An investigation of fractiure criteria for predicting serface fracture in paste extrusion

Y uniaxial yield stress N m−2z axial coordinate in cylindrical polar system m Greek letters ˙i normal strain rate component i = r, or z or 1, 2 or 3 s−1 ˙K equivalent strain rate [2=3˙2

An investigation of fracture criteria for predicting surface

fracture in paste extrusion Annette T.J Domantia, Daniel J Horrobinb, John Bridgwatera; ∗

a Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK

b Department of Mathematics & Statistics, The University of Melbourne, VIC 3010, Australia

Received 27 February 2001; received in revised form 20 May 2002

Abstract

In the extrusion of pastes, fractures may be found on the surface of product Such fractures compromise strength and are often unacceptable aesthetically Here some theoretical criteria for predicting the onset of surface fracture, using the elastic–plastic /nite element method are evaluated and the success of these criteria

in predicting recent observations is assessed Two criteria based on stress /elds successfully predicted an increase in the depth of fracture cracks with extrusion ratio However these criteria, which are dependent

on the deforming zone stresses and extrudate residual stresses, respectively, do not successfully predict the increase in fracture with increasing die entry angle observed experimentally Three criteria based on ductile fracture are also investigated and di2culties associated with their accurate evaluation in extrusion problems highlighted However, all three successfully predict the increase in fracture with increasing die entry angle In considering the e3ect of extrusion ratio on surface fracture, two of these criteria should be at least qualitatively correct while for the third this is unlikely.

? 2002 Elsevier Science Ltd All rights reserved.

Keywords: Paste; Surface fracture; Extrusion; Defects; Fracture criteria; Modelling; ABAQUS

1 Introduction

Extrusion has long been used in the metals industry to make bars, tubes, wires and strips withsigni/cant attempts being made to describe its occurrence in fundamental terms Polymer extrusion

is of enormous industrial signi/cance and commands a very substantial literature on the origins of

a wide range of surface instabilities but the origin of the behaviour of important materials such

as high-density polyethylene remains very incomplete Pastes consist of mixtures of /ne powders

∗Corresponding author Tel.: +44-1223-334798; fax: +44-1223-334796.

E-mail address: john bridgwater@cheng.cam.ac.uk (J Bridgwater).

0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd All rights reserved.

PII: S 0020-7403(02)00047-4

Y uniaxial yield stress (N m−2)

z axial coordinate in cylindrical polar system (m)

Greek letters

˙i normal strain rate component (i = r,  or z or 1, 2 or 3) (s−1)

˙K equivalent strain rate [
2=3(˙2

1+ ˙2

2+ ˙2

3)1=2] (s−1)

˙rz shear strain rate component in the r–z plane (s−1)

˙ non-negative scalar describing the magnitude of the plastic strain rate (N−1 m2s−1)

 Poisson’s ratio (dimensionless)

 angular coordinate in cylindrical polar system (m)

 dimensionless radial coordinate (dimensionless)

ij stress component (either normal, in which case i=j, or shear, in which case i
= j) (N m −2)

i normal stress component (i = r,  or z or 1, 2 or 3) (N m−2)

rz shear stress component in the r–z plane (N m−2)

! rate of rotation of a material element (s−1)

Subscripts

1 major principal component

2 intermediate principal component

3 minor principal component

r radial component

 circumferential component

z axial component

Superscripts

◦ Jaumann stress rate (corrected for rotational motion)

· material stress rate (corrected for linear motion)

Extrusion direction

Fig 1 Surface fracture for an alumina paste extrudate [1].

and complexliquids which, during extrusion, are subject to a range of Low conditions Shaping bypaste extrusion now forms an important route for manufacturing an ever-increasing number of com-mon materials including foods, chemicals, catalyst substrates and pharmaceutics, as well as engineparts made from advanced materials, but the literature is sparse During extrusion, imperfections

in the quality of the extrudate may arise, ranging from a rough or uneven surface to a completeseverance of the extrudate Historically a trial-and-error method has been used for forming extrusionproducts of su2cient quality, a costly, uncertain and time-consuming practice The ability to iden-tify and predict these defects is crucial to modern practice and is challenging fundamentally Smallexternal or internal defects may also be potential sources of weakness in the /nal product

