...........
`Volume 52
`
`BULLETIN OF THE UNIVERSITY OF UTAH
`October 1961
`
`No. 29
`
`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
`
`/
`
`SNF Exhibit 1012, Page 1 of 322
`
`
`Volume 52
`
`BULLETIN OF THE UNIVERSITY OF UTAH
`October 1961
`
`No. 29
`
`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
`
`/
`
`SNF Exhibit 1012, Page 1 of 322
`
`
`
`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 compre
`hensive 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 plas
`ticity 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
`
`SNF Exhibit 1012, Page 2 of 322
`
`
`
`calculations.
`The careful reader of the author’s previous reports and papers on
`the subject of flow of bulk solids will notice substantial modifica
`tions 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 recommenda
`tions 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 pre
`paration [22, 23]. It is shown that, provided the slopes of the walls
`of a converging channel are sufficiently steep and mathematically
`
`SNF Exhibit 1012, Page 3 of 322
`
`
`
`continuous, the stress pattern in the neighborhood of the vertex of
`the channel is, primarily, a function of the slope and of the fric
`tional conditions of the walls at the vertex, with the influence of
`the top boundary of the channel vanishing at the vertex . The parti
`cular 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
`
`SNF Exhibit 1012, Page 4 of 322
`
`
`
`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 opera
`tion 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
`
`SNF Exhibit 1012, Page 5 of 322
`
`
`
`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 import
`ant 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
`
`SNF Exhibit 1012, Page 6 of 322
`
`
`
`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 Univer
`sity 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.
`
`SNF Exhibit 1012, Page 7 of 322
`
`
`
`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
`
`SNF Exhibit 1012, Page 8 of 322
`
`
`
`CONTENTS
`
`PART I - THE YIELD FUNCTION
`Introduction
`The coordinate system
`Effective yield locus
`Stresses and density during flow
`Yield locus
`Time yield locus
`Stresses during failure
`Flow-function
`Flowfactor
`Wall yield locus
`PART II - STEADY STATE FLOW
`General equations
`Stress field
`Velocity field
`Superposition
`Physical conditions
`Grids, special lines and regions
`Converging channels
`Equations of stress
`Radial stress field
`Derivation
`Solutions of the radial stress field
`Resultant vertical force
`Stresses at the walls
`Influence of compressibility
`General stress field
`Proof of convergence to a radial
`stress field at the vertex
`Boundaries
`
`viii
`
`1
`1
`5
`9
`10
`15
`22
`22
`24
`26
`28
`35
`35
`35
`37
`40
`40
`42
`57
`57
`59
`59
`63
`68
`84
`84
`107
`
`107
`114
`
`SNF Exhibit 1012, Page 9 of 322
`
`
`
`Radial velocity field
`Vertical channels
`Stress field
`Velocity field
`PART III - INCIPIENT FAILURE
`General equations
`Stress field
`Initial acceleration field
`Superposition
`Physical conditions
`Grids and special lines
`Doming
`Piping
`PART IV - FLOW CRITERIA
`Introduction
`No-doming
`Plane and axial symmetry
`Plane asymmetry
`Flowfactor plots
`Influence of compressibility
`No-piping
`PART V - TESTING THE FLOW PROPERTIES OF BULK SOLIDS
`Apparatus
`Testing
`Continuous flow
`(a) Representative specimen
`(b) Uniform specimen
`(c) Flow
`(d) Shear
`Example
`
`ix
`
`119
`124
`128
`132
`135
`135
`137
`138
`143
`143
`143
`145
`148
`156
`156
`156
`157
`158
`160
`160
`176
`182
`182
`186
`186
`186
`188
`190
`195
`198
`
`SNF Exhibit 1012, Page 10 of 322
`
`
`
`'
`
`. Time effect
`Density
`Plots of flow properties
`Angle of friction
`PART VI - DESIGN
`Introduction
`Flow properties of bulk solids
`Limitations of the analysis
`Types of flow
`Mass flow
`Hopper& with one vertical wall
`P uig f low
`Calculations of the dimensions of the outlet
`(a) Doming
`(d) Piping
`'
`(c) Particle interlocking
`Influence of dynamic over-pressures
`- Flow ptomor.ing devices
`Examples of design for, flow
`Feeders
`Feeder Loads
`Belt feeder
`Side-discharge reciprocating feeder
`Segregation and blending in flow
`Flooding
`Heat transfer
`Gas counterflow
`Structural problems
`Stresses acting on hopper walls
`Bin failures
`
`Ore
`.
`
`Broken rock
`
`x
`
`’
`
`.
`
`202
`204
`204
`206
`208
`208
`209
`217
`218
`219
`228
`230
`231
`231
`234
`236
`236
`242
`248
`268
`272
`274
`278
`282
`286
`288
`288
`292
`292
`292
`294
`294
`
`SNF Exhibit 1012, Page 11 of 322
`
`
`
`Coarse ore
`Block-caving
`REFERENCES
`
`301
`304
`307
`
`xi
`
`SNF Exhibit 1012, Page 12 of 322
`
`
`
`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, fric
`tional, 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-
`
`SNF Exhibit 1012, Page 13 of 322
`
`
`
`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 princi
`ple 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-
`
`SNF Exhibit 1012, Page 14 of 322
`
`
`
`Fig. 1
`Yield surface
`
`-3 -
`
`SNF Exhibit 1012, Page 15 of 322
`
`
`
`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
`=0, cr = a, = a( 1 + sin 6)
`a = a = ct0 = a(l - sin 6), t
`x
`y
`2
`
`xy
`(X
`1
`
`-4-
`
`SNF Exhibit 1012, Page 16 of 322
`
`
`
`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 pseudo
`cohesion 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-
`
`SNF Exhibit 1012, Page 17 of 322
`
`
`
`coefficient m to distinguish between the two systems. Coefficient
`m — 0
`
`applies to plane strain, and
`
`m = 1
`
`(1)
`
`(2)
`
`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 circumfer
`ential 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
`
`(3)
`
`'
`and of the ray r.
