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Effect of interparticle cohesive forces on the flow behaviour of powders
Powder Technology. 20 (1978) 161 - 178 @ Eisevier Sequoia S-A., Lausanne - P&-ted
Effect
of Interparticle
Cohesive
161 in the Netherlands
Forces on the Flow
Behaviour
of Powders
0. MOLERUS Lehrstuhl
fiir Mechanische
Verfahrenstechnik
(Received
March 18,1977;
in revised form December
der UniuersitCt Erlangen. :Viirnberg (F-R-G.)
In einer zuvor in dieser Zeitschrift verijffentlichten Arbeit [l] hat der Verfasser ein theoretisches lModel1 des mechanischen Verhaltens kohasiver Schiittgiiter abgeleitet, welches vcn den in den Partikelkontakten iibertragenen KrZten ausgeht Diese Theorie beriicksichtigt insbesondere das Anwachsen der HetkrZfte mit zunehmender Verfestigung. Eine Ubervereinfachung in den der Theorie zugrundeliegenden Voraussetzungen fiihrt jedoch zu falschen Vorhersagen iiber die Druckfestigkeit kohasiver Schiittgiiter. In der vorliegenden Arbeit wird dieser Fehler beseitigt. Eine Weiterentwicklung der Theorie fiihrt zu einer anschaulichen Darstellung der der Theorie zugrundeliegenden Parameter_ Diese k6nnen jetzt unmittelbar durch Schertests an kohiisiven Giitem ermittelt werden. Die Brauchbarkeit der verbesserten Theorie wird an Hand der Darstellung von Druckfestigkeftskenniinien verschiedener Pulver nachgewiesen _
19,1977)
development of the theory results in a graphic representation allowing immediate visualization of the basic theoretical parameters. Determination of the latter is possible directly, by means of shear tests on cohesive powders. The usefulness of this improved theory is demonstrated through the representation of unconfined yield stress characteristics of various powders.
l_ INTRODUCTION l_ l_ Previous work In a previous paper
[ 11, the author described a theory for the yielding of cohesive powders which takes into account the forces acting at inter-particle contacts. In particular, the theory considers the increase in the adhesive forces transmitted at interparticle contacts with increasing consolidation. This theory gives the effective yield locus for steady-state flow and gives individual yield loci dependent on previous consolidation. UP to this point the theory is in accordance with experimental evidence.
SUMMARY
In a paper previously published in this journal [ 1) , the author derived a theoretical model of the mechanical behaviour of cohesive powders which takes into account the forces acting at interparticle contacts In particuIar, this theory considers the increase in the adhesive forces transmitted at interparticle contacts with increasing consolidation. An over-simplification inherent in the assumptions’ underlying the theory results, however. in incorrect predictions concerning the unconfined yield stress of a cohesive powder. In the present paper, this weakness in the previous theory is eliminated_ Further
1.2. Shortcomings of the previous theory From a practical point of view, a theory should accurately describe those parts of the behaviour of real materials which must be known for practical applications. A severe shortcoming of the theory presented in the above-mentioned paper is found in the field of practical applications_ The final result of the calculations described in [1] is a normalized representation of the individual yield loci (cf_ [l], Fig. 8). This result means that in a dimensionless representation of the individual yield loci using a T/S&%, a/C, coordinate system, alI individual yield loci should coincide, Z&r being that
162
of-the stress which corresponds to the centre of the appropriate end iMohr circle. (Capital Greek Ietters here and in the following refer to the effective yield locus.) One condition for a family of yield loci to converge onto a single curve in a dimensionless representation is that the length ratios of two arbitrary rays fi and OB to two arbitrary yield loci Y-L-1 and Y-L-2 must be related by (see Fig. 2) value
OA,
OB,
OA,=OSa The stress ox of an arbitrary ray m may be used as common denominator in the nondimensional representation of a family of yield loci by a single curve. Given the general possibility of nondimensionaI representation, the normal stresses uL corresponding to the respective end points E of the individual yield loci find particular use as denominators_ Following this approach, Williams and Birks [ 21 concluded from esperimental results that tnis nondimensional representation of a family of yield loci by one curve was approrimati ly validIf this nondimensional representation of a family of yield loci by one curve is valid, the curve connecting the end points E of the individual yield loci must- be a straight line through the origin of the u, r-axes_ From Fig. 1 another property can be read off directly_ From the simihuity of the triangles OA,B, and 0A2B2, it follows that
Fig_ l_
E.‘ormalization
of yield
loci.
the tangents at the points of the different yield loci situated on the same arbitrary ray z have the same slope- Since the end Mobr circles are tangents to the respective individual yield loci, the angles x1, x2 in Fig. 1 are identical. From the similarity of the triangles OEICJrl and OE2Z3r~, it follows that -OOLl wOOL2
_ O~bXl OX&I2
From this result it is apparent that if the yield loci coincide in a nondimensional representation as proposed by Wiliiams and Birks [2], they also coincide in the nondimensional representation derived in the previous paper CllSchwedes [3] brought to the author’s attention that this property, postulated by Williams and Birks from their experiments and derived theoretically in the previous paper [I], means that the unconfined yield stress f, as a function of the consolidating stress Z 1 is a straight line through the origin, in contradiction to common experimental experience (Fig_ 2) _ The intersection between the experimental fc-curve and the straight ff-line yields the critical hopper outlet dimensions. in accordance with Jenike’s theory [4] _ The normalized representation by one curve thus predicts that a material will either flow for all outlet diameters or not flow at all, depending on the slope of the theoretically straight f,-line_ Such
163
comparison with the cohesive forces in the consolidated material_ This argument, although true for sufficiently strong consolidation, is responsible -for unrealistic modelling of material behaviour with respect to a problem of practical importance, namely hopper design. The object of the present work is to overcome this weakness in the previous paper. Fig. 2. Unconfined experiment_
yield stress f, : theory
[ 1 ] and
material behaviour is obviously rather unusual. In terms of the nondimensional representation described here, this unusual material behaviour means that the unconfmed yield stress f, may itself be used as denominator in the nondimensional representation. 1.3. Physical reasons for the theoretical shortcomings The reasons for the shortcomings of the previous theory may be seen from Fig_ 2_ The function f, = f,(C I) for a given powder cannot be evaluated, from measurements, with sufficient accuracy for values of Z, approaching zero. Extrapolation, however, of the measured curve f,(C,) for a cohesive powder to C1 = 0 yields the dotted part of the f,-line in Fig. 2. This result means that such a powder exhibits a small but nonvanishing, unconfined yield stress f, > 0 for si -+ 0. The theory derived in [l] is based on the consideration of forces transmitted at interparticle contacts. Thinking in terms of these forces, the property f, > 0 for C, + 0 means that adhesive forces greater than zero exist in a powder bulk not consolidated by external forces. Following the theory derived in [ 11, persistent, Le. plastic, deformation at contacts results in an increase in the adhesive forces between adjacent particles, depending on previous consolidation of the powder bulk.
Considering, for example, material data on fine-grained limestone particles, it can be shown that the cohesive forces at the contacts of an unconsolidated powder are small in comparison with the cohesive forces in the consolidated powder. From this result, the cohesive forces in the unconsolidated material were considered to be negligibly small in
2. E_XPERIMENTXL
RESULTS
2. I. Xormalized
yield loci As an example of the behaviour of a cohesive powder, Fig_ 3 shows the yield loci of a barytes powder as measured [5] in a Jenike shear cell. Consolidation included the second step, i.e. shear until steady-state yielding was achieved. Each yield locus is the regression Iine of 15 shear tests separated into 5 groups, one group consisting of 3 repeated
measurements for one fked normal stress. The lines drawn are merely those parts of the respective yield loci which are measurable with high accuracy in the Jenike shear cell. The end LMohr circles of the individual yield loci represent the respective states of consolidation. The yield loci are numbered 1 to 5_ The centres of the end Mohr circles belonging to the respective yield loci are accordingly labelled XXX, to ZXf5_ The nondimensional normalized yield loci, described in Section 1, are depicted in Fig. 4. The numbering of the normalized yield loci corresponds to that of the Individual yield loci depicted in Fig. 3_ As may be seen by comparison of Figs. 3 and 4, the arrangement of the various yield
Fig. 3. Measured yield !oci of a barytes powder.
164
0
/
0
I
o-2
i
‘_
OL
05
08
L%4
loci is inverted in the normalized representation, i.e. the yield locus corresponding to the lo_=;er consolidation lies above that corresponding to the higher consolidation. This result is in accordance with the findings of Stainforth et aZ_ [6], who concluded from their experiments that in the normalized representation proposed by WiiIiams and Birks 121, a family of yield loci does not converge on to one nondimensional ykid locus. Inspection of Fig. 5 in the above-rrenConed paper IS] reveals that in the normalized representation the relative position of the respective yield loci is also inverted_
forces
t
pressng
force IN]
Fig_ 5. Adhesive forces between limestone particles and a metal surface; influence of previous compression [il.
