The coincidence of an acceleration characteristic with a streamline leads to the relation V2 ym (l + k sin Cj (146) along the wall. This relation is readily obtained from, say, eq. (143) with v^ = V, v2 = V sin */>. Evidently V can be zero along such a wall, which means that a field can develop without an acceleration discontinuity along the rough wall. When the solid fails along a weak wall, equations (79) and (80) apply along the wall with substituted for 5 , and an acceleration discontinuity is mandatory since a zero acceleration along the wall would enforce a zero acceleration within a region of the plastic field. 7. A stress discontinuity may occur only along a trajectory for the reason given in steady state flow, point 11. 8. Stress singularities are inadmissible for the reason given in steady state flow, point. 12. Doming Consider the pressures acting in a dome which has formed across a channel of width B, Fig. 73. It is assumed that the dome will fail if the bottom layer of the dome fails. The bottom layer is taken of a -145- unit thickness, measured in the vertical direction. Evidently, the pressures in the layer are at a minimum when the surface of the dome is smooth and regular and, barring the possibility of tensions, when the mass above does not bear down on the layer. The latter implies zero stress between the layer and the mass above. Under these conditions the layer can be thought of as a self-supporting dome, geometrically and stress-wise independent of the vertical coordinate. The major pressure acts in the plane of the dome. If the vertical axis of the dome is drawn through its highest point, then at a distance y from the axis -O - 2 T v sin 2,to = -- ■- L. 1 1 + m For failure to occur ^ fc, or f 2y ^ (1 + m) ^ sin 2a>. (g) I For symmetrie channeIs, 2y = B and the largest dome occurs for cd' = 3^/4 a.nd co" = Jt/4. For such a dome to fail, it is necessary that f B > (1 + m) y . (147) For asymmetric plane strain with one wall vertical, 0*= 0, eq.(g) gives different relations for each abutment. The conditions on the sloping wall, 0=0', will be denoted by a prime, while the conditions at the vertical wall will be denoted by the superscript v. The rela­tions for failure at the two abutments are f1 s i n 2co1 fV s in 2u)V B - a > - ~~f-f-- -- and a > - - - -- (148) The above formulas (147) and (148) assume that the wall is sufficiently -146- Doming strong to support the major stress at an angle = n/4, for the symmetric dome, and at angles and tu" for asymmetric plane strain. In view of the uncertainty of the value of <t>' in incipient failure, these assump­tions appear reasonable and on the safe side. Piping The conditions leading to the incipient failure of a solid around a vertical, circular hole, or pipe is analysed in this section. The unconfined yield pressure f and the angle of friction are assumed constant throughout the failing region. The stress field is assumed axi-symmetric with a traction-free inner boundary of constant diameter D, Fig. 74. These assumptions imply independence of the vertical coordinate x as well as the circumferential coordinate Ci. A relation will be derived between the yield properties of the solid (fc,^) and the smallest diameter D at which a bounded plastic field can develop. The equations of equilibrium (48) and (49) now reduce to T da a dy + y cr.a = 0. Ratio (149) is introduced and the equations of equilibrium transform to (a) Bounded field Fig. 74 Piping -149- da a - a _J£ + ^ L _ dn 2n + -*r---^ = 0. The substitution of expressions (30) , (31) , (37) (with k = +1), and their appropriate derivatives yields ■4^ + 2 cot 2co dto - -^7- -jv- dn = 0 , a 2n(n-l) 5 ^■=(1 - sin 5 cos 2co) + 2 sin sin 2oo dco - 4^ sin ^(1 + cos 2co) = 0. a zn a is now eliminated between these two equations and, after transforma­tion, the following differential equation is obtained. dm _ JL ___sin 2co , 1 - sin j> cos 2oo 1 + sin Msn'i dn 2. cos 2,60 - sin 6 ' n - 1 2n ‘ The solution of this equation produces a family of curves co = cd (n) Fig. 75, passing through the point (n - 1, 00 = 0). Two types of solutions are distinguished: (a) bounded fields. The bound of the plastic field is provided by an envelope of the second family of characteristics. The envelope occurs for 0) = ic/4 - i/2, and along it dco/dn-*■«*. (b) Un­bounded fields, for which the term within brackets, and with it dco/dn, change sign for to less than rt/4 - 6/2. The physical field extends without bound as n increases. Since the plastic field is independent of the vertical coordinate^ a field (b), unbounded in the horizontal direction, is unbounded in all directions. Such a field does not represent a real situation and, hence, only the bounded fields (a) are accepted as solution to piping. The question is now