The effect of deformations passes on the extrusion pressure in axi-symmetric equal channel angular extrusion A.R. Eivani a , S. Ahmadi b , E. Emadoddin c , S. Valipour d , A. Karimi Taheri a, * a Department of Materials Science and Engineering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9466, Tehran, Iran b Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA c Department of Materials, Faculty of Engineering, Semnan University, Molavi Boulevard, Semnan, Iran d Department of Mathematics, School of Science, Razi University, Kermanshah, Iran article info Article history: Received 6 May 2008 Received in revised form 17 July 2008 Accepted 23 July 2008 Available online 9 September 2008 PACS: 45. 45.20.-d 45.20.dg Keywords: Axi-symmetric ECAE Deformation pass Upper bound Outer curved corner Friction coefficient abstract The most applicable configuration of the equal channel angular extrusion (ECAE) dies is the axi-symmetric one. However, most of the previous analytical solutions are focused on the plane strain conditions. In this research, an upper bound model is used to investigate the deformation of the material during axi-symmet- ric ECAE. The analysis considers the effect of die angle, friction between the sample and the die walls, and the angle of the outer curved corner of the die, on the extrusion pressure. It is found that increasing the die angle and outer curved corner angle and decreasing the friction coefficient results in decreasing extrusion pressure. The proposed model is verified using two dies of the same die angle and different outer curved corner angles. The applicability of the solution in the ECAE process with more than one pass is investigated and the difference between the theoretical and experimental results are discussed. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Equal channel angular extrusion (ECAE) is a processing method in which the material is subjected to an intense plastic straining through simple shear without any corresponding change in the cross-sectional dimension of the sample [1]. During the ECAE pro- cess, the grain refinement occurs together with significant strain hardening resulting in remarkable enhancement of strength in many engineering materials [2–6]. This process could be performed in two different designs; plane strain and axi-symmetric. However, referring to the literature [7–13] it will be appreciated that in con- trast to numerous publications on deformation mode, extrusion force, and strain of plain strain ECAE, few papers has been published on the mechanics of axi-symmetric ECAE especially using the upper bound method. For example Lee [7] has analyzed the stresses and strains in channel angular deformation (CAD), in which two chan- nels are not equal in cross-section. He has considered ECAE as a spe- cial form of CAD in his upper bound analysis. Alkorta and Sevillano [8] have analyzed the pressure needed for non-friction ECAE of per- fectly plastic and strain hardening materials using an upper bound and a FEM solution. They have compared the results achieved from these solutions. Luis Perez [9] has analyzed a configuration of ECAE dies called equal fillet radii angular extrusion (EFRAE) being slightly different from the general ECAE dies. Altan et al. [10] have analyzed the deformation of the material in a 90° ECAE die using the upper bound theorem. Their model includes the effect of friction between the sample and the die walls, the radius of inner corner of the die, and the dead metal zone on the deformation pattern during ECAE. More- over, the same authors [11] have performed a comprehensive study on the ECAE process using upper bound method considering the ef- fects of die geometry and friction coefficient on the total strain and extrusion pressure. A deep study has also been performed by the same authors on the total strain using a new method dividing the outer curved corner of the die to infinite number of sub-dies [12], based on shear and principal strains [13] and considering the forma- tion of dead metal zone [14]. However, all of the mentioned studies are in the case of plane strain or rectangular cross-section while the axi-symmetric one is more applicable from the practical aspects for producing nano-structured materials [15–17]. Therefore, further investigations are of interest to assess the effect of process parame- ters on the mechanics of axi-symmetric ECAE. Moreover, there has 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.07.025 * Corresponding author. Address: Department of Materials Science and Engi- neering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9466, Tehran, Iran. Tel.: +0031 62 823 03 15; fax: +0031 15 278 67 30. E-mail address: [email protected](A. Karimi Taheri). Computational Materials Science 44 (2009) 1116–1125 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The effect of deformations passes on the extrusion pressure in axi-symmetricequal channel angular extrusion
A.R. Eivani a, S. Ahmadi b, E. Emadoddin c, S. Valipour d, A. Karimi Taheri a,*
a Department of Materials Science and Engineering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9466, Tehran, Iranb Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USAc Department of Materials, Faculty of Engineering, Semnan University, Molavi Boulevard, Semnan, Irand Department of Mathematics, School of Science, Razi University, Kermanshah, Iran
a r t i c l e i n f o
Article history:Received 6 May 2008Received in revised form 17 July 2008Accepted 23 July 2008Available online 9 September 2008
The most applicable configuration of the equal channel angular extrusion (ECAE) dies is the axi-symmetricone. However, most of the previous analytical solutions are focused on the plane strain conditions. In thisresearch, an upper bound model is used to investigate the deformation of the material during axi-symmet-ric ECAE. The analysis considers the effect of die angle, friction between the sample and the die walls, andthe angle of the outer curved corner of the die, on the extrusion pressure. It is found that increasing the dieangle and outer curved corner angle and decreasing the friction coefficient results in decreasing extrusionpressure. The proposed model is verified using two dies of the same die angle and different outer curvedcorner angles. The applicability of the solution in the ECAE process with more than one pass is investigatedand the difference between the theoretical and experimental results are discussed.
