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J . R . J O H A N S O NMechani sms D i v i s i on ,A p p l i e d R e s e a r c h L a b o r a t o r y ,Un i ted S ta te s S tee l Corpora t i on ,Monroev i l l e , Pa .A s s o c . M e m . A S M E

S t r e s s a n d V e l o c i t y F i e l d s i n t h eG r a v i t y F l o w o f B u l k S o l i d sT his paper presents the mathematical theory of stress' and velocity field s for the steadygravi ty flow of bulk solid s in converging channels. T he basic equati ons based on theconcepts of soil mechani cs and pl asticity are presented al ong wi th the appropr ia teboundar y conditi ons. .1 simpl ifi ed stress field (call ed the "rad ia l stress field" byJ enike), compatible wi th straight-wall ed converging channels, is shown by calcul atedand experi mental evidence to approximate closely the physical stress fi eld s occurr ingin the deforming portions of the soli d i n plane-strain and axisymmetric straight-wall ed-converging channels.

T he gravi ty flow of bulk solids such as ore, coal,concentrates, chemicals, flour, and sugar has been studied forsome tim e. H owever, the mathemat ical computat ion of stressand velocity fields for the steady flow of such solids in convergingchannels is of recent origin . T he method used in this paper isbased on the concepts of soil mechanics and plasticity adapted byJ enike and Shield [l] 1 to permit the description of steady flow offrictional, cohesive solids.

T he first porti on of the paper deals with th e theory of steadyflow as it appli es to convergin g channels.T he second porti on is concerned wi th a specific solu tion calledradial stress. I t is shown by calculated and experimental ex-amples that radial stress applies to flowing portions of the solidin plane-strain and axisymmetric straight-walled convergingchannels.

T h e o r y o f S t e a d y F l o wT he stress and veloci ty fields discussed here appl y to condit ionsof steady flow. T hi s means tha t the stress and velocity at anypoint are unchanged with tim e. T he boundaries are also as-

sumed to be un changed wi th tim e. A s the solid flows out of achannel, the channel is refilled at the top in such a way as to1 N um ber s in brackets designate References at end of pa per.Pr esented at the Su mmer Confer ence of the Appl ied M echani csDi v is ion, Boulder , Colo. , J une 9-11, 1964, of TH E AMERICAN SOCIETY

O F M E C H A N I C A L E N G I N E E R S .Di scussion of this paper should be addressed to the E dit ori al D e-par t ment , A SM E , Un i t ed Eng ineer ing Cent er , 345 E as t 47t h S t reet ,N ew Y ork , N. Y . 10017, and wil l be accepted until Oct ober 10, 1964.Di scussion received after the closing date wil l be retur ned. M an u-scr ipt received by A S M E Ap pl i ed M echani cs D i v is ion, M ay 20, 1963;f inal draf t , October 28, 1963. Paper N o. 64 A P M -16.

T

maintai n constant boundari es. T hi s condition is rarely achievedin practice. H owever, if the change wi th time is small and con-tin uous, the steady-flow fields calculated fr om the boundary con-ditions at a given ti me will closely approximate the real fields.Steady flow in a converging channel is characterized by a con-tinu ous deformati on wi thout change in the stress at a given poin t.I t appears from experimental evidence that this type of flowoccur s onl j' for certain stress conditi ons. R esults of the shear-test procedure proposed by J enike [2] show that M olir circlesrepresenting a steady-flow stress are approximately tangential to astraight line through the point of zero stress, F ig. 1. D e J ong

N o m e n c l a t u r ex, y, z = rectangular coordinates

r, 6, = spherical coordinates7 = bu lk densityS = effective angle of friction

of a solid angle of i ntern al fr ict ionof a solid= angle of fri ction between aflowing solid and a wall6' = angle of in clin ati on of wallof a channel measuredfrom vertical axisfj. = 7r/4 8/ 2 = angle be-

tween directions of ma-jor pressure and ofstress characteristicsacute angle between tan-gent to a wall and direc-tion of major principalpressureangle between z-axis anddirection of major pres-sure measured counter-clockwise from z-axispressure components in x,y, and z-directions

0i, cr2

1 z y l ' z z l ' y zX I , 0

major and minor principalpressures in x-y or r-0-planes

mean norm al pressure =| (o"i + a-,) = i (cr x +

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[3] shows the same results in a lest in whi ch cri +

