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

International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 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 ÿnite element method are evaluated and the success of these criteria in predicting recent observations is assessed Two criteria based on stress ÿelds 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, not successfully predict the increase in fracture with increasing die entry angle observed experimentally Three criteria based on ductile fracture are also investigated and di culties 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 e ect 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 Introduction Extrusion has long been used in the metals industry to make bars, tubes, wires and strips with signiÿcant attempts being made to describe its occurrence in fundamental terms Polymer extrusion is of enormous industrial signiÿcance 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 ÿne 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 0 - ( ) 0 - 1382 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Nomenclature CL D0 D E GW H p P R r si t vi Y z Cockcroft and Latham fracture integral (N m−2 ) Barrel diameter (m) die diameter (m) Young’s modulus (N m−2 ) generalised work fracture integral (N m−2 ) hybrid fracture integral (N m−2 ) pressure (negative average normal stress) [ 13 ( + + )] (N m−2 ) extrusion pressure (N m−2 ) extrusion ratio [D02 =D2 ] (dimensionless) radial coordinate in cylindrical polar system (m) deviatoric normal stress component (i = r,  or z or 1, or 3) [ i + p] (N m−2 ) time (s) velocity component (i = r or z) (m s−1 ) uniaxial yield stress (N m−2 ) axial coordinate in cylindrical polar system (m) Greek letters ˙i normal strain rate component (i = r,  or z or 1, or 3) (s−1 ) ˙ equivalent strain rate [ 2=3( ˙21 + ˙22 + ˙23 )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 m2 s−1 ) Poisson’s ratio (dimensionless)  angular coordinate in cylindrical polar system (m) dimensionless radial coordinate (dimensionless) stress component (either normal, in which case i=j, or shear, in which case i = j) (N m−2 ) ij normal stress component (i = r,  or z or 1, or 3) (N m−2 ) i shear stress component in the r–z plane (N m−2 ) rz ! rate of rotation of a material element (s−1 ) Subscripts major principal component intermediate principal component 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) A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1383 Extrusion direction · mm Fig Surface fracture for an alumina paste extrudate [1] and complex liquids which, during extrusion, are subject to a range of ow conditions Shaping by paste extrusion now forms an important route for manufacturing an ever-increasing number of common materials including foods, chemicals, catalyst substrates and pharmaceutics, as well as engine parts 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 complete severance of the extrudate Historically a trial-and-error method has been used for forming extrusion products of su cient quality, a costly, uncertain and time-consuming practice The ability to identify and predict these defects is crucial to modern practice and is challenging fundamentally Small external or internal defects may also be potential sources of weakness in the ÿnal product Fig 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 mm and die land length mm, the system studied by Domanti and Bridgwater [1] The most noticeable feature is the presence 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 may compromise the strength and performance of the extruded product Similar surface fracture has been observed for various materials other than ceramic pastes For example, Domanti and Bridgwater also discuss surface fracturing in the extrusion of soap The phenomenon is well known in the industry but the sole study in which some of the important parameters are varied systematically is the recent one of these two workers Although pastes may contain 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 related phenomena have been examined, where several explanations for its cause have been proposed, along with criteria for predicting its occurrence Hence the present purpose is to examine criteria that have appeared in the metal forming literature and to evaluate how well these predict the experimental observations of Domanti and Bridgwater 1.1 Theoretical approaches to fracture Fracturing of solids occurs in many situations, and theories for describing phenomena have been extensively developed during the last century Conventional fracture mechanics [2,3] deals with macroscopic cracks in bodies that would often otherwise be loaded to within their elastic limit By examining the stress and deformation ÿelds in the vicinity of the crack, it is determined whether the crack will grow, resulting in fracture 1384 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 In contrast, when investigating defects that arise in forming processes, it is not usual to assume pre-existing macroscopic cracks in the workpiece Instead, strategies for predicting defects have been based on the stress and=or deformation histories of the entire, initially uncracked body, which inevitably undergoes substantial inelastic deformation forming Among those adopting this approach are Clift et al [4], Ko et al [5], and DeLo and Semiatin [6] The ÿrst of these is particularly extensive, investigating nine di erent ductile fracture criteria applied to three di erent metal-forming operations, of which extrusion is one Two of these, the Generalised Work Criterion and the Cockcroft and Latham Criterion, are two of the three ductile fracture criteria discussed here At the heart of a typical calculation procedure is a numerical, usually ÿnite element, model that takes 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 material and 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 procedure has been followed here, although temperature and deformation rate are not involved, as we have assumed that the extrusion process is isothermal, an excellent