� 2008 Elsevier B.V. All rights reserved.
1. Introduction [8] have analyzed the pressure needed for non-friction ECAE of per-
Equal channel angular extrusion (ECAE) is a processing method inwhich the material is subjected to an intense plastic strainingthrough simple shear without any corresponding change in thecross-sectional dimension of the sample [1]. During the ECAE pro-cess, the grain refinement occurs together with significant strainhardening resulting in remarkable enhancement of strength inmany engineering materials [2–6]. This process could be performedin two different designs; plane strain and axi-symmetric. However,referring to the literature [7–13] it will be appreciated that in con-trast to numerous publications on deformation mode, extrusionforce, and strain of plain strain ECAE, few papers has been publishedon the mechanics of axi-symmetric ECAE especially using the upperbound method. For example Lee [7] has analyzed the stresses andstrains in channel angular deformation (CAD), in which two chan-nels are not equal in cross-section. He has considered ECAE as a spe-cial form of CAD in his upper bound analysis. Alkorta and Sevillano
ll rights reserved.
Materials Science and Engi-enue, P.O. Box 11365-9466,
278 67 30.eri).
fectly plastic and strain hardening materials using an upper boundand a FEM solution. They have compared the results achieved fromthese solutions. Luis Perez [9] has analyzed a configuration of ECAEdies called equal fillet radii angular extrusion (EFRAE) being slightlydifferent from the general ECAE dies. Altan et al. [10] have analyzedthe deformation of the material in a 90� ECAE die using the upperbound theorem. Their model includes the effect of friction betweenthe sample and the die walls, the radius of inner corner of the die, andthe dead metal zone on the deformation pattern during ECAE. More-over, the same authors [11] have performed a comprehensive studyon the ECAE process using upper bound method considering the ef-fects of die geometry and friction coefficient on the total strain andextrusion pressure. A deep study has also been performed by thesame authors on the total strain using a new method dividing theouter curved corner of the die to infinite number of sub-dies [12],based on shear and principal strains [13] and considering the forma-tion of dead metal zone [14]. However, all of the mentioned studiesare in the case of plane strain or rectangular cross-section while theaxi-symmetric one is more applicable from the practical aspects forproducing nano-structured materials [15–17]. Therefore, furtherinvestigations are of interest to assess the effect of process parame-ters on the mechanics of axi-symmetric ECAE. Moreover, there has
been no study considering the second pass of ECAE process regard-ing to extrusion force. However, as ECAE is a sever plastic deforma-tion method which is generally used in more than one pass, theextension and verifying of the solutions to the second and the thirdpasses of ECAE is very important.
In the present study, an upper bound solution is presented toconsider the effects of die geometry such as die angle, outer curvedcorner angle, sample dimensions and also friction coefficient onthe extrusion pressure in axi-symmetric ECAE. Geometrical differ-ences between plane strain and axi-symmetric dies result in somemathematical complications in the solution and some differencesin results which are challenged and solved in this study. To verifythe reliability of the model, extrusion force is measured in two diesof the same angle but of different outer curved corner angles andthe results are compared with the results of the theoretical model.Moreover, the application of the model in the second and thirdpass of ECAE is compared with experimental results.
