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Dislocation density-based equal channel angular pressing

Dislocation density-based modeling of deformation beha v ior of

aluminium under equal channel angular pressing

Seung Chul Baik a ,Yuri Estrin b,*,Hyoung Seop Kim c ,Ralph Jo ¨rg Hellmig b

Technical Research Laboratory,Pohang Iron and Steel Co.Ltd.,Pohang 790-785,South Korea

Institut fur Werkstoffkunde und Werkstofftechnik,Technisch Uni v ersitat Clausthal,Agricolastrasse 6,D-38678Clausthal-Zellerfeld,Germany

Department of Metallurgical Engineering,Chungnam National Uni v ersity,Daejeon 305-764,South Korea

Recei v ed 4June 2002;recei v ed in re v ised form 18October 2002

Abstract

In this study,the deformation beha v ior of aluminium during equal channel angular pressing (ECAP)was calculated on the basis of a dislocation density-based model.The beha v ior of the material under ECAP,including the dislocation density and cell size e v olution as well as texture de v elopment,was simulated using the finite element method (FEM).The simulated stress,strain and cell size were compared with the experimental data,which were obtained by ECAP for se v eral passes in a modified Route C regime.Good agreement between simulation results and experimental data,including strain distribution,dislocation density and cell size e v olution,strain hardening and texture de v elopment was obtained.As concerns the general trends,the stress was found to increase rapidly in the first ECAP pass,the strain-hardening rate then dropping from the second pass on.Calculations showed a non-uniform strain distribution e v ol v ing in the course of ECAP.The simulated cell size is also in good agreement with the experiment,particularly with the obser v ed rapid decrease of the cell size during the first pass slowing down from the second pass https://www.sodocs.net/doc/3c18398091.html,rger cells were found to form in the upper and the lower parts of the workpiece where the strain is smaller than in the middle part.Due to the accumulation of strain throughout the workpiece and an o v erall trend to saturation of the cell size,a decrease of the difference in cell size with the number of passes was predicted.#2002Published by Elsevier Science B.V.

Keywords:Equal channel angular pressing;Se v ere plastic deformation;Dislocation density;Constituti v e modeling;Finite element analysis;Dislocation cell size;Aluminium

1.Introduction

Due to their unusual mechanical and physical proper-ties,ultra-fine-grained materials are currently attracting a great deal of attention [1].Of particular interest is grain refining by se v ere plastic deformation that makes is possible to produce bulk materials with fine grain structure.Extremely large strains required for that cannot be achie v ed v ia con v entional processing routes,such as rolling or wire drawing,whereby one or more of the workpiece dimensions are continuously reduced with strain.Accordingly,in these processes,the large strain required can only be produced in foils or filaments [2].

In recent years,a number of techniques were de v eloped,by which metallic materials can be deformed to ex-tremely large plastic strains with minimal changes in their net dimensions [2].Of those methods,equal channel angular pressing (ECAP)[3á5]is considered to be the most promising for the preparation of bulk,fully dense,relati v ely uniform ultra-fine-structured ma-terials [6].Importantly,there is little shape change (notably almost no cross-sectional change)during ECAP,allowing repeated pressing [7].

A typical ECAP die consists of two intersecting channels of identical cross-section [8á10].A billet of material is placed in one of the channels and forced by a plunger into the second channel.Due to the geometric constraints of the die,the billet deforms in shear within a small area at the intersection between the two channels.The material is thus deformed by almost pure shear [5].The mechanical properties and the

*Corresponding author.Tel.:'49-5323-722004;fax:'49-5323-723148

E-mail address:juri.estrin@tu-clausthal.de (Y.Estrin).

Materials Science and Engineering A351(2003)86á

www.else v https://www.sodocs.net/doc/3c18398091.html,/locate/msea

0921-5093/02/$-see front matter #2002Published by Elsevier Science B.V.PII:S 0921-5093(02)00847-X

microstructure of materials deformed using ECAP were reported in se v eral publications[11á14].The effect of die shape on the deformation was also studied[1,15,16] in order to optimize the ECAP process.As the plunger can be retracted and the sample remo v ed,and then re-pressed se v eral times,the effect of the pass number and the role of specimen rotation between consecuti v e passes were also looked into[5,17,18].

