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Compression cycles in tableting
Powder Technology, 26 (1980) 199 - 204 @ E!sevier Sequoia S-A., Lausann e - Printed in the Netherlands
Compression
Cycles in Tableting
J. T. CARSTENSEN
and PIERRE
School
University
of Pharmacy.
(Received
199
June 28,1979;
TOURE of Wisconsin.
Madison,
in revised form February
SUMMARY
When powders are compressed to make a tablet, the powder is placed within a die and compressed by two punches. In general the act of compression is considered to occur in five stages: (a) first the powder rearranges to its cIosest packing, then (b) the particles deform elastically. (c) Beyond the elastic limits the particles will either ticture or deform plastically, and either of these processes leads to interparticulate bonding, i-e. gives rise to a tablet. (d) After release of the upper punch there is a relaxation of stresses in the bonded mass, and this is followed by (e) ejection of the tablet from the die. It is conventional to describe these processes by monitoring either the die wall pressure (F) or the lower punch pressure (Ps) as a function of the upper punch pressure (P). These two types of cycles are conventionally described in the literature by assuming the solid to be a non-porous body. The cycles exhibit a hysteresis loop, and it is shown here that the consequence of considering the body nonporous throughout is that the hysteresis area of the cycles is either a quadratic or a linear function of the maximally applied pressure. INTRODUCTION
Compression of powders and granules in tablet presses is of pharmaceutical importance and the physics of this process has been the subject of a fair amount of study 11 - 281. Making reference to Fig. 1, powder is fed into a die and compressed between two snugly fitting punches. The processes involved in the complete cycle of making a tablet are [5] : (a) rearrangement of the powder to its closest packing,
WI 53706
(U.S.A.)
19,198O)
(b) elastic deformation of the particles, (c) beyond the elastic limit, either particle fracture or plastic deformation, (d) stress relaxation after release of the upper punch, and (e) ejection. If the elastic limit of a solid is not exceeded in compression, then no bonding occurs, le. a tablet is not formed, and only steps (a), (b) and (e) occur. In step (a) the apparent density rises from the cascaded value to the tapped density value. In steps (b), (c) and (d) it is generally considered [ll, 121 acceptable to draw the analogy with a solid block and the Poisson relation is applied to step (b): Y = lateral strain/longitudinal strain
(1)
where Y is the Poisson ratio. In a tablet (Figs. 1 and 2) au applied (longitudinal) pressure, P, causes a (lateral) die wall pressure, F, and a pressure transmitted to the lower punch, Pp. Within the elastic limit the strain for a nonporous solid is proportional to the stress (Hooke’s law) so that F=vP
(2)
It is recognized that the analogy with a nonporous solid is dubious, but since it has been applied consistently in the past since Long introduced it [ll, 121, it is instructive to draw the conclusions of its application. Much attention has been given to the actual process of bonding (step c), i.e. to whether brittle fracture or plastic deformation is the process by which bonding takes place. What follows will show that if the stated analogy with a non-porous solid is applied then the areas of the hysteresis loops of die-wall pressure versus applied force will be either a quadratic or linear function of the maximally applied upper punch pressure. This in turn
200
two mechanisms ahuded to is apphcable to the compaction of a particular solid based on its compression cycle features. If the apphed force exceeds point B (whichis the elastic limit of the material), then in the case of constant yield in stress the powder wiU behave like a Bingham body. If the yield value is denoted S, then the line BC wiU obey the equation
Fig_ I_ Forces in a tablet die. Fig. 2. Compression cycle. Ordinate is die-wall pressure, abscissa is applied (upper punch pressure). A similar diagram ensues when Iower punch pressure is pIotted us_ upper punch pressure.