2. Analysis
In this research, in order to develop an upper bound solution, asimple deformation model considering the effect of outer curvedcorner angle, introduced by Alkorta and Sivillano [8] and then usedby Altan et al. [10], and Eivani and Karimi Taheri [11], in squarecross-section ECAE dies is extended, and utilized in the case ofaxi-symmetric ECAE dies. In this deformation model, the ECAEdie is divided into three regions as shown in Fig. 1. In Region I,
Fig. 1. The deformation model used
the material moves rigidly downward with a velocity of V0. RegionII, called the ‘‘deformation zone”, is where the material undergoescontinuous plastic deformation.
It is assumed that in this region the material moves along theconcentric circles with center at O. Considering that in any partof the die, the cross-section should not be smaller than the initialcross-section of the die, one can conclude that / + w 6 p, where/ is the die angle and w is the outer curved corner angle of thedie (deformation zone). In Region III the material moves outwardthe die without any further deformation. Region II is separatedfrom Region I by the entry surface of the deformation zone, Ci,and from Region III by the exit surface of the deformation zone,Co. The origin of the rectangular coordinate system is taken aspoint O shown in Fig. 1. The x-axis is taken positive to the left,and the y-axis is positive down. Cylindrical coordinates (r,h), de-fined with the origin at O, are also utilized when it is needed.The angle between the entry surface and the velocity in Region I,and between the exit surface and the velocity in Region III are as-sumed to be the same and presented by u.
The material in Region II, the deformation zone, is moving witha constant velocity of V0cosu. Using the cylindrical coordinates, thevelocity field in this region can be expressed as
tr ¼ 0; th ¼ V0 cos u; tz ¼ 0 ð1Þ
where ti(i = r,h,z) is the velocity field components in the deforma-tion zone (Region III) and z is the axis of the cylindrical coordinatesystem.
for ECAE process in this study.
Fig. 2. Velocity discontinuities on the (a) entry and (b) exit surfaces of the deformation zone.
At the entry and exit surfaces the velocity undergoes a suddenchange. The kinematical relations based on the principle of conser-vation of mass present the velocity discontinuity on these surfaces as
jtij ¼ jtoj ¼ V0 sinu ð2Þ
where jtij is the velocity discontinuity on the entry surface and jtojis the velocity discontinuity on the exit surface of the deformationzone.
The values of velocity discontinuities on the entry (Ci) and exit(Co) surfaces are shown in Fig. 2a and b, respectively.
It should be mentioned that in this model the non-homogeneityof deformation has been ignored to simplify the analytical solution.However, it is clear that it decreases the precision of the modelwhich the reason will be discussed later.
The friction force along all of the surfaces where the material isin contact with the die is modeled by ms0 where m is the frictioncoefficient and s0 is shear yield stress. The components of thestrain rates in the deformation zone are obtained from the velocityfield [18] given in Eq. (1) as
e� ¼ �1
2V0 cos h
rð3Þ
In this equation e�
rh is the none-zero strain rate component inthe deformation zone (Region III). The other strain rate compo-nents are equal to zero. Hence, one can easily verify that the veloc-ity field satisfies the incompressibility conditions given by
e�
rr þ e�hh þ e
�zz ¼ 0 ð4Þ
where in this equation e�
iiði ¼ r; h; zÞ are the principal strain ratecomponents.
2.1. Extrusion pressure
The power supplied to deform the material in the ECAE process,J, is given by
J ¼ FV0 ¼ Ppa2V0 ð5Þ
where F and P are the force and pressure applied by the ram, respec-tively and a is the radius of the extrusion channel. The power dissi-pated during ECAE can be expressed as
W�
tot ¼W�
d þW�
i þW�
o þW�
m þW�
WðACÞ þW�
WðBDÞ þW�
lðiÞ
þW�
lðoÞ ð6Þ
In this equation W�
tot;W�
d;W�
i;W�
o;W�
m;W�
WðACÞ;W�
WðBDÞ;W�
lðiÞ andW�
lðoÞ are the total power dissipated during ECAE, power dissipatedin the deformation zone, power dissipated on the entry surface,
power dissipated on the exit surface, power dissipated on the con-tact surface between the material and the die wall in the deforma-tion zone, power dissipated on the die wall AC, power dissipatedon the die wall BD, power dissipated on the surface between thematerial and the die wall before point A in Fig. 1, and power dissi-pated on the surface between the material and the die wall afterpoint B in Fig. 1, respectively. The detailed expressions of the indi-vidual terms in Eq. (6) are presented in Appendix A. Substitutingthe terms of Eq. (6) with the appropriate expression from theappendix, the total power dissipated is
W�
tot ¼ s0V0 Idw sin/þ w
2
� �þ 4a2mwcosec
/þ w2
� �EðfÞ
�
þ2pa2ð1þ 2mÞ cot/þ w
2
� �þ 2pamðli þ loÞ
�ð7Þ
where li and lo are instant length of the specimen in the entry channeland exit channel before point A and after point B in Fig. 1, and Id is
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � x sin /þw
2
� �� a
� �2q
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2 � x sin /þw
2
� �� a
� �2q
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � x sin /þw
2
� �� a
� �2q
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2 � x sin /þw
2
� �� a
� �2q
dx
To investigate the effect of the die angles, / and w, dependenceof the pressure on the material properties is eliminated with con-sidering normalized pressure parameter as P
s0, and a new parameter
which does not consider the effect of sample dimensions is intro-duced as P0
It should be noted that in this parameter the effect of the con-tact length of the sample and the die entry and exit channels isnot considered. In the other words, as in different dies the lengthof the entry and exit channels differs, P0 has been calculated whichis not a function of the entry and exit lengths.