posed: What is the smallest diameter D at which incipient -150- CD Jl jt/4-^/2 (a) bounded fields * (b) unbounded fields 0 1 2 3 n n e Fig. 75 Function cd ■ co(n) -151- I failure will occur in a bounded field? To answer this question (dco/dn) is taken as the parameter of the family co = co(n) , and a relation is found between this parameter and the diameter D. E q „(h) is integrated = D n ~ ^ V ' r 4n% ‘ Expression (31) is substituted for TXy> leading to i _ _ 4n a sin j sin 2co . r(n - 1) ' In the limit as n-»l, co-?0, a^>0, and, in accordance with e q . (38), 2a sin $-»fc . The expression for the diameter becomes f . „ „ . . „ c , . s m 2co D = 2 - lim ------7. T . n - 1 n»l The application of l'Hospital's rule yields f , n , c ,do\ ° 4 r dn n=l" (151) It is now evident that the required smallest diameter D is determined by the smallest value of the parameter (dco/dn)n_^ which causes a bounded field. The corresponding curve co = co(n) is shown in Fig. 75. The curve intersects the line co = Jt/4 - i /2 for a value of n = n , hence, the e i; outer diameter of the plastic field is Dn . e The ratio n is obtained directly by placing co = rt/4 - i /2 in the e expression within brackets of eq. (150), and equating the expression to zero, n^ = (2 sin i - (152) e This relation is plotted in Fig, 76; n^ increases rapidly as <t> decreases -152- Fig. 76 Function n^ = f(8) e 30 40 50 60 i 70 Fig. 77 Function (do) dn) , = g(6) n = 1 -153- toward 30° . For fa = 30° , = °°. Solids whose angle fa approaches the low value of 30° also have low values of f and stable piping is not observed in these solids. Hence, the analysis loses its practical value as that value of fa is approached. However, it is of interest to note that, for values of fa > 19.5° (sin fa = 1/3), both types of fields, (a) and (b) , exist. For fa < 19.5°, only type (a) remains, because the expression within brackets of e q . (150) does not change sign and dco/dn is positive for all values of n > 1. Values of (dco/dn) , for the smallest diameter D at which incipient n-1 failure must occur in a bounded field have been computed numerically and are plotted in Fig. 77. The diameter D is then obtained for any material of known ratio f /T from eq.(151). c It is interesting to note that investigations of slope stability in axial symmetry [22] lead to profiles which all converge to a vertical hole, Fig. 78. Within the accuracy of numerical calculations, the diameter of that hole seems to be independent of the starting diameter at the surface and seems to equal D, the diameter computed above for the smallest bounded field independent of the vertical coordinate. An acceleration field for piping is given in reference [22]. Fig..78 Slope stability in axial symmetry -155- PART IV FLOW CRITERIA Introduction The flow of a solid in a converging channel is subject to two typical obstructions: doming and piping. The flow criteria are based on the assumption that the solid consolidates under the pressures prevailing during steady state flow, develops an unconfined yield pressure f^, in accordance with its flow-function, and flows if the unconfined yield pressure is insufficient to support a stable dome or a stable pipe. Thus, the flow criteria relate the major consolidating pressure of steady state flow with the pressures of incipient failure, defining minimum ratios a-^/£ ~ called critical flow-f actors, which the solid must exceed for flow to start and to continue. No-doming Doming usually originates at the outlet of a channel where a radial stress field is closely approached during flow, and that field is assumed in the derivation of the flow criteria. The dome formulas (147) and (148) contain the unconfined yield pressure f at the abutments of the dome, i.e. at the end points of the walls. The major consolidating pressure is given by eq. (20) . With substitution (102) for cr in radial flow, the major pressure at a wall is -156- cr| = r x s(0')(l + sin 5). a| is eliminated by means of the flowfactor ff, eq,(41) thus ff = r s(0') (1 + sin 5) . (a) c Plane and axial symmetry, A channel, Fig. 79 (a), is symmetric for the purpose of this analysis if its walls are symmetric with respect to the vertical axis in the neighborhood of the vertex, hence, -0" =0', Then, from the figure, r = B/2 sin 0' and eq.(a) becomes _ _ B_j s (01)(1 + sin S) i t " r o • r 1 * f 2 sin 0 c Substitution for B y/f^ from the eq.(147) produces the following value of the critical flowfactor ff (153) 2 sin 0 This is the no-doming formula for symmetric channels. For the particular case of vertical chanels, 0' = 0" = 0, an expression for the flowfactor is obtained in a closed form by substitut­ing expression (126) for a in e q . (20), thus a = .. r y (1 + sin 5) ^ 1 (1 + m) sin 5 sin 2CO is eliminated by means of eq,(41) and, since at a wall y = y 1 = -B/2, the expression for the flowfactor becomes B x ________ 1 + sin 5_____________ ff = f 2(1 + m) sin 6 sin (2co' - rt) Substitution for By/f from the eq.