shouId be a means of distinguishing between the two bonding mechanisms, Le. whether fmcture or deformation has occurred. In the past, eqn. (2) has been employed in a modified form [ll, 121 T F = [V/(1 -
,)]P
(3)
On instrumented presses it is possible to monitor applied and sensed pressures”, and the initiaI part of the cycle (AI! in Fig. 2) adheres to eqn. (3) to scme extent. Soft materials wih have a large value of Y. Hard materials wiII have lower values, and Y wiII approach 0.3. At low v-values if is therefore to be expected that bonding wrII occur by brittle fracture 1141, and at high values of u plastic deformation is to be expected to be the bonding mechanism. Two distinct types of cycles are usually encounterA [ 53 , one where the material has a constant yield in stress (frequently occurring in the case of a plastic deformation) and one where the material behaves Iike a Mohr body (frequently equivalent to brittle fracture). The features of these wiII be elaborated on below. Distinction between the two *es of behavior is most often made based on sIopes and intercepts in the compression cycIe graphs, e.g. [ 181 a large slope of AB is associated with plastic deformation. Slope and intercept determinations in this type of graph are, however, not exacting and the purpose of this writing is to suggest an alternative method for deciding which of the
‘The treatment here uses pressures. A treatment could equally well he derived from forces. Knowledge of punch areas, of course, allows conversion of one to the other.
F=P-S
(4)
If the solid acts like a Mohr body, then the shearing stress (denoted 7,) at failure is (to an approximation) a Iinear function of the normaI stress, en. so that Tn =C+pa,
(5)
where C is cohesion and p is a frictional coefficient. To develop the relationships between F and P, the following symbols wiI.Ibe used: N=v/(l-V)
(6)
M = (1 - p)/(l Q = 2c/(l+ K = 2C/(l
+ W)
(7)
LL)
(8)
-_cl)
(9)
If the stress increases and then exceeds C (eqn. 5), then failure occurs. rn (in the plane of shear) at this point equals (a - 7)/2 [ ‘71 and the normal stress is equal to u, = ((T + r)/2, and inserting these two values in eqn- (5) then yields (a -
r)/2 = c + [p(a + r)/2]
(10)
or 7 = C(1 -!J)/(l
+ wL)lcJ - c=/u
+ PII
(11)
Using the nomenclature in eqns. (7) and (8), this can be written F=MP-Q
(12)
for the tableting situation. Point B in Fig. 1 is the intersect of AB with SC, and its abscissa, x’(B) is therefore the root of: N%‘(B) = M&(B) - Q, Le., x’(B)
=
Q/M --N)
(13)
Point C is associated with the maximum pressure, II, and it has the coordinates PointC=(II,MII
-Q8)
(14)
202 TABLE
1
Hysteresis Ioop areas [S] pressure Loop area, z (1O-3 N=)
I
I
lo
Is
MCXiMLw
i
20
APPLlED
FORCE.
25 Ti.
LN
Fig_ 3. Data from Leigh et ni_ [11] of hysteresis area (z) as a function of maximum applied force, II_ The ordinate is (z + 34.5)/H (eqn. 22).
282 174 88 29 *Obtained coefficient
as a function
Maximum
of applied
(2 + 34.5)1x
force, x (hN)
0.32x=
25.6 20.2 14.8 9.9
282 174 88 29
12.4 10.3 8.2 6.4
by multiple regression with a correlation of r2 = 1 - (9 X lo-‘).
cycle area is proportional to bottom punch cycle areas. Pn versus P cydes for starch granules have been reported by MangenotCruaud and her data are shown in Fig. 4. Fessi [3] tested a series of plastics (polyvinyl chloride, PVC, polyvinyl alcohol, PVA, and PVA/PVC copolymer). His data are reproduced in Fig. 5.
Fig. 4. Data from Mangenot-Cruaud [133 of lower us_ upper applied punch pressures (eqn. 30). The two curves represent different manufacturing processes, one giving rise to a hard, non-porous granule, the other giving rise to a porous, fragile granule-
_/s+-
II-
-,
o-
_/y
CL
=z u
9-
5
8-,/ it5
-
_/_&’ 5
g5
6
Inii KviPd
Fig_ 5. Data from Fessi [33 treated according to eqn. (22) assuming the leading term to predominate. The slope is cIose to 2 as predicted.