3. Experimental procedure
A commercial grade aluminum alloy (AA 6070) was used in theECAE tests carried out in this work. The composition of the alloy isshown in Table 1. Rods of the above material were received in hotextruded condition. To obtain uniform metallurgical properties inall samples, they were annealed at 550 �C for two hours and cooledin the furnace. Then, the Cook and Lark compression testing [19]was conducted to determine the stress–strain curve of the alloy.Specimens of three different height to diameter ratios (h/d = 1.5,1.0 and 0.5) were tested at room temperature using a hydraulicpress with a strain rate of approximately 0.05 s�1. Three specimensfor each h/d ratio were tested using an industrial oil as thelubricant.
The dies used for ECAE consisted of two channels with circularcross-sections of 14 mm diameter, intersecting at a die angle of 90�and two outer corner angles of zero and 30�. A split die was used toavoid stress concentration at the corners and to facilitate easy re-moval of the ECA extruded specimen (Fig. 3).
Specimens for extrusion were machined to 13.95 mm diameterand 80 mm length to a visibly good finish. As it will be explainedthe initial configuration of the specimen in the die is effective onthe extrusion force and therefore it is shown in Fig. 4. Specimenswere ECA extruded at room temperature in one pass. All extrusionswere conducted using a hydraulic press of 100 t capacity. As thepress was operated manually, it was not possible to apply a con-stant strain rate during the test. The actual strain rate varied be-tween 0.3 and 0.5 s�1. The force-stroke diagram was recordedduring the extrusion.
Fig. 3. The die used in this study with / = 90 and w = 30�.
Table 1The chemical composition of the Al alloy used in this study
Element Al Si Mg Mn Cu Fe Other elements
wt.% Base 1.4 0.9 0.8 0.3 0.4 0.2
During the ECAE tests, it was observed that deforming thematerial without back pressure is not possible. In other words,the material breaks as a result of large cracks occurring on the sur-face of the sample. Performing the second and third passes of ECAEwith the aim of receiving a sound product was possible only whena back pressure was used. Therefore, to have the same condition forall of the tests, a back pressure of 50 kN was also used in the firstpass and then was deducted from the final extrusion force.
4. Results and discussion
4.1. Work hardening constants and friction coefficient determinations
From the force-stroke data recorded during compression testingof the initial material, the true stress and true strain values werecalculated and the flow curve was determined which is shown inFig. 5. The strain hardening exponent, n, and the strength coeffi-cient, K, were determined by fitting the data to the equationr = Ken which are presented on the graph. Decreasing the n valuefrom 0.26 to 0.04 by increasing the number of deformation passesmeans that the rate of strain hardening decreases as the number ofpasses increases. It is believed that this increase is due to the sat-
Fig. 5. Stress–strain curves of the samples without deformation, after one pass andafter two passes of ECAE.
uration of the structure with dislocations which decreases the pos-sible amount of work hardening [20]. It is also clear that with con-ducting ECAE, the strength coefficient increases which is also dueto the work hardening occurring in the structure during deforma-tion [20,21]. With respect to the extruded material and the diematerial and using the pin-on-disk method [22], the friction coef-ficient, m, was estimated as 0.18.