10
Fig. 4. Measured yield loci of a barytes powder: normalized representation_
2.2. Dependence of the adhesive previous compression
j
00s 10’~
Oi
/
1
0
OR
The previous theory [l] was based on theoretical considerations of the dependence of the adhesive forces at inter-particle contacts on previous consolidating forces. In the meantime, however, direct experimental evidence has become avaiIabIe_ Schiitz and Schubert [7] investigated the adhesion of previously compressed, almost spherical limestone particles with sizes in the region of 60 pm to polished, fiat metal surfaces. Both compression and separation were performed by means of centrifuges_ Figure 5 depicts the experimental results in the low range of compressive normal forces, i-e- in the region where compressive forces occur at the contact points of adjacent particies in cohesive powder bulks. Even if Fig. 5 does not reveal the behaviour at inter-particle contacts, it seems reasonable to
formulate the dependence of the cohesive force H on the previous compressive normal force N as H=H,
+-KN
For a given cohesive material with a given particle size distribution, the adhesive force Ho at an unconsohdated contact and the dimensionless factor K in eqn. (1) may be regarded as material constants. In the case of the investigated contact between limestone particles and a metal surface, the values H,, = 1 X lo-’ N and K 2~ 0.3 can be read off from Fig. 5. 2_3_ Comparison with theory It is interesting to compare these experimental results with the theoretical considerations contained in the previous paper Cl] _ Based on the interaction of the adhesive force, the external, compressive normal force and the flattening at the point of contact due to irreversible plastic deformation, these theoretical considerations resulted in the equation (cf_ [ 11, eqn. (12(a))
H=
-46 12Z,2
(2)
Insertion of material data reveals that the last term in the denominator of eqn. (2) is rather small. The simplified expression (3) can therefore be used with sufficient accuracy
165
to replace eqn_ (2)_ The first term on the right-hand side of eqn. (3) is simply the adhesive force between rigid particles, while the dimensionless factor A/(&Z&) describes the ratio of the van der Waals adhesive pressure A/(6&) [S] to the plastic yield pressure pi of the material of the particles_ Numerical values of the Hamaker constant A for different minerals reported by Dahneke [9] seem to indicate that a Hamaker constant A = 15 X lo-*’ J will be of the right order of magnitude for the interaction between limestone and metals. Schijnert and Steier [lo] concluded the plastic yield pressure to be pf = 5 X 10’ Pa from their experiments concerning the limit of brittle fracture of limestone particles in the 1 - 20 pm size range. With the characteristic distance of adhesion z. = 4 A = 4 X lO-‘O m, a comparison of eqns. (1) and (3) for the system investigated yields
A
15
x lo-=
= 0.249
K=~I=6ir-43-10-30-5XI08
This result is in quite reasonable accordance with the experimental vaiue K = O-3. Comparison of eqns. (i) and (3) Gows that, in fitting experimental results, freedom to adjust the geometric mean 6 = 6, 62/ (6 1 + 6z) of the generally unknown diameters 6 1, 6 Z of the contacting surfaces will exist. For a polished metal surface which is hard in comparison with the limestone surface, it seems justifiable to consider the contact diameter of the metal surface 6 q + that of the surface of the limestone p&cles 6 1 _ Even for almost spherical particles, it is doubtful whether limitation of the direct neighbourhood of the contact area to one diameter of curvature of the contact region of the particle is permissible. On the other hand, the nominal particle diameter overestimates the characteristic diameter of the contact area. Bearing in mind this uncertainty, it seems reasonable to calculate the numerical value 61 of the particle contact area using the measured adhesive force and the available material data. For the system investigated, comparison of eqns. (lj and (3) then yields s,~~6=LL-x=-
12H&Z,2 A
= 1.28
12 r 10-T 15x
X 10m6 m = 1.28
x 42 x lo-” lo-20 pm
=
This result appears to describe the contact area of limestone particles of nominal diameter d = 60 pm quite reasonably-
3. THEORY
3.1. Fundamentals The theoretical foundations of the present paper are generally the same as in [l], which may therefore be consulted for details of the derivation_ Here the main points are summarized: (i) Monosized spherical particles of diameter d are assumed. (ii) A simple model is used for the transmission of external stresses through a lattice of solid particles, whereby contacting points at which forces can be transmitted are assumed to exist on the surface of the particles. The coordiuation number lz is assumed to be a unique function of the pack’s porosity, i.e. I: = k(e). The result of these assumptions may be summarized as follows: The relation between the nomlal stress D (shear stress 5) and the normal force N (tangential force T) at a contact lying in the imagined plane in which the above-mentioned stresses act is (3
=
(1 -E)k(E)
7
T;d*
N I T
Thus nominal stresses and the corresponding contact forces are simply related by a factor [(l - E)/z(E)] /zd*_ (iii) Frictional behaviour at the interpartitle contacts is assumed, ibe_ breakdown of adhesion at a contact is assumed to occur if the ratio of the magnitude of the tangential force T to the resulting compressive force C, transmitted at the contact, reaches a critical value I
ITI
-
166 c7 0=
(I-
E)WE) H aI*
0
(6)
these relationships the relation
into condition
(5) yields
describing the cohesion in an unconsolidated powder bulk has to be taken into account. 3.2. Aim ofthe theory Thinking in terms of interparticle contacts, there are three parameters which describe the behaviour of a cohesive powder bulk: the adhesive force H,, at the contacts in an unconsolidated powder bulk, the constant K, which describes, according to eqn. (l), the increase in the adhesive forces between adjacent particles due to consolidation, and the angle p of internal friction of the powder bulk (eqn- (5)).A quantitative theory of the yielding of cohesive powders should Link together these three p ammeters, which are evident from a microstructural point of view and which involve characteristic data, measurable in shear tests on a powder bulk Furthermore, if these are the three essential p ammeters, they should describe the behaviour of a powder bulk in sufficient detail. 3_3_ Theeffecfiue yield locus Steady-state motion over considerable distances (several metre-) with continuous deformation of the powder bulk occurs for example in the convergent part of a hopper_ Such steady-state deformation can obviously take place only by continuous loosening of momentary contacts Steady&ate yielding of a powder bulk therefore proceeds in such a way that breakdown of adhesion is just occurring at points of contact due to the interaction between external forces transmitted there and instan’kmeous cohesive forces generated by these external forces. According to the procedure described in [I] , the normai force arising from the external stress and acting at a contact which is orientated at an angle 0 to the direction of maximum principal stress is described by N(o ) = F,,
-+ FR
cos
C(G) = Ho + (1 -F K)(F>l + Fn cos 20) The magnitude of the shear force at a contact is IT( = F&in 201. Insertion of
tan +,
=(l+K;tZUlp
modifies
(8)
eqn. (7) to
Fn(sin
20 -tan
tan@,
CP, cos 20) d +H,
tanp
(92
0)
(9)
The left-hand side of this inequality depends on the angle o_ The only difference between eqn. (9) and eqn. (14b) of the previous paper [l] is the additional term H, tan p _ Therefore the results concerning the left-hand side of eqn. (9) remain unchanged. In particular, the condition that the left-hand side is at a maximum for
(10) remains valid_ Insertion of eqn relation (9) yields Fn = sin +,
(10)
and equality
into
HO
+ 1+ii
Using eqn. (4) and eqn. (6), the contact forces may be converted into stresses_ Radius ZR and centre XL1 o f a Mohr circle describing steady-state yielding are therefore related by
XR
00
-I
l+K
’
(11)
Substituting for the factor K in eqn. (ll), the definition (8) of the effective angle of friction yields the final result
W!
$29
Followirrg eqn- (l), a cohesive force H(Q) Ho f KN(O) is generated_ Thus the resulting compressive force is
Since isin 01, cos o arc even functions, the relation may be restricted to o > O_ Defining the effective angle of friction 9, as in [l] (cf- [l], eqn. 15),
=
As may be seen easily from Fig. 6, the effective yield locus, regarded as an envelope of the Mohr circles defined by eqn- (12). is a straight line which includes an angle +‘e with the o-axis. The effective yield locus intersects the r-axis at a distance u. tan p from the origin. This means that the effective yield
167
Fig_ 6. Effective
yield
locus.
locus is, in gener& a straight line. but net, hs described in the literature, passing through the origin In determining the angle oe from experiments, the effective yield locus is usually drawn, by definition, through the origin of the u. T-axes [4] _ If eqn. (12) is vaiid, the experimental values of ~1’~ should decrease with increasing consolidating stress X1_ inspection of carefully measured effective yield loci confirms this theory. Measurements by Kun and Miinz 1111, for example, show decreasing values of Ge with increasing values of s1 for all the cohesive powders investigated. From the experimental data (I>, = +,(X1) given in [ll] , the effective yield loci can be recalculated easily according to the definition given here. The effective yield loci for the most cohesive limestone powders investigated (fractions 1, 6 and 9 of the above-mentioned paper) are depicted in Fig. 7. For better legibility, the three different effective yield
10-5 iPo1 1‘
Fig.