assumption for pastes The material is described by a perfect plasticity model with Tresca boundary conditions This bears some resemblance to a continuum damage mechanics analysis It is widely believed that on a microscopic level, fracture in metals occurs by nucleation, growth and coalescence of voids Damage mechanics [8] seeks to model the overall e ect of an evolving population of such microdefects 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 the local stress and deformation ÿelds 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 until the material is said to have lost all structural integrity, or at least the appearance of a macroscopic crack is predicted However, the key di erence between a typical damage mechanics analysis and the calculations here is that the former takes into account the e ect on the overall mechanical properties, such as Young’s modulus and yield stress, of the growing microdefects Rather than the one-way coupling between the stress=strain history calculation and the fracture calculation, a more complicated, fully coupled analysis is performed Such analyses have become increasingly popular in recent years for investigating a range of failure phenomena that arise in solid mechanical applications [9] 1.2 The current work A schematic diagram of a ram extruder used to produce the extrudate in Fig is shown in Fig The die comprises two sections: (i) the die entry, the region where deformation occurs immediately upstream from the re-entrant corner, and (ii) the die land, the parallel-sided section downstream 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 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1385 Fig 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 A conical entry is shown in the ÿgure, the entry angle being that the die face makes with the axial ◦ direction For a square entry die, the entry angle is 90 so the die face is perpendicular to the axial direction and the contraction is abrupt The other parameter used to deÿne geometry is the reduction in cross-sectional area, expressed as an extrusion ratio deÿned as the ratio of the ÿnal to the initial cross-sectional area or the square of the diameter ratio, if both barrel and die-land are circular We evaluate the e ect of the die entry geometry on the level of fracture The key conclusions from Domanti and Bridgwater relevant to this objective are: (i) the level of fracture decreases with increasing extrusion ratio, for extrusion ratios in the range 4.34 – 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 the extrudate diameter since the frequency (i.e spacing of cracks) was found to be equal to the extrudate radius for all entry angles and reduction ratios A commercial ÿnite element package (ABAQUS) is used to simulate the extrusion of a paste using an elastic–plastic material model This provides the detailed information about the stresses and strains within the deforming material required for evaluation of the fracture criteria Two classes of fracture criteria are considered: (i) those based on the current stress ÿeld alone, and (ii) those based on the entire stress and deformation histories The ÿrst class is often regarded as being appropriate for describing brittle fracture, and the second class for describing ductile fracture Here, the ÿrst 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 Pugh suggested that cracking resulted from insu cient compressive stress in the deforming zone, while Fiorentino et al argued that the cause lay with tensile residual stresses 1386 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 We examine three ductile fracture criteria: the Generalised Work and Cockcroft and Latham criteria considered by Clift et al [4], and a new one that can be viewed as a combination of the ÿrst two, which we call the Hybrid Criterion In all three cases, the rate of change of damage in a material element is given by a quantity that has dimensions of rate of energy dissipation per unit volume For the Generalised Work Criterion, this quantity is simply the usual expression for the rate of energy dissipation per unit volume in a deforming material, ˙1 + ˙2 + ˙3 , which does not distinguish between energy dissipated under tensile and compressive stresses The other two use some measure of the rate of energy dissipation in the presence of tensile stresses alone Model It is widely recognised that pastes with a wide range of formulations exhibit a yield stress, or at least an apparent yield stress, and thus a plasticity-based model is appropriate The stresses within a deforming paste also can exhibit a dependence on strain rate, which is neglected here However, ceramic pastes of the type studied by Domanti and Bridgwater show little rate dependence The details of the plastic response of typical pastes, i.e the yield criterion and ow rule followed, remain largely unresolved For this work, the von Mises yield criterion and associated ow rule have been adopted, mainly for reasons of convenience However, experiments with various solid metals [14,15] indicate that the von Mises criterion and its associated ow rule describe deformations rather better than the Tresca criterion and its associated ow rule In the absence of alternative information, it is reasonable to assume pastes behave in a similar way If the predominantly plastic nature of paste ow is widely accepted, the response below the yield point is much less clear It is often assumed that when subjected to small stresses the material deforms elastically, with Young’s modulus that is large compared with the yield stress, so that the elastic strains are always small This is analogous to the behaviour of solid metals, ignoring the phenomenon of creep The elastic–plastic approach is followed here, with the elastic response described by the linear Hookean model The combined uniaxial stress–strain relationship therefore has 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 mechanics are required For paste ows, as in metal forming, body forces are usually neglected; typical ow velocities are small enough for inertial e ects to be