an experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic...

Journal of Non-Newtonian Fluid Mechanics, 51 (1994) 61-78 Elsevier Science B.V.

61

An experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic fluids

M.J. Adams *, B. Edmondson, D.G. Caughey and R. Yahya

Unilever Research, Port Sunlight Laboratory, Bebington, Wirral L63 3JW (UK)

(Received January 14, 1993; in revised form August 8, 1993)

Abstract

Lubrication theory is commonly employed to analyse the squeeze-film flow of plastic fluids under no-slip wall boundary conditions. Solutions exist for both Bingham and Herschel-Bulkley fluids but they infer that there exists a rigid or unyielded core and flow zones adjacent to the platens; it has been recognised previously that such a velocity field is kinematically inconsistent. Furthermore, the pressure boundary condition at the edge of the platens is conventionally set to zero which is inconsistent with experimental data presented here for a model Herschel-Bulkley fluid (Plasticine). An attempt has been made to examine the lubrication theory in more detail by a comparison with equilibrium stress analysis for rigid-plastic solids. Squeeze-film measurements were carried out using a model Herschel- Bulkley fluid and the results were consistent with the theory, suggesting that it is a useful first approximation. Nevertheless, the approach does not resolve the kinematic inconsistency resulting in lubrication theory.

Keywords: elastoplastic fluids; Herschel-Bulkley fluid; lubrication theory; Plasticine; squeeze- fihn flow

1. Introduction

The idea of a yield criterion for ductile solids, such as metals, is well established. It defines a critical combination of stresses at which elastic deformation terminates and plastic deformation is initiated. Certain fluids, such as concentrated suspensions, also exhibit yield phenomena. Oldroyd [l] recognised that such fluids may be treated in a similar way to ductile solids

* Corresponding author.

0377-0257/94/%07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0377-0257(93)01190-F

62 M.J. Adams et al. / J. Non-Newtonian Fluid Mech. 51 (1994) 61- 78

except that, at the onset of yield, the fluid will undergo viscous flow rather than plastic deformation. He formulated a constitutive relationship involv- ing post-yield Newtonian flow. This may be written in invariant form for the more general case of Ostwald-deWaele flow, as follows:

z* = [z&f: @‘2( + kJ(@: 6)(“-‘)‘21]d (la)

for +r*: r* 2 zi,

o* = Ge for $*: r* < ri, (lb)

where Z* is the deviatoric stress tensor, z. is the shear yield stress, d is the rate of deformation tensor, k is the shear flow consistency, n is the shear flow index, G is the shear elastic modulus and e is the strain tensor. These expressions assume that the fluid is subject to the von Mises yield criterion which is defined by the following equality

Tt 1 *: z* =T2

07 (2)

where $r*: z* is the second invariant of the deviator+ stress tensor. It is now known that fluids showing plastic behaviour are often viscoelastic prior to the onset of yield (e.g. Ref. 2) rather than perfectly elastic as implied by eqn. (1).

In order to evaluate the material parameters in eqn. (l), it is necessary to measure the material response when subject to a well-prescribed velocity field. This usually requires simple geometric configurations. For example, a procedure has been developed that is based on the compression of cylindrical specimens for determining the mechanical properties of metals; this technique is known as ‘upsetting’ [see Ref. 31. A similar technique, known as ‘squeeze-film rheometry’ is employed to study plastic fluids and is the subject of the current paper.

While uniaxial compression is a nominally simple procedure, the resulting velocity fields for plastic solids, such as metals, are extremely complex [4] except for the practically unrealisable case of frictionless wall stress boundary conditions when a specimen undergoes homogeneous extensional deformation. A significant wall traction at the platens results in the formation of conical unyielded zones adjacent to the platens. These effects may be minimised by lubricating the specimens. Nevertheless, this is difficult to achieve effectively, and the inhomogeous deformation, or ‘redundant work’ as it is termed in the mechanical engineering field, results in an overestima- tion of the yield stress.