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along itself. On e can also visual ize combinat ions of these con-ditions. T he top boundary of the bulk solid is assumed tr action-free. I f in addi tion to being traction- free the surface also satis-fies the E Y L conditi on for steady flow, the stress state of thesurf ace is defined by p oin t 0 i n F ig. 1 and is therefore stress-free.T he M ohr circle for this condi tion degenerates to a poin ta = r = 0. T he dir ection of prin cipal stress co, however, can bedetermined [6] as a function of the angle of inclination e of thestress-free boundary as

sin e2co, = e + sin 1 - - + I Tsin o ( 4 )T he subscri pt s in dicates the values along a stress-free bounda ry.I t should be noted here that for < = 0, all terms in equations (2)that in volve m disappear; hence equation (4) holds for both axialsymmetry and plane strain.

When the solid flows on rigid side walls, the normal and shearstresses along the wall are found experimentally to lie approxi-mately on a straight li ne as shown in F ig. 1 by the "wall yieldlocus ," W Y L . T he angle is said to be the angle of fri ction be-tween the rigid wall and the solid under steady flow. T he anglebetween the direction of the major principal stress and the slopeof the wall is called v'. F rom t he geometr y of F ig. 1 comes therelation2v' -1 S 'n ^ A*'IT Sin 1 r ffl ( 5 )

Thus v' is a function of the solid property S and the angle of fric-tion between th e soli d an d th e wall is the angle of internal friction. F i g . 6 H o r i z o n t a l s t r e s s - f r e e b o u n d a r y

P r o p e r t i e s o f S t r e s s a n d V e l o c i t y S o l u t i o n sWithout actually solving the equations, it is possible to drawsome general conclusions about the solutions in certain regions oralong certain lines.Owing to the uniqueness of solution of hyperbolic-type equa-tions and the continuous dependence of the solution on the initialdata [7], any well-posed boundary-val ue problem with boundaryvalues identical to or approximately equal to a known solution ofthe equations will define a solution (within the region of unique-ness of the pr oblem ) equal t o or approxim ately equal to the k nownsolution.T hi s property of hyperboli c systems yields some interestingresults when applied to the stress and velocity equations. T hecondition u = c, v = (1 m)d, where c and d are constants, is a

solution to equations (3). H ence, if condit ion u = c, v = 0 is as-signed along the noncharacteristic top boundar y AB of F ig. 5, theresulting boundar y-value pr oblem enforces this soluti on down to afirst characteristic CD where C is the lowest point at which thewall is vertical.Suppose velocity was not assigned along AB but it was ob-served that u = c along the portion of the vertical w all BE .This would enforce the solution u = c, v = 0 in region A BCD.T he conclusion fr om this is that if the velocity along a sufficientportion of a vertical wall is observed to be constant, the velocityfield must be constant down to the point at which the wall is nolonger vert ical . T he length of the observed porti on must be suchthat the unique field determined by it includes at least one point

along the centerli ne. I f u is appr oximat ely constant along sucha section, then the field determined would approximate u = c,v = 0.T he solut ion of equati ons (2) for co = i r/ 2 and cr = a(x) is

do ydx 1 sin S

(7 Oo = 1 sin d (X .To) (6 )A n example of a boundar y definin g such a field is the hori zontalstress-free bound ary sliowm in F ig. 6. Si nce e = 0, equation (4)implies co = T T / 2 , whil e cr = 0 along x = 0; hence

yx1 sin S

in region ABC. If the z-axis is the centerline, enforcing co = T T / 2along it, then this solution occurs in the region AC D also.R a d i a l S t r e s s S o l u t i o n

T he more importan t applications of the flow of bulk solids occurfor chann els wi th straight sloping walls. I t has been shown [4]that near th e in tersection of the side wall s, poin t .4 in F ig. 3, inplane strain or axial symmetry, the stress field approaches a cer-J o u r n a l o f A p p l i e d M e c h a n i c s S E P T E M B E R 1 9 6 4 / 5 0 1