4.2. Effects of die geometry on the extrusion pressure (theoreticalresults)
According to Eq. (9) four parameters consisting of die angle,outer curved corner angle, friction coefficient and die geometry af-fect the extrusion pressure. Fig. 6a–c show the effect of the outercurved corner angle on the normalized extrusion pressure for threedifferent friction coefficients, m, of 0, 0.5 and 1 considered for dif-ferent die angles, /, of 90, 105, 120 and 135�.
Referring to these figures, it is found that the normalized extru-sion pressure decreases with increasing the die angle as well aswith the outer curved corner angle of the die. The effect of die an-gle is more dominant than the effect of the other parameters. Asmall change in die angle has a large effect on the extrusion pres-sure. According to these figures, friction coefficient affects notice-ably the extrusion pressure. The pressure increases with
Fig. 6. The effect of outer curved corner angle on the normalized extrusio
increasing the friction coefficient and this effect is reduced withincreasing outer curved corner angle because the contact surfaceof the die with the material increases with decreasing outer curvedcorner angle. Therefore, using a die with a large outer curved cor-ner is recommended in the case of large friction coefficient.
4.3. Force-punch displacement curve during ECAE
Typical ECAE force versus punch displacement curves areshown in Fig. 7a and b for two different die configurations, onewithout outer curved corner and the other with a curved cornerangle of 30�. Referring to this figure, the extrusion pressure in-creases gradually with increasing the ram movement (O to A inthe figure). This increase is due to the initial easy movement ofthe specimen in the die, Fig. 4. Then, the rate of increase in pressureintensifies (A to B) until the pressure reaches a local maximum (B),after which it reaches point C. The reason to this behavior is therestriction of the exit channel of the die that leads to the forgingof the billet. Obviously, at the start of the material movement a sta-tic frictional condition is applied, but as the deformation proceeds,a dynamic frictional condition is prevailed resulting in a slightreduction in force (B to C). From point C the extrusion pressure in-creases again (C to D), but with a slow rate and it continues to theend of the process. The larger flow stress of the work-hardened
n pressure for various die angles, (a) m = 0, (b) m = 0.5 and (c) m = 1.
Fig. 7. Typical axi-symmetric ECAE force-stroke diagrams at two different outer curved corners (a) w = 0 and (b) w = 30�.
material in the exit channel against the un-deformed material inthe entry channel imposes the increasing of the extrusion pressurewith the development of the ECAE process (D to E).
It should be noted that in the die with larger outer curved cor-ner angle, the maximum value (point B) is not easily observableand differentiable from the rest of the curve being due to thecurved corner of the die. A die with a curved corner induces aweaker constraint against the material movement compared witha die of sharp corner. In other words, the material moves easilythrough the curved corner and results in a smaller extrusionpressure.
The effect of the second and the third pass of ECAE process onthe force vs punch displacement data are also shown in Fig. 7. Tocompare the amount of increase in the force due to the secondand the third pass of ECAE, the extrusion force pertaining to pointE and the percentile increase from point C to E was calculated andpresented in Table 2.
As shown in Fig. 5, with increasing the number of passes theflow stress of the material increases. For example in die 1, theextrusion force increases from 74.3 to 106.8 kN from the first passof deformation to the third pass. As shown in Fig. 5, with increasingthe number of passes, the strength of the material increases. Withincreasing the flow stress, further deformation of material requireslarger force which results in an increase in the extrusion pressure.
From Table 2, it is observed that in the first pass of ECAE theextrusion force from point C to E increases about 25.9% while inthe third pass it increases close to 2.9%. Previously it was discussedthat the increase of extrusion force from point C to E is due to thelarger flow stress of the work-hardened material in the exit chan-nel against the un-deformed material in the entry channel whichimposes the increasing of the extrusion force with the develop-
Table 2The magnitudes of extrusion force and the amount of increase in the extrusion forceduring deformation
Dienumber
Passnumber
Extrusion force(kN)
Difference from point C to E (%)
Die 1 First pass 74.3 25.9Second pass 102.2 10.5Third pass 106.8 2.9
Die 2 First pass 65.9 29.1Second pass 91.7 11.2Third pass 99.5 3.2
ment of the ECAE process. However, as it was discussed, afterone or two passes of ECAE (shown in Fig. 5), the work hardeningof the material decreases. Therefore, the flow stress of the materialin the entry and the exit channels does not increase significantlyafter passing the deformation zone in the ECAE process. Conse-quently, the frictional force in the entry and exit channel is not dif-fering significantly and the extrusion force will be almost constant.