insigniÿcant, and the yield stress is usually large enough for gravitational e ects to be ignored To illustrate how they are written for an axisymmetric problem, the equations underlying the model are listed in Table Further details can be found in standard texts, e.g [17] The rest of the formulation concerns the boundary conditions at solid surfaces Unlike viscous uids, elastic and=or plastic solids are usually regarded as being capable of slipping at such surfaces In the theory of metal forming, a Coulomb friction boundary condition is thought to be most realistic However, this is sometimes substituted, for computational An alternative formulation for numerical plastic ow calculations exists, where the material is e ectively treated as a highly viscous uid 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 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1387 Table Equations of motion for an axisymmetric problem referred to an Eulerian cylindrical polar coordinate system Basic equations Force equilibrium equations (body forces neglected): @ r @ rz r −  + + =0 @r @z r @ z @ rz rz + + =0 @z @r r Von Mises yield criterion: ( r − Â) +(  − z) (1a) +( z − r) 2 rz +6 2Y (1b) Prandtl–Reuss ow rule, incorporating Hooke’s Law and the von Mises associated ow rule: ( ˙r − E ˙0 = ( ˙ − E ˙z = ( ˙z − E 1+ ˙rz = ˙rz E ˙r = ˙ − ˙z ) + (2 r −  − z) ˙; ˙z − ˙r ) + (2  − z − r) ˙; ˙r − ˙ ) + (2 z − r − Â) ˙; +3 rz ˙: (1c) 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 ˙r = @vr ; @r ◦ where ◦ r ij ˙ = vr ; r ˙z = @vz ; @z ˙rz = @vr @vz + @z @r : (1d) is the Jaumann stress rate with components = ˙r − ◦ rz !;  ◦ = ˙ ; z = ˙z + ◦ rz !; rz = ˙rz + ( r − z )!; (1e) where ! is the rate of rotation of a material element != @vr @vz − @z @r (1f) and ˙ij is the material stress rate ˙ij = @ ij @ ij @ ij + vr + vz : @t @r @z (1g) (ii) ˙ is a non-negative scalar describing the magnitude of the plastic strain rate It is zero where the yield criterion (1b) is satisÿed as a strict inequality (below the yield point), and can be non-zero if (1b) is satisÿed as an equality (at the yield point) 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 di erentiating (1b) ( r − ◦  )( r − ◦ Â) +(  − ◦ z )(  − ˙z ) + ( z − ◦ r )( z − ◦ r) +6 ◦ rz rz = 0: (1h) 1388 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig Example ÿnite 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 the material 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-rich layer is thought to form where slip occurs at a surface, which then lubricates the interior ow 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 We have 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 pastes studied Solution procedure The equations of motion were solved using the ABAQUS ÿnite 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 meshes were constructed from irregular quadrilateral elements with widely varying sizes, small and large elements 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 The smallest elements are required 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 ÿllet radius at the corner The ÿllet radius is a small proportion (typically around 1%) of the barrel radius, and so the solution obtained from the analysis is expected to be close to the solution for a die with perfectly sharp corners Using meshes of the form shown in Fig 3, the solutions were found to be insensitive to the precise details of the mesh construction Young’s modulus was chosen to be two or three orders of magnitude larger than the uniaxial yield stress so that elastic strains were small, and the solution corresponded closely that for the rigid-plastic limit Here the stresses in the deforming material should not be sensitive to the precise values of Young’s modulus and Poisson ratio However, a value for the Poisson ratio must be speciÿed, the value selected, somewhat arbitrarily, being 0.49 This is close to 1=2, and so the elastic component A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1389 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 stresses outside the deforming region could still be sensitive to the Poisson ratio This could be important since we need the residual stress ÿeld 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 ratio 9.77, using Poisson ratios of 0.49 and 0.10, the di erences in the residual stress ÿelds were found to be insigniÿcant Similarly, duplicate analyses for the same die geometry with Young’s moduli 400 times and 1000 times the uniaxial yield stress produced no signiÿcant di erences ABAQUS implements an updated Lagrangian ÿnite element formulation where, as the calculation proceeds, the mesh deforms following the deformation of the material This causes di culty when the deformation is large, as in the problems studied here, since the elements quickly become grossly distorted 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 the current solution is mapped from the old mesh to the new mesh The calculation is then restarted using the mapped solution to provide the initial conditions for the new analysis step The rezoning procedure 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 the procedure can be found in Ref [18] and in more detail in Ref [19] Results: deforming zone stresses Pugh and Gunn [11], as reported by Pugh and Low [20], suggested that the cause of surface fracture was the insu ciency of the ‘total hydrostatic compressive stress existing in regions in which the material is being deformed’ Experiments to test this hypothesis were carried out by Pugh and Gunn, and earlier by Pugh and Green [10] They extruded brittle metals with apparatus that allowed the extrudate to emerge into a pressurised liquid, and determined the uid back pressure required to suppress surface cracking Pugh’s hypothesis is examined, using results from the ÿnite element calculations 4.1 Implementation In the absence of detailed information about the stress ÿeld in the deforming material, Pugh suggested that a rough measure of the hydrostatic compressive stress in the deforming zone was the average stress on the die face, calculated