Oscillatory Residual Stresses in Steady Angular Channel Extrusion
OOscillatory Residual Stresses in SteadyAngular Channel Extrusion
Arunava Ray , Pritam Chakraborty , and AnindyaChatterjee Mechanical Engineering, Indian Institute of TechnologyKanpur, India Aerospace Engineering, Indian Institute of TechnologyKanpur, India * Corresponding author: Address: 210/F NWTF, IndianInstitute of Technology Kanpur, UP-208016, India. email:[email protected] a r X i v : . [ c s . C E ] J a n bstract Angular channel extrusion has evolved as processes that can in-duce significant strengthening of the formed product through grainrefinement. However, significant residual stresses are developed in theextruded product whose quantification is necessary for accurate pro-cess design and subsequent heat treatment. Experimental evaluationof residual stress provides the through thickness (normal) variationat chosen sampling points on the formed product and may provideinaccurate estimates if variations along the extrusion (longitudinal)directions are present. Process models can complement the exper-imental measurements and improve the estimates of residual stressdistribution. While models of this process have been developed, veryfew of them have been applied to understand the variation of resid-ual stress in the formed products. The present work aims to addressthis limitation by providing a complete map of residual stress distri-bution in angular extrusion process through numerical simulations.Interestingly, our simulations show that the angular channel extrudedproduct can have significant longitudinal variation of residual stressdepending on the extrusion ratio and strain hardening rate. Detailedanalyses of the process reveals that these spatial oscillations occur dueto cyclic movement of the contact location between the die and thetop-billet surface in the exit channel. The outcome of this study sug-gests that accurate measurement technique of residual stress field inangular channel extruded products should consider the possibility oflongitudinal variations. The findings can be extended to other contin-uous forming processes as well.
Keywords: residual stress, angular channel extrusion, finite element method,elasto-plasticity.
Angular channel extrusion is a manufacturing process where the workpieceis pressed through a die consisting of two channels intersecting at some angle[Segal et al., 1981]. The process is associated with severe plastic deformationleading to grain refinement and subsequent strengthening of the formed prod-uct. However, as with other forming processes, significant residual stressesare developed for this process as well, which necessitate appropriate char-acterization to ensure that the performance of the formed products are not2etrimentally affected by these locked-in stresses [Schajer, 2013]. Contourmethod and X-ray diffraction technique have been employed in Khanlari1and Honarpisheh1 [2020], and Romero-Resendiz et al. [2020], respectively, toquantify the through thickness variation of residual stress in angular channelextruded products. However, owing to the complexity of these methods, vari-ation of residual stress along the thickness of the formed product at few cho-sen locations could only be obtained. Typically full-field experimental mea-surement of residual stress is extremely difficult and the through thicknessvariations at certain sampling points are usually accepted as representative.Such a consideration is accurate if variations in the extrusion (longitudinal)directions are absent. However, these variations may be present depending onthe process and material parameters. Thus, full-field map of residual stressis necessary to disallow any error due to oversight and numerical analysescan augment the experimental measurements in this respect.Various analytical and numerical models of angular channel extrusionhave been developed over the years. In the earliest work, Segal [1995] gavean analytical estimation of the total strain as a function of the die angle ofa test sample which was pressed through a die containing two equal chan-nels. Frictionless contact between die and billet, fully filled die channel byworkpiece, and sharp corners were assumed in the model. Iwahashi et al.[1996] extended the model proposed by Segal [1995] for a multiple pass an-gular channel extrusion process. Finite Element Method (FEM) analyses(FEA) of the process by assuming plane strain conditions and consideringthe effect of friction was first presented by Prangnell et al. [1997]. It wasshown that with the inclusion of friction, the FEM predictions are in agree-ment with the upper bound solution. DeLo and Semiatin [1999] presented2-D and 3-D non-isothermal FEA of the process. The results, such as loadversus stroke length of the punch, were compared with experiments for aTi-6Al-4V alloy and a reasonable agreement was obtained. Kim et al. [2000]used two different material models (strain hardening and perfectly plastic)with nearly equal yield stress values to explain the die-billet gap formationat the outer