Another difference is also observed in the second and the thirdpass of ECAE when compared with the first pass which is the dis-appearing of the peak in the force-stroke plot pertaining to pointB. As it was mentioned, this force drop is because of sticking fric-tion. Optical observations show that after the first pass of deforma-tion the surface quality of material increases which decreases thefriction coefficient of the material. However, producing a samplewith the same surface quality to verify the friction coefficient isnot possible. Therefore, just an optical observation was used.
Moreover, it was also discussed that increasing the deformationforce before the material enters the deformation zone is due toupsetting deformation as a result of discrepancy of the channel sizeand the initial material. However, after the first pass of ECAE, thematerial is quite fitted in the entry channel. Therefore, an increasein the extrusion force during deformation as a result of upsetting isnot expected.
4.4. Comparison between the theoretical and experimental results
The maximum extrusion force pertaining to point E in Fig. 7 waschosen as the experimental force for deformation. This extrusionforce was compared with the theoretical value calculated fromEq. (9). Using this equation, the extrusion forces relating to two dif-ferent die conditions in this study which their specifications areshown in Table 3 were evaluated. According to the results of thestress–strain curve shown in Fig. 5 and the amount of strain in eachpass of deformation, the average shear flow stress of the material iscalculated and together with the final extrusion force from Eq. (9)
Table 3The specifications of the dies used in this study and resulting effective parameters onthe extrusion force
is presented in Table 4. The detailed expressions of the individualterms in Tables 3 and 4 are presented in Appendix B.
Fig. 8 illustrates a comparison between the theoretical andexperimental results of extrusion pressure at w = 0 and w = 30�after one, two and three passes of deformation. As can be seen fromFig. 8, the experimental results show a good agreement with theresults of the upper bound solution. From Fig. 8 it is clear thatthe theoretical value is larger than the experimental value. Thisdiscrepancy is simply related to the origin of the upper bound the-orem. The upper theorem is based on evaluating a larger value fordeformation force than the actual amount [18,23]. Therefore, thetheoretical value is larger than the experimental one.
It is also clear that the difference between the experimental andtheoretical extrusion force increases as the number of ECAE passesincreases. However, there is still a reasonable agreement betweenthem as the variation is close to 20%. This difference could be re-lated to the fact that using the back pressure could change thedeformation mode of the material. During the ECAE processingwithout back pressure, large cracks formed on the surface of mate-rial which is a result of tensile stresses on the surface of the mate-rial [24]. With conducting the ECAE process with back pressure, thecracks are not observed which it means that the stress state duringdeformation has changed. According to Von-Mises yield criterion[18], the change of state of stress could change the required stressfor yielding of material and therefore the required deformationforce. However, as in the first pass of deformation with or withoutback pressure, no crack forms, therefore, the application of back
pressure does not change the deformation mode. In this case, theeffect of back pressure will be omitted only with deducting fromthe extrusion force.
The other effective parameter which could result in this dis-crepancy is that during axi-symmetric deformation processes, thedeformation is not homogenous [25,26]. This non-homogeneityshould be considered in the deformation model. However, it makesthe solution very complicated and consequently the analyticalsolution will be impossible. Therefore, the non-homogeneity ofdeformation has been neglected in the deformation model whichresults in a difference in the experimental and theoretical results.
5. Conclusions
An upper-bound analysis was carried out in order to investigatethe plastic deformation behavior of the material during axi-sym-metric ECAE process with outer curved corner. From the resultsof the analysis the following conclusions are made
(1) Extrusion force or pressure in equal channel angular defor-mation is a function of die angle, outer curved corner ofthe die, friction coefficient and dimensions of the die.Decreasing the die angle has a larger effect on increasingextrusion pressure than has the outer curved corner angleof the die. Increasing the friction coefficient leads to a largeincrease in the extrusion pressure. Therefore, it is stronglyrecommended to keep the friction coefficient as small aspossible in the ECAE process. Increasing the outer curvedcorner angle of the die restricts the effect of friction becauseit reduces the contact surface between the material and thedie walls.