as the extrusion load divided by the area of contact between the billet and die (i.e die face area) This quantity has been determined Finite element analyses allow the stress ÿeld in the deforming material to be investigated in more detail, and hence several 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 1390 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig Deforming zone as deÿned by integration points where there is active yielding These two possibilities are also examined In either case, it is necessary to have a suitable deÿnition of ‘deforming material’, and we have taken it to be the region where the plastic strain rate is non-zero In ABAQUS, a ag variable is used to label element integration points as ‘actively yielding’ if the plastic strain increment is non-zero during a given analysis increment, and this provides a convenient way to identify the deforming zone Fig shows the locations of the actively yielding integration points for a single increment in an analysis for a square entry die with barrel diameter 25 mm and die diameter mm The integration points are particularly closely spaced near the re-entrant corner, where the elements are small to accommodate the high strain rates that occur there, and some integration points in this vicinity have been omitted for clarity A fourth stress quantity that has been examined, namely the extrusion pressure This is a somewhat loose interpretation of ‘average hydrostatic compressive stress in the deforming zone’, but has the advantage of convenience, as it is the most likely stress parameter to be recorded 4.2 E ect of extrusion ratio Square entry dies: Analyses were performed for extrusion through square entry dies with barrel diameter 25 mm and die diameters between and 24 mm, corresponding to extrusion ratios between 156 and 1.09 Fig shows the maximum and average deforming zone pressures, the average die face pressure and the extrusion pressure, all plotted against the extrusion ratio The latter two pressures have been plotted from results given in Ref [18], and are not strictly average pressures, but rather average stresses normal to the die face and ram, respectively 1396 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig 10 Longitudinal stress versus radial coordinate for extrusion through tapered entry dies with various entry angles and an extrusion ratio of 9.77 5.2 E ect of taper angle Results were obtained for tapered entry dies with barrel diameter 25 mm, die diameter mm, ◦ ◦ corresponding to an extrusion ratio of 9.77, and six entry angles between 15 and 90 We have ◦ ◦ already seen that for this extrusion ratio the longitudinal stress proÿles for 45 and 90 entry angle dies are approximately linear, the stress being compressive near the axis and tensile near the surface The longitudinal stress proÿles for these two cases and the four further entry angles are shown in Fig 10 The proÿles are rather similar and, where the variation is greatest near the axis, there is no clear trend in the stress with entry angle It is possible that the larger entry angles result in slightly more compressive stresses near the axis and slightly more tensile stresses near the surface, but it is not possible to be certain These results are in agreement with the experimental results of Osakada et al [21] who determined that the entry angle had no e ect on the surface residual stresses for entry angles of ◦ ◦ ◦ 10 , 30 and 45 Conversely, they contradict the experimental results of Polyakov et al [22] who observed an increase in tensile stress with increasing entry angle However, the latter authors ex◦ ◦ amined the very small extrusion ratio of 1.06 (and small entry angles of and 10 ), and we have seen previously (Fig 7) that the longitudinal stress proÿle changes markedly at small extrusion ratios For all entry angles, the surface tensile stress is about the same, and the change from compressive to tensile stresses occurs at about = 2=3 Therefore, whether we assume that either (i) the depth of fracture corresponds to the magnitude of the surface stress, or (ii) the depth of fracture corresponds to the depth to which tensile stresses occur, we conclude that the depth of fracture for a given extrusion ratio should be relatively independent of entry angle However, for all the pastes they studied, Domanti and Bridgwater found that the depth of fracture increased substantially as the entry ◦ ◦ ◦ angle increased from 15 to 45 ; above 45 the fracture behaviour remained about the same These A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1397 experiments were conducted with 12 mm diameter dies (extrusion ratio 4.34), while the numerical analyses were performed for an extrusion ratio of 9.77 As seen in Fig 7, there are small di erences between the stress proÿles for R = 4:34 and 9.77, but the general behaviour is similar We therefore conclude that the surface residual stresses are unlikely to be the sole cause of the observed surface fracture Also, for the tapered entry dies the surface stress continues to become more tensile down to the smallest value of R examined, although we might anticipate that this trend would be reversed if the extrusion ratio were small enough 5.3 Die exit stress ÿeld Experiments conducted by Domanti [25] using Perspex dies with a rectangular cross-section indicated that crack opening occurred as the extrudate left the die land, and so the stress ÿeld in this region is of interest Fig 11 shows the minor principal stress ÿeld in the extrudate for a square entry die with an extrusion ratio of On the extrudate surface (both within and beyond the die land) and on the axis, the minor principal stress is the radial stress, and it is presumably close to the radial stress at other locations within the extrudate Within the die land the normal stress on the extrudate surface is non-zero, as the wall prevents the extrudate from expanding elastically At the die exit the normal stress is suddenly relieved, as the contours in Fig 11 indicate There is no observable die swell, since Young’s modulus is large compared with the change in normal