Fig 1 shows a ceramic alumina paste formed into a long rod by extrusion from a cylindrical barrel

of 25 mm diameter through an axisymmetric die of square entry with diameter 6 mm and die landlength 2 mm, the system studied by Domanti and Bridgwater [1] The most noticeable feature is thepresence of regular breaks in the surface, roughly encircling the extrudate and therefore described

as circumferential cracks Such surface fracturing is an example of an extrusion defect, which maycompromise the strength and performance of the extruded product

Similar surface fracture has been observed for various materials other than ceramic pastes Forexample, Domanti and Bridgwater also discuss surface fracturing in the extrusion of soap Thephenomenon is well known in the industry but the sole study in which some of the importantparameters are varied systematically is the recent one of these two workers Although pastes maycontain two or more phases, processing is carried out under conditions such that the phase separation

is minimal and the material overall exhibits a yield stress Thus it is in metal forming where relatedphenomena have been examined, where several explanations for its cause have been proposed, alongwith criteria for predicting its occurrence Hence the present purpose is to examine criteria that haveappeared in the metal forming literature and to evaluate how well these predict the experimentalobservations of Domanti and Bridgwater

1.1 Theoretical approaches to fracture

Fracturing of solids occurs in many situations, and theories for describing phenomena have beenextensively developed during the last century Conventional fracture mechanics [2,3] deals withmacroscopic cracks in bodies that would often otherwise be loaded to within their elastic limit Byexamining the stress and deformation /elds in the vicinity of the crack, it is determined whether thecrack will grow, resulting in fracture

At the heart of a typical calculation procedure is a numerical, usually /nite element, model thattakes information about (i) the geometry of the die and workpiece, (ii) operating variables such

as temperature and the rate of deformation, and (iii) the bulk constitutive response of the materialand the interaction with solid boundaries The stresses and strains are then calculated as functions

of time from which predictions regarding the occurrence of fracture are obtained This procedurehas been followed here, although temperature and deformation rate are not involved, as we haveassumed that the extrusion process is isothermal, an excellent assumption for pastes The material isdescribed by a perfect plasticity model with Tresca boundary conditions

This bears some resemblance to a continuum damage mechanics analysis It is widely believedthat on a microscopic level, fracture in metals occurs by nucleation, growth and coalescence ofvoids.1 Damage mechanics [8] seeks to model the overall e3ect of an evolving population of suchmicrodefects through a macroscopic ‘damage’ variable, or variables, that can be loosely interpreted as

a porosity A law is introduced that describes how the damage changes with time in response to thelocal stress and deformation /elds This law may be based on micromechanical arguments, or may

be postulated without detailed reference to the microstructure An analysis is then performed similar

to that above, in the sense that the existence of macroscopic cracks is not assumed, and the stress,strain and damage histories are computed throughout the specimen The damage accumulates untilthe material is said to have lost all structural integrity, or at least the appearance of a macroscopiccrack is predicted However, the key di3erence between a typical damage mechanics analysis andthe calculations here is that the former takes into account the e3ect on the overall mechanicalproperties, such as Young’s modulus and yield stress, of the growing microdefects Rather thanthe one-way coupling between the stress=strain history calculation and the fracture calculation, amore complicated, fully coupled analysis is performed Such analyses have become increasinglypopular in recent years for investigating a range of failure phenomena that arise in solid mechanicalapplications [9]

1.2 The current work

A schematic diagram of a ram extruder used to produce the extrudate in Fig 1 is shown in Fig 2.The die comprises two sections: (i) the die entry, the region where deformation occurs immediatelyupstream from the re-entrant corner, and (ii) the die land, the parallel-sided section downstream

1 Similar void growth has been observed in deforming Plasticine [7], a dispersion of clay particles in mineral oil that

is often used as a model material for studying metal-forming processes, and which is perhaps not dissimilar to a ceramic paste.

Fig 2 A simple ram extruder with a conical entry die.

from that corner Outside the die entry, both upstream in the barrel and downstream in the die land,the material is essentially rigid; pastes and other soft solids observed slip at solid boundaries Aconical entry is shown in the /gure, the entry angle being that the die face makes with the axialdirection For a square entry die, the entry angle is 90◦ so the die face is perpendicular to theaxial direction and the contraction is abrupt The other parameter used to de/ne geometry is thereduction in cross-sectional area, expressed as an extrusion ratio de/ned as the ratio of the /nal tothe initial cross-sectional area or the square of the diameter ratio, if both barrel and die-land arecircular

We evaluate the e3ect of the die entry geometry on the level of fracture The key conclusionsfrom Domanti and Bridgwater relevant to this objective are:

(i) the level of fracture decreases with increasing extrusion ratio, for extrusion ratios in the range4.34–69.4, and

(ii) the level of fracture increases with increasing entry angle, for entry angles in the range 15–45◦

,and remains reasonably constant for entry angles above 45◦

The term ‘level of fracture’ refers primarily to the depth of cracks in the extrudate relative to theextrudate diameter since the frequency (i.e spacing of cracks) was found to be equal to the extrudateradius for all entry angles and reduction ratios

A commercial /nite element package (ABAQUS) is used to simulate the extrusion of a pasteusing an elastic–plastic material model This provides the detailed information about the stresses andstrains within the deforming material required for evaluation of the fracture criteria Two classes offracture criteria are considered:

(i) those based on the current stress /eld alone, and

(ii) those based on the entire stress and deformation histories

The /rst class is often regarded as being appropriate for describing brittle fracture, and the secondclass for describing ductile fracture

Here, the /rst class is represented by the hypotheses of Pugh (presented in Pugh and Green [10],Pugh and Gunn [11], and Pugh [12], and Fiorentino et al [13]) These hypotheses were developed

to explain cracking in extrusion of brittle metals such as bismuth, magnesium and beryllium Pughsuggested that cracking resulted from insu2cient compressive stress in the deforming zone, whileFiorentino et al argued that the cause lay with tensile residual stresses

If the predominantly plastic nature of paste Low is widely accepted, the response below the yieldpoint is much less clear It is often assumed that when subjected to small stresses the materialdeforms elastically, with Young’s modulus that is large compared with the yield stress, so thatthe elastic strains are always small This is analogous to the behaviour of solid metals, ignoringthe phenomenon of creep The elastic–plastic approach is followed here, with the elastic responsedescribed by the linear Hookean model.2 The combined uniaxial stress–strain relationship thereforehas a very simple form, with the stress being proportional to strain until the yield point is reached,and being constant thereafter.

In addition to the constitutive model, the usual force equilibrium relationships of continuum chanics are required For paste Lows, as in metal forming, body forces are usually neglected; typicalLow velocities are small enough for inertial e3ects to be insigni/cant, and the yield stress is usu-ally large enough for gravitational e3ects to be ignored To illustrate how they are written for anaxisymmetric problem, the equations underlying the model are listed in Table 1 Further details can

me-be found in standard texts, e.g [17] The rest of the formulation concerns the boundary conditions

at solid surfaces Unlike viscous Luids, elastic and=or plastic solids are usually regarded as beingcapable of slipping at such surfaces In the theory of metal forming, a Coulomb friction boundarycondition is thought to be most realistic However, this is sometimes substituted, for computational

2 An alternative formulation for numerical plastic Low calculations exists, where the material is e3ectively treated as a highly viscous Luid below the yield point [16] This formulation, which has some computational advantages, was developed

to approximate the true behaviour of solid metals However, for pastes and other soft solids it may prove to be more realistic than the elastic–plastic model.

Notes regarding the ow rule:

(i) The stress and strain rates must correspond to real material deformation, and must therefore vanish in the case of rigid body motion ˙ ij is the true strain rate with components

Strictly speaking, ˙ can only be non-zero if the material is at the yield point and the rate of change of stress is such that it remains at the yield point The latter condition can be expressed mathematically by di3erentiating (1b)

r  )(◦r− ◦  z )(◦ z z r )(◦z− ◦r ) + 6 rz◦ rz = 0: (1h)

Fig 3 Example /nite element mesh showing variation in element size and rounding at re-entrant corner.

convenience, by a Tresca boundary condition, where the wall shear stress is constant, provided thematerial is slipping or on the point of slipping This parameter is usually expressed as a fraction

of the shear yield stress of the material For pastes, which are solid–liquid mixtures, a liquid-richlayer is thought to form where slip occurs at a surface, which then lubricates the interior Low.Depending on the rheology of this layer, the wall shear stress may then be approximately constant(the Tresca condition), or may increase from zero as the slip velocity increases More generally,

it may combine both features, increasing with slip velocity from an initial non-zero value Wehave chosen to use the simplest Tresca boundary condition, where the wall shear stress is zero,corresponding to a well-lubricated interface; this is a reasonable approximation for ceramic pastesstudied