In order to estimate the magnitude of the strain introduced in the samples and to understand the deformation beha v ior in this process,v arious calcula-tions were performed.For two channels of equal cross-section intersecting at an angle and with an outside cur v ature,the strain as a function of the number of ECAP passes can be calculated analytically[19].The strain rate in a material undergoing ECAP was e v al-uated by a geometrical approach[20].Computer simu-lations of deformation using the finite element method (FEM)were also carried out to analyze the effect of different processing conditions[1,6,21á25]on material flow.Howe v er,none of the pre v ious analyses ha v e taken into account the strain hardening of the material by considering the microstructure e v olution during ECAP.Indeed,grain refinement by se v ere plastic deformation is related to the e v olution of subgrain (dislocation cell)boundaries[26]and the texture de v el-opment,although the mechanisms of grain refinement are not fully understood yet.Hence,to understand the microstructure refinement and the macroscopic defor-mation beha v ior during ECAP,it is necessary to analyze the dislocation density e v olution and the v ariation of the dislocation cell size during ECAP.

Estrin et al.[27]proposed a model that predicts the strain-hardening beha v ior of dislocation cell-forming crystalline materials at large strains.All stages of strain hardening starting from Stage II up to stress saturation at the end of Stage V were reproduced correctly within the unified model.It was shown that the predicted dislocation densities and the strain-hardening cur v es were in good agreement with the experimental data for torsion deformation of Cu.As the model was deri v ed to include large deformations,it is well suited for predict-ing strain-hardening beha v ior and the e v olution of the cell size in the course of se v ere plastic deformation, particularly ECAP.The original model[27]that refers to a two-dimensional cell structure was recently general-ized[28]for the three-dimensional case.

The aim of this study is to analyze the deformation beha v ior of polycrystalline aluminium under ECAP by FEM in connection with the three-dimensional v ersion of the dislocation density-based model[28].The out-comes include a description of the dislocation density e v olution and the cell size v ariation,along with texture e v olution.2.Dislocation density-based strain-hardening model Here we gi v e a brief re v iew the three-dimensional v ersion of the dislocation density-based strain-harden-ing model[27,28]used in the present simulations.The dislocation densities are introduced as scalar internal v ariables of the model whose e v olution with strain determines the o v erall strain-hardening beha v ior.This approach is combined with the crystal plasticity con-siderations,thus making it possible to simultaneously trace strain hardening and texture e v olution.The dislocation population is partitioned into dislocations forming a cell structure and those contained within the cell interiors.This gi v es rise to the notion of a‘two-phase material’:a cell interior with a relati v ely low dislocation density r c,and cell walls of width w with a higher dislocation density r w.These two distinct dis-location densities are the internal v ariables of the model. The total dislocation density,r t,is gi v en by a rule of mixtures:

0f r

'(1(f)r

;(1) where f denotes the v olume fraction of the cell walls.An important element of the model is the consideration of the e v olution of the v olume fraction of the cell walls, which is based on the experimental obser v ations. According to the pre v ious reports[29,30],f decreases with strain monotonically.The e v olution of f was approximated in[27]by the following empirical func-tion:

f0f

'(f

)exp

(g r

?g r

;(2)

where f0is the initial v alue of f,f its saturation v alue at large strains and the quantity?g r describes the rate of v ariation of f with resol v ed shear strain g r.The fact that f is significantly smaller than f0(cf.[27]and Table1) implies that the subgrain walls become sharper in the

Table1

The parameter v alues used in the simulations

r t00

(m(2) 1.0)1013

r t00

(m(2) 1.0)1014

f00.25

f 0.06

?g r 3.2

˙g r

1.0

m100

n67

a0.25

G(GPa)26.3

b(m) 2.86)10(10

K30

a*0.0024

b*0.0054

k0 3.22

S.C.Baik et al./Materials Science and Engineering A351(2003)86á9787

course of straining.The a v erage cell size d is directly related to the total dislocation density through

?????r

p;(3)

where K is a proportionality constant and also decreases with the accumulation of the total dislocation density in the course of ECAP.