7. Measured
I
effective
yield
loci
[ 11 ]_
loci and their respective u, r-axes are shifted horizontally_ As would be postulated according to the theory derived here, the effective yield loci are approximately linear and clearly intersect the r-axes above their respective origins. This result is also in accordance with the finding of Stainforth ef al. 163 , who concluded from their experimental results that the continuous flow locus is a straight line intersecting the T-axis above the origin. From the above derivation of the effective yield locus, its separrting function is easily uSerstood_ The Mohr circles, whose envelope is the effective yield locus, describe those states of stress where breakdown of adhesion at the points of contact is just occurring, with cohesive forces generated by the instantaneous stress. Mohr circles lying below the effective yield locus cannot therefore result in breakdown of adhesion, but only - if achieved for the first time - in consolidation of the powder bulk, since the tangential forces generated at the contacts are obviously less than those postulated by equality in relation (5). On the other hand, Mohr circles exceeding the effective yield locus are realized only if stronger cohesive forces have been generated, due to previous consolidation. 3.4. Prerequisites for the derivation of individual yield loci In the general case, any state of stress described by Mohr circles lying below the effective yield locus must be admitted as a consolidating state of stress. As discussed in detail and experimentally demonstrated in Cl] , a
168
consolidated po%vder must generally be regarded as anisotropic. From this point of view, it does not seem promising to the present author to search for an equation describing the individu& yield loci corresponding to an arbitrary loading history. Fortunately, from a practical point of view, a restricted analysis may be sufficient for most applications_ In hopper design, for example, the loading history of the contents of the hopper with respect to flow upon reopening the outlet is previous, steady-state flow. Similarly, in other technical apphcations, previous steady-state flow may be regarded as the loading history of the powder bulk, Le. the consolidation is characterized by the effective yield locus. Furthermore, in mass fIow hopper design the directions of the maximum principal stresses may be assumed almost to coincide with flow, and with bridging in the case of blocking_ As described in detaiI in [ 11, these conditions are simulated in the Jenike shear tester when the “second step of consolidation” is apFZied_ As a restricted case, but one which is of principaI, practical interest, an equation describing individual yield loci is derived under the two prerequisites: (i) coincidence of the directions of the maximum principaI stresses during consolidation and during incipient yielding, (ii) characterization of consolidation by the effective yield locus. 3-h
Calculation of the individual yield loci Departing from coincidence of the direc‘lions of the maximum principal stresses with consolidation and with incipient yield, the cohesive force at a contact orientated at an angle 0 to the direction of maximum principal stress is given, according to eqn. (l), by w(o)
= He + H&l f Ha cos 29
whereas, due to the instan*taneous stress, a normal force: N(Q) = F,,
f FR cos a
is transmitted at the same contact_ Thus, the resulting compressive force at the contact is C(Q) =
c,, + c,
cos
20
with CM = FM i- %I
(13)
+ Ho
and CR=FR+HR
(14)
respectively. The magnitude the contact is ~T(o)i = Fa
of the tangential
force
at
isin 201
Insertion of the preceding equations into condition (5) for breakdown of adhesion at a contact yields FR isin 201 G tan p (C,,
+ CR cos 20)
Since Isin 20i and cos 20 are even functions, the relation may be restricted to Q 2 0, i-e. to the inequality F,sin2$-CR
tanpcos2#
tanp
(15)
Comparison of inequality (15) with inequality (22) in the previous paper fl] reveals that the only difference consists of the additional term He in C,,, as defined by eqn (13). Since the left-hand side of eqn. (15), which depends on the angle 0, is identical to the lefthand side of eqn. (22) in [ 11, all conclusions concerning the orientation of the contacts which undergo breakdown of adhesion with incipient yieIding hold true and therefore are not repeated here. The orientation 0 of these contacts is given by ,+Z_ 2 with a resulting from tanrr=+np
(16) R
Insertion of result (16) into (15) and setting equality yield, after some rearrangement, the interrelation between the forces acting at a contact undergoing breakdown of adhesion I FR = sinp
(17)
cos’p --I& sin sin P I&‘,, + HM + Ho) 2 -H& The only difference from the previously derived eqn (25a) in [l] is the additional term we in (17). Using eqns. (l), (4) and (6), eqn. (17) can be converted into an equation combining stresses instead of contact forces:
P]
168 OR
=
SiIlp[
+UO)~-K~U$R
(ah% +KU~I -KCTVR
Sin
COS’P
p]
W)
Equation (18) relates the centre uM and radius ca of a Mohr circle describing incipient yielding to the centre uvM and radius uva of the Mohr circle describing previous consolidation. The only prerequisite for the validity of eqn. (18) is that laid down in Section 3.4 (i), namely coincidence of the directions of the maximum principal stresses during consolidation and during incipient yielding. 3-6. Individual yield loci with a loading history of steady-state yielding In order to fulfil prerequisite (ii) of Section 3.4, the quantities valid for the effective yield locus can be substituted for the data characterizing the consolidating stress. Thus according to eqn. (12):
UVR--R -7
=sins,
CM
f-
tanP tan
yield loci of which are depicted in Fig_ 3_ Insertion of the numerical values chosen for @= and p into eqn. (8) yields K = 0.33, which is close to the value K = 0.3 derived from the measurements of Schiitz and Schubert [7] on the adhesion of limestone particles to a metal surface. The numerical value chosen for the cohesive stress u. of the unconsolidated powder bulk is u. = 2.68 X lo3 Pa. Applying eqn. (6) and assuminge = 0.6, k(e) = 3/s = 5, it follows that the average adhesive force at a contact in the unconsolidated powder bulk is Ho = 9.5 X lo-’ N, for a mean particle size of about 15 pm, as in the barytes powder investigated. This result is of the same order of magnitude as the value Ho z 1 X 10m7 N derived from the measurements of Schiitz and Schubert_ Yield loci calculated on the basis of the chosen numerical values for a.=, p and u. are depicted in Fig. 8. In accordance with experimental results, the yield loci are curved
uo
at,
Inserting these quantities into eqn. (18) and considering the relation (8) between K, to ‘De and p leads after some rearran gement
Fig. 8. Calculated
I4 l--
x&r-W
tan *‘e i=lP
_
CM + 00
yield loci.
2
-sin2(~e
--p)
1
Equation (19) contains three parameters which may be regarded as material constants: the angle p of internal friction of the powder, the angle ae describing the slope of the effective yield locus and the isostatic cohesive stress u. of the unconsolidated powder. The previous consolidation is represented by the centre ZM of the end Mohr circle of the respective individual yield locus. A numerical example show: the usefulness of eqn. (19) The numerical value-s 4, = 38.5” and p = 30.8O are close to the experimental data for the barytes powder investigated, the
- .sin(*,
--p)
tan p
(19)
I at negative u-values, Le. in the region of tensile stress_ The intersection of the yield locus with the abscissa is the tensile strength uZ3, characterized by three equal principal stresses. In tensile tests, however, it is always the smaller uniaxial tensile strength uZ1 which is measured. For values of u > 0, the deviations of the individual yield loci from straight lines are negligible_ The calculated yield loci are numbered 1 to 5. Dividing eqn. (19) by Csx yields the normalized representation of the individual yield loci:
170
sin(~e
The same yield loci as depicted in Fig. 8 are represented in Fig. 9 in the normalized formIn accordance with the experimental results depicted in Figs. 3 and 4, the arrangement of the several calculated yield loci is inverted in the normalized representation of Fig. 9 (compare numbers 1 to 5). From a comparison of the pureIy experimental yield loci in the usual and in the normalized representations (Figs 3 and 4) with the corresponding representations of calculated yield loci (Figs. 8 and 9), it wouId appear that eqn. (19) permits a quantitative description of cohesive powder behaviour.
Fig_ 9_ Calculated
yield loci; normalized
representation.
(i) From eqn. (6), the isostatic tensile stress a,-, represents the cohesive forces acting at the contacts of an unconsolidated powder. (ii) According to eqn. (5), the angle P describes the frictional behaviour at inter-particle contacts (iii) From eqn. (S), the ratio tan +&an p > 1 of the angles 9, and p describes the increase in the cohesive forces which can be transmitted at interparticle contacts with increasing consolidation of the powder bulk. In particular, a, = p means that no increase in the cohesive forces results from increasing compression_ From these physicaI origins of the material constants in eqn. (lS), the following theoretical conclusions concerning different types of powder flow behaviour may be drawn. 4_ I _ Cohesionkss material A cohesionless material is characterized by no cohesive forces acting at the contacts in the unconsolidated powder bulk, i-e. by ue = 0, and by no increase in the cohesive forces with increasing consolidation, i-e_ by Qp, = p _ Equation (19) then leads to the ckrssical Coulomb criterion: - OR
4. TYPES
OF POWDER
FLOW
BEHAVIOUR
Equation (19) is derived from theoretical considerations of the forces acting at interparticle contacts in powder bulks- Thus this equation does not represent merely a formal description based on the shape of measured curves, but links together different physical quantities_ Furthermore, through the use of cohesive powder properties eqn. (I9), deducible from interparticle contact behaviour are translated into properties which are measurable with the aid of commonIy used shear tests. From the derivation of eqn. (19), the physical meaning of the three material corstants oe, p and +, is apparent:
(20)
--mP
(21)
0x1
i-e. one straight line through the origin of the 0, r-axes characterizing incipient as weli as steady-state yiekling (Fig_ 10).
Fig. 10. Coulomb
criterion.
Since cohesive forces between adjacent particles are disregarded not only for the unconsolidated but also for the consolidated
l-i1
material, eqn. (21) describes a theoretical limit, which is more or less approached by real materials.
(ii) The formation of solid bridges between adjacent particles, due to crystallization of previously dissolved matter dsuring drying.
4.2. Cohesive Compression
(iii) Sintering at the contacts of a soft material with a relatively low melting point. From the nature of the processes summarized above, it follows that they may occur during the storage of a powder, thus resulting in an increase in powder strength. The role of the external stress resulting, for example, from gravitational loading thus lies only in the fact that the powder is kept at rest in a state characterized by a certain average coordination number. It must therefore be expected that materials for which eqn. (23) holds show time-dependent behaviour, Le. the value o. may change with storage time.
forces
independent
of
From the theoretical eqn. (lS), one type of powder flow may be deduced for which
cohesive forces at the interparticle contacts are not negligibly small, but are, on the other hand, independent of compression. In this case it is true that co > 0, but Qk, = p and eqn. (19) therefore results in CR = sin P(uhX + 0s)
(22)
As is depicted in Fig. 11, the intersection of the yield locus with the taxis, usually called the cohesion, c, is in this case related to the material data introduced here via c=uo
tanp
(23)
Following the derivation given here, the meaning of this result is apparent_ The cohesion c is the limiting shear stress which can be transmitted in a plane on which no external normal stress is acting_ The cohesion c > 0 results from the fact that, due to cohesive forces at the interparticle contacts, an isostatic tensile stress u0 existsThe theoretical considerations described in this paper give some impression of the type of material for which a yield criterion according to eqn. (22) may be valid. If the cohesive forces at the contacts are, on the one hand, not negligibly small but are, on the other hand, not influenced by ,ompression, it is to be expected that mechanisms other than the interaction between van der Waals forces and deformation in the contact area dominate. These mechanisms may be, for example, (i) The build-up, through humidity, of liquid bridges, which cover the interparticle contacts.
Fig. 11. Yield compression.