corner radius of the process using FEA. The gap between thebillet and outer die radius was found larger for higher strain rate sensitivematerials due to the relatively softer outer part. From the FEA, only the vonMises stress at the steady state deformed zone were presented. Kim [2001],Kim et al. [2001] and Li et al. [2004] separately showed the effect of outercorner radius and friction on the extrusion load and effective plastic strain.Nagasekhar et al. [2009] simulated the angular channel extrusion process in3BAQUS/Explicit [Dassault Systemes, 2016] for different friction conditionsand showed that the extrusion load versus displacement curve from analysishave the same trend as with experiments on a copper billet. Lee et al. [2017]showed that the residual stress distribution from 3-D FEM simulations inABAQUS was in good agreement with the experimental measurement usingneutron diffraction. They reported the radial variation of residual stresses,but the variation of the residual stress along longitudinal direction (LD) wasnot studied.As is evident from the above review, a majority of the modeling workrelated to angular channel extrusion has been focused on understanding theeffect of various process parameters on the deformation of the extruded prod-uct. Limited research activity can be found on residual stress analyses, andthese only provide comparative studies on the variation along the throughthickness or normal direction (ND). In the present work, 2D plane strainFEA of quasi-static angular extrusion for different die geometries and mate-rial properties is performed to investigate the variation of residual stresses inboth the ND and LD. Interestingly, spatially oscillating residual stress pro-files are clearly observed in the extruded component after single pass angularextrusion. Depending on the process parameters (such as extrusion ratio andhardening constants), the variation of the normal stress along the LD of theextruded billet can have either periodic or aperiodic oscillations, or no oscil-lations at all. The observed longitudinal variations can be physically relatedto the movement of the contact location between the die and billet in the exitsection. These observations of oscillatory fields bring forth new possibilitiesof longitudinal variations of residual stresses in angular channel extrusionand other continuous forming processes, which so far has been assumed tobe absent.The organization of the paper is as follows: In section 2, the finite elementmodel of the angular extrusion process is presented. The details of the meshsensitivity study using different values of ER are also shown. In section 3,the results of the parametric study using different values of ER and strainhardening rate are shown. The residual stress distribution for the differentparameters is discussed in detail. The cause of residual stress variation alongLD as observed in section 3 is explained in detail in section 4. The conclusionsfrom this study is presented in section 5.4
Numerical Model of the Angular ExtrusionProcess
The details of the FEM model of the angular extrusion process and the elasto-plastic material parameters are presented in this section. Furthermore, theprocedure to determine the size of elements that provides convergent solutionis described.
Two dimensional plane strain simulations of the angular extrusion processare carried out in the FEM software ABAQUS. An isotropic elasto-plasticmaterial model with linear strain hardening has been used in the simula-tions. The linear strain hardening response is a simplistic representation ofstage III hardening in most metals/alloys and thus considered in the study.The hardening rate has been varied in the parametric study to analyze itsinfluence on residual stress distribution. The values of Young’s Modulus (E),Poisson’s ratio ( ν ) and yield stress are chosen as 200GPa, 0.3, and, 400 MPa,respectively, and kept constant in all the simulations.The schematic of the die geometry used in the simulations is shown inFig. 1. In the figure, W and W are the widths of the inlet and the exitchannel, respectively, with ER defined as W / W . The corner angle (Ψ) anddie angle ( φ ), as shown in the figure, are both 90 ◦ . R and R are the cornerradii of the right and left wall of the die, respectively, while L and L arethe lengths of entry and the exit channels, respectively. In this study theER is varied by changing W and keeping W fixed at 0.025 m. All othergeometric parameters are normalized with respect to W and kept constantin the simulations. These parameters are shown in Table 1.5igure 1: Schematic of the die. The extrusion ratio ER=W /W .Geometric parameters L L W R R Normalized value with respect to W