The approach taken follows the classical framework of dislocation density-based modeling first de v eloped by Kocks[31]and subsequently modified by Mecking and Estrin[32á34].The kinetic equations relate the resol v ed shear stress t r to the resol v ed plastic shear rate˙g r:The two different dislocation densities gi v e rise to two

distinct(scalar)stresses,t r

c an

d t r

;in the cell interiors

and the cell walls,respecti v ely:

t r c 0a Gb

?????r

˙g r

˙g

1=m

;(4)

t r w 0a Gb

??????r

˙g r

˙g

1=m

;(5)

where˙g r

c and˙g r

denote the shear rates in the cell

interiors and the cell walls,respecti v ely,G is the shear

modulus,b is the magnitude of the Burgers v ector,˙g r

0is

a reference strain rate,1/m is the strain rate sensiti v ity parameter and a is a constant,typically about0.25.The o v erall beha v ior of the composite structure is defined by a scalar quantity,t r;that is obtained using the rule of mixtures applied to the two‘phases’:

t r0f t r

w '(1(f)t r

:(6)

The e v olution of the dislocation density in the cell interior is go v erned by the following equation[28]:

˙r c 0a1

???

??????r

˙g

(b1

6˙g

bd(1(f)

˙g

(1=n

˙g

:(7)

The three terms on the right-hand side of Eq.(7)are the contributions from different dislocation mechanisms. The first one describes the rate of generation of cell interior dislocations generated by Frank-Read sources at the interface.The second term represents the loss of the cell interior dislocations that mo v e into the walls to become part of wall structure.The geometry para-meters,a*and b*,are considered to be constant.The last term describes mutual annihilation of cell interior dislocations go v erned by cross-slip.Here k0is taken to be a constant,while the parameter n characterizing the strain rate sensiti v ity of the annihilation process is in v ersely proportional to the absolute temperature and increases with the stacking fault energy.The e v olution of the cell wall dislocation density is gi v en by ˙r

6b1˙g

(1(f)2=3

bdf

???

b1˙g

(1(f)

??????r

˙g

(1=n

˙g

:(8)

Here the first term represents the dislocation density gain in the walls corresponding to the loss of cell interior dislocations.The second term expresses the increase of dislocation density in the wall due to the acti v ation of Frank-Read sources at the interface by dislocations coming from the cell interior.Finally,the third term accounts for annihilation of cell wall dislocations in v ol v ing cross-slip.

In order to satisfy strain compatibility along the interface between cell interiors and cell walls,the following relation is imposed:

˙g r

0˙g r

0˙g r:(9)

3.Application of dislocation density-based strain-hardening model to a polycrystalline material

In crystal plasticity,se v eral slip systems are acti v e with different slip rates,and the resol v ed shear stress also v aries form one slip system to another.For simplicity,it is assumed that the dislocation cells within a grain are identical and that their mechanical response can be characterized by a unique resol v ed shear strain rate˙g r:Misorientations between cells are disregarded, while the indi v idual orientation of each grain is con-sidered.Assuming that the shear resistance is the same for all slip systems,˙g r was deri v ed in Ref.[28]as follows:

˙g r0

X N

s01

˙g

m'1

1'm

:(10)

Here N is the number of slip systems and˙g r

denotes the strain rate for the s th slip system.In this study,texture e v olution was included based on the full-constraint Taylor model[35].The shear rates in the acti v e slip systems of the grains of a polycrystalline aggregate can be calculated using the random choice method with linear programming[36]*a procedure that was adopted in the present work.An ensemble of300grains was used in the present simulation.

4.Experimental

ECAP experiments were conducted using samples of pure(99.99%)aluminium.Samples were cut with dimensions of12mm)12mm)60mm.The initial a v erage grain size of the material was about2mm2.

S.C.Baik et al./Materials Science and Engineering A351(2003)86á97 88

Prior to ECAP,the samples were polished using 1200-

grit SiC paper.