3 Solution procedure

The equations of motion were solved using the ABAQUS /nite element package, version 5.5(Hibbitt, Karlsson and Sorensen, Inc., 1995) As the problems considered here were all axisymmetric,two-dimensional meshes, covering the radial and longitudinal directions, were used These mesheswere constructed from irregular quadrilateral elements with widely varying sizes, small and largeelements being employed in regions where the rates of strain were large and small, respectively

A typical mesh, in this case for a square entry die, is shown in Fig 3 The smallest elements arerequired near the die corners, especially the re-entrant corner The corners are also rounded slightly,and the local element size is chosen to be slightly smaller than the /llet radius at the corner The/llet radius is a small proportion (typically around 1%) of the barrel radius, and so the solutionobtained from the analysis is expected to be close to the solution for a die with perfectly sharpcorners Using meshes of the form shown in Fig 3, the solutions were found to be insensitive tothe precise details of the mesh construction

Young’s modulus was chosen to be two or three orders of magnitude larger than the uniaxial yieldstress so that elastic strains were small, and the solution corresponded closely that for the rigid-plasticlimit Here the stresses in the deforming material should not be sensitive to the precise values ofYoung’s modulus and Poisson ratio However, a value for the Poisson ratio must be speci/ed, thevalue selected, somewhat arbitrarily, being 0.49 This is close to 1=2, and so the elastic component

of the deformation is at approximately constant volume; the plastic component is constant volume.

It might be anticipated that even with a large Young’s modulus to yield stress ratio, the stressesoutside the deforming region could still be sensitive to the Poisson ratio This could be importantsince we need the residual stress /eld in the extrudate when we come to the hypothesis of Fiorentino

et al However, when duplicate analyses were carried out for a square entry die with extrusion ratio9.77, using Poisson ratios of 0.49 and 0.10, the di3erences in the residual stress /elds were found to

be insigni/cant Similarly, duplicate analyses for the same die geometry with Young’s moduli 400times and 1000 times the uniaxial yield stress produced no signi/cant di3erences

ABAQUS implements an updated Lagrangian /nite element formulation where, as the calculationproceeds, the mesh deforms following the deformation of the material This causes di2culty whenthe deformation is large, as in the problems studied here, since the elements quickly become grosslydistorted leading to errors This was overcome by rezoning at frequent intervals In this procedure,when the distortion has increased to an unacceptable level, the calculation was stopped A new mesh,comprising well-shaped elements, is then constructed within the boundary of the old mesh, and thecurrent solution is mapped from the old mesh to the new mesh The calculation is then restartedusing the mapped solution to provide the initial conditions for the new analysis step The rezoningprocedure also enables small elements to be retained in the regions where the rate of strain is large,and large elements in regions where the rate of strain is small Further information regarding theprocedure can be found in Ref [18] and in more detail in Ref [19]

4 Results: deforming zone stresses

Pugh and Gunn [11], as reported by Pugh and Low [20], suggested that the cause of surfacefracture was the insu2ciency of the ‘total hydrostatic compressive stress existing in regions inwhich the material is being deformed’ Experiments to test this hypothesis were carried out by Pughand Gunn, and earlier by Pugh and Green [10] They extruded brittle metals with apparatus thatallowed the extrudate to emerge into a pressurised liquid, and determined the Luid back pressurerequired to suppress surface cracking Pugh’s hypothesis is examined, using results from the /niteelement calculations

4.1 Implementation

In the absence of detailed information about the stress /eld in the deforming material, Pughsuggested that a rough measure of the hydrostatic compressive stress in the deforming zone wasthe average stress on the die face, calculated as the extrusion load divided by the area of contactbetween the billet and die (i.e die face area) This quantity has been determined Finite elementanalyses allow the stress /eld in the deforming material to be investigated in more detail, and henceseveral alternative interpretations of ‘total hydrostatic compressive stress’ are possible For example,this term could refer to:

• the volume average pressure in the deforming material (where the pressure at a point is the

negative average of the normal stress components), or

• the maximum pressure in the deforming material.

Fig 4 Deforming zone as de/ned by integration points where there is active yielding.