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tain conditi on called radial stress. T hi s soluti on [8] is suchthat the direction of principal stress is constant along anyradial line from the intersection of the walls, making it compati blewith the straight side walls with a constant frictional conditionalong them. H owever, the question of the validit y of the radialstress field at points away from the intersection of the walls re-mains. F ollowi ng are examples of calculated stress fields and ex-perimental evidence that radial stress is closely approachedwhenever flow occurs along straight walls with constant frictionalproperties.C a l c u l a t e d E x a m p l e s o f C o n v e r g e n c e t o R a d i a l S t r e s s F i e l d

The following examples of plane-strain stress fields, as calcu-lated by the method of characteristics [5, 6, 7] for boundary condi-tions chosen at random, may or may n ot represent actual physicalcondit ions. I n each example the radial field compati ble with thegiven straight-wall conditions is closely approached in regionsaway fr om the stress-free top boundary. I t must be rememberedthat, although certain regions may be assigned as flowing or asrigid, the actual size of these regions is governed by more thanmere equilibrium of forces, and in practice may differ from thoseassigned in the examples. I t wil l be shown in a later section thatthe actual physical boundary is such as to approach radial stresspossibly closer than in the examples given here. Since theseexamples are calculated by the method of characteristics [5],they not only show the convergence tendency to radial stress, butthey also serve as a check on the validity of the calculated radialstress fields and on the method of characteristics calculations.I n the followi ng examples, a is shown to approach r adial stressvalue at the centerline and at the wall. T o show that the stressfields in the given examples approach radial stress, it is sufficientto show that the value of a along the centerline approaches radialstress. T his follows from the continuous dependence of the solu-tion on the boundary conditions in the region of the unique solu-ti on defined by such condit ions. I n these examples, the center-lin e has 0) = 7t/2, which is one condi ti on necessary for symmetr icradial stress. I f

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W A L L

F i g . 1 2 R a d i a l s t r e s s a n d v e l o c i t y c h a r a c t e r i s t i c s ; S = 5 0 , 0 ' = 2 9 0 ' = 2 4

F ig. 12 shows one of the famil y of characteristics as calcul atedfor co, given by the radial stress field for the boundary conditions6' = 29, = 24 , and 8 = 50. T his characteristic was deter-mined by a graphical solution of the velocity-characteristic equa-tion

dx = ta" (" - I )for th e appropriate radial stress solution, as is shown in F ig. S ofJ enike's publi cation [8]. T he other members of the famil y ofsolutions are of exactly the same shape with respect to 6 and dif-fer only by a scale factor in r, because co = co(0) for r adial st ress.F ig. 13 shows the same view as F ig. 11, except that a centerline isdrawn in and the radial stress velocity characteristic is plottedstarting at the same point on the centerline as the velocity charac-teristi cs observed in the model. T he higher chara cteristic ob-served in the model on the left deviates from the radial charac-teristic as it approaches the wall. H owever, the lower charac-teristic at the right coincides with the radial characteristic asclosely as can be determin ed from the photograph. F rom this itis observed that the radial stress is closely approached, for ifco = co(0) is given as radial , then th e stress field is rad ial [8],T he deviation fr om the radial characteristic exhibited by t heupper left characteristic is to be expected since it starts at thecenterl ine very close to the top stress-free bound ary . (A hori -zontal stress-free boundary with x measured from it requirescr = yx/ (l sin 8).)

y i

F i g . ] 1 P l a n e - s t r a i n m o d e l w i t h a h o r i z o n t a l t o p b o u n d a r y ; h = 5 0 ,e' = 2 9 , < p ' = 2 4

V E L O C I T YC H A R A C T E R I S T I C

F i g . 1 0 E q u i p m e n t f o r p l a n e - s t r a i n t es t i n g

wall conditions were produ ced. So that the flow patterns could beobserved and photographed, it was necessary to place thin hori-zontal layers of white powdered clay against the glass on the frontside, F ig. 11. T hese layers then deformed, showi ng the flowpattern. Cau tion should be used in interpreting the deforma-tions of the layers, because the result observed is the integratedeffect of the flow patt ern on the layers. F or this reason, only th einitial deformations are considered significant.F ig. 11 shows the plane-strain model filled with the solid andwith a horizontal top bound ary. T he walls are inclin ed from thevertical at 6' = 29 with ' = 24. T he entire solid was movin g,including the upper side regions that appear undisturbed in thephotograph. T he observed breaks in the whi te lines are regionsof rapid changes in velocity and not velocity discontinuities.This is evidenced from the fact that flow did occur across thebreaks. Cr oss flow would not be possible in the case of a velocit ydiscontinuity because of the infinite acceleration of the particlesin volved. Si nce the velocity field across these breaks is con-tinu ous, the break line must be a velocity characteristic. T hi sfollows from the role that characteristics play in determiningregions of unique solution for hyperbolic-type partial differentialequations [5].