The ECAP die was made from tool steel (X38CrMoV 51),which was hardened to 47HRC after milling out the channel.Fig.1shows the dimensions of the die with a split design in which one side contains the whole channel,while the other part is used for closure.The channel was rectangular in cross-section.The angle F between the entrance and the exit channels was 908and die corner angle,c ,208.The dimensions of the entrance channel,12.5mm )12.5mm,were a little larger than that of the exit channel,v iz.12mm )12mm.The die was designed for the con v enient repetiti v e pressing

because the section of a sample after each pressing becomes larger than the dimensions of the exit channel due to elastic spring-back.The die was placed in an INSTRON 8502machine with a maximum applicable load of 200kN.The pressing speed for the aluminium specimens was 4mm min (1.Repetiti v e pressing was performed up to four passes following a modified Route C,referred to as Route C r ,which is shown schematically in Fig.2.Between two consecuti v e pressings,the sample was rotated by 1808about its long axis and turned around in such a way that its rear end became its front end.

The a v erage cell size of the material deformed by ECAP was measured using transmission electron micro-scopy (Philips CM200).The yield strength was deter-mined by the standard tensile tests on specimens that were cut from the workpiece (with their axis parallel to the pressing direction)after v arious numbers of ECAP passes.The gauge length of the tensile specimens with 3mm diameter was 12mm.

5.Details of ?nite element analysis

Fig.3is the flow chart to describe the procedure of finite element simulation using the abo v e strain-hard-ening model.Gi v en the material parameters of the

model,deformation increment matrix is extracted for each element at time t .The resol v ed shear strain rates,˙g r s ;in all slip systems s of e v ery grain are calculated using the deformation increments,and the unique resol v ed strain rate,˙g r ;is obtained by substituting the resol v ed shear strain rates ˙g r s in Eq.(10).The rates of v ariation of the dislocation density in cell interiors and cell walls are calculated by substituting ˙g r in Eqs.(7)and (8).The resol v ed shear stress,t r ,is then obtained using Eqs.(4)áFig.1.Schematic diagram of ECAP rig.

(6),and the a v erage cell size at t'D t is calculated from Eqs.(1)á(3).The equi v alent stress,s,is introduced on the basis of work consistency to update equi v alent strain v s.equi v alent stress cur v es using the relation

t r˙g r0s˙o;(11) where˙o is the equi v alent strain rate,which is obtained by FEM calculation.

The deformation gradients of each element are obtained from the FEM results.The deformation gradient history yields the v elocity gradients from which strain increments are calculated for each time increment. The strain increment history of each element thus obtained is used to calculate crystallographic orienta-tions based on the full-constraint Taylor model and the orientations of grains are updated as well.The updated data are used in the FEM calculations of deformation at the next step.The calculation is carried out up to the final time step t f.Finally,the deformation beha v ior of the material under ECAP,including strain hardening, texture and cell size e v olution,can be obtained from the FEM calculation.

In the simulation focusing on f.c.c.materials in general and Al in particular,the{111} 110 slip systems were considered to be acti v e.As an initial condition,each element was assumed to consist of300 randomly oriented crystallites.Friction between the die and the workpiece was neglected.The simulations were performed using the ABAQUS software[37],the calcula-tion for crystallographic orientations and strain hard-ening being performed with the user subroutine UMAT [37].

All parameters used in the present strain-hardening model are summarized in Table1.The v alues of the shear modulus,G,and the Burgers v ector,b,were taken from Ref.[38],while for the parameters for the cell wall v olume fraction,f0,f and?g r;the v alues for Cu from Ref[27]were adopted for want of measured v alues for Al.The authors realize,of course,that this may impair the quantitati v e accuracy of the simulation,but belie v e that the qualitati v e trends predicted are not affected. The parameters related to rate sensiti v ity,m and n,were chosen from literature[39,40].The initial v alues taken

for the dislocation densities in the cell walls,r t00

w ;and

cell interiors,r t00

c ;were1014and1013m(2,respecti v ely.

The remaining parameters for dislocation density e v olu-tion,a*,b*and k0,were adjusted to get a best-fit to the experimental data in the analysis of the stressástrain cur v es,and the parameter K was determined by comparing the calculated cell sizes with the experimental ones.

Two-dimensional calculations were carried out as-suming plane strain conditions.Fig.4shows the initial mesh used in the finite element analysis,where each element had four nodes and four integration points.In order to check the sensiti v ity of the results to the mesh size,the calculations of strain were repeated using a finer mesh.Fig.5shows the deformed meshes and the strain distribution calculated with the reference mesh used throughout this paper(Fig.4),v is-a`-v is the results obtained with a mesh size being one-quarter of the reference one.The fact that both results differ only insignificantly demonstrates that the reference mesh can be used reliably.