The above phenomena have been observed by flow visualisation experiments with Plasticine [ 51; the extrusion experiments reported in Ref. 5 and in Section 5 suggest that this material is a Herschel-Bulkley fluid. In the flow visualisation experiments, the conical zones were observed to flatten gradu-

M.J. Adams et al. 1 J. Non-Newtonian Fluid Mech. 51 (1994) 61-78 63

ally with increasing strain as would be expected. Peek [6] attempted to model the squeeze-film flow of plastic fluids by considering the flow in the plastic shear bands between the unyielded regions, but this approach has been largely abandoned by subsequent workers except for some refinement by Scott [7]. Generally, more recent work (e.g. Refs. 8 and 9) has been based on lubrication theory which was first employed by Scott [lo] and assumes a radial pressure flow under no-slip wall boundary conditions; this solution will be discussed in more detail later. The flow field is analogous to that along a pipe with a central unyielded zone and flow zones adjacent to the walls. The previous visualisation studies [5] suggest that this is the flow field developed at large uniaxial stress whereas the formation of conical unyielded zones considered by Peek [6] corresponds to smaller deformations.

In order to obtain useful closed-form solutions for procedures such as upsetting and squeeze-film, it is necessary to make a number of approxima- tions. While there are some close analogies between the plasticity and lubrication theories, it would appear that there are fundamental differences. A major aim of the current work was to rationalise these differences.

Metals are commonly characterised by simple empirical expressions derived directly from lubricated uniaxial compressive or tensile measurements. At ordinary temperatures, metals often show strain hardening behav- ior while, at higher temperatures, strain rate hardening may be observed [ 31. The flow stress Q is often represented by an expression of the following form

Q = BE”?’ (3)

where B is a material constant, E and i are the uniaxial strain and strain rate respectively, and z and m are termed the strain and strain rate exponents respectively and are functions of temperature. It is interesting to note that the treatment proposed by Oldroyd [l] to describe the strain rate sensitivity of plastic fluids has not been applied to hot metals for which strain rate hardening is much more pronounced than strain hardening [3].

A knowledge of the strain and strain rate hardening as expressed by eqn. (3) together with a yield criterion is sulhcient for a numerical analysis of deformation processes provided that the wall boundary conditions can be specified. However, prior to the advent of readily available numerical techniques, such as finite element analysis, a number of approximate methods have been developed, for example, equilibrium stress analysis, limit methods and slip line field [3]. These methods rely on neglecting the elastic deformation at low strains so that the material is treated as a rigid-plastic. The results of applying equilibrium stress analysis to upsetting will be considered later.

Under steady flow conditions, the: elastic deformation of plastic fluids is also commonly neglected to derive analytical solutions. In addition, many

64 M.J. Adams et al. 1 J. Non-Newtonian Fluid Mech. 51 (1994) 61-78

flow fields such as that associated with squeeze-film may be analysed using a one-dimensional constituitive relationship. For this rigid-viscous flow case, eqn. (1) reduces to the Herschel-Bulkley equation [ 111, thus

(da)

where i, is the shear strain rate. For a Bingham fluid, the flow index is unity. As described previously, lubrication theory has been employed to analyse

steady state squeeze-film flow between parallel plates for both Herschel- Bulkley and Bingham fluids under no-slip wall boundary conditions. The most comprehensive treatment is due to Covey and Stanmore [8] and further details will be given in the next Section. A useful review of the subject is given in a recent paper by Sherwood et al. [9]. Here, a comparison will be made of a result obtained from lubrication theory for plastic fluids with that from equilibrim stress analysis for rigid-plastic solids. This will exemplify the conflict between the two approaches. For the squeeze-film flow of plastic fluids, Scott [lo] obtained the following result for the critical mean pressure j required to plastically deform a specimen of radius R and height h between parallel platens

2z0R P=x. (5)

Strictly, here and elsewhere in this paper, pressure refers to the wall normal stress rather than a hydrostatic pressure. The importance of this distinction will be evident later.