6-
%I
-6.3 locus
for cohesive
forces
independent
of
4.3. General case: linearized yield loci As shown in Section 3.6, the deviations of the yield loci of a cohesive powder from straight lines are hardly discernible in the region u > O_ Therefore it appears justifiable to iinearize eqn. (19). A reasonable linearization can be performed in such a way that the linearized yield loci represent the end Mohr circles and therefore the effective yield locus exactly_ The representation of the individual yield loci in eqn. (19) is writ&m in such a form that the necessary procedure is apparent_ In describing individual yield loci, eqn. (19) relates radii CT~ to the centres n&Z of those Mohr circles whose envelope is an individual yield locus for a given fixed value E, of the centre of the end hlohr circle. On the other hand, putting ubI = Cnr and therefore aa = C a, eqn. (19) describes the end Mohr circles whose envelope is the effective yield locus. In this case the square root in eqn. (19) simply becomes cos(+= - p)_ Thus it follows from eqn. (19) that
This latter equation can easily be transformed into the form of eqn. (12), which describes the effective yield locus_ Linearization of the square root in eqn. (19) with respect to the difference CM --- ohI therefore yields the desired form of the linearized yield loci. Simple computations give
a~=sinp
r
L
o&f+
sin ae
( sinp
-1
)
Chit-
cos ‘P,
cosp
00
1
(24) Putting oM = X,, and therefore (JR = ZR in eqn. (24) immediately results in eqn. (12), describing the effective yield locus. Equation (24) contains the additional assumption that the difference between the effective angle of friction @, and the angle of internal friction p is small enough that, with sufficient accuracy, I l_ The values a, = 38.5” and cos(ae -p) p = 30_S” in the numerical example of Section 3.6 yield COS(+~ - p) = 0.991, thus confirmkg the assumption_ Equation (24), which describes different individual yield loci depending on th= value XX1 of the centre of the end Mohr circle, can be obtained directly from Fig. 12_ The main advantage of the linearized version, eqn_ (24), lies in the fact that the material constants oo, p and =,derived from theoretical considerations. are directly apparent from the measured yield loci. The dotted parts of the linearized yield loci in the region CI< 0 describe those areas in which deviations from the exact solution of eqn_ (19) become significant_ The angle of internal friction p is the angle included between m individual yield locus and the u-axis. The angle +, is given by the angle betweerthe effective yield locus and the u-axis, as described in Section 3.3. The isostatic tensile stress uo, characterizing the adhesive forces acting at the contacts cf an unconsolidated powder, is found from
-+g*-‘!‘~-*Go
IL------
Fig_12. Linearizedyield loci_
-54
-----
the intersection u. tan p of the effective yield locus with the ~-axis. 5. EXPERIhlENTS Starting from theoretical considerations, three different types of powder behaviour were discerned in the preceding section. If these considerations are more than mere speculation, they should be provable by experiment_ in order to ensure the necessary accuracy, experimental yield loci represented below are regression lines resulting from the statistical treatment of repeated measurements in a Jenike shear tester. Details are reported in [ 53 _ 5-l_ Cohesionless material The measured yield loci of a polyvinyl chloride powder as representative of a cohesionless material are depicted in Fig. 13. The powder consisted of spherical particles, most of them in the size range 120 - 200 pm. Figure 13 reveals the nature of eqn. (21) as a theoretical limit. The careful measurements make it possible to discern different, but very closely neighbouring, yield loci. From a practical point of view, however, these may be condensed onto one straight line through the origin of the u, r-axes, giving the yield locus, eqn. (Zl;, of a cohesionless material. The cohesionless nature of the powder investigated may be attributed to elastic behaviour prevailing in connection with the rather narrow range of particle size.
173
Fig. 13. Yield lotus of a polyvinyl chloride powder.
Fig. 15. Yield loci of an adipic acid powder after 24 hours consolidation.
5 2_ Cohesive forces independent of compression Yield loci as predicted by eqn. (22) are depicted in Figs. 14 and 15. These yield loci belong to an adipic acid powder, the crystalline particles of which fit mainly into the size range 30 - 150 pm. The nature of the cohesive forces in this case can, from theory, be attributed to mechanisms other than the interaction of cohesive forces and irreversible deformation of the contact area and is understandable from the material properties of adipic acid. Adipic acid exhibits moderate solubility in water. The cohesion at interparticle contacts in the powder may therefore be affected to a considerable extent by the humidity of the atmosphere_ On the other hand, due to the low melting point of the material, sintering at the interparticle contacts may be significant even at room temperature_ Since these processes, which can be regarded as responsible for the yield properties of the
powder, are time dependent, time dependence of the yield properties themselves has to be expectedFigure 14 depicts the yield loci resulting from shearing immediately after consolidation. Careful measurements give different yield loci, which however lie closely enough together to be represenwd as one yield locus for practical purposes_ Evaluation of Fig_ 14 yields the numerical values ‘P, = p = 33.7” and = 431 Pa. 00 Figure 15 shows the yield loci of the same material kept at rest under consolidating conditions for 24 hours. Evaluation of Fig_ 15 yields +, = p = 35.2” and u. = 551 Pa. Comparison with the data resulting from Fig_ 14 therefore reveals an increase in the isostatic tensile st-ength uo, representing adhesive forces transmitted at interparticle contacts, of about 28% over a period of 24 hours.
Fig. 14. Yield loci of an adipic acid powder after 0 hours consolidation.
5-3. General case: linearized yield loci The experimen+al results described in Section 2.1 and depicted in Fig. 3 concerning a barytes powder with particles in the broad size range of about 1 I.rrn up to 100 ,um demonstrate cohesive powder behaviour as predicted by theory. The individual yield loci as well as the effective yield locus are straight lines in the measurable regime- It therefore seems reasonable to consider the different individual yield loci as repeated measurements of the angle p of internal friction. The effective yield locus then defines the effective angle of friction @, and the isostatic tensile the adhesive forces stress oo. charactxizing acting at the interparticle contacts of the unconsolidated powder bulk. Evaluation of
174
Fig_ 3 as proposed in Section 4.3 yields p = 30”, CD,= 39-3” and ue = 287 X lOa Pa_
In the case of cohesive forces independent of compression, i.e. += = P, the simple equation
6. &‘PLIC_ATION
f, =
OF THE THEORY
As demonstrated in the preceding section, the types of flow predicted by theory are discernible by experiment. This fact seems to indicate that the material properties p , ~1’~and ue postulated in the theory are significant for the behaviour of cohesive powders Until now in hopper design, for example, measured yield loci have been used - possibly with the aid of formal calculations - to construct the dependence of the unconfined yield stress f, on the maximum principal consolidattig stress Z1_ In the following, an alternative is proposed_ It is suggested that the material data p, 9, and (TVbe evaluated from measured yield loci and the dependence f, = f,(C >) ke then calculated from the Linearized yield loci, eqn_ (24). This procedure at least leads to more confidence in the results obtained, since, in many cases, extrapolation of measured data is inevitable in determining the dependence f, = f,(C 1)_ 6.1. Calculation yield loci
off,
= fdS,)
from
2 sin p
a0
(29)
1 -sin p follows from eqn- (28). Evaluation of measurements on the three powders investigated yields the results depicted in Fig. 16. The solid lines represant direct evaluation of the measurements, whereas the straight dotted lines g5ve the results from application of the theory using the experimental values p, Q, and u. for the respective powders The hatched region in Fig. 16 depicts that part of the f, Z ,-plane containing the mass Row hopper ff-lines according to [4] for the materials investigated here-
Zirrearized I
The unconfined yield stress f, is characterized by
I
0
05
10
15
20
30
LO
50 10eL 1. I Pal
;fc
=
OR
=
(25)
uXI
Insertion of condition (25) into eqn. (24) describing linearized yield loci gives $1
-sin
F) = (sin Qe -sin
uc
(26)
From eqn. (121, it follows that for the maxinium principal stress at consolidation: r,=HhItCR
=(l+sin4r,)Z~~
+
+cos+‘, tanp co (27) Replacing Z:hr in (26) by Z I from eqn. (27) ghes the final result: f, = (I
+ sin a,)(1
+- 2cosG, 1+sincP,
-sin
pj
OF SYhIBOLS
f
1+sinp I-sinp
This result gives an impression of the of the theory in practice. In most cases the interesting intersection between the hopper ff-lines and the powder f,-lines is ascertained either from only one measured point on the fc-line or solely by extrapolation of measured data. Regarding the measurements as yielding the material data, p , (P, and (ro, however, results in improved confidence in the design procedure-
IX.-
S(sin a.= -Grlp)Z,
yield stresses of the powders
applicability
P)Z>X +
+cos9,tanp
Fig. 16. Unconfined investigated_
-P
=o
(2s)
A Hamaker constant c cohesion, defined by eqn. (23) C, C(o) resulting compressive force at a contact
175
C, , Ca quantities d :
fc
Fhl, Fa
H, H(e) ET,,, Efa Ho k ki3. AT,N(o)
Pi
T. T(o)
determining the resulting pressures at a contact particle diameter end point of a yield locus flow factor according to Jenike unconfined yield stress quantities determining the contact force cohesive force quantities determining the cohesive force cohesive force without consolidation coordination number elastic properties of two contacting particles normal force at a contact plastic yield pressure of the material tangential force at a contact constant in Hamaker’s equation angle defined by eqn- (16) geometrical mean of the radii of asperities at a contact radii of asperities at a contact porosity factor defined by eqn. (1) coefficient of friction angle of internal friction normal stress (>0 when compressive) isostatic tensile stress defined by eqn. (6) normal stresses at a ray OA (see Fig_ 1) normal stress at the end point of a yield locus centre of a Mohr circle radius of a Mohr circle centre of the Mohr circle of a consolidating stress
radius of the Mohr circle of a consolidating stress maximum tensile strength tensile strength with three equal principal stresses centre of an end Mohr circle radius of an end Mohr circle maximum principal stress with consolidation shear stress angle of orientation at a contact measured between the normal to the particle surface and the major principal stress direction angle of effective friction angles, see Fig_ 1
REFERENCES
6 7 8 9 10 11
0. Molerur, Powder Tecbnol.. 12 (1975) 259 275. J. C. Williams and A_ H. Birks, Powder Technol., l(1967) 199 - 206. J. ‘Schwedes, personal communication_ A. W. Jenike. Storage and flow of solids, Utah Univ. Eng. Exp. St;_ Bull. 123 (1964). H. Kohmann, Diplomarbeit am Lehrstuhl fiir 1Mechanische Verfabrenstechnik der UniversitZt Erlangen, Niirnberg, 1976. P. T. Stainforth, R. C. Ashley and 3. N. B. Morley, Powder Technol., 4 (1970/71) 250 - 256W. Schiitz and H. Schubert, Chem. Ing. Tech., 48 (1976) 567. II. Krupp, Adv. Colloid Interf: :e Sci_, 1 (1967) 111. B. D&eke. J. CoiIoid Interface Sci., 40 (1972) 1 - 13. K. Schanert and K. Steier, Chem. Ing. Tech., 43 (1971) 773 - 777. H. P. Km-z and G. hIiinz, Powder Technol., 11 (1975) 37 - 40.