In this calculation,1á4pressings(Route C r)were simulated.For the simulation of multiple pressings,the end v alues of equi v alent strains and equi v alent stresses, as well as of r c,r w and f,for a particular pressing were adopted as the initial v alues in the simulation of the subsequent pressing.In this procedure,only data obtained in the uniformly deformed region were adopted in the next pressing as indicated in Fig.6.For comparison with experimental data,the v alues of the equi v alent stress and the cell size calculated for elements located in the middle of a uniformly deformed portion of the workpiece were taken.

6.Results and discussion

6.1.Stresses and strains

The yield strength of Al deformed by1á4passes of ECAP was measured using uniaxial tensile testing.Fig. 7shows the measured stressástrain cur v es and the yield strength after each https://www.sodocs.net/doc/3c18398091.html,ing the test data,the experimental stress v s.strain points were obtained by ‘gauging’the strain:the calculated increment of the equi v alent strain of1.02in the middle of the

specimen Fig.4.Initial mesh for the simulation of ECAP.

S.C.Baik et al./Materials Science and Engineering A351(2003)86á97 90

after an ECAP pass was set equal to that obtained in experiment.Fig.8shows a simulated equi v alent-stressáequi v alent-strain cur v e,along with the experimental data points taken at the end of each of the four consecuti v e ECAP passes.The stress is seen to rapidly increase during the first pressing.Thereafter,the strain-hardening rate decreases appreciably,and little hard-ening occurs during the third and fourth pressings.The trend obser v ed in Fig.8confirms the results of an earlier report[15].It should be noted that the cur v e shown in Fig.8was obtained by combining the indi v idual equi v alent-stressáequi v alent-strain results calculated for each pressing pass.The agreement between the model predictions and the experimental data seen in Fig.8is remarkably good.The distributions of equi v a-lent strain after each pressing obtained by FEM are shown in Fig.9.The strain is seen to rapidly increase when the material passes the shear area where the two channels meet.The equi v alent strain gained in a single pressing in the middle part of the specimen is almost the same for each pressing step.It suggests that the equi v alent strain in the middle of the specimen can readily be calculated as a function of the number of passes.Howe v er,the strain near the specimen surface differs from that in the middle part.For the first pressing,the strain in the lower region is smaller than that in the middle.It is known that the strain in the lower region of the specimen is dependent on the

angle Fig.6.Schematic diagram showing the updating procedure for simulation of repetiti v e pressings.

S.C.Baik et al./Materials Science and Engineering A351(2003)86á9791

on the outer corner of the die,c,which defines the arc of cur v ature at the outer point of intersection of the two channels.The result[21]of FEM calculation showed a non-uniform deformation in the lower specimen part for the case of large c,while strain was uniform throughout the workpiece thickness for a die with c008under zero friction condition.As the workpiece is deformed along the outside cur v ature of the die,the strain distribution v aries with the angle c.Howe v er,the non-uniform deformation in the lower region for the first pressing shown in Fig.9(a)cannot be explained in terms of the angle of the outer cur v ature.Indeed,due to a corner gap the bottom surface does not contact the outer corner.It was reported pre v iously[23,41]that a larger corner gap was formed in the case of a material with a higher strain-hardening rate.For a non-hardening material,the workpiece deformed more uniformly,and the die corner was filled.On the other hand,for a strain-hardening material,the inside region of the workpiece within the deforming zone,which recei v ed more se v ere deforma-tion,was harder than the outside region of the deform-ing zone.The outside region of the workpiece could thus flow faster to the exit channel.The outside surface of the strain-hardening workpiece went through a shorter distance and,accordingly,its bottom surface showed a lower shear deformation than for a non-hardening workpiece.Following Ref.[23],it can be concluded that the formation of a gap seen in the simulation of

the

first pressing is due to a high strain-hardening rate.With a decrease in the strain-hardening rate shown in Fig.8, the gap is diminished from second pressing on(cf.Fig.