stress (which is of the order of the yield stress) so that the elastic strains are small In principle, there is a singularity in the stress ÿeld at the die exit, the stress being multi-valued The contours in Fig 11 that intersect the surface near the exit not meet at a point because the elements in the ÿnite element mesh were too large in this region to allow the singularity to be accurately represented In conventional fracture mechanics, singular stress ÿelds are shown to occur at crack tips in elastic materials, and the intensity of the singularity is a key factor in determining whether the crack will grow, resulting in fracture [3] It seems plausible, therefore, that in extrusion the intensity of the singularity at the die exit might in uence whether fracture occurs The singularity will be more intense, if the die land wall is rough, as the surface shear stress, in addition to the normal stress, will suddenly change from non-zero to zero at the die exit Horrobin [19] conducted ÿnite element analyses very similar to those here and found that for rough dies it was di cult to get the numerical algorithms to converge reliably, unless the die land was made long so that no material was extruded This might indicate that under some circumstances it is not possible to obtain a steady solution for extrusion from a rough die, without an event occurring, possibly fracture that cannot be accommodated by the numerical analyses Results: ductile fracture criteria Three ductile fracture criteria, two of which have been previously proposed in the literature as indicators of cracking, have been evaluated These were assessed for an extrusion ratio R = 9:77 and various entry angles The ÿrst examined, termed the Generalised Work Criterion, states that fracture occurs in a material element when the rate of plastic energy dissipation reaches a critical value when integrated with 1398 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig 11 Minor principal stress ÿeld in extrudate for a square entry die with extrusion ratio respect to time, following the element as it travels through the die The integrated plastic energy dissipation GW is the total accumulated plastic work GW = ( ˙1 + ˙2 + ˙3 ) dt; (2) where ; and are the principal stresses and ˙1 , ˙2 and ˙3 are the corresponding principal strain rates In determining this quantity from the results using ABAQUS, the overall principal strain rates A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1399 were used, which include elastic as well as plastic components However, the contribution from the elastic strain rates was considered to be negligible, due to the large ratio of Young’s modulus to yield stress Of the criteria investigated, the Generalised Work Criterion was found by Clift et al [4] to be the only one that accurately predicted the site of fracture initiation for all three metal-forming processes considered: upsetting, extrusion, and strip deformation The second criterion examined is due to Cockcroft and Latham [26], and states that the integral of the major principal stress multiplied by the strain rate magnitude should be the fracture indicator, but that only regions where is tensile should contribute to the integral CL = max( ; 0) ˙21 + ˙22 + ˙23 dt: (3) Thus, the integral only increases signiÿcantly when a material element passes through a region where there is a signiÿcant strain rate in conjunction with a tensile stress Clift et al [4] found this criterion accurately predicted the site of fracture initiation for extrusion, but not always for upsetting or strip deformation Ko et al [5] successfully used it to predict the site of fracture initiation, and the level of subsequent deformation, in the axisymmetric extrusion of an aluminium alloy A possible criticism of the Cockcroft and Latham Criterion is that it does not take into account the direction in which the material is being strained CL will increase even when this direction does not coincide with the direction in which the tensile stress acts We therefore suggest a third criterion that overcomes this di culty It is a combination of the previous two criteria, and is termed the Hybrid Criterion Here, products of the principal stresses and principal strain rates are integrated, as in the Generalised Work Criterion, but each term in the integral is only included if the individual principal stress is tensile H= {max( ; 0) ˙1 + max( ; 0) ˙2 + max( ; 0) ˙3 } dt: (4) 6.1 Implementation Now that integrals with respect to time must be evaluated, application of the fracture criterion is now more complicated The method of implementation was as follows: (i) a mesh containing of about 500 elements initially positioned within the barrel was displaced axially towards the die face, (ii) values of the fracture integrals at each node in the mesh (initially all zero) were updated after every analysis increment, depending on the stress and strain rate components at the node during the increment, (iii) at the end of the ÿrst analysis step, when it was decided that rezoning was necessary, a new mesh was constructed and the values of the fracture integrals at the new nodes obtained by interpolation, (iv) steps (ii) and (iii) were repeated for each subsequent analysis step until the end of the analysis 6.2 Errors Fig 12(a) shows the major principal stress ÿeld near the re-entrant corner for extrusion through ◦ a die with entry angle 30 and extrusion ratio 9.77; the region where the major principal stress is 1400 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 ◦ Fig 12 Stress and strain rate contour plots for extrusion through a tapered entry die with entry angle 30 and extrusion ratio R = 9:77: (a) major principal stress ÿeld near re-entrant corner of die; and (b) equivalent strain rate ÿeld near re-entrant corner of die tensile lies between the white dashed lines The corresponding strain rate ÿeld, shown in Fig 12(b) with the dashed lines from Fig 12(a) reproduced, is dominated by the singularity arising at the corner, where the ow abruptly changes direction By choosing contour levels appropriate for the strain