Effect
of Interparticle
Cohesive
161 in the Netherlands
Forces on the Flow
Behaviour
of Powders
0. MOLERUS Lehrstuhl
fiir Mechanische
Verfahrenstechnik
(Received
March 18,1977;
in revised form December
der UniuersitCt Erlangen. :Viirnberg (F-R-G.)
In einer zuvor in dieser Zeitschrift verijffentlichten Arbeit [l] hat der Verfasser ein theoretisches lModel1 des mechanischen Verhaltens kohasiver Schiittgiiter abgeleitet, welches vcn den in den Partikelkontakten iibertragenen KrZten ausgeht Diese Theorie beriicksichtigt insbesondere das Anwachsen der HetkrZfte mit zunehmender Verfestigung. Eine Ubervereinfachung in den der Theorie zugrundeliegenden Voraussetzungen fiihrt jedoch zu falschen Vorhersagen iiber die Druckfestigkeit kohasiver Schiittgiiter. In der vorliegenden Arbeit wird dieser Fehler beseitigt. Eine Weiterentwicklung der Theorie fiihrt zu einer anschaulichen Darstellung der der Theorie zugrundeliegenden Parameter_ Diese k6nnen jetzt unmittelbar durch Schertests an kohiisiven Giitem ermittelt werden. Die Brauchbarkeit der verbesserten Theorie wird an Hand der Darstellung von Druckfestigkeftskenniinien verschiedener Pulver nachgewiesen _
19,1977)
development of the theory results in a graphic representation allowing immediate visualization of the basic theoretical parameters. Determination of the latter is possible directly, by means of shear tests on cohesive powders. The usefulness of this improved theory is demonstrated through the representation of unconfined yield stress characteristics of various powders.
l_ INTRODUCTION l_ l_ Previous work In a previous paper
[ 11, the author described a theory for the yielding of cohesive powders which takes into account the forces acting at inter-particle contacts. In particular, the theory considers the increase in the adhesive forces transmitted at interparticle contacts with increasing consolidation. This theory gives the effective yield locus for steady-state flow and gives individual yield loci dependent on previous consolidation. UP to this point the theory is in accordance with experimental evidence.
SUMMARY
In a paper previously published in this journal [ 1) , the author derived a theoretical model of the mechanical behaviour of cohesive powders which takes into account the forces acting at interparticle contacts In particuIar, this theory considers the increase in the adhesive forces transmitted at interparticle contacts with increasing consolidation. An over-simplification inherent in the assumptions’ underlying the theory results, however. in incorrect predictions concerning the unconfined yield stress of a cohesive powder. In the present paper, this weakness in the previous theory is eliminated_ Further
1.2. Shortcomings of the previous theory From a practical point of view, a theory should accurately describe those parts of the behaviour of real materials which must be known for practical applications. A severe shortcoming of the theory presented in the above-mentioned paper is found in the field of practical applications_ The final result of the calculations described in [1] is a normalized representation of the individual yield loci (cf_ [l], Fig. 8). This result means that in a dimensionless representation of the individual yield loci using a T/S&%, a/C, coordinate system, alI individual yield loci should coincide, Z&r being that
162
of-the stress which corresponds to the centre of the appropriate end iMohr circle. (Capital Greek Ietters here and in the following refer to the effective yield locus.) One condition for a family of yield loci to converge onto a single curve in a dimensionless representation is that the length ratios of two arbitrary rays fi and OB to two arbitrary yield loci Y-L-1 and Y-L-2 must be related by (see Fig. 2) value
OA,
OB,
OA,=OSa The stress ox of an arbitrary ray m may be used as common denominator in the nondimensional representation of a family of yield loci by a single curve. Given the general possibility of nondimensionaI representation, the normal stresses uL corresponding to the respective end points E of the individual yield loci find particular use as denominators_ Following this approach, Williams and Birks [ 21 concluded from esperimental results that tnis nondimensional representation of a family of yield loci by one curve was approrimati ly validIf this nondimensional representation of a family of yield loci by one curve is valid, the curve connecting the end points E of the individual yield loci must- be a straight line through the origin of the u, r-axes_ From Fig. 1 another property can be read off directly_ From the simihuity of the triangles OA,B, and 0A2B2, it follows that
Fig_ l_
E.‘ormalization
of yield
loci.
the tangents at the points of the different yield loci situated on the same arbitrary ray z have the same slope- Since the end Mobr circles are tangents to the respective individual yield loci, the angles x1, x2 in Fig. 1 are identical. From the similarity of the triangles OEICJrl and OE2Z3r~, it follows that -OOLl wOOL2
_ O~bXl OX&I2
From this result it is apparent that if the yield loci coincide in a nondimensional representation as proposed by Wiliiams and Birks [2], they also coincide in the nondimensional representation derived in the previous paper CllSchwedes [3] brought to the author’s attention that this property, postulated by Williams and Birks from their experiments and derived theoretically in the previous paper [I], means that the unconfined yield stress f, as a function of the consolidating stress Z 1 is a straight line through the origin, in contradiction to common experimental experience (Fig_ 2) _ The intersection between the experimental fc-curve and the straight ff-line yields the critical hopper outlet dimensions. in accordance with Jenike’s theory [4] _ The normalized representation by one curve thus predicts that a material will either flow for all outlet diameters or not flow at all, depending on the slope of the theoretically straight f,-line_ Such
163
comparison with the cohesive forces in the consolidated material_ This argument, although true for sufficiently strong consolidation, is responsible -for unrealistic modelling of material behaviour with respect to a problem of practical importance, namely hopper design. The object of the present work is to overcome this weakness in the previous paper. Fig. 2. Unconfined experiment_
yield stress f, : theory
[ 1 ] and
material behaviour is obviously rather unusual. In terms of the nondimensional representation described here, this unusual material behaviour means that the unconfmed yield stress f, may itself be used as denominator in the nondimensional representation. 1.3. Physical reasons for the theoretical shortcomings The reasons for the shortcomings of the previous theory may be seen from Fig_ 2_ The function f, = f,(C I) for a given powder cannot be evaluated, from measurements, with sufficient accuracy for values of Z, approaching zero. Extrapolation, however, of the measured curve f,(C,) for a cohesive powder to C1 = 0 yields the dotted part of the f,-line in Fig. 2. This result means that such a powder exhibits a small but nonvanishing, unconfined yield stress f, > 0 for si -+ 0. The theory derived in [l] is based on the consideration of forces transmitted at interparticle contacts. Thinking in terms of these forces, the property f, > 0 for C, + 0 means that adhesive forces greater than zero exist in a powder bulk not consolidated by external forces. Following the theory derived in [ 11, persistent, Le. plastic, deformation at contacts results in an increase in the adhesive forces between adjacent particles, depending on previous consolidation of the powder bulk.
Considering, for example, material data on fine-grained limestone particles, it can be shown that the cohesive forces at the contacts of an unconsolidated powder are small in comparison with the cohesive forces in the consolidated powder. From this result, the cohesive forces in the unconsolidated material were considered to be negligibly small in
2. E_XPERIMENTXL
RESULTS
2. I. Xormalized
yield loci As an example of the behaviour of a cohesive powder, Fig_ 3 shows the yield loci of a barytes powder as measured [5] in a Jenike shear cell. Consolidation included the second step, i.e. shear until steady-state yielding was achieved. Each yield locus is the regression Iine of 15 shear tests separated into 5 groups, one group consisting of 3 repeated
measurements for one fked normal stress. The lines drawn are merely those parts of the respective yield loci which are measurable with high accuracy in the Jenike shear cell. The end LMohr circles of the individual yield loci represent the respective states of consolidation. The yield loci are numbered 1 to 5_ The centres of the end Mohr circles belonging to the respective yield loci are accordingly labelled XXX, to ZXf5_ The nondimensional normalized yield loci, described in Section 1, are depicted in Fig. 4. The numbering of the normalized yield loci corresponds to that of the Individual yield loci depicted in Fig. 3_ As may be seen by comparison of Figs. 3 and 4, the arrangement of the various yield
Fig. 3. Measured yield !oci of a barytes powder.
164
0
/
0
I
o-2
i
‘_
OL
05
08
L%4
loci is inverted in the normalized representation, i.e. the yield locus corresponding to the lo_=;er consolidation lies above that corresponding to the higher consolidation. This result is in accordance with the findings of Stainforth et aZ_ [6], who concluded from their experiments that in the normalized representation proposed by WiiIiams and Birks 121, a family of yield loci does not converge on to one nondimensional ykid locus. Inspection of Fig. 5 in the above-rrenConed paper IS] reveals that in the normalized representation the relative position of the respective yield loci is also inverted_
forces
t
pressng
force IN]
Fig_ 5. Adhesive forces between limestone particles and a metal surface; influence of previous compression [il.