9).The sensiti v ity of the results to the strain-hardening rate shows that proper account of this property is v ery important for a correct prediction of the deformation beha v ior under ECAP.Fig.9(b)shows the strain distribution in an Al specimen that underwent two ECAP passes.It is seen that the strains in both the upper and the lower regions are smaller than in the middle of the specimen.As the lower part of the specimen in the first pressing becomes its upper part in the second one due to rotation by1808in between,the relati v ely small strain in the lower region results from an interplay between the strain distributions formed in the two pressings.For the subsequent passes from the second pressing onwards,the bottom surface was in contact with the die at the corner,so that in addition to the effect of strain hardening,the effect of the angle c should also be considered to explain the non-uniform strain in the lower part.Due to the gap formation and the cur v ature of the outer corner,the strain in the lower part of the specimen decreases as compared with that in the middle part for each pressing.The small non-uniform strain in the lower part is retained when it becomes the upper part for the next pressing,and the strains in both the upper and the lower parts of the specimen that went through multiple pressings pro v e to be smaller than the strain in its middle part.

The v ariation of strain through the thickness of workpiece is shown in Fig.10,where the abscissa,s, represents the distance from the bottom of workpiece, normalized with respect to its thickness.The difference in strain between the middle part and the bottom surface is seen to increase with the number of passes.The strain in the bulk of the material,for s in the range0.2á0.8,is relati v ely uniform,its magnitude growing after each

pressing.

The sensiti v ity of deformation beha v ior to the loca-tion within the workpiece can be followed in more detail in Fig.11,where the v ariation of strain rate components with the position is presented for all four pressings.The flow lines calculated for s00.1,0.5and0.9are shown in Fig.11(a).Three non-zero strain rate components,˙o

;

˙o

and˙o

;rele v ant for the plane strain condition,are plotted as a function of the angle,u,as defined in Fig. 11(a),in Fig.11(b)á(d).The results for the middle part (Fig.11(c))show that the u dependence of the strain rate

components˙o

and˙o

is represented by a function symmetrical with respect to u0458*the angle corre-sponding to the line where the two channels meet.Fig. 11(d)shows that this symmetry also holds in the upper part of the workpiece.By contrast,this symmetry is broken in the lower part of the workpiece(Fig.11(b)), the magnitude of the strain rate components being smaller than in the middle or the upper part.This is in agreement with the obser v ation that the difference in strain between the bulk and the near-surface regions results from the different deformation pattern in the lower part of the workpiece.

6.2.Cell size

Pre v ious studies[42,43]demonstrated that ECAP is capable of producing an ultra-fine grain size in poly-crystalline materials.Under se v ere plastic deformation during ECAP,the grain size is decreased when strain is accumulated[15].As the primary aim of ECAP is grain size refinement,it is important to trace the grain size e v olution during deformation by modeling and simula-tion.Howe v er,the mechanisms of grain refinement are not fully understood as yet,and no pro v ision for updating the population of the grains was made in the model used here.In this study,the focus was rather on the v ariation of the a v erage dislocation cell size,d,along with the e v olution of the dislocation density.Fig.12 shows the cell structures obser v ed using transmission electron microscopy in Al specimens after different numbers of pressings.The a v erage cell size was mea-sured by transmission electron microscopy with no account of misorientation between cells made,so that the definition of d used in this study is the same as the definition of subgrains in pre v ious reports[44].Fig.13 shows the results of the calculation of d as a function of the equi v alent strain v is-a`-v is experimental data.A good correspondence between the calculated results and the measured v alues is seen.A rapid decrease of d during the first pressing slows down in the subsequent passes, from second pressing on.This tendency parallels the decrease of strain hardening discussed in the pre v ious section.The results of simulation and experiment shown are consistent with the obser v ation[43]that the

subgrain

size decreases rapidly and approaches saturation after the first pass of ECAP.