rate values near the corner, as is the case in this ÿgure, detail is lost elsewhere Upstream from the corner lies the plastically deforming region, where the strain rate is signiÿcantly non-zero, but small on this scale, while downstream the strain rate quickly goes to zero in the rigid extrudate A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1401 Fig 13 CL values along the extrudate surface in the die land region Comparing Figs 12(a) and (b), the regions where tensile stresses and large strain rates exist are almost mutually exclusive, but there is a very small region at the downstream boundary of the deforming zone where these coincide Material elements receive contributions to the CL and H integrals as they pass through this region, and the element that receives the biggest overall contribution determines, in principle, whether fracture occurs or not However, it is di cult to determine accurately the ÿnal values of the integrals because the region is so small, and the strain rate ÿeld is so rapidly varying that it cannot be represented very accurately in the model This di culty is illustrated in Fig 13, which shows CL values sampled at positions on the extrudate surface within the die land, at an instant during the analysis The curve deÿned by the points is piecewise linear, with the junctions between the linear segments corresponding to element boundaries; since we are sampling along the extrudate surface, the junctions correspond to nodes, but this will not generally be the case if we sample along a parallel line within the extrudate Close to the re-entrant corner, on the left-hand side of the diagram, the linear segments are very short due to the very small elements used in this region The initial CL value at the re-entrant corner is very small, indicating that the contribution to the integral from the deformation upstream is negligible, as we would anticipate from Fig 12(a) The integral increases rapidly just after the die corner, where a tensile principal stress and a large strain rate are simultaneously encountered The subsequent values exhibit large variations between nodes, although maintaining the same order of magnitude In principle, these values should be identical, since all correspond to material elements that have followed the same path through the die (i.e along the wall) and have therefore been subjected to the same deformation history Having noted that it is di cult to evaluate the CL integral accurately, we can at least take an average of the values along the die land and estimate the error in this average When taking this average it is important not to give too great a weighting to the longer linear segments in Fig 13, which comprise many points, but are in fact deÿned only by the two end values The results and the associated error bars are therefore based just on the values at the junctions of the linear segments This was considered to be a representative average, although it leads to a larger apparent error for 1402 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig 14 GW values across the extrudate for di erent entry angles the integral at the very surface of the extrudate, since the sampling line along the surface passes through fewer element boundaries than a parallel line slightly within the material 6.3 Results 6.3.1 Generalised Work Criterion Fig 14 shows how the longitudinally averaged GW integral varies across the extrudate for di erent entry angles Error bars are shown where the error bounds are larger than the symbols used to represent the data points In general, the observed variation in GW in the longitudinal direction is less than for CL and H , because GW depends on the deformation that material elements experience while traversing the entire deforming zone A much larger region therefore contributes to GW (note that GW is about an order of magnitude greater than CL or H ), and throughout most of this region the strain rate ÿeld is represented more accurately than it is near the re-entrant corner ◦ ◦ When the entry angle is small (15 and 30 ), GW shows little variation across the extrudate, ◦ although there is a slight increase with increasing radial coordinate in the 15 case and a slight ◦ decrease in the 30 case For the other four entry angles there is a pronounced increase in GW with radial coordinate, although the average slopes of the curves vary somewhat erratically The ◦ ◦ most surprising feature of Fig 14 is that the curves for the largest entry angles (75 and 90 ) are ◦ clearly lower than the other four curves, with the 90 values being lowest of all This cannot be correct, because at steady state the integral of GW over the extrudate cross-section should be a simple multiple of the extrusion pressure, and the extrusion pressure increases monotonically with entry angle Speciÿcally, the relationship is P=2 GW d ; (5) A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1403 Fig 15 Comparison between the integrated GW values and the extrusion pressure where P is the extrusion pressure and is the dimensionless radial coordinate The right-hand side of this relationship has been evaluated for the di erent entry angles investigated, and the results (indicated by crosses) are compared with the extrusion pressure in Fig 15 For the four smaller entry angles the integrated GW values slightly overestimate the extrusion ◦ ◦ pressure, but the di erence is not too large For the 75 and 90 entry angles the integrated GW values are very much less than the extrusion pressure A likely cause is that in the steady state the ÿnite element analyses were not continued for long enough for a true result to have been established When the entry angle is large the steady-state deforming zone lies mostly upstream from the die face, as indicated schematically in Fig 16(a) At the start of the analysis, the mesh representing the material initially located in the barrel region moves rigidly towards the die face until it touches the salient corner of the die At this point deformation begins at the salient corner, and the steady-state deforming zone is quickly established, very