`
`co = e + t,
`where \|r is the angle between the directions of
`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
`
`-6 -
`
`SNF Exhibit 1012, Page 18 of 322
`
`
`
`Fig. 2
`Coordinate systems in plane strain
`and in the meridian plane of axial symmetry
`
`SNF Exhibit 1012, Page 19 of 322
`
`
`
`The assumed yield function is of the following general form
`
`(4)
`
`(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 consolidat
`ed 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 con
`solidation 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 temp
`erature T, and the moisture content H, thus
`T = T(J15t,T,H).
`
`(6)
`
`SNF Exhibit 1012, Page 20 of 322
`
`
`
`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,
`(7)
`H = constant.
`t = 0, T = constant,
`Density then becomes a single-valued function of the major con
`solidating 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 .
`1 - sin 5
`
`w
`
`ct2
`
`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-
`
`SNF Exhibit 1012, Page 21 of 322
`
`
`
`Qi x 2V +S
`sin 5 = v ' * J -------(11)
`J
`+ a
`y
`
`1 x
`1
`
`'
`<i
`
`or
`
`p
`
`li
`
`r - v2 + 4t2r 9
`
`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 +an
`1 2
`x y
`r Q
`a .+ a „
`a + a
`.
`
`'
`
`"
`
`,
`
`a -
`The component stresses can now be expressed in the plane-Cartesian/
`polar-cylindrical coordinates by
`a = a(l + sin 8 cos 2oo) ,
`
`(13)
`
`(14)
`
`a = a(l - sin 5 cos 2a>) ,
`T
`= o sin & sin 2a>:
`xy
`’
`
`(15)
`(16)
`v '
`
`-10-
`
`SNF Exhibit 1012, Page 22 of 322
`
`
`
`Fig. 3
`Effective yield surface
`
`-11-
`
`SNF Exhibit 1012, Page 23 of 322
`
`
`
`and in the polar/spherical coordinates by
`cr^ = a(l + sin 6 cos
`
`,
`
`Oq = a(l - sin 6 cos 2v|/) ,
`
`T Q = o sin 8 sin
`rt7
`The principal pressures are
`= a(1 + sin 5),
`
`.
`
`a2 = o(l - sin 8),
`
`Oq, = a( 1 + k sin 8),
`
`(17)
`
`(18)
`
`(19)
`
`(20)
`
`(21)
`
`(22)
`
`where
`
`(23)
`k = + 1,
`for converging flow, locates the stresses on the edge OA of the pyramid,
`Fig. 3, while
`
`(24)
`k = -1,
`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) .
`This relation has been found experimentally to be well represented
`by the equation
`
`(25)
`
`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-
`
`SNF Exhibit 1012, Page 24 of 322
`
`
`
`Fig. 4
`Effective yield locus
`
`Fig. 5
`Yield loci
`-13-
`
`SNF Exhibit 1012, Page 25 of 322
`
`
`
`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.
`
`* £ • -
`
`........ i .... .
`
`Table 1
`
`Solid
`Adipic acid
`Soybean oil meal (dry)
`Gocoa powder
`Light soda ash
`Foundry sand
`Iron ore (6% H^O)
`Taconite concentrate
`Taconite concentrate
`Copper concentrate (dry)
`Copper concentrate (57o H?0)
`Feed granules
`Feed granules
`
`P
`.026
`.010
`.096
`.017
`.009
`.076
`.029
`.036
`.009
`.055
`.017
`.020
`
`Eq. (26) leads to awkward mathematical expressions and in parts of
`the analysis will be replaced by
`(27)
`...........
`T “ TocP,
`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-
`
`SNF Exhibit 1012, Page 26 of 322
`
`
`
`Yield locus (YL)
`The yield function defined by conditions (7) and eq. (8) is repre
`sented 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 plast
`icity, 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
`-(#,/4 - i/2) to the direction of the major pressure G^, and the solid
`deforms. These two planes are called the slipplanes, and are represent
`ed 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
`
`_
`
`-j-
`
`-15-
`
`SNF Exhibit 1012, Page 27 of 322
`
`
`
`
`
`ae
`
`Fig. 6
`
`Yield locus
`
`Yield locus
`
`-16-
`-$NF Exhibit 1012, Page 28 of 322
`
`SNF Exhibit 1012, Page 28 of 322
`
`
`
`Fig. 7
`Sliplines
`
`Unconfined yield pressure, f
`
`-17-
`
`SNF Exhibit 1012, Page 29 of 322
`
`
`
`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 compo
`nent 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
`
`CT1 = fc”
`°2 ~ O’
`f is called the unconfined yield pressure and is obtained by inscrib
`ing 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-
`
`SNF Exhibit 1012, Page 30 of 322
`
`
`
`Fig. 9
`Unlikely shape of the yield locus
`
`Fig. 10
`Tensile (brittle) failure
`
`-19-
`
`SNF Exhibit 1012, Page 31 of 322
`
`
`
`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-
`
`SNF Exhibit 1012, Page 32 of 322
`
`
`
`2nd
`
`slipline;2:‘/
`
`
`lst
`
`slipline
`
`. Fig. 11
`
`a Fig. 11
`
`Failure of a dome
`
`Failure of a dome
`
`-21-
`-21-
`SNF Exhibit 1012, Page 33 Of 322
`
`SNF Exhibit 1012, Page 33 of 322
`
`
`
`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)
`If flow is stopped for an interval of time t, the consolidating
`.• j
`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
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