10
Fig. 4. Measured yield loci of a barytes powder: normalized representation_
2.2. Dependence of the adhesive previous compression
j
00s 10’~
Oi
/
1
0
OR
The previous theory [l] was based on theoretical considerations of the dependence of the adhesive forces at inter-particle contacts on previous consolidating forces. In the meantime, however, direct experimental evidence has become avaiIabIe_ Schiitz and Schubert [7] investigated the adhesion of previously compressed, almost spherical limestone particles with sizes in the region of 60 pm to polished, fiat metal surfaces. Both compression and separation were performed by means of centrifuges_ Figure 5 depicts the experimental results in the low range of compressive normal forces, i-e- in the region where compressive forces occur at the contact points of adjacent particies in cohesive powder bulks. Even if Fig. 5 does not reveal the behaviour at inter-particle contacts, it seems reasonable to
formulate the dependence of the cohesive force H on the previous compressive normal force N as H=H,
+-KN
For a given cohesive material with a given particle size distribution, the adhesive force Ho at an unconsohdated contact and the dimensionless factor K in eqn. (1) may be regarded as material constants. In the case of the investigated contact between limestone particles and a metal surface, the values H,, = 1 X lo-’ N and K 2~ 0.3 can be read off from Fig. 5. 2_3_ Comparison with theory It is interesting to compare these experimental results with the theoretical considerations contained in the previous paper Cl] _ Based on the interaction of the adhesive force, the external, compressive normal force and the flattening at the point of contact due to irreversible plastic deformation, these theoretical considerations resulted in the equation (cf_ [ 11, eqn. (12(a))
H=
-46 12Z,2
(2)
Insertion of material data reveals that the last term in the denominator of eqn. (2) is rather small. The simplified expression (3) can therefore be used with sufficient accuracy
165
to replace eqn_ (2)_ The first term on the right-hand side of eqn. (3) is simply the adhesive force between rigid particles, while the dimensionless factor A/(&Z&) describes the ratio of the van der Waals adhesive pressure A/(6&) [S] to the plastic yield pressure pi of the material of the particles_ Numerical values of the Hamaker constant A for different minerals reported by Dahneke [9] seem to indicate that a Hamaker constant A = 15 X lo-*’ J will be of the right order of magnitude for the interaction between limestone and metals. Schijnert and Steier [lo] concluded the plastic yield pressure to be pf = 5 X 10’ Pa from their experiments concerning the limit of brittle fracture of limestone particles in the 1 - 20 pm size range. With the characteristic distance of adhesion z. = 4 A = 4 X lO-‘O m, a comparison of eqns. (1) and (3) for the system investigated yields
A
15
x lo-=
= 0.249
K=~I=6ir-43-10-30-5XI08
This result is in quite reasonable accordance with the experimental vaiue K = O-3. Comparison of eqns. (i) and (3) Gows that, in fitting experimental results, freedom to adjust the geometric mean 6 = 6, 62/ (6 1 + 6z) of the generally unknown diameters 6 1, 6 Z of the contacting surfaces will exist. For a polished metal surface which is hard in comparison with the limestone surface, it seems justifiable to consider the contact diameter of the metal surface 6 q + that of the surface of the limestone p&cles 6 1 _ Even for almost spherical particles, it is doubtful whether limitation of the direct neighbourhood of the contact area to one diameter of curvature of the contact region of the particle is permissible. On the other hand, the nominal particle diameter overestimates the characteristic diameter of the contact area. Bearing in mind this uncertainty, it seems reasonable to calculate the numerical value 61 of the particle contact area using the measured adhesive force and the available material data. For the system investigated, comparison of eqns. (lj and (3) then yields s,~~6=LL-x=-
12H&Z,2 A
= 1.28
12 r 10-T 15x
X 10m6 m = 1.28
x 42 x lo-” lo-20 pm
=
This result appears to describe the contact area of limestone particles of nominal diameter d = 60 pm quite reasonably-
3. THEORY
3.1. Fundamentals The theoretical foundations of the present paper are generally the same as in [l], which may therefore be consulted for details of the derivation_ Here the main points are summarized: (i) Monosized spherical particles of diameter d are assumed. (ii) A simple model is used for the transmission of external stresses through a lattice of solid particles, whereby contacting points at which forces can be transmitted are assumed to exist on the surface of the particles. The coordiuation number lz is assumed to be a unique function of the pack’s porosity, i.e. I: = k(e). The result of these assumptions may be summarized as follows: The relation between the nomlal stress D (shear stress 5) and the normal force N (tangential force T) at a contact lying in the imagined plane in which the above-mentioned stresses act is (3
=
(1 -E)k(E)
7
T;d*
N I T
Thus nominal stresses and the corresponding contact forces are simply related by a factor [(l - E)/z(E)] /zd*_ (iii) Frictional behaviour at the interpartitle contacts is assumed, ibe_ breakdown of adhesion at a contact is assumed to occur if the ratio of the magnitude of the tangential force T to the resulting compressive force C, transmitted at the contact, reaches a critical value I
ITI
-
166 c7 0=
(I-
E)WE) H aI*
0
(6)
these relationships the relation
into condition
(5) yields
describing the cohesion in an unconsolidated powder bulk has to be taken into account. 3.2. Aim ofthe theory Thinking in terms of interparticle contacts, there are three parameters which describe the behaviour of a cohesive powder bulk: the adhesive force H,, at the contacts in an unconsolidated powder bulk, the constant K, which describes, according to eqn. (l), the increase in the adhesive forces between adjacent particles due to consolidation, and the angle p of internal friction of the powder bulk (eqn- (5)).A quantitative theory of the yielding of cohesive powders should Link together these three p ammeters, which are evident from a microstructural point of view and which involve characteristic data, measurable in shear tests on a powder bulk Furthermore, if these are the three essential p ammeters, they should describe the behaviour of a powder bulk in sufficient detail. 3_3_ Theeffecfiue yield locus Steady-state motion over considerable distances (several metre-) with continuous deformation of the powder bulk occurs for example in the convergent part of a hopper_ Such steady-state deformation can obviously take place only by continuous loosening of momentary contacts Steady&ate yielding of a powder bulk therefore proceeds in such a way that breakdown of adhesion is just occurring at points of contact due to the interaction between external forces transmitted there and instan’kmeous cohesive forces generated by these external forces. According to the procedure described in [I] , the normai force arising from the external stress and acting at a contact which is orientated at an angle 0 to the direction of maximum principal stress is described by N(o ) = F,,
-+ FR
cos
C(G) = Ho + (1 -F K)(F>l + Fn cos 20) The magnitude of the shear force at a contact is IT( = F&in 201. Insertion of
tan +,
=(l+K;tZUlp
modifies
(8)
eqn. (7) to
Fn(sin
20 -tan
tan@,
CP, cos 20) d +H,
tanp
(92
0)
(9)
The left-hand side of this inequality depends on the angle o_ The only difference between eqn. (9) and eqn. (14b) of the previous paper [l] is the additional term H, tan p _ Therefore the results concerning the left-hand side of eqn. (9) remain unchanged. In particular, the condition that the left-hand side is at a maximum for
(10) remains valid_ Insertion of eqn relation (9) yields Fn = sin +,
(10)
and equality
into
HO
+ 1+ii
Using eqn. (4) and eqn. (6), the contact forces may be converted into stresses_ Radius ZR and centre XL1 o f a Mohr circle describing steady-state yielding are therefore related by
XR
00
-I
l+K
’
(11)
Substituting for the factor K in eqn. (ll), the definition (8) of the effective angle of friction yields the final result
W!
$29
Followirrg eqn- (l), a cohesive force H(Q) Ho f KN(O) is generated_ Thus the resulting compressive force is
Since isin 01, cos o arc even functions, the relation may be restricted to o > O_ Defining the effective angle of friction 9, as in [l] (cf- [l], eqn. 15),
=
As may be seen easily from Fig. 6, the effective yield locus, regarded as an envelope of the Mohr circles defined by eqn- (12). is a straight line which includes an angle +‘e with the o-axis. The effective yield locus intersects the r-axis at a distance u. tan p from the origin. This means that the effective yield
167
Fig_ 6. Effective
yield
locus.
locus is, in gener& a straight line. but net, hs described in the literature, passing through the origin In determining the angle oe from experiments, the effective yield locus is usually drawn, by definition, through the origin of the u. T-axes [4] _ If eqn. (12) is vaiid, the experimental values of ~1’~ should decrease with increasing consolidating stress X1_ inspection of carefully measured effective yield loci confirms this theory. Measurements by Kun and Miinz 1111, for example, show decreasing values of Ge with increasing values of s1 for all the cohesive powders investigated. From the experimental data (I>, = +,(X1) given in [ll] , the effective yield loci can be recalculated easily according to the definition given here. The effective yield loci for the most cohesive limestone powders investigated (fractions 1, 6 and 9 of the above-mentioned paper) are depicted in Fig. 7. For better legibility, the three different effective yield
10-5 iPo1 1‘
Fig.