soon after material ÿrst ows past the re-entrant corner Some material elements are in the middle of the eventual deforming zone before experiencing any deformation The plastic work these elements accumulate by the time they reach the extrudate (i.e the GW value) is therefore less than for elements traversing the entire deforming zone after it has been established The situation is di erent when the entry angle is small, because the steady-state deforming zone does not extend far, if at all, upstream from the die face, as shown in Fig 16(b) Elements cannot, therefore, travel into the steady-state deforming zone at the start of the analysis without experiencing some deformation We cannot say that the deformation that the ÿrst material undergoes is necessarily similar to that experienced by material crossing the entire deforming zone when steady state has been reached This might account for the discrepancies between the integrated GW values and P for the smaller entry angle dies, and for the somewhat erratic variation of the average slopes of the curves in Fig 15 However, the accumulated plastic work in the ÿrst part of the extrudate is unlikely to be substantially less than at steady state, as is inevitable if the entry angle is large 1404 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig 16 Schematic deforming zones for large and small entry angle dies: (a) large entry angles; and (b) small entry angles The consequence is that we must mostly disregard the results for the larger entry angle dies as far as GW is concerned, although it is quite likely that the increase in GW with radial coordinate is correct The results for CL and H from the same analyses are probably more trustworthy, as these quantities only depend on the deformation taking place close to the downstream boundary of the deforming zone 6.3.2 Cockcroft and Latham Criterion and Hybrid Criterion Fig 17 shows the longitudinally averaged CL values across the extrudate for the di erent entry angles For clarity, the error bars are shown only on one side of each data point, the error in the ◦ ◦ opposite direction being equal In this case the results for the 75 and 90 dies seem more consistent with those for the smaller entry angles, although we cannot make any statements about the area under the curve as we could for GW All the curves have a minimum at some radial coordinate ◦ ◦ between and 1, and local maximum values at either end For the 15 and 30 dies the maximum on the axis is greater than the maximum at the surface; the axial maximum could be connected with ◦ the central bursting phenomenon observed in metal extrusion For the 45 die the CL value on the axis is only slightly greater than the minimum at = 0:3 This might be an error, as the other ÿve curves are much steeper near the axis, although there is a decrease in the axial value between the ◦ ◦ ◦ 15 and 30 dies For entry angles of 45 and above, the largest CL value occurs at the surface, and this, in principle, is related to surface fracture The CL values increase with entry angle at all radial ◦ ◦ ◦ ◦ coordinates for entry angles between 45 and 75 There is a slight decrease between 75 and 90 , but for these larger entry angles the variation in CL along the extrudate is very large, as indicated by the large error bars, and so the drop cannot be considered signiÿcant A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1405 Fig 17 CL values across the extrudate for di erent entry angles Fig 18 CL to H ratio across the extrudate for di erent entry angles The results for the Hybrid Criterion are very similar to those shown in Fig 17 for the Cockcroft and Latham Criterion, although all the values are slightly lower It is more revealing to plot the ratio √ of CL to H as shown in Fig 18 The data then show that for all entry angles CL=H is about at the extrudate surface, and decreases slightly towards the axis, where it is about 3=2 It can be shown (see the appendix) that CL=H must lie between these limits, if we assume that wherever CL and H are increasing: (i) the elastic strain rates are, on average, small compared with the plastic strain rates, 1406 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Fig 19 GW , CL and H values at the extrudate surface as a function of entry angle (ii) there is one and only one tensile principal stress, and (iii) the intermediate principal stress is not closer to the major principal stress than to the minor principal stress The lower limit corresponds to a uniaxial deformation, and the upper limit corresponds to a pure shear or plane strain deformation Therefore, the results in Fig 18 indicate that, where deformation occurs with a tensile principal stress, the deformation undergone by material travelling along the axis is approximately uniaxial, and that undergone by material travelling along the wall is approximately pure shear These observations are to be expected Firstly, from symmetry considerations, on the axis r must equal  and rz must be zero and therefore the stress state and deformation are uniaxial Secondly, if we visualise approaching the re-entrant corner (in reality a circular edge since the die is axisymmetric) ever more closely, its curvature becomes less apparent and therefore close to the singularity the edge appears approximately straight and the local deformation is approximately plane strain These restrictions on the deformations at the axis and the corner may permit a wholly or partly analytical approach to calculating CL and H , which would be valuable given the di culties encountered when trying to accurately determine these quantities based on the results from a numerical analysis alone Also, the constant ratios between CL and H at the axis and at the surface indicate that if we are only concerned with surface fracture, or we are only concerned with defects that might arise at the axis, the predictions from the two criteria will be identical However, if we wish to treat defects both at the surface and at the axis with a single criterion, their predictions will be slightly di erent Fig 19 shows the average surface values