These two possibilities are also examined In either case, it is necessary to have a suitable de/nition

of ‘deforming material’, and we have taken it to be the region where the plastic strain rate is non-zero

In ABAQUS, a Lag variable is used to label element integration points as ‘actively yielding’ if theplastic strain increment is non-zero during a given analysis increment, and this provides a convenientway to identify the deforming zone

Fig 4 shows the locations of the actively yielding integration points for a single increment in ananalysis for a square entry die with barrel diameter 25 mm and die diameter 8 mm The integrationpoints are particularly closely spaced near the re-entrant corner, where the elements are small toaccommodate the high strain rates that occur there, and some integration points in this vicinity havebeen omitted for clarity

A fourth stress quantity that has been examined, namely the extrusion pressure This is a somewhatloose interpretation of ‘average hydrostatic compressive stress in the deforming zone’, but has theadvantage of convenience, as it is the most likely stress parameter to be recorded

4.2 E6ect of extrusion ratio

Square entry dies: Analyses were performed for extrusion through square entry dies with barreldiameter 25 mm and die diameters between 2 and 24 mm, corresponding to extrusion ratios between

156 and 1.09 Fig 5 shows the maximum and average deforming zone pressures, the average die facepressure and the extrusion pressure, all plotted against the extrusion ratio The latter two pressureshave been plotted from results given in Ref [18], and are not strictly average pressures, but ratheraverage stresses normal to the die face and ram, respectively

Fig 5 Various measures of pressure versus extrusion ratio for square entry dies.

According to Pugh’s hypothesis, a higher level of fracture is expected when the average hydrostaticcompressive stress in the deforming zone is small The average deforming zone pressure and theextrusion pressure both increase monotonically with extrusion ratio R, while the maximum deformingzone pressure and the average die face pressure both exhibit minima when R is about 2 to 3.Considering that the maximum deforming zone pressure occurs at the die face for smooth-walleddies, it is not surprising that this quantity is closely related to the average die face pressure Thepressure increases when R is small and decreasing, indicating that as R tends to zero the extrusionpressure approaches zero more slowly than the die face area approaches zero

Pugh [12], discussing the magnesium rod extrusions carried of Pugh and Gunn [11], also identi/ed

a minimum in the average die face pressure He suggested that where the minimum occurs ataround R=2, large back pressures would be required to prevent surface fracture Equivalently, whenextruding into atmospheric pressure, the level of fracture would be expected to be highest at around

R = 2

The paste extrusion experiments indicated that the relative depth of fracture increases with diediameter in the range D = 3–12 mm, or equivalently decreases with increasing extrusion ratio in therange R = 4:34–69.4 This correlates well with the increase in all four quantities plotted in Fig 5when the extrusion ratio is large and increasing At present, no experimental results are availablefor smaller extrusion ratios Such data would allow the suitability of the di3erent deforming zonepressures to be better assessed, and Pugh’s hypothesis to be interrogated more thoroughly

45◦

tapered entry dies Analyses were also performed for extrusion through tapered entry dies with

45◦ entry angles, barrel diameter 25 mm and die diameters between 8 and 20 mm, corresponding

Fig 6 Various measure of pressure versus entry angle for dies with extrusion ratio R = 4:34.

to extrusion ratios between 9.77 and 1.56 These produced very similar relationships between thedeforming zone pressures and extrusion ratio The conclusions are thus similar

4.3 E6ect of taper angle

Analyses were performed for extrusion through tapered entry dies with barrel diameter 25 mm,die diameter 12 mm (corresponding to an extrusion ratio 4.34), and seven entry angles between 5◦

and 90◦ Fig 6 shows the four dimensionless pressures plotted against entry angle for these dies.The extrusion pressure and average die face pressure are again based on Ref [18] Some scatter

is apparent in the maximum deforming zone pressures, reLecting the di2culty of determining thisquantity accurately These points relate to single outlying stress values, whereas the other quantitiesare all a3ected by many stress values The general trend exhibited by all four quantities, however,

is of a moderate increase with entry angle as the entry angle increases from 5◦

to 90◦

.Applying Pugh’s hypothesis to these results suggests that smaller entry angles are more likely togive rise to fracture This contradicts the experimental results of Domanti and Bridgwater None ofthe four pressure quantities shows a clear decrease with increasing entry angle, and so in this casePugh’s hypothesis appears to fail However, Pugh appreciated that the e3ect of entry angle had notbeen adequately incorporated into the theory

5 Results: extrudate residual stresses

Fiorentino et al [13] suggested that the longitudinal stress at the extrudate surface determineswhether the material will fracture, fracture being associated with a tensile value Both Osakada et al.[21] and Polyakov et al [22] experimentally evaluated stresses in metal extrudates and found thatthese were tensile at the surface and compressive in the centre Rudkins et al [23] performed /niteelement analyses that supported the hypothesis of Fiorentino et al

Fig 7 Calculated longitudinal stress versus radial coordinate for square entry dies with various extrusion ratios: (a) larger extrusion ratios; and (b) smaller extrusion ratios.