(2) A peak in the force-punch displacement curves occurs whichis a result of upsetting deformation during ECAE and thesticking friction before the deformation initiates. However,after the first pass, the billet will accurately fit the dieentrance channel. Consequently, no upsetting deformationat the initiation of ECAE is expected. Moreover, the surfacequality of the material increases which decreases the frictioncoefficient. Therefore, in the second and the third pass ofdeformation, the occurrence of the peak is not expected.
(3) The extrusion force in the second and the third pass showssmaller agreement with the experimental results which isbecause of the fact that using the back pressure changesthe deformation mode of the material.
Acknowledgements
The authors would like to thank the Iranian National ScienceFoundation (INSF) and Sharif University of Technology, Tehran,Iran, for the financial support and research facilities used in thiswork.
Appendix A
A.1. Power dissipated in the deformation zone
The velocity field in the deformation zone is
tr ¼ 0; th ¼ V0 cos u; tz ¼ 0 ðA:1Þ
Consequently, the components of the strain rate field in cylin-drical coordinate system are
and the other strain rate components are zero. The equivalent strainis
e� ¼ 2ffiffiffi
3p
ffiffiffiffiffiffiffiffiffiffiffiffi12
e�
ije�
ij
rðA:3Þ
where
e�
ije�
ij ¼ 2e�2
rh ðA:4ÞTherefore
e� ¼ 1ffiffiffi
3p
rV0 cos u ¼ 1ffiffiffi
3p
rV0 cos
/þ w2
� �ðA:5Þ
As it is clear from Fig. A.1
dV ¼ xdx � dy � dh ðA:6Þ
The ellipse in this figure is the OM cross-section of the materialin the deformation zone. The following equation shows the math-ematical relationship of the ellipse in the new rectangular coordi-nate system shown in Fig. A.1 by X and Y
x� bb
� �2
þ ya
� 2¼ 1 ðA:7Þ
which in this equation
b ¼ acos u
ðA:8Þ
Therefore, one can easily conclude that
y ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � ðx cos u� aÞ2
qðA:9Þ
For a Von-Mises material the power dissipated in the deforma-tion zone is
W�
d ¼ 2s0
ZVd
ffiffiffiffiffiffiffiffiffiffiffiffi12
e�
ije�
ij
rdV ðA:10Þ
Using Eqs. (A.5,A.6) and (A.9) one results in
W�
d ¼ s0
Z uþw
u
Z 2b
0
Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�ðx cos u�aÞ2p
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�ðx cos u�aÞ2p
xV0 cos uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 þ y2Þ
p dydxdh ðA:11Þ
which by simple mathematical approaches can be simplified to
W�
d ¼ s0V0w cos uZ 2a
cos u
0x
� ln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � ðx cos u� aÞ2
qþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2 � ðx cos u� aÞ2
q�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � ðx cos u� aÞ2
qþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2 � ðx cos u� aÞ2
q
dx
ðA:12Þ
Fig. A.1. Side and normal cross-section of the deformation zone and an element i
This integral can not be evaluated by the elementary tech-niques, though numerical methods can be applied to find appropri-ate values for any given value of x. For more details of how to solvethe integral one can refer to [27].
A.2. Power dissipated on the entry surface, Ci
The velocity discontinuity on (Ci) was found as V0 sinu. Theshear surface is an ellipse the same as shown in Fig. A.1 with smalland great radiuses of a and b, respectively. Therefore
W�
i ¼ pa2s0V0 tan u ¼ pa2s0V0 cot/þ w
2
� �ðA:15Þ
A.3. Power dissipated on the exit surface, Co
As exit surface is the same as entry surface, power dissipated onthe exit surface is the same as that on the entry surface
W�
i ¼W�
i ðA:16Þ
A.4. Power dissipated on the contact surface between the material andthe die wall in the deformation zone, Cm
The velocity discontinuity on this surface is Dtm = V0 cosu. Sinceboth the velocity discontinuity and the flow stress in shear are con-stant, the power dissipated can be calculated by integration of
W�
m ¼Z
Cm
sDtm ds ðA:17Þ
Regarding to Fig. A.2 for calculating ds
ds ¼ xdldh ðA:18Þ
and
n the mentioned zone, in the left figure, Y-axis is perpendicular to the sheet.
Fig. A.2. side and normal cross-section of the contact surface between material and the die in the deformation zone and an element in the mentioned zone, in the left figure,Y-axis is perpendicular to the sheet.