In most reported investigations, the cycles are not carried out at different maximum pressures_ A few investigators have done so [3, 11, 13, 25] and their data have been subjected to the analysis suggested by t?ne treatment leading to eqns. (22) and (30). The data by Leigh deal with the compression of sodium chloride and are shown in Fig. 3. Tour6 [25] has shown that die-wall pressure
RESULTS
AND
DISCUSSION
Figures 3, 4 and 5 show that in two of these cases (Figs. 3 and 5) eqn. (22) holds, Le. the powder acts like a Mohr body, and in the other case (Fig. 4) the relation follows eqn. (30), Le. if the analogy with a non-porous solid holds, then the starch granules experience constant yield in stress. Some further comments are in order at this point. It is usually assumed that P = T and that P = a. In this case the slopes of lines BC and DE should equal unity for the case of constant shear in stress. This has been pointed out by Obiorah [18], who investigated cycles of several substances (including sodium chloride) and found that “none of the materials gave a pressure cycle similar to that of a body with constant yield stress, since the value of the second upward slope of the line would... equal one for such a material.” This is different from the finding by Leigh ef al. [ 111, who found sodium chloride to act with constant yield stress in shear. The requirement of unity slope has been relaxed by certain authors and the criterion reduced to a requirement that the slopes be identical. Leigh ef 32. [ll] , for instance, found the two slopes to be about 0.8 for sucrose crystals and 0.4 for sucrose granules
203
and concluded that sucrose behaved with constant yield stress in shear. In this case one assumes proportionality of the type F=ar
(31)
P=@J
(32)
This is reasonable in light of the equation of Shaxby and Evans [29], that the transmitted pressure, F’, at a distance of L below the applied pressure (PO) will be given by F’ = PO exp(-4XL/D)
(33)
where K is a constant and D is the die diameter. This 1301 has been claimed to apply to the ratio of transmitted radial pressure as well, Le., F, = PO exp(-4&v/D)
(34)
where ,Qis the frictional coefficient. Hence F is not a constant number, but rather an average over the part of the die-wall length which is exposed_ It should be pointed out that Strijbos et al. [ 241 found eqn. (33) not to apply. There is the further complication that the initial phase of the compression is a particle rearrangement, whereas the final phase is a fairly non-porous compact. Finally it should be mentioned (in relation to the failure criteria leading to eqn. 11) that Strijbos et al [ 241 have shown that the powder yield locus lies above the wall yield locus. In any event, if eqns. (31) and (32) are accepted, then eqns. (6) - (9) and (12) - (30) will still be correct except that now the capital letter symbols assume other values. Hence eqns. (22) and (30) can still be used to distinguish mechanisms. Now, of course, v-values reported from slopes of AB cannot be expected to be the same for each investigator*. This is indeed the case, and for instance for sodium chloride it has been reported to be 0.3 in one case [ll] and 0.2 (calculated) in another comparable case [18] _ Obiorah, in fact, refrains from calculating v and states it to be proportiond to slope AS. The above limits interpretation of the other slopes as well. If the slope of line BC were, e.g., 0.43, it might be *As pointed out by Leigh et al_ make a difference_
[ll]
. the die will
concluded from eqn. (7) that p = 0.4. With F = CYTand P = /3a the value would have to be
adjusted by a factor of o/p. It should further be noted that slope determinations of compression cycles in general are not very exact. The location of the point B can, for instance, be quite ambiguous. The method for distinguishing between the two types of behavior described in this note would therefore seem to offer some advantages. The method simply requires that the cycle be determined at several maximum pressures and that the areas of the loops be measured_ A plot of S uewus Il will then show the rheology in question, in that it will follow either eqn. (30) or eqn. (22). It has been emphasized above that the analogy with a non-porous body is dubious. There are a host of other assumptions made. Brittle fracture is assumed to create bonding by way of fresh surfaces, and is equated with a Mohr body type behavior. Secondly, the behavior beyond point D (eqn 18) is open to question. It has been used here simply to show the consequences of its application. The nomenclature used in the article has been kept close to that of previous literature (e-g- r,, 7, S and C), although for instance a definite distinction between S and C may not necessarily exist. It is also, as it has been in most past literature, been assumed that the die and lower punches are incompressible_
CONCLUSION It has been shown that with the assumptions made in past literature for compression cycles, a Mohr body should give hysteresis areas that are quadratic in the maximum applied pressure, and for solids with constant yield in stress the areas should be linearly related to the constant applied upper punch pressure. Examples from literature treated in this way have been presented.