Measurements of a model plastic fluid (Plasticine) show that the mean pressure scales linearly with the aspect ratio (2R/h) at least for a relatively small range of aspect ratios, about 3-7, depending on the strain [ 121. At higher aspect ratios, there were deviations from linearity. However, the data in the lower aspect ratio range extrapolated to a positive mean pressure at a zero aspect ratio and not to a zero pressure as implied by eqn. (5). It should be emphasised that this is an extrapolation; the lubrication approximation is not valid in the limit of R/h tending to zero.

The equilibrium stress analysis for the upsetting of rigid-plastic solids is due to Siebel (see Ref. 3). He considered Coulombic wall boundary conditions where the interaction at the platen walls is characterised by a coefficient of friction ,u. The method assumes homogeneous deformation and, consequently, the yield stress will be overestimated as discussed previously. This assumption is based on plane strain deformation which is equivalent to the lubrication approximation for fluids. The solution for the critical deformation pressure in this case is


p=o, I+%, ( >

where o. is the uniaxial yield stress. The value of the coefficient of friction under no-slip wall boundary conditions depends on the a von Mises solid, the shear yield stress is given by rro/ 4

ield criterion. For 3 and hence p takes

a value of 0.577 (see Ref. 12). The above solution predicts that the mean pressure extrapolated to a zero aspect ratio is non-zero and equal to the uniaxial yield stress, this is consistent with the experimental results referred to above [ 121.

A major difference between the above solutions is that the limit pressure at the edge of the specimen is set effectively to zero (actually atmospheric pressure) in lubrication analysis and it is taken as being equal to the uniaxial yield stress in the equilibrium stress analysis which is in accord with the experimental data for Plasticine described previously [ 121. The introduction of this latter condition and other modifications of the solution obtained by Covey and Stanmore [8] are described in the next Section. In addition, the analysis is extended to obtain a solution for the case where one of the platens has a spherical rather than planar geometry. This was carried out because of the experimental dilhculties encountered in ensuring that two flat platens are perfectly parallel; such difficulties have been reported previously [9]. The problem was identified in the current work because large errors in the viscosities of silicone fluid standards were observed when the equipment was used with parallel platens. An accuracy of f 5% was achieved with the spherical platen. It should be emphasised that the problems experienced by the present authors using parallel platens were due to the use of a driven upper platen. Parallel platens have been employed by many authors in a creep mode when self-alignment of the platens is possible (e.g. Ref. 13).

In order to assess the solution obtained in the current work for squeeze- film flow, measurements were made with Plasticine which is a model Herschel-Bulkley fluid [5]. A particular advantage of this material is that it is possible to ensure no-slip wall boundary conditions by roughening the platens [5]. In principle, it would be possible to introduce slip boundary conditions into the current analysis but at the expense of considerable complexity.

The analysis of the squeeze-film experimental data using the solution developed here is significantly more accurate if the flow index can be determined independently. The value of this index was obtained using orifice flow measurements which were analysed using an appropriate empirical solution which will be described later. This solution is expressed in terms of the velocity of the material through the orifice and, hence, can not be used to obtain the flow consistency.

66 M.J. A&ms et al. j J. Non-Newtons Fluid Mech. 51 (‘1994) 61-78

2. squeeze-* Bnalysis

2.1 Parallel platens

Covey and Stanmore [8] showed that the lubrication analysis for the squeeze-film flow of a Herschel-Bulkley fluid under no-slip wall boundary conditions leads to the following equation which is applicable for a general gap profile h = h(r) between the platens

(7)

where r is a radial coordinate with an origin at the centre of the platens. The parameter X is the dimensionless pressure gradient and is given by

where p =-p(r) is the local pressure. The dimensionless parameter S is the plasticity number which is given by

s = R&k Iin

Approbate closed-fob solutions for eqn. (7) were obtain by Covey and Stamnore [8] but were applicable only to small and large values of S. Here, an asymptotic analysis will be described. For X >> 1, eqn. (7) may be written in the following approximate form if only the leading terms are considered:

S(-X-l)~_~(-x-l)(~+l)~~=o, .;l ; 0

which leads to

(10)

(11)