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F i g . 1 3 P l a n e - s t r a i n m o d e l w i t h r a d i a l v e l o c i t y c h a r a c t e r i s t i c s

F i g . 1 4 P l a n e - s t r a i n m o d e l ; S = 5 0 , 6' = 2 9 , < f > ' = 2 6 . 5 T o give an indi cation of the effect that deviation fr om radialstress has on the shape of the characteristics, a comparison be-tween two stress characteristics in F ig. 7 is given. I n the figure,AB is the characteristic computed from the boundary conditionsas shown, and AC is the stress characteristic for the radial stressfield, compati ble with the wall conditions. One observes thatthere is a noticeable difference in the two characteristi cs. T hecalculated value of cr differs hy about 15 percent fr om the radialvalue.F ig. 14 shows the plane-strain model w ith 6 ' = 29 but with the

walls slightly r ougher than in the previous ease. As before, theappropriate radial velocity characteristic is plotted and goodagreement occurs between the radial velocity characteristic andthe observed characteristic.Convergence to radial stress is not limited to horizontal stress-free boundaries. As was indicated in the calculated fields, con-vergence also occurs in a straight wall section below a verticaltr ansi tion. F ig. 15 shows such a case for 6' = 29 and 0 ' = 24.As before, there is good agreement between the radial character-istics and the observed characteristics.

A x i a l S y m m e t r y T e s t sThe axial-symmetry equipment consists of right circular cones

F i g . 1 5 P l a n e - s t r a i n m o d e l w i t h a v e r t i c a l s e c t i o n ; S = 2 4

5 0 , 6 ' = 2 9

L I M I T O F R A D I A L S T R E S S

R A D I A L S T R E S S R E G I O N

made of galvani zed metal. A fter being constr ucted, these conesare reinforced by metal bands and are then cut in half along avertical plane of symmetr y, as shown in F ig. 17. F or the experi-ment the cone is assembled in the vertical position. T hi n l ayersof white clay are used as markers, as in the plane-strain model.Some of the solid is allowed to flow slowl y out of the cone. T helevel of the solid is then made even with the top of the cone andboth top and bottom are sealed. T he cone is then placed withits axis horizontal, and the upper half of the cone is removed.T he solid is scraped off to about 1/16 in. above the bottom sectionof the cone. T he remainder of the solid is removed by gentl yblowing it away. T he whi te layers of clay have sufficient co-hesion to remain intact while the dark material, essentially co-hesionless, is removed. W it h this technique, the results of theaction inside the solid can be observed w ith out actually distur bingthe test.

I n axial symmetr y, radial fields occur onl y for certain boundar ycondit ions, as shown in F ig. 16. F or example, if the wall has anangle of fri ction = 20 for a solid w ith 8 = 50, F ig. 16 showsthat 6' 26 is the maximum angle the half-cone can have andstill develop a radial stress field. T o investigate the flow pattern

F i g . 16

to 20 30 40B'L i m i t o f r a d i a l s t r e s s s o l u t i o n ; 5 =

50

5 0

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F i g . 2 0 A p p r o x i m a t e s t r es s f i e ld f o r c o n v e r g i n g c h a n n e l s

F i g . 1 7 A x i a l s y m m e t r y m o d e l ; 5 = 50, 0 ' = 1 5 , 0 ' = 2 4

F i g . 1 8 A x i a l s y m m e t r y m o d e l ; S = 50 , 0 ' = 2 1 , < t > ' = 2 4

F i g . 19 A x i a l s y m m e t r y m o d e l ; 5 = 50 , 0 ' = 30 , < t > ' = 2 4

J o u r n a l o f A p p l i e d M e c h a n i c s

in axial symmetry, cones with three different angles were used.T he fri ction of the cones was changed by coveri ng th em withsandpaper and high gloss paper. T he points plotted in F ig. 9are actual test point s. T he circled point s indi cate that flow alongthe walls was observed, and the square points indicate no flowalong the walls.