5.1 E6ect of extrusion ratio

Square entry dies Fig 7 shows the longitudinal stress in the die land as a function of theradial coordinate  (made dimensionless dividing by the die land radius) for sixof the 12 extru-sion ratios examined For clarity the curves are plotted on two separate diagrams: Fig 7(a) showsthe results for the three larger extrusion ratios, and Fig 7(b) the results for the three smaller ex-trusion ratios The values were obtained by averaging in the longitudinal direction within the dieland; the stresses are observed to vary little in this direction The results can be summarised asfollows:

• For the largest extrusion ratios (R = 156 and 9.77) the longitudinal stress pro/le is approximately

linear, with the stress being compressive (negative) on the axis and tensile at the surface Both

Fig 8 Surface longitudinal stress versus extrusion ratio for square entry dies.

central and surface stresses increase in magnitude as R decreases, although the pro/le for R=6:25(not shown) is quite similar to that for R = 9:77

• For intermediate extrusion ratios (R = 4:34 and 2.44) the pro/le is curved, being Latter near the

axis and steeper near the surface There is a slight increase in the surface tensile stress comparedwith the larger extrusion ratios, but there is now a reduction in the compressive stress on theaxis

• For the smallest extrusion ratios (R = 1:56 and 1.09) the longitudinal stress on the axis has

been reduced to such an extent that it has become tensile The pro/le for R = 1:09, which hasthe greatest tensile stress on the axis, also shows a reversal in the surface stress, which is nowcompressive

The linear pro/les for large extrusion ratios are similar to that observed by Osakada et al [21].The longitudinal stress is compressive for  less than about 2=3, which corresponds to about halfthe extrudate cross-section This obeys the requirement of zero net force acting on the radial cross-section

The occurrence of tensile longitudinal stresses on the axis was reported by Fiorentino et al [13]and Rudkins et al [23], who suggested central bursting or chevron cracking as a cause No evidence

of this form of cracking in pastes was observed during the experimental work

Fig 8 shows the surface longitudinal stress plotted against extrusion ratio The equivalent diediameters are indicated on the top axis The surface tensile stress increases rapidly with increasingextrusion ratio up to about R = 2, beyond which it decreases steadily Fiorentino et al reported thatBuhler and Pieter [24] observed similar behaviour experimentally for rod and wire drawing of steel

If the magnitude of the tensile stress is related to the likelihood of fracture, then extrusion through

an 18 mm diameter die should be most likely to result in fracture

Fig 9 Longitudinal stress versus radial coordinate for tapered entry dies with a 45◦ entry angle and various extrusion ratios.

Paste extrusion experiments were carried out with dies of diameter 3, 6 and 12 mm; it wasfound that the relative depth of fracture decreased with increasing extrusion ratio This is consistentwith Fig 8, if we interpret the larger surface tensile stresses to imply a greater depth of frac-ture The pro/le in Fig 8 is also in qualitative agreement with the measurements of Pugh andGunn [11], for the critical Luid back pressure required to suppress surface cracking in magnesiumrods Their results, which cover a smaller range of extrusion ratios than that examined currently,showed that the greatest Luid back pressure is required for an extrusion ratio of about 2 Thiscoincides with the ratio at which the largest surface tensile stress occurs in Fig 8 Similarly, theresults of Pugh and Gunn show a decrease in the Luid back pressure required at very small ex-trusion ratios, which is consistent with Fig 8, where the surface tensile stress is reduced whenthe extrusion ratio is small, ultimately becoming compressive The results of Pugh and Gunn sug-gest that at extrusion ratios above four no back pressure is required to prevent fracture In Fig 8,the surface tensile stress does not reduce to zero, but it does decrease with increasing extrusionratio

R = 3:19 and 1.56) the pro/le is no longer linear, the compressive stress on the axis being reducedand the tensile stress at the surface being increased For R = 1:56, the compressive stress on the axis

is not reduced su2ciently to become tensile, unlike for the square entry die with the same reductionratio (Fig 7(b))

Rudkins et al [23] used ABAQUS to determine the longitudinal stresses resulting from extrusion

of an elastic–plastic material through a tapered entry die with entry angle 30◦ and extrusion ratio1.54 Their longitudinal stress pro/le is similar to the curve in Fig 9 for entry angle 45◦

andextrusion ratio 1.56