This solution is asymptotically correct for large and small @S/R); it is exact at S = 0. An advantage compared with Covey and Stanmore’s [8] result is that it is a single solution which reduces, in these limits, to the analytical solutions for viscous fluids and plastic solids as described later. Figure 1 shows plots of -X as a function of rS/R calculated from the solution for different values of the flow index-n. On this scale, the plots are indistinguish- able from those calculated from the exact relationship (7). However, at small values of rS/R the errors become more significant with increasing values of the flow index as shown in Fig. 2.

n=0.7

n=0.5

__________------- _____I__--------

67

Fig. 1. A plot of the dimensionless pressure gradient -X as a function of the d~ensi~nl~ term (rjR)S calculated using eqn. (11) for different values of the flow index.

Fig. 2. The errors in the dimensionless pressure gradient calculated from eqn. (II) (--X0) compared with the values obtained from eqn. (7) ( - X,) as a function of (r/R)S for diierent values of n; the error is expressed as [( -X,) - ( -X,)1/( -X,).

An expression for the radial pressure gradient may be obtains by rearranging (111, thus

68 M..?. Aa’ams et al. f J. ~~n-~ew~5~~ Fluid Mech. 51 (1994) 6I-78

9

8-

c ‘\_ *- ix

g 3 ‘2.. - x. n=0.2

n=O.I

3 2-

l-

0 / I 1 I I 0 20 40 60 80 100

S Fig. 3. The errors in the mean pressure calculated from eqn. (14) ii, compared with the values obtained from eqn. (7) pa as a function of S for different values of n; the error is expressed as f&--j&)/j&.

The radial pressures distribution may be obtained by integration subject to p = a0 at r = R as discussed in the previous Section. For parallel platens, this yields

(13)

This expression may be further integrated to obtain the mean pressure, thus

(141

Figure 3 shows the errors associated with eqn. f 14) for different values of n; these errors were calculated on the basis of polynomial representations of solutions to (7) which were integrated to obtain nominally exact mean pressures. The error is zero at a zero value of S and increases to a maximum at small values ( < 1). The magnitude of the error depends on n and is greatest for a value of about 0.4 which corresponds to a maximum error of 7.4%; the error is zero for n = 1.

For a rigid-plastic solid (S = 0), eqn. (14) reduces to the following expression

The ratio ~o/a~ is equal to the coefficient af friction under no-slip wall boundary conditions, that is when the material fully yields at the wall. Consequently, we have the same result as (6) under the special condition that no-slip occurs at the wall. For viscous flow, the shear yield stress is zero and the edge pressure is also zero. Setting Z, and o, in (13) to zero results in the ~~ati~uship obtained by Scott fK!j fur ~stwald-deWaele fluids.

For a platen with a spherical geometry and a large radius of curvature a against a planar platen, the gap h = h(r) may be approximated by a parabolic relationship, thus

h =h,+$, where ho is the ~i~~~ gap distance at r = 0, An expressiun for the mean pressure may be found by su~tit~tiu~ of f 16) intu (12) and integrating as before, this leads to

where

and

The integral 1, has to be evaluated u~e~~~~~ except for the special case of a Bingham fiuid,

70 M.J. Adams et al. 1 J. Non-Newtonian Fluid Mech. 51 (1994) 61-78

3. Experimental

3.1 Material

Plasticine (Peter Pan Playthings 1983 Ltd.) was homogenised in a Z-blade mixer. Discs of 50 mm diameter and about 3 mm thickness were prepared by compression in a cylindrical die using an Instron universal testing machine (model 6022). In order to facilitate removal of the sheets from the die, waxed paper inserts were used. The specimens were thermally equilibrated for at least 2 h prior to the measurements. This was carried out in an environmen- tally controlled laboratory, maintained at 21S”C, which housed the Instron.

3.2 OriJice $0~

Orifice flow measurements were made using a plastometer (A. Macklow- Smith Ltd.). The ram diameter in this equipment was 64 mm and five orifice diameters of l-4 mm were employed. Steady state pressures were measured for ram velocities in the range ( 1 S- 12) x 10m3 m s- ’ and after arresting the displacement of the ram.