REFERENCES C. J. deBlaey and J. Polderman, Pharm. Neekbl.. IO5 (1971) 57 - 65. C. J. deBIaey, W. deRijk and J. Polderman, Phnrm_ Ind.. 33 (1971) 697 - 900. H. Fessi and J. T. Carstensen, Int. J. Phnrm.. 4 (1980), in press.
204
4 5 6
7 8 9 10 11 12 13 14 15
16
C. Fiihrer, Dfsch. Apoth. Ztg.. 105 (1965) 1150 - 1153. J- GilIard, P. Tome and M. RoIand, Pharm. Acta Helu., 51 (1976) 226 - 231. J- T_ Carstensen. Rate Processes and Mechanical Properfies in Pharmacc&Lal Solids, Academic Press, New York, 1980, in press_ E_ N. Hiesttd and C. B. Peot, J. Pharm. Sci., 63 (19743 605 - 612_ E. N. Hiestand, J- E. Wells, C. B. Peot and J_ F. Ochs, J_ Pharm. Sci_, 66 (1977) 510 - 519. T. Higuchi, L. N. Elowe and L. Busse, J. Am. Pharm. Assoc.. Sci. Ed.. 43 (1954) 685 - 689. T. Higuchi, T. Shimamoto, S. P. Eriksen and T. Yashiki, J_ Pharm- Sci.. 54 (1965) 111 - 118. VS.Leigh, J_ E. Carless and B. W. Burt, J. Pharm. Sci_. 57 (1967) 888 - 892. TV. M. Long, Powder MetaZZ., 6 (1960) 73 - 87. 0. Mangenot-Cruaud. These de Docteur esSciences Ph-aceutiques, Univ- Paris-Sud, 1979_ K. Marshall, J. Pharm. Pharmacol, I5 (1963) 413 - 421. K. Marshall, The Physics of Tablet Compression, 12th Arden House Conference, Feb. 2. I977, Colorco?, West Point, PA, Pp. 15 - 16. E. Nelson, S. M. Naqui, L. W. Busse and T. Higuchi, J. Am. Pharm. Assoc., Sci. Ed., 43 (1954) 596 - 602.
17
E. Nelson, L. W. Busse and T. Higuchi, J. Am. Pharm. Assoc., Sci. Ed.. 44 (1955) 223. 18 B. A. Obiorah. Int. J. Pharm., I (1978) 249 - 256. 19 W. Parmentier, Ph.D. Thesis. Tech. Univ. CaroloWilhemina. Brauuschweig, 1974. 20 S. Shlanta, Jr. and G. Mi’osovich, J. Pharm. Sci.. 562 - 564_ and D. Ganderton. J. Pharm. 21 E. Shotton Pharmacol-. 12 (1960) 93T - 96T. 22 E_ Shotton and D. Gander-ton, J. Pharm. Pharmacol. Z3 (1961) 144T - 152T. 23 E. Shotton and B. A. Obiorah. L Pharm. Pharmacol.. 25 (1973) 37P - 43P. 24 S. Strijbos, P. J. Rankin, R. J. Klen Wessink, J. Bannink and G. J. Oudemans, Powder Technol. 18 (1977) 187 - 200_ 25 P. Tour& These de Docteur es-Sciences Pharmaceutiqu&, Univ. Paris-Sud, 1979. 26 D- Train. Trans. Inst. Chem. E~P.. _. 35 (1957) 258 - 284. 27 J_ Varsano and L. Lachman, J. Phnm. Sci_. 55 (1966) 1128 - 1133. 28 J. J_ Windheuser, J. Misra, S. P_ Eriksen and T. Higuchi, J. Pharm. ScL, 52 (1963) 767 - 772_ 29 J_ is_ Sbaxby and J_ C. Evans, Trans. Faraday Sot.. I9 (1923) 60 - 72. I8 (1945) 30 H. Unckel, Arch. Eisenhiittenru.. 161- 167.