for GW , CL and H plotted against entry angle; the ◦ ◦ GW values for the 75 and 90 dies have been omitted as these results have been shown to be unreliable All the curves increase within the range of entry angles for which data is available CL ◦ ◦ and H show a slight decrease between 75 and 90 , but as already indicated this decrease is not really signiÿcant given the size of the errors The fact that there is not a signiÿcant di erence between ◦ ◦ ◦ the values for 60 , 75 and 90 is in good agreement with the experimental results of Domanti A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1407 ◦ and Bridgwater, who found that for entry angles above 45 di erences in fracture were di cult to discern The ductile fracture criteria therefore show promise in terms of correctly predicting the e ect of entry angle on fracture It remains to be seen whether they can also correctly predict the e ect of extrusion ratio on fracture However, in the case of the Generalised Work Criterion we can anticipate that this will not be the case We have already shown that the average value of GW across the extrudate is a simple multiple of the extrusion pressure, which increases monotonically with increasing extrusion ratio However, the level of fracture decreases with increasing extrusion ratio, at least for larger extrusion ratios To reconcile this with the behaviour of the average value of GW , the accumulated work done would have to become highly concentrated at the extrudate axis as the extrusion ratio increases, in order for the surface value of GW to fall This seems unlikely On the other hand, we have already seen how the magnitude of the tensile stress on the extrudate surface decreases with increasing extrusion ratio, supporting the hypothesis of Fiorentino et al [13] It would not be surprising if the Cockcroft and Latham Criterion and the Hybrid Criterion, which are closely linked to this tensile stress, also correctly predict the e ect of extrusion ratio on fracture Conclusions The conclusions from this work can be summarised as follows: (i) The stress ÿeld based fracture criteria, i.e those due to Fiorentino et al [13] and Pugh [12], give rise to similar predictions, making it di cult to distinguish between them in practice (or more optimistically indicating that either could potentially provide a satisfactory fracture indication) (ii) These criteria predict an increase in the level of fracture with increasing extrusion ratio, for extrusion ratios up to about two, and a decrease thereafter The behaviour for large extrusion ratios is consistent with the experimental results of Domanti and Bridgwater Small extrusion ratios have yet to be investigated experimentally (iii) The hypothesis of Fiorentino et al based on the residual stress ÿeld, does not predict any signiÿcant change in the likelihood of fracture with entry angle Pugh’s hypothesis, based on the deforming zone stress ÿeld, predicts a slight decrease in the level of fracture with entry angle Both predictions contradict the experimental results of Domanti and Bridgwater, who found that the level of fracture increases with entry angle (iv) The integrals associated with the two ductile fracture criteria that depend only on the regions where tensile stresses exist (i.e the Cockcroft and Latham Criterion, and Hybrid Criterion) are di cult to evaluate accurately from the numerical results This arises since the regions that contribute to the integrals are very small, and the strain rates there are large and rapidly varying and are therefore hard to represent within a discretised model (v) All the ductile fracture criteria show promise with regard in predicting the increase in the level of fracture with entry angle, although accurate results for the Generalised Work integral for large entry angles were not obtained The e ect of extrusion ratio has yet to be investigated for the ductile fracture criteria, although it seems likely that the Cockcroft and Latham Criterion and the Hybrid Criterion will again provide qualitatively correct predictions while the Generalised Work Criterion will not 1408 A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 Acknowledgements We are indebted to the EPSRC, the British Council and the Commonwealth Scholarship Commission for ÿnancial support Appendix A The relationship between the Cockcroft and Latham Criterion and Hybrid Criterion We assume that where CL and H are increasing (i) the elastic strain rates can be neglected, (ii) there is one and only one tensile principal stress, i.e ¿0¿ ¿ (A.1) and (iii) there is one and only one tensile principal deviatoric stress, i.e s1 ¿ ¿ s2 ¿ s3 (A.2) which implies that is not closer to than it is to This is certainly true on the axis and at the re-entrant corner, where equals and (1=2)( + ), respectively From the ÿrst assumption, the principal strain rates are related to the principal stresses solely through the plastic ow rule, which, in terms of the deviatoric principal stresses, is ˙1 = 3s1 ˙; ˙2 = 3s2 ˙; ˙3 = 3s3 ˙: (A.3) Therefore, from assumption (iii) we have ˙1 ¿ ¿ ˙2 ¿ ˙3 : (A.4) Furthermore, since the principal strain rates must sum to zero in order to satisfy incompressibility implicit in Eq (A.3) − ˙1 =2 ˙2 0: (A.5) Now, from the deÿnitions of CL and H , and using assumption (ii), dCL = dt dH = dt ˙1 : ˙21 + ˙22 + ˙23 (A.6) and (A.7) A.T.J Domanti et al / International Journal of Mechanical Sciences 44 (2002) 1381 – 1410 1409 Fig 20 ˙21 + ˙1 ˙2 + ˙22 versus ˙2 Dividing (A.6) and (A.7), and employing the incompressibility constraint again we have dCL = dH 2( ˙21 + ˙1 ˙2 + ˙22 ) : ˙1 (A.8) Fig 20 shows the quadratic expression ˙21 + ˙1 ˙2 + ˙22 plotted against ˙2 Within the feasible region deÿned by inequalities (A.5), ˙21 + ˙1 ˙2 + ˙22 lies within the range ˙21 =4 ˙21 + ˙1 ˙2 + ˙22 ˙21 (A.9) and therefore dCL=dH lies within the range dCL √ 3=2 (A.10) dH or on integration CL √ 2: 3=2 (A.11) H The maximum occurs when ˙2 = (corresponding to a uniaxial deformation) and the minimum occurs when ˙2 = − ˙1 =2 (corresponding to a 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