(147) yields -157- ££ _ ______1 + sin S . (154) 2 sin 6 sin (2031 - it) Plane asymmetry. This channel, Fig. 79(b), has one wall sloping at an angle 0' and the other wall vertical,0" = 0. Here the conditions at each wall are different, and are considered separately. At the slop­ing wall r 1 = B sin 0' and eq.(a) becomes ffi - B T s ( 0 ')(1 + sin 5) " f' sin 0' ‘ c Substitution for y f^ from the first of equations (148) yields B sin 2co' s(0')(l + sin 5) , s ff = " B^a 2 sin 0' * (c) At the vertical wall rV = B cot 0' + b - c, eq„(a) becomes ffv «. (B cot 0' + b - c)r s(0V)(l + sin 5) fV ' c and substitution for y f^ from the second of equations (148) produces r f~v B cot 0' + b - c sin 2coV ,„vv ,, . . ^ f f = ------------------ - ----- s (0 ;(1 + sin 8). (d) a I From equilibrium it follows that the assumed dome has the shape of the parabola ? b (a + y) / \ x = - ----- (e) (B - a) The parameters a, b, c and the functions sin 2o>' and sin 203V in equation (c) and (d) are now replaced by tan 03' and tan 03V , leading to ff i _ tan coV - tan Q)1 - tan Q31 s (0 ') (1 + sin 5) f ,,sv , 2 , sin 0' ' K tan 03 1 + tan 03' -158- (a) (b) Fig. 79 Outlet conditions for no doming -159- „„v (tan co1 - tan cov ) ( 2 cot 0 ' tan co' tan oov - tan ojV - tan c o ' ) ff = --------------------------------------------------------------- Sc 2 tan os'(1 + tan co ) x s(0V ) (1 + sin 5). (156) The acceptable range for angle (jo' is 3fl/4 < co' < i t . Within this range dffV/cko' > 0, while c i f f ' / c k o ' < 0, and there is one value of co' for which ffV = ff' = ff. This value of the flowfactor is maximized with v respect to co , yielding the critical flowfactor. Flowfactor plots. Contours of constant values of the critical flowfactors are plotted for 6 = 30°, 40°, 50°, 60° and 70° for plane symmetry in Figures 80 to 84, and for axial symmetry in Figures 85 to 89. For plane asymmetry, the flowfactor is plotted in Figures 90 to 92 for 5 = 50° and for an angle of friction at the vertical wall 0V = 20°, 30°, and 40°. Influence of compressibility. Figures 93 and 94 show a comparison of the flowfactors for a compressible solid with P = 0.10, with those of an incompressible solid. The plots are for 5 = 50°. Figure 93 shows symmetric plane strain, and Figure 94 axial symmetry. The contin­uous lines are for the compressible solid, and the dashed lines for the incompressible solid. The latter are copied from Figures 82 and 87. It is evident that even the large degree of compressibility corresp­onding to (3 = 0.10 does not appreciably affect the no-doming criterium. Hence, the compressibility of the solids may be neglected. -160- 30 20 10 20 30 40 50 Fig . 80 Critical flowfactor ff, S = 30° Plane symmetry (symmetric plane flow) -161- 60 d 1 70 0 10 20 30 40 50 60 &' 70 Fig. 81 Critical flowfactor ff, 8 = 40° Plane symmetry (symmetric plane flow) -162- 70 . 90° 60 50 40 30 20 10 0 E 50 40 30 - 20 - 10 30 Fig. 82 Critical flowfactor ff, 5 = 50° Plane symmetry (symmetric plane flow) -163- 60 - 50 40 - 30 - 20 - 10 30 40 Fig. 83 Critical flowfactor ff, S = 60° Plane symmetry (symmetric plane flow) -164- Critical flowfactor ff, 6 = 70° Plane symmetry (symmetric plane flow) Fig. 85 Critical flowfactor ff, 5 * 30° Axial symmetry (conical flow) -166- 50 90° 40 30 20 10 10 20 30 40 Fig. 86 Critical flowfactor ff, 8 = 40° Axial symmetry (conical flow) -167- 50 60 0' 70 30 40 Fig. 87 Critical flowfactor ff, S = 50° Axial symmetry (conical flow) -168- - 40 i' - 30 - 20 - 10 Fig. 88 Critical flowfactor ff, 6 = 60° Axial symmetry (conical flow) -169- Fig. 89 Critical flowfactor ff, 5 = 70° Axial symmetry (conical flow) -170- Fig. 90 Critical flowfactor ff, 5 = 50°, = 20° Plane asymmetry (plane flow-one vertical wall) -171- Fig. 91 Critical flowfactor ff, 5 = 50°, 6V = 30° Plane asymmetry (plane flow-one vertical wall) -172- Fig. 92 Critical flowfactor ff, 8 = 50°, = 40° Plane asymmetry (plane flow-one vertical wall) -173- 10 20 30 40 50 60 0' 70 Fig. 93 Critical flowfactor ff, 5 = 50°, p = 0.10 Plane symmetry (symmetric plane flow) -174- Fig. 94 Critical flowfactor ff, 5 <* 50°, p = 0.10 Axial symmetry (conical flow) -175- No-piping A tendency for piping exists in all channels in which the solid flows within rough walls. It was shown in the analysis of axi-symmetric radial stress fields that such channels are very steep or, indeed, vertical. It is observed in experiments that these channels circumscribe the outlet and, hence, that the largest dimension of the outlet defines the diameter of the channel. In the derivation of the formula for no-piping it