Compression
Cycles in Tableting
J. T. CARSTENSEN
and PIERRE
School
University
of Pharmacy.
(Received
199
June 28,1979;
TOURE of Wisconsin.
Madison,
in revised form February
SUMMARY
When powders are compressed to make a tablet, the powder is placed within a die and compressed by two punches. In general the act of compression is considered to occur in five stages: (a) first the powder rearranges to its cIosest packing, then (b) the particles deform elastically. (c) Beyond the elastic limits the particles will either ticture or deform plastically, and either of these processes leads to interparticulate bonding, i-e. gives rise to a tablet. (d) After release of the upper punch there is a relaxation of stresses in the bonded mass, and this is followed by (e) ejection of the tablet from the die. It is conventional to describe these processes by monitoring either the die wall pressure (F) or the lower punch pressure (Ps) as a function of the upper punch pressure (P). These two types of cycles are conventionally described in the literature by assuming the solid to be a non-porous body. The cycles exhibit a hysteresis loop, and it is shown here that the consequence of considering the body nonporous throughout is that the hysteresis area of the cycles is either a quadratic or a linear function of the maximally applied pressure. INTRODUCTION
Compression of powders and granules in tablet presses is of pharmaceutical importance and the physics of this process has been the subject of a fair amount of study 11 - 281. Making reference to Fig. 1, powder is fed into a die and compressed between two snugly fitting punches. The processes involved in the complete cycle of making a tablet are [5] : (a) rearrangement of the powder to its closest packing,
WI 53706
(U.S.A.)
19,198O)
(b) elastic deformation of the particles, (c) beyond the elastic limit, either particle fracture or plastic deformation, (d) stress relaxation after release of the upper punch, and (e) ejection. If the elastic limit of a solid is not exceeded in compression, then no bonding occurs, le. a tablet is not formed, and only steps (a), (b) and (e) occur. In step (a) the apparent density rises from the cascaded value to the tapped density value. In steps (b), (c) and (d) it is generally considered [ll, 121 acceptable to draw the analogy with a solid block and the Poisson relation is applied to step (b): Y = lateral strain/longitudinal strain
(1)
where Y is the Poisson ratio. In a tablet (Figs. 1 and 2) au applied (longitudinal) pressure, P, causes a (lateral) die wall pressure, F, and a pressure transmitted to the lower punch, Pp. Within the elastic limit the strain for a nonporous solid is proportional to the stress (Hooke’s law) so that F=vP
(2)
It is recognized that the analogy with a nonporous solid is dubious, but since it has been applied consistently in the past since Long introduced it [ll, 121, it is instructive to draw the conclusions of its application. Much attention has been given to the actual process of bonding (step c), i.e. to whether brittle fracture or plastic deformation is the process by which bonding takes place. What follows will show that if the stated analogy with a non-porous solid is applied then the areas of the hysteresis loops of die-wall pressure versus applied force will be either a quadratic or linear function of the maximally applied upper punch pressure. This in turn
200
two mechanisms ahuded to is apphcable to the compaction of a particular solid based on its compression cycle features. If the apphed force exceeds point B (whichis the elastic limit of the material), then in the case of constant yield in stress the powder wiU behave like a Bingham body. If the yield value is denoted S, then the line BC wiU obey the equation
Fig_ I_ Forces in a tablet die. Fig. 2. Compression cycle. Ordinate is die-wall pressure, abscissa is applied (upper punch pressure). A similar diagram ensues when Iower punch pressure is pIotted us_ upper punch pressure.