A lower bound solution for the extrusion of a rigid-plastic material through a sharp-edged orifice has been extended, on an empirical basis, for Herschel-Bulkley fluids [ 141. This solution relates the extrusion pressure j& to the mean flow velocity u in the orifice as follows:

j& = 2(0, + Au’) In&/d), (20)

where d is the diameter of the orifice and 4 is the diameter of the ram extruder. The parameter 1 is a material constant which should be related to the extensional flow consistency. It is a reasonable assumption that the exponent 5 is equal to the flow index [5].

In order to analyse experimental data using eqn. (20), it is convenient to express it in terms of a reduced pressure j& as follows:

- - - A = 2;;(doFd) = w (21)

where PO is the mean extrusion pressure at a zero flow rate. This latter quantity was measured after arresting the motion of the ram; the pressure relaxes to a steady state value after a few minutes. The value of 5 was then obtained from the linear regression gradient of a double logarithmic plot of the reduced pressure as a function of the orifice flow velocity u.

3.3 Squeeze-film

Figure 4 shows a schematic diagram of the apparatus which was designed as an attachment to the Instron. The upper platen had a diameter of 43 mm

71

Spring loaded LVDT clamped to piston

Framework bolted to of lnstron

\

Fig. 4. A schematic diagram of the squeeze-film attachment to an Instron Universal tester where LVDT refers to a linear variable diRerentia1 transformer.

and a radius of curvature of 260 mm. It was attached to a lo-kN load cell. The lower platen was planar with a diameter of 44 mm and was mounted on the base plate of the Instron. Both plates were sand-blasts to an R, of about 1 pm to prevent wall slip [5]. The displacement and imposed velocity were measured using an LVDT. Loads were measured for gaps in the range 0.5-2 mm at velocities in the range 0.03- 1 mm s-i.

It was found that the most accurate method of determining the terms Ii and I, in eqn. ( 17) was by linear regression of the measured mean pressure Is: (at some fixed gap) as a function of h’” where the flow index was determined independently using orifice extrusion as indicated previously. In order to obtain the value of the flow consistency from the term I, in eqn. ( 17), the integral was evaluated numerically.

4. Results

Figure 5 shows the Iinearised orifice extrusion data for all the orifice diameters and extrusion velocities investigated; these data points are reduced to a single line within experimental error as predicted by eqn. (21). The uniaxial yield stress calculated from the zero flow extrusion pressure, j0 is 0.22 MPa and the flow index, from the slope of the line in Fig. 5, is 0.40.

The typical force-displacement characteristics observed in the squeeze- film experiments are illustrated in Fig. 6. This set of data corresponds to the upper curved platen coming into contact with a Plasticine disc at a separation distance &, of about 3.4 mm. The linear increase in the force with reducing separation distance for a travel of about 0.8 mm results from the initial penetration into the test specimen. At smaller separation distances,

Fig. 5. A double logarithmic plot of the reduced mean orifice pressure as a function of the mean orifice flow velocity.

Fig. 6. A typical force displacement curve obtained during the squeeze-film measurements.

when the platen is fully in contact with the specimen, the data became non-linear and analysis was confined to this region.

Figure 7 shows the measured mean compressive pressure as a function of the approach velocity raised to the power n which was taken as 0.40 from the orifke flow expe~ents. Linear regression lines are shown for each data set which correspond to different separation distances. The linearity of the

M.J. Adams et al. / J. Non-Newtonian Fluid iUech. 51 (1994) 61- 78 73

0.01 0.02 0.03 0.04 0.05 0.06 0 (Velocity, m/s) h n

)7

Fig. 7. The mean squeeze-film pressure as a function of the velocity raised to the power n ( =0.4) for the separation distances (mm) shown.

data apparently supports the contention that the orifice extrusion technique provides a sensible value of the flow index. However, it was found that linearity is an insensitive criterion for determining an accurate flow index using this type of plot; the uncertainties involved would be at least f25%.