22 6 ER 0.8 1.8Table 1: Normalized geometric parameters of the die.In all the simulations, both the die and punch are considered rigid. Thedeformable billet is extruded through the die by first displacing the punchdownwards with a constant velocity of 0.005 m/s through the vertical part.Later, when the bend is reached, points on the left face of the billet arepushed along appropriate arcs of circle until the bend has been fully traversed.Finally, the billet is pulled gently out using forces on the right face (this lastphase is accompanied by very small deformations, because frictionless contactbetween the die, billet and punch is used). The billet is discretized using 4node bilinear plane strain quadrilateral elements with reduced integrationand hourglass control. 6 .2 Convergence Analysis of Mesh
In the FEM simulations, the initial rectangular billet geometry is discretizedusing equisized 4-node quadrilateral elements. A sufficiently small elementsize is needed for accurately capturing the spatial gradients of the displace-ment and stress fields during the extrusion process. An h -refinement strategyis adopted whereby the domain is successively discretized with elements ofthe same family but of smaller size, followed by comparing responses along achosen material line. In the absence of singularities, the differences betweensuccessive solutions should decrease with the reduction in element size. Oncean acceptably small difference between the two solutions is obtained, the so-lutions are deemed to have converged and the coarser of the last two meshesis selected for subsequent simulations.Using the above strategy, the simulation cases 1 and 7 (the extreme caseswithin Table 2) are considered to determine the appropriate element size.FEM simulations are performed using three different element sizes of 1 mm,0.758 mm and 0.5 mm, with the aspect ratio about 1 in each case. Localconvergence of the computed solution is determined by comparing the inter-polated nodal values of σ xx along a particular material line. To constructthat material line, two points A and B are selected in the undeformed con-figuration of the billet as shown in Figs. 2(a) and 2(b). The nodal σ xx valuesare extracted from material points on the straight line joining points A andB in the final deformed configuration (at the end of the simulation) as shownin Figure 2(c). Case No. 1 2 3 4 5 6 7Extrusion Ratio (ER) 1 0.9 0.8 0.7 0.75 0.6 0.5Table 2: The simulation cases considered in the FEA where the hardeningrate is fixed at 5 MPa and the extrusion ratio is varied.Results of the comparison are shown in Fig. 3. For simulation case 7(Fig. 3(a)), the stress variations are almost identical along A-B. However,for case 1 (Fig. 3(b)), a deviation can be observed near point A for thecoarsest mesh, while results are almost overlapping for element sizes 0.758and 0.5 mm, respectively. Based on this analysis, it can be concluded thatthe element size of 0.758 mm provides a reliable solution and hence this sizeis used in all subsequent simulations. 7igure 2: (a) Initial configuration of the angular extrusion process. (b)Magnified view showing points A and B. (c) Magnified view of the finaldeformed configuration showing the material line on which σ xx is obtained.Figure 3: Comparison of σ xx along material line shown in Fig. 2(c) for threemesh sizes (0.5 mm, 0.758 mm, 1 mm) and simulation cases (a) 7 and (b) 1(from Table 2). 8 Parametric Study and Distribution of Resid-ual Stress
A parametric study is now reported wherein the extrusion ratio (ER) and thehardening rate are varied. In the first seven simulation cases the hardeningrate is kept constant at 5 MPa and the ER is varied from 1 to 0.5 (Table 2).The variation of residual stress for these cases are shown in section 3.1. Inthe next seven simulation cases, the ER is kept constant at 0.75 and thehardening rate is varied (Table 3). The results of these cases are discussedin section 3.2. Specifically the stress component σ xx along both the ND andLD are presented for the different simulation cases.Case No. 8 9 10 11 12 13 14Hardening Rate (MPa) 20 60 160 260 360 460 560Table 3: Simulation cases considered in the FEA where the ER is fixed at0.75 and the hardening rate is varied.The variation of σ xx along ND is obtained at multiple sections along thelength of the extruded billet. These sections are constructed such that theyare approximately perpendicular to the top and bottom surface of the billet(Fig. 4). The values of σ xx are then interpolated on these sections from theGauss-point values. Fig. 4 shows a typical extruded 2D billet along with thesectional line A-B used to extract the ND variation of σ xx .Variations of σ xx along LD are then evaluated from several sections likeA-B separated by ≈ σ xx is first evaluated on all these sections and the variationsalong LD are then reported on curve C-D that connects all these points asshown in Fig. 4. For all the simulation cases, the curve CD remains nearlyparallel to the bottom surface of the extruded billet. However, the distanceof the curve from the bottom surface ( h b ) changes with ER and is shown inTable 4. From the table it can be observed that the maximum σ xx is belowthe medial axis ( h b /h =0.5) for ER ≥ σ xx is evaluated, and curve C-D obtained by joining all thepoints with maximum sectional σ xx .ER h (m) h b /h h b to h (see Fig. 4) for different ERs. As seen in Figs. 5