shouId be a means of distinguishing between the two bonding mechanisms, Le. whether fmcture or deformation has occurred. In the past, eqn. (2) has been employed in a modified form [ll, 121 T F = [V/(1 -
,)]P
(3)
On instrumented presses it is possible to monitor applied and sensed pressures”, and the initiaI part of the cycle (AI! in Fig. 2) adheres to eqn. (3) to scme extent. Soft materials wih have a large value of Y. Hard materials wiII have lower values, and Y wiII approach 0.3. At low v-values if is therefore to be expected that bonding wrII occur by brittle fracture 1141, and at high values of u plastic deformation is to be expected to be the bonding mechanism. Two distinct types of cycles are usually encounterA [ 53 , one where the material has a constant yield in stress (frequently occurring in the case of a plastic deformation) and one where the material behaves Iike a Mohr body (frequently equivalent to brittle fracture). The features of these wiII be elaborated on below. Distinction between the two *es of behavior is most often made based on sIopes and intercepts in the compression cycIe graphs, e.g. [ 181 a large slope of AB is associated with plastic deformation. Slope and intercept determinations in this type of graph are, however, not exacting and the purpose of this writing is to suggest an alternative method for deciding which of the
‘The treatment here uses pressures. A treatment could equally well he derived from forces. Knowledge of punch areas, of course, allows conversion of one to the other.
F=P-S
(4)
If the solid acts like a Mohr body, then the shearing stress (denoted 7,) at failure is (to an approximation) a Iinear function of the normaI stress, en. so that Tn =C+pa,
(5)
where C is cohesion and p is a frictional coefficient. To develop the relationships between F and P, the following symbols wiI.Ibe used: N=v/(l-V)
(6)
M = (1 - p)/(l Q = 2c/(l+ K = 2C/(l
+ W)
(7)
LL)
(8)
-_cl)
(9)
If the stress increases and then exceeds C (eqn. 5), then failure occurs. rn (in the plane of shear) at this point equals (a - 7)/2 [ ‘71 and the normal stress is equal to u, = ((T + r)/2, and inserting these two values in eqn- (5) then yields (a -
r)/2 = c + [p(a + r)/2]
(10)
or 7 = C(1 -!J)/(l
+ wL)lcJ - c=/u
+ PII
(11)
Using the nomenclature in eqns. (7) and (8), this can be written F=MP-Q
(12)
for the tableting situation. Point B in Fig. 1 is the intersect of AB with SC, and its abscissa, x’(B) is therefore the root of: N%‘(B) = M&(B) - Q, Le., x’(B)
=
Q/M --N)
(13)
Point C is associated with the maximum pressure, II, and it has the coordinates PointC=(II,MII
-Q8)
(14)
202 TABLE
1
Hysteresis Ioop areas [S] pressure Loop area, z (1O-3 N=)
I
I
lo
Is
MCXiMLw
i
20
APPLlED
FORCE.
25 Ti.
LN
Fig_ 3. Data from Leigh et ni_ [11] of hysteresis area (z) as a function of maximum applied force, II_ The ordinate is (z + 34.5)/H (eqn. 22).
282 174 88 29 *Obtained coefficient
as a function
Maximum
of applied
(2 + 34.5)1x
force, x (hN)
0.32x=
25.6 20.2 14.8 9.9
282 174 88 29
12.4 10.3 8.2 6.4
by multiple regression with a correlation of r2 = 1 - (9 X lo-‘).
cycle area is proportional to bottom punch cycle areas. Pn versus P cydes for starch granules have been reported by MangenotCruaud and her data are shown in Fig. 4. Fessi [3] tested a series of plastics (polyvinyl chloride, PVC, polyvinyl alcohol, PVA, and PVA/PVC copolymer). His data are reproduced in Fig. 5.
Fig. 4. Data from Mangenot-Cruaud [133 of lower us_ upper applied punch pressures (eqn. 30). The two curves represent different manufacturing processes, one giving rise to a hard, non-porous granule, the other giving rise to a porous, fragile granule-
_/s+-
II-
-,
o-
_/y
CL
=z u
9-
5
8-,/ it5
-
_/_&’ 5
g5
6
Inii KviPd
Fig_ 5. Data from Fessi [33 treated according to eqn. (22) assuming the leading term to predominate. The slope is cIose to 2 as predicted.