F ig. 17 shows the flow pattern in a cone with 6' = 15 and < t > '= 24. I n this case the 6', < t > ' point in F ig. 16 is well withi n theradial stress region. T he radial stress veloci ty characteristi c isplotted in the photograph and agrees with the actual velocitycharacteri stic. F ig. 18 shows the flow pattern for a cone wi th6' = 21 , c> 24. T hi s poin t lies ju st on the criti cal line ofF ig. 16. M ost of the flow occurred in the mi ddle of the cone.H owever, th e sides did flow somewhat , as is evidenced by theupward curvature of the white lines along the wall of the cone.I t is of i nterest to note th at for th e same cone wit h = 23 th eflow along the walls was much more rapid than for the case pic-tured in Fi g. 18 where (p' = 24. F ig. 19 shows a cone wit h 6' =30 and = 24, wh ich poin t lies outside the radial stress regionin F ig. 16. I t can be seen that no flow occurr ed along the edgesof the cone but occurr ed only in a central plu g. T he remainder ofthe cases plott ed in F ig. 16 are not pi ctur ed because they exhibi tedthe behavior as the three points already discussed. T hus it isseen that if the 6', ^'-values lay in the radial stress region, flowoccurred along the walls. I f the point was outside the radialstress region, only a central core flowed.

T he conclusions from thi s evidence are as follows:1 T he radial stress field is a basic element in axisymmetri c andplane-symmetric gravity flow of bulk solids.2 W henever possible, the stress field wil l appr oximat e theradial stress field.3 W henever the radial stress field is not possible, the flow willbe somewhat restricted.4 T he stress field in a straight-w alled, converging channel is

closely approximated by the combination of fields as determinedfrom the stress-free top boundary and the radial stress in the lowerregion of the chann el. F ig. 20 il lustr ates such a combin ati on forthe case of a horizontal top boundar y. T he poin t B of transitionfrom the top stress field to radial stress at the centerline is givenfrom the compati bil ity of the two fields byxr = (1 - sin S)Rs"

where s" is the valu e of s at the centerl ine. T he region AC O forthis case must be ri gid or elast ic; however, the boundar y AC be-tween the rigid and deforming solid is not defined by the stressfields. I n Fi g. 13, the points A, C, and 0 are shown in the ex-S E P T E M B E R 1 9 6 4 / 5 0 5

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perimenta l setup and the region A C O appears to be ri gid as itmust be for the proposed stress field to be closely approximated.

R e f e r e n c e s1 A . W . J enike and R. T. Shield, "O n the P l ast ic F low of C oul omb

S o l i d s B e y o n d O r i g i n al F a i l u r e , " J O U R N A L O F A P P L I E D M E C H A N I C S ,vol . 26, TRAN S. A S M E , vol . 81, Ser ies E , 1961, p. 559.2 A . W . J en ike, P . J . E l sey, and R . H . Woo l l ey , "F l ow P roper t i eso f Bu lk So l i ds , " Pr oceedin gs, A meri can S ociety for T esting M ateri als,vol. 60, 1960, pp. 1168-1190.3 G . D e J . D e J ong, "Stat ics and K inemat ics in the F ai lable Zone

of a G ran u lar M at er i a l , " Wa l t man -D e l f t , Th es i s f o r P hD , F ebruary18, 1959.4 A . W . J en ike, "G r av i t y F low o f Bu lk So l i ds , " U t ah E ng ineer ingE xperim ent Stat ion Bu l let in N o. 108, vol . 52, October, 1961.5 J . R . J ohanson, "St r ess and V eloc i ty F ields in Gr av i ty F l ow ofBu lk So l i ds , " Thes i s f or Ph D , J une , 1962.6 V . V . Sokolovski , Statics of Soil M edia, But t erw or t hs Sc ient i f i cPubl icat ions . 1960.7 I . G. P et rovsky , Par tial Differential E quations, I ntersc iencePu bl ishers , N ew Yor k, N . Y . , 1954, 1957.8 A . W . J en ike, "S t eady G rav i t y F low of F r i c t i ona l -Cohesiv eS o l i d s i n C o n v e r g i n g C h a n n e l s , " J O U R N A L OF A P P L I E D M E C H A N I C S .vol . 31, TRA NS. A S M E vol . 84, Ser ies E, 1964, pp. 5- 11.

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