7. Measured
I
effective
yield
loci
[ 11 ]_
loci and their respective u, r-axes are shifted horizontally_ As would be postulated according to the theory derived here, the effective yield loci are approximately linear and clearly intersect the r-axes above their respective origins. This result is also in accordance with the finding of Stainforth ef al. 163 , who concluded from their experimental results that the continuous flow locus is a straight line intersecting the T-axis above the origin. From the above derivation of the effective yield locus, its separrting function is easily uSerstood_ The Mohr circles, whose envelope is the effective yield locus, describe those states of stress where breakdown of adhesion at the points of contact is just occurring, with cohesive forces generated by the instantaneous stress. Mohr circles lying below the effective yield locus cannot therefore result in breakdown of adhesion, but only - if achieved for the first time - in consolidation of the powder bulk, since the tangential forces generated at the contacts are obviously less than those postulated by equality in relation (5). On the other hand, Mohr circles exceeding the effective yield locus are realized only if stronger cohesive forces have been generated, due to previous consolidation. 3.4. Prerequisites for the derivation of individual yield loci In the general case, any state of stress described by Mohr circles lying below the effective yield locus must be admitted as a consolidating state of stress. As discussed in detail and experimentally demonstrated in Cl] , a
168
consolidated po%vder must generally be regarded as anisotropic. From this point of view, it does not seem promising to the present author to search for an equation describing the individu& yield loci corresponding to an arbitrary loading history. Fortunately, from a practical point of view, a restricted analysis may be sufficient for most applications_ In hopper design, for example, the loading history of the contents of the hopper with respect to flow upon reopening the outlet is previous, steady-state flow. Similarly, in other technical apphcations, previous steady-state flow may be regarded as the loading history of the powder bulk, Le. the consolidation is characterized by the effective yield locus. Furthermore, in mass fIow hopper design the directions of the maximum principal stresses may be assumed almost to coincide with flow, and with bridging in the case of blocking_ As described in detaiI in [ 11, these conditions are simulated in the Jenike shear tester when the “second step of consolidation” is apFZied_ As a restricted case, but one which is of principaI, practical interest, an equation describing individual yield loci is derived under the two prerequisites: (i) coincidence of the directions of the maximum principaI stresses during consolidation and during incipient yielding, (ii) characterization of consolidation by the effective yield locus. 3-h
Calculation of the individual yield loci Departing from coincidence of the direc‘lions of the maximum principal stresses with consolidation and with incipient yield, the cohesive force at a contact orientated at an angle 0 to the direction of maximum principal stress is given, according to eqn. (l), by w(o)
= He + H&l f Ha cos 29
whereas, due to the instan*taneous stress, a normal force: N(Q) = F,,
f FR cos a
is transmitted at the same contact_ Thus, the resulting compressive force at the contact is C(Q) =
c,, + c,
cos
20
with CM = FM i- %I
(13)
+ Ho
and CR=FR+HR
(14)
respectively. The magnitude the contact is ~T(o)i = Fa
of the tangential
force
at
isin 201
Insertion of the preceding equations into condition (5) for breakdown of adhesion at a contact yields FR isin 201 G tan p (C,,
+ CR cos 20)
Since Isin 20i and cos 20 are even functions, the relation may be restricted to Q 2 0, i-e. to the inequality F,sin2$-CR
tanpcos2#
tanp
(15)
Comparison of inequality (15) with inequality (22) in the previous paper fl] reveals that the only difference consists of the additional term He in C,,, as defined by eqn (13). Since the left-hand side of eqn. (15), which depends on the angle 0, is identical to the lefthand side of eqn. (22) in [ 11, all conclusions concerning the orientation of the contacts which undergo breakdown of adhesion with incipient yieIding hold true and therefore are not repeated here. The orientation 0 of these contacts is given by ,+Z_ 2 with a resulting from tanrr=+np
(16) R
Insertion of result (16) into (15) and setting equality yield, after some rearrangement, the interrelation between the forces acting at a contact undergoing breakdown of adhesion I FR = sinp
(17)
cos’p --I& sin sin P I&‘,, + HM + Ho) 2 -H& The only difference from the previously derived eqn (25a) in [l] is the additional term we in (17). Using eqns. (l), (4) and (6), eqn. (17) can be converted into an equation combining stresses instead of contact forces:
P]
168 OR
=
SiIlp[
+UO)~-K~U$R
(ah% +KU~I -KCTVR
Sin
COS’P
p]
W)
Equation (18) relates the centre uM and radius ca of a Mohr circle describing incipient yielding to the centre uvM and radius uva of the Mohr circle describing previous consolidation. The only prerequisite for the validity of eqn. (18) is that laid down in Section 3.4 (i), namely coincidence of the directions of the maximum principal stresses during consolidation and during incipient yielding. 3-6. Individual yield loci with a loading history of steady-state yielding In order to fulfil prerequisite (ii) of Section 3.4, the quantities valid for the effective yield locus can be substituted for the data characterizing the consolidating stress. Thus according to eqn. (12):
UVR--R -7
=sins,
CM
f-
tanP tan
yield loci of which are depicted in Fig_ 3_ Insertion of the numerical values chosen for @= and p into eqn. (8) yields K = 0.33, which is close to the value K = 0.3 derived from the measurements of Schiitz and Schubert [7] on the adhesion of limestone particles to a metal surface. The numerical value chosen for the cohesive stress u. of the unconsolidated powder bulk is u. = 2.68 X lo3 Pa. Applying eqn. (6) and assuminge = 0.6, k(e) = 3/s = 5, it follows that the average adhesive force at a contact in the unconsolidated powder bulk is Ho = 9.5 X lo-’ N, for a mean particle size of about 15 pm, as in the barytes powder investigated. This result is of the same order of magnitude as the value Ho z 1 X 10m7 N derived from the measurements of Schiitz and Schubert_ Yield loci calculated on the basis of the chosen numerical values for a.=, p and u. are depicted in Fig. 8. In accordance with experimental results, the yield loci are curved
uo
at,
Inserting these quantities into eqn. (18) and considering the relation (8) between K, to ‘De and p leads after some rearran gement
Fig. 8. Calculated
I4 l--
x&r-W
tan *‘e i=lP
_
CM + 00
yield loci.
2
-sin2(~e
--p)
1
Equation (19) contains three parameters which may be regarded as material constants: the angle p of internal friction of the powder, the angle ae describing the slope of the effective yield locus and the isostatic cohesive stress u. of the unconsolidated powder. The previous consolidation is represented by the centre ZM of the end Mohr circle of the respective individual yield locus. A numerical example show: the usefulness of eqn. (19) The numerical value-s 4, = 38.5” and p = 30.8O are close to the experimental data for the barytes powder investigated, the
- .sin(*,
--p)
tan p
(19)
I at negative u-values, Le. in the region of tensile stress_ The intersection of the yield locus with the abscissa is the tensile strength uZ3, characterized by three equal principal stresses. In tensile tests, however, it is always the smaller uniaxial tensile strength uZ1 which is measured. For values of u > 0, the deviations of the individual yield loci from straight lines are negligible_ The calculated yield loci are numbered 1 to 5. Dividing eqn. (19) by Csx yields the normalized representation of the individual yield loci:
170
sin(~e
The same yield loci as depicted in Fig. 8 are represented in Fig. 9 in the normalized formIn accordance with the experimental results depicted in Figs. 3 and 4, the arrangement of the several calculated yield loci is inverted in the normalized representation of Fig. 9 (compare numbers 1 to 5). From a comparison of the pureIy experimental yield loci in the usual and in the normalized representations (Figs 3 and 4) with the corresponding representations of calculated yield loci (Figs. 8 and 9), it wouId appear that eqn. (19) permits a quantitative description of cohesive powder behaviour.
Fig_ 9_ Calculated
yield loci; normalized
representation.
(i) From eqn. (6), the isostatic tensile stress a,-, represents the cohesive forces acting at the contacts of an unconsolidated powder. (ii) According to eqn. (5), the angle P describes the frictional behaviour at inter-particle contacts (iii) From eqn. (S), the ratio tan +&an p > 1 of the angles 9, and p describes the increase in the cohesive forces which can be transmitted at interparticle contacts with increasing consolidation of the powder bulk. In particular, a, = p means that no increase in the cohesive forces results from increasing compression_ From these physicaI origins of the material constants in eqn. (lS), the following theoretical conclusions concerning different types of powder flow behaviour may be drawn. 4_ I _ Cohesionkss material A cohesionless material is characterized by no cohesive forces acting at the contacts in the unconsolidated powder bulk, i-e. by ue = 0, and by no increase in the cohesive forces with increasing consolidation, i-e_ by Qp, = p _ Equation (19) then leads to the ckrssical Coulomb criterion: - OR
4. TYPES
OF POWDER
FLOW
BEHAVIOUR
Equation (19) is derived from theoretical considerations of the forces acting at interparticle contacts in powder bulks- Thus this equation does not represent merely a formal description based on the shape of measured curves, but links together different physical quantities_ Furthermore, through the use of cohesive powder properties eqn. (I9), deducible from interparticle contact behaviour are translated into properties which are measurable with the aid of commonIy used shear tests. From the derivation of eqn. (19), the physical meaning of the three material corstants oe, p and +, is apparent:
(20)
--mP
(21)
0x1
i-e. one straight line through the origin of the 0, r-axes characterizing incipient as weli as steady-state yiekling (Fig_ 10).
Fig. 10. Coulomb
criterion.
Since cohesive forces between adjacent particles are disregarded not only for the unconsolidated but also for the consolidated
l-i1
material, eqn. (21) describes a theoretical limit, which is more or less approached by real materials.
(ii) The formation of solid bridges between adjacent particles, due to crystallization of previously dissolved matter dsuring drying.
4.2. Cohesive Compression
(iii) Sintering at the contacts of a soft material with a relatively low melting point. From the nature of the processes summarized above, it follows that they may occur during the storage of a powder, thus resulting in an increase in powder strength. The role of the external stress resulting, for example, from gravitational loading thus lies only in the fact that the powder is kept at rest in a state characterized by a certain average coordination number. It must therefore be expected that materials for which eqn. (23) holds show time-dependent behaviour, Le. the value o. may change with storage time.
forces
independent
of
From the theoretical eqn. (lS), one type of powder flow may be deduced for which
cohesive forces at the interparticle contacts are not negligibly small, but are, on the other hand, independent of compression. In this case it is true that co > 0, but Qk, = p and eqn. (19) therefore results in CR = sin P(uhX + 0s)
(22)
As is depicted in Fig. 11, the intersection of the yield locus with the taxis, usually called the cohesion, c, is in this case related to the material data introduced here via c=uo
tanp
(23)
Following the derivation given here, the meaning of this result is apparent_ The cohesion c is the limiting shear stress which can be transmitted in a plane on which no external normal stress is acting_ The cohesion c > 0 results from the fact that, due to cohesive forces at the interparticle contacts, an isostatic tensile stress u0 existsThe theoretical considerations described in this paper give some impression of the type of material for which a yield criterion according to eqn. (22) may be valid. If the cohesive forces at the contacts are, on the one hand, not negligibly small but are, on the other hand, not influenced by ,ompression, it is to be expected that mechanisms other than the interaction between van der Waals forces and deformation in the contact area dominate. These mechanisms may be, for example, (i) The build-up, through humidity, of liquid bridges, which cover the interparticle contacts.
Fig. 11. Yield compression.