The values of the intercepts Ii (see eqn. ( 17)), at different separation distances, obtained from the linear regression analyses of the data in Fig. 7 are plotted as a function of I3 (see eqn. ( 18)) in Fig. 8. These data are linear as predicted by eqn. ( 18) and the intercept and slope correspond to uniaxial and shear yield stress values of 0.194 MPa and 0.116 MPa respectively.

Figure 9 shows a plot of I2 as a function of I4 which is linear within the experimental scatter as predicted by eqn. (19). The value of the flow consistency obtained from the regression line through the origin is 24.9 kPa.s”. Thus, in summary, the current results suggest that the Herschel-Bulkley equation for Plasticine may be written as

z = 1.2 x lo5 + 2.5 x 104jo.@ Pa for 1~1 > 1.2 x lo* Pa, (22a)

JJ = 0 for 1~1 < 1.2 x lo5 Pa, (22b)

where JJ has units of s-i.

5. Discussion

A comparison has been made between the conventional lubrication and plasticity analyses of the &axial compression of plastic materials. It was concluded that a major difference between the two approaches was the assignment of the value of the local wall normal stress at the edge of a

14 M.J. Adams et al. / J. Non-Newtonian Fluid Mech. 51 (1994) 61-78

, ,

9 10 11 12 13 14 15 ' 13

5

Fig. 8. The intercepts Z, (defined in eqn. (17)) from the data in Fig. 7 as a function of the geometric term Z, (defined in eqn. (18)).

30 40 50 60 70 60 ! 14 b-“1

0

Fig. 9. The slopes I, (defined in eqn. (17)) from the data in Fig. 6 as a function of the term Z4 (defined in eqn. (19)).

specimen. This is considered to be effectively zero in the lubrication analysis while it is taken as equal to the uniaxial yield stress in the plasticity treatment. The analysis of the experimental data described in the current work supports this latter contention. More specifically, a plot of the plasticity term I, as a function of the geometric term I3 in (18) produces a positive intercept at I3 = 0 which is equal to the uniaxial yield stress according to this equation. The conventional lubrication analysis predicts that the intercept should be zero.


The origin of the difference between the two approaches is that, for rigid-phase solids, each point in the solid is subject to a yield criterion including points adjacent to the platen wall and at the edge of the specimen. In the equilibrium stress analysis of upsetting, the circumferential stress be, the radial stress o, and the axial stress o, ( = -p) are taken as principal stresses. It is also assumed that 6, = ran, consequently the yield criterion may be written as

c7,-cz =a(), (23)

where 6, is taken as zero at r = R. Hence, in the case of the von Mises yield criterion, p = J 32,, at r = R. For viscous fluids (z,_, = 0), gr is also taken as zero at r = R and hence o, is zero at this radial coordinate. The error in the lubrication analysis arises because it has been assumed that this result is also valid when r. is not zero as in the case of plastic fluids.

Actually, the assumption that o, at the edge is zero has also been questioned in the context of specimens of radii smaller than the platens [ 151. It was correctly pointed out that strictly the boundary condition should be described in terms of a vanishing of the stresses on the free surface of the specimen. For the present case, where the platens are smaller than the specimens, there will be an equivalent argument for some internal plane connected at the periphery of the smaller platen. However, the approximation made here seems to be consistent with the experimental data.

Further support for the present analysis was obtained from parallel work which was concerned with a range of experimental configurations [5]. This included the measurement of the pressure distribution of thicker Plasticine discs in upsetting between parallel platens. It was found that the data were well described by the relationship developed here (eqn. ( 13)) which predicts that the local wall edge pressure should be equal to the uniaxial yield stress. The shear flow consistency obtained by fitting ( 13) to the wall pressure distributions measured for different uniaxial strains was actually identical to the value derived here from the squeeze-film flow data. However, this close similarity of these values may be fortuitous since different batches of Plasticine were employed. In the previous work, the flow index and yield stress measured using orifice flow were 0.45 MPa and 0.18 MPa respectively compared with 0.40 and 0.22 MPa found in the current work.