In most reported investigations, the cycles are not carried out at different maximum pressures_ A few investigators have done so [3, 11, 13, 25] and their data have been subjected to the analysis suggested by t?ne treatment leading to eqns. (22) and (30). The data by Leigh deal with the compression of sodium chloride and are shown in Fig. 3. Tour6 [25] has shown that die-wall pressure
RESULTS
AND
DISCUSSION
Figures 3, 4 and 5 show that in two of these cases (Figs. 3 and 5) eqn. (22) holds, Le. the powder acts like a Mohr body, and in the other case (Fig. 4) the relation follows eqn. (30), Le. if the analogy with a non-porous solid holds, then the starch granules experience constant yield in stress. Some further comments are in order at this point. It is usually assumed that P = T and that P = a. In this case the slopes of lines BC and DE should equal unity for the case of constant shear in stress. This has been pointed out by Obiorah [18], who investigated cycles of several substances (including sodium chloride) and found that “none of the materials gave a pressure cycle similar to that of a body with constant yield stress, since the value of the second upward slope of the line would... equal one for such a material.” This is different from the finding by Leigh ef al. [ 111, who found sodium chloride to act with constant yield stress in shear. The requirement of unity slope has been relaxed by certain authors and the criterion reduced to a requirement that the slopes be identical. Leigh ef 32. [ll] , for instance, found the two slopes to be about 0.8 for sucrose crystals and 0.4 for sucrose granules
203
and concluded that sucrose behaved with constant yield stress in shear. In this case one assumes proportionality of the type F=ar
(31)
P=@J
(32)
This is reasonable in light of the equation of Shaxby and Evans [29], that the transmitted pressure, F’, at a distance of L below the applied pressure (PO) will be given by F’ = PO exp(-4XL/D)
(33)
where K is a constant and D is the die diameter. This 1301 has been claimed to apply to the ratio of transmitted radial pressure as well, Le., F, = PO exp(-4&v/D)
(34)
where ,Qis the frictional coefficient. Hence F is not a constant number, but rather an average over the part of the die-wall length which is exposed_ It should be pointed out that Strijbos et al. [ 241 found eqn. (33) not to apply. There is the further complication that the initial phase of the compression is a particle rearrangement, whereas the final phase is a fairly non-porous compact. Finally it should be mentioned (in relation to the failure criteria leading to eqn. 11) that Strijbos et al [ 241 have shown that the powder yield locus lies above the wall yield locus. In any event, if eqns. (31) and (32) are accepted, then eqns. (6) - (9) and (12) - (30) will still be correct except that now the capital letter symbols assume other values. Hence eqns. (22) and (30) can still be used to distinguish mechanisms. Now, of course, v-values reported from slopes of AB cannot be expected to be the same for each investigator*. This is indeed the case, and for instance for sodium chloride it has been reported to be 0.3 in one case [ll] and 0.2 (calculated) in another comparable case [18] _ Obiorah, in fact, refrains from calculating v and states it to be proportiond to slope AS. The above limits interpretation of the other slopes as well. If the slope of line BC were, e.g., 0.43, it might be *As pointed out by Leigh et al_ make a difference_
[ll]
. the die will
concluded from eqn. (7) that p = 0.4. With F = CYTand P = /3a the value would have to be
adjusted by a factor of o/p. It should further be noted that slope determinations of compression cycles in general are not very exact. The location of the point B can, for instance, be quite ambiguous. The method for distinguishing between the two types of behavior described in this note would therefore seem to offer some advantages. The method simply requires that the cycle be determined at several maximum pressures and that the areas of the loops be measured_ A plot of S uewus Il will then show the rheology in question, in that it will follow either eqn. (30) or eqn. (22). It has been emphasized above that the analogy with a non-porous body is dubious. There are a host of other assumptions made. Brittle fracture is assumed to create bonding by way of fresh surfaces, and is equated with a Mohr body type behavior. Secondly, the behavior beyond point D (eqn 18) is open to question. It has been used here simply to show the consequences of its application. The nomenclature used in the article has been kept close to that of previous literature (e-g- r,, 7, S and C), although for instance a definite distinction between S and C may not necessarily exist. It is also, as it has been in most past literature, been assumed that the die and lower punches are incompressible_
CONCLUSION It has been shown that with the assumptions made in past literature for compression cycles, a Mohr body should give hysteresis areas that are quadratic in the maximum applied pressure, and for solids with constant yield in stress the areas should be linearly related to the constant applied upper punch pressure. Examples from literature treated in this way have been presented.
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