6-
%I
-6.3 locus
for cohesive
forces
independent
of
4.3. General case: linearized yield loci As shown in Section 3.6, the deviations of the yield loci of a cohesive powder from straight lines are hardly discernible in the region u > O_ Therefore it appears justifiable to iinearize eqn. (19). A reasonable linearization can be performed in such a way that the linearized yield loci represent the end Mohr circles and therefore the effective yield locus exactly_ The representation of the individual yield loci in eqn. (19) is writ&m in such a form that the necessary procedure is apparent_ In describing individual yield loci, eqn. (19) relates radii CT~ to the centres n&Z of those Mohr circles whose envelope is an individual yield locus for a given fixed value E, of the centre of the end hlohr circle. On the other hand, putting ubI = Cnr and therefore aa = C a, eqn. (19) describes the end Mohr circles whose envelope is the effective yield locus. In this case the square root in eqn. (19) simply becomes cos(+= - p)_ Thus it follows from eqn. (19) that
This latter equation can easily be transformed into the form of eqn. (12), which describes the effective yield locus_ Linearization of the square root in eqn. (19) with respect to the difference CM --- ohI therefore yields the desired form of the linearized yield loci. Simple computations give
a~=sinp
r
L
o&f+
sin ae
( sinp
-1
)
Chit-
cos ‘P,
cosp
00
1
(24) Putting oM = X,, and therefore (JR = ZR in eqn. (24) immediately results in eqn. (12), describing the effective yield locus. Equation (24) contains the additional assumption that the difference between the effective angle of friction @, and the angle of internal friction p is small enough that, with sufficient accuracy, I l_ The values a, = 38.5” and cos(ae -p) p = 30_S” in the numerical example of Section 3.6 yield COS(+~ - p) = 0.991, thus confirmkg the assumption_ Equation (24), which describes different individual yield loci depending on th= value XX1 of the centre of the end Mohr circle, can be obtained directly from Fig. 12_ The main advantage of the linearized version, eqn_ (24), lies in the fact that the material constants oo, p and =,derived from theoretical considerations. are directly apparent from the measured yield loci. The dotted parts of the linearized yield loci in the region CI< 0 describe those areas in which deviations from the exact solution of eqn_ (19) become significant_ The angle of internal friction p is the angle included between m individual yield locus and the u-axis. The angle +, is given by the angle betweerthe effective yield locus and the u-axis, as described in Section 3.3. The isostatic tensile stress uo, characterizing the adhesive forces acting at the contacts cf an unconsolidated powder, is found from
-+g*-‘!‘~-*Go
IL------
Fig_12. Linearizedyield loci_
-54
-----
the intersection u. tan p of the effective yield locus with the ~-axis. 5. EXPERIhlENTS Starting from theoretical considerations, three different types of powder behaviour were discerned in the preceding section. If these considerations are more than mere speculation, they should be provable by experiment_ in order to ensure the necessary accuracy, experimental yield loci represented below are regression lines resulting from the statistical treatment of repeated measurements in a Jenike shear tester. Details are reported in [ 53 _ 5-l_ Cohesionless material The measured yield loci of a polyvinyl chloride powder as representative of a cohesionless material are depicted in Fig. 13. The powder consisted of spherical particles, most of them in the size range 120 - 200 pm. Figure 13 reveals the nature of eqn. (21) as a theoretical limit. The careful measurements make it possible to discern different, but very closely neighbouring, yield loci. From a practical point of view, however, these may be condensed onto one straight line through the origin of the u, r-axes, giving the yield locus, eqn. (Zl;, of a cohesionless material. The cohesionless nature of the powder investigated may be attributed to elastic behaviour prevailing in connection with the rather narrow range of particle size.
173
Fig. 13. Yield lotus of a polyvinyl chloride powder.
Fig. 15. Yield loci of an adipic acid powder after 24 hours consolidation.
5 2_ Cohesive forces independent of compression Yield loci as predicted by eqn. (22) are depicted in Figs. 14 and 15. These yield loci belong to an adipic acid powder, the crystalline particles of which fit mainly into the size range 30 - 150 pm. The nature of the cohesive forces in this case can, from theory, be attributed to mechanisms other than the interaction of cohesive forces and irreversible deformation of the contact area and is understandable from the material properties of adipic acid. Adipic acid exhibits moderate solubility in water. The cohesion at interparticle contacts in the powder may therefore be affected to a considerable extent by the humidity of the atmosphere_ On the other hand, due to the low melting point of the material, sintering at the interparticle contacts may be significant even at room temperature_ Since these processes, which can be regarded as responsible for the yield properties of the
powder, are time dependent, time dependence of the yield properties themselves has to be expectedFigure 14 depicts the yield loci resulting from shearing immediately after consolidation. Careful measurements give different yield loci, which however lie closely enough together to be represenwd as one yield locus for practical purposes_ Evaluation of Fig_ 14 yields the numerical values ‘P, = p = 33.7” and = 431 Pa. 00 Figure 15 shows the yield loci of the same material kept at rest under consolidating conditions for 24 hours. Evaluation of Fig_ 15 yields +, = p = 35.2” and u. = 551 Pa. Comparison with the data resulting from Fig_ 14 therefore reveals an increase in the isostatic tensile st-ength uo, representing adhesive forces transmitted at interparticle contacts, of about 28% over a period of 24 hours.
Fig. 14. Yield loci of an adipic acid powder after 0 hours consolidation.
5-3. General case: linearized yield loci The experimen+al results described in Section 2.1 and depicted in Fig. 3 concerning a barytes powder with particles in the broad size range of about 1 I.rrn up to 100 ,um demonstrate cohesive powder behaviour as predicted by theory. The individual yield loci as well as the effective yield locus are straight lines in the measurable regime- It therefore seems reasonable to consider the different individual yield loci as repeated measurements of the angle p of internal friction. The effective yield locus then defines the effective angle of friction @, and the isostatic tensile the adhesive forces stress oo. charactxizing acting at the interparticle contacts of the unconsolidated powder bulk. Evaluation of
174
Fig_ 3 as proposed in Section 4.3 yields p = 30”, CD,= 39-3” and ue = 287 X lOa Pa_
In the case of cohesive forces independent of compression, i.e. += = P, the simple equation
6. &‘PLIC_ATION
f, =
OF THE THEORY
As demonstrated in the preceding section, the types of flow predicted by theory are discernible by experiment. This fact seems to indicate that the material properties p , ~1’~and ue postulated in the theory are significant for the behaviour of cohesive powders Until now in hopper design, for example, measured yield loci have been used - possibly with the aid of formal calculations - to construct the dependence of the unconfined yield stress f, on the maximum principal consolidattig stress Z1_ In the following, an alternative is proposed_ It is suggested that the material data p, 9, and (TVbe evaluated from measured yield loci and the dependence f, = f,(C >) ke then calculated from the Linearized yield loci, eqn_ (24). This procedure at least leads to more confidence in the results obtained, since, in many cases, extrapolation of measured data is inevitable in determining the dependence f, = f,(C 1)_ 6.1. Calculation yield loci
off,
= fdS,)
from
2 sin p
a0
(29)
1 -sin p follows from eqn- (28). Evaluation of measurements on the three powders investigated yields the results depicted in Fig. 16. The solid lines represant direct evaluation of the measurements, whereas the straight dotted lines g5ve the results from application of the theory using the experimental values p, Q, and u. for the respective powders The hatched region in Fig. 16 depicts that part of the f, Z ,-plane containing the mass Row hopper ff-lines according to [4] for the materials investigated here-
Zirrearized I
The unconfined yield stress f, is characterized by
I
0
05
10
15
20
30
LO
50 10eL 1. I Pal
;fc
=
OR
=
(25)
uXI
Insertion of condition (25) into eqn. (24) describing linearized yield loci gives $1
-sin
F) = (sin Qe -sin
uc
(26)
From eqn. (121, it follows that for the maxinium principal stress at consolidation: r,=HhItCR
=(l+sin4r,)Z~~
+
+cos+‘, tanp co (27) Replacing Z:hr in (26) by Z I from eqn. (27) ghes the final result: f, = (I
+ sin a,)(1
+- 2cosG, 1+sincP,
-sin
pj
OF SYhIBOLS
f
1+sinp I-sinp
This result gives an impression of the of the theory in practice. In most cases the interesting intersection between the hopper ff-lines and the powder f,-lines is ascertained either from only one measured point on the fc-line or solely by extrapolation of measured data. Regarding the measurements as yielding the material data, p , (P, and (ro, however, results in improved confidence in the design procedure-
IX.-
S(sin a.= -Grlp)Z,
yield stresses of the powders
applicability
P)Z>X +
+cos9,tanp
Fig. 16. Unconfined investigated_
-P
=o
(2s)
A Hamaker constant c cohesion, defined by eqn. (23) C, C(o) resulting compressive force at a contact
175
C, , Ca quantities d :
fc
Fhl, Fa
H, H(e) ET,,, Efa Ho k ki3. AT,N(o)
Pi
T. T(o)
determining the resulting pressures at a contact particle diameter end point of a yield locus flow factor according to Jenike unconfined yield stress quantities determining the contact force cohesive force quantities determining the cohesive force cohesive force without consolidation coordination number elastic properties of two contacting particles normal force at a contact plastic yield pressure of the material tangential force at a contact constant in Hamaker’s equation angle defined by eqn- (16) geometrical mean of the radii of asperities at a contact radii of asperities at a contact porosity factor defined by eqn. (1) coefficient of friction angle of internal friction normal stress (>0 when compressive) isostatic tensile stress defined by eqn. (6) normal stresses at a ray OA (see Fig_ 1) normal stress at the end point of a yield locus centre of a Mohr circle radius of a Mohr circle centre of the Mohr circle of a consolidating stress
radius of the Mohr circle of a consolidating stress maximum tensile strength tensile strength with three equal principal stresses centre of an end Mohr circle radius of an end Mohr circle maximum principal stress with consolidation shear stress angle of orientation at a contact measured between the normal to the particle surface and the major principal stress direction angle of effective friction angles, see Fig_ 1
REFERENCES
6 7 8 9 10 11
0. Molerur, Powder Tecbnol.. 12 (1975) 259 275. J. C. Williams and A_ H. Birks, Powder Technol., l(1967) 199 - 206. J. ‘Schwedes, personal communication_ A. W. Jenike. Storage and flow of solids, Utah Univ. Eng. Exp. St;_ Bull. 123 (1964). H. Kohmann, Diplomarbeit am Lehrstuhl fiir 1Mechanische Verfabrenstechnik der UniversitZt Erlangen, Niirnberg, 1976. P. T. Stainforth, R. C. Ashley and 3. N. B. Morley, Powder Technol., 4 (1970/71) 250 - 256W. Schiitz and H. Schubert, Chem. Ing. Tech., 48 (1976) 567. II. Krupp, Adv. Colloid Interf: :e Sci_, 1 (1967) 111. B. D&eke. J. CoiIoid Interface Sci., 40 (1972) 1 - 13. K. Schanert and K. Steier, Chem. Ing. Tech., 43 (1971) 773 - 777. H. P. Km-z and G. hIiinz, Powder Technol., 11 (1975) 37 - 40.