The relative values of the uniaxial and shear yield stresses obtained in the current work are very similar to those expected for a material obeying the von Mises yield criterion where co = &. It has already been pointed out that the mean pressure relationship ( 15) derived for squeeze-film flow is exactly equivalent to the homogeneous plastic deformation relationship (6) under no-slip boundary conditions. Consequently, it might be expected that the uniaxial flow stress derived from the mean pressure measurements

76 M.J. Adams et al. /J. Non-Newtonian Fluid Mech. 51 (1994) 61-78

should be constrained value of the yield stress as discussed previously. This is certainly the case for the value of the flow stress obtained from the orifice extrusion data which is greater than the uniaxial yield stress from the squeeze-film experiments. In orifice extrusion, inhomogeneous flow is mani- fested as a central flow region converging towards the exit [ 121. This is due to the formation of an unyielded or dead zone which leads to internal shear bands at the interface between the static and flowing regions and also a component of bulk shear in the flow field rather than only pure extension. These redundant work components result in a flow stress which is greater than the true uniaxial flow stress.

In order to examine further the nature of the lubrication solution obtained for squeeze-film flow, it is useful to recall an analysis due to Lipscomb and Denn [ 161. On the basis of kinematic considerations, they showed that local yielding in any region is only possible after yielding of the whole region. That is, the coexistence of flow adjacent to the walls and a rigid or unyielded core is kinematically inconsistent. This inconsistency has recently been examined by Wilson [ 171. He considers the biviscosity constitutive relationship for which the Bingham fluid may be regarded as a limiting case when the high viscosity applicable prior to yield becomes singular. The advantage of such a relationship is that a radial plug flow is possible in the unyielded core. Wilson correctly argued that a three- dimensional yield criterion must be introduced to determine the yield surface. He employed von Mises’ criterion which reduces to eqn. (23) under the assumptions applied in equilibrium analysis. Using eqn. (ll), the velocity field in the unyielded core calculable from the Covey and Stanmore [8] approach is as follows:

(24)

Thus in the so-called ‘unyielded’ region, the velocity is a function of r and hence the material undergoes extensional flow. The velocity field in the ‘flow zone’ is a function of both r and z. Thus, in addition to shear flow, there is an extensional component that is compatible with the flow in the ‘unyielded region.

In summary, the yield surface in the Covey and Stanmore [8] solution represents a boundary at which shear flow ceases. The inconsistency is that radial flow occurs in the core which violates the yield criterion as pointed out by Lipscomb and Denn [ 161. For the limiting case of a rigid-plastic solid, which corresponds to a zero value of the plasticity number, eqn. (24) shows that the radial velocity is zero. This is the same inconsistency that arises from equilibrium stress analysis with stick boundary conditions. In fact, (11) reduces to a pressure gradient, a~/&, equal to -22,/h for S = 0


which is the relationship obtained from equilibrium stress analysis. In practice, the specimen will then deform by barrelling and folding [ 121. It seems for the compression of a viscoplastic fluid, under no-slip boundary conditions, that homogeneous extension is possible in the central region with the quasi-lubrication layers adjacent to the platens. The freedom to develop extensional flow throughout the core of the specimen may account for the apparently unconstrained value of the uniaxial yield stress obtained in the current work. That is, although the approach adopted in the current work is not rigorously correct, it does appear from the consistency of the experimental data and the theory that the approach leads to a useful first-order approximation.

The lubrication theory ascribes the viscous work to shear flow and it might be concluded from the velocity fields that the flow consistency obtained is a lumped value of shear and extension. However, measurements of lubricated samples of Plasticine have shown that the energy associated with extensional viscous flow is negligible compared with that arising from plastic deformation [5]. This is consistent with an analysis of the squeeze- film flow of Newtonian fluids [ 131; it was shown that the compressional pressures for lubricated flows are more than an order of magnitude less than for unlubricated flows.

Acknowledgement

This work was carried out as part of a MAFF/DTI funded LINK scheme on ‘Solid Food Processing Operations’.

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an experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic...

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