As the microstructure refinement is strain-dependent, the pattern of cell size distribution(Fig.14)follows that of strain(cf.Fig.9)albeit with an‘in v ersion’:a smaller cell size corresponds to a larger strain.Hence,d is larger in the lower part of the workpiece than in the middle part,where a larger strain pre v ails.Due to workpiece rotation by1808between the pressings,the differences in d between the upper and the lower parts of the work-piece nearly le v el-off after multiple pressings,lea v ing a coarser cell structure in both near-surface regions of the workpiece.The quantitati v e difference in the a v erage cell size between the bulk and the near-surface regions is seen in Fig.15that shows the v ariation of d with the position within the workpiece,s.The cell size is uniform in the bulk of the workpiece,in the range of s roughly between0.2and0.8,where the strain is uniform as well (cf.Fig.10).It should be noted that although the difference in strain between the bulk and the near-surface regions increases with the number of passes,the difference in d diminishes.This is a consequence of the tendency to saturation of the dislocation density and the a v erage grain size inherent in the mechanism underlying the microstructure e v olution.E v en though the strain builds up at a different pace in the bulk and at

the

periphery of the workpiece,the o v erall accumulation of strain with the number of passes led to con v erging v alues of d throughout the workpiece.

7.Conclusions

In this paper,the deformation beha v ior of Al during ECAP was analyzed on the basis of the three-dimen-sional v ersion of a dislocation-based strain-hardening model implemented in a finite element code.The strain-hardening model [28]used accounts for dislocation density e v olution and the concomitant v ariation of the a v erage cell size during se v ere plastic deformation.

Fig.12.The cell structures obser v ed in the ND áTD planes (as de?ned in Fig.1)by transmission electron microscopy after (a)one pass,(b)two passes and (4)four passes of

ECAP.

order to consider the grain orientations in the calcula-tion of dislocation density e v olution,the texture de v el-opment was also calculated as part of the FEM simulation.The simulated stress,strain and a v erage cell size were compared with the experimental data for Al obtained in ECAP tests using1á4passes in the Route C r regime.These results led us to the following conclu-sions.

The equi v alent stress rapidly increases in the first pressing,but the growth of stress slows down from second pressing on,the material showing only little strain hardening during the third and fourth pressings. The stressástrain cur v e obtained by finite element analysis is in good agreement with the experimental data.

Due to the formation of a gap in the outside die corner,which is sensiti v e to the strain-hardening rate, strain non-uniformity de v elops in the lower part of the workpiece during the first pressing.The strain le v el in the lower part of the workpiece is smaller than in the middle part of the workpiece.This result demonstrates the importance of adequately accounting for the strain-hardening rate for a reliable prediction of the deforma-tion beha v ior under ECAP.

The strain in both the upper and the lower parts of the workpiece that underwent multiple pressings is lower than in the core of the workpiece.This is a feature of Route C r ECAP,in which the(non-uniform)small strain produced in the lower part of the workpiece is retained in the upper part for the next pressing. Consistent with the strain distribution pattern,the a v erage cell size d decreases rapidly during the first pressing,but le v els-off in the further course of ECAP. The simulated v ariation of d is in good agreement with the experimental data obtained by transmission electron microscopy.A finer cell structure forms in the bulk of the workpiece than at its periphery,but the differences between these regions decrease with the number of ECAP passes when saturation in d is approached.

At present,the model does not include an ingredient important for ECAP modeling,v iz.a pro v ision for updating the grain structure.In particular,the possibi-lity of‘con v ersion’of cells to grains with accumulation of misorientation between adjacent cells,as well as nucleation and growth of new grains within the micro-structure need to be considered.Further work is definitely necessary to pro v ide the model with the facility to include these factors.Howe v er,as the grain structure is not deemed to seriously influence the de v elopment of the cell structure under the conditions considered,it is belie v ed that the model is adequate for calculating the e v olution of the dislocation density and the a v erage cell size.A good agreement of the simulation results with experimental data on both the strain-hardening beha v ior and the e v olution of the cell size during ECAP appears to justify this approach that the authors hope to refine in the future.

With the crystal plasticity model used,a correct account of the orientation of the grains is essential. Texture e v olution must therefore be an integral part of the simulation.Texture de v elopment was included in the calculations reported here.Details of texture analysis, supported by texture measurements on Al and Cu,will be gi v en elsewhere.

Acknowledgements

The authors acknowledge useful discussions with L.S. To′th,H.-G.Brokmeier,https://www.sodocs.net/doc/3c18398091.html,po v ok and P.Thompson. One of the authors(SCB)gratefully acknowledges support from the Ministry of Science and Culture of Lower Saxony for a research fellowship.HSK acknowl-edges support by Korea Research Foundation through Grant KRF-2001-041-E00418.

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