Substituting x � b = u and then u = bsint in this integral result in
W�
m ¼ 4a2ms0V0w
cos u
Z p2
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2 sin2 t
pdt ðA:23Þ
where
e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � a2
pb
¼ sin u ðA:24Þ
f is the eccentricity of the ellipse. Note that 0 < f < 1 and thefunction E(f), that is defined by Eq. (A.25) is called the completeelliptic integral of the second kind.
EðfÞ ¼Z p
2
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2 sin2 t
qdt ðA:25Þ
The integral can not be evaluated by the elementary techniquesfor general f, though numerical methods can be applied to findappropriate values for any given value of f. Tables of values ofE(f) for various values of f can be found in collection of mathemat-ical tables. Therefore, power dissipated on the contact surface be-tween the material and the die wall in the deformation zone canbe written as
W�
m ¼ 4a2ms0V0wcosec/þ w
2
� �EðfÞ ðA:26Þ
A.5. Power dissipated on the surfaces AC and DB
Since there is friction on these surfaces, the dissipated power is
W�
WðACÞ ¼W�
WðDBÞ ¼ 2pa2ms0V0 tan u
¼ 2pa2ms0V0 cot/þ w
2
� �ðA:27Þ
A.6. Power dissipated on the surface between the material and the diewall before and after the deformation zone
The dissipated power is
W�
lðiÞ ¼ 2pams0V0li ðA:28Þ
and
W�
lðoÞ ¼ 2pams0V0lo ðA:29Þ
Appendix B
B.1. Details of the parameters used in Tables 3 and 4
As the radius of the extrusion channel is a, and l is the length ofthe part of the specimen out of the deformation zone (in the entryand exit channel, l = li + lo) and as it is clear from Fig. 3 that thelength of the material in the deformation zone is equal to the ra-dius of the extrusion channel
l ¼ L� a ðB:1Þ
where L is the total length of the specimen. Therefore, in this studyl = 66 mm; To calculate the amount of the total strain [12,13]
etot ¼1ffiffiffi3p 2 cot
/þ w2
� �þ w
� �ðB:2Þ
In die 1, etot = 1.155 and in die 2, etot = 0.969.To calculate the average yield strength, �r0
�r0 ¼1
eb � ea
Z eb
ea
Kende ðB:3Þ
and the average shear strength
�s0 ¼�r0ffiffiffi
3p ðB:4Þ
So in die 1, �s0 ¼ 85:29 MPa and in die 2, �s0 ¼ 81:48 MPa.To calculate the extrusion force using Eq. (9) we haveFor die 1, F = 80.28 kN, and for the other die, F = 70.55 kN.
[15] R. Lapovok, C. Loader, F.H. Dalla Torre, S.L. Semiatin, Mater. Sci. Eng., A 425(1–2) (2006) 36–46.
[16] Y.W. Tham, M.W. Fu, H.H. Hng, M.S. Yong, K.B. Lim, J. Mater. Process. Technol.192–193 (2007) 575–581.
[17] T. Mukai, M. Kawazoe, K. Higashi, Nanostruct. Mater. 10 (5) (1998) 755–765.[18] R. Hill, The Mathematical Theory of Plasticity, first ed., Oxford University Press,
New York, 1967.[19] M. Cook, E.C. Lark, J. Inst. Met. 71 (1945) 371–390.[20] S. Poortmans, B. Diouf, A.. Marie Habraken, B. Verlinden, Scripta Mater. 56 (9)
(2007) 749–752.[21] L. Dupuy, E.F. Rauch, Mater. Sci. Eng., A 337 (1–2) (2002) 241–247.[22] ASTM Standard, G99-05, Standard Test Method for Wear Testing with a Pin-
on-Disk Apparatus.[23] B. Avitzur, Metal Forming: Processes and Analysis, first ed., McGraw-Hill, New
York, 1968.[24] J.R. Bowen, A. Gholinia, S.M. Roberts, P.B. Prangnell, Mater. Sci. Eng., A 287 (1)
(2000) 87–99. 15 July.[25] B.S. Moon, H.S. Kim, S.I. Hong, Scripta Mater. 46 (2) (2002) 131–136.[26] A.V. Nagasekhar, Yip Tick-Hon, H.P. Seow, J. Mater. Process. Technol. 192–193
(2007) 449–452.[27] A.R. Eivani, M.Sc. Thesis, Sharif University of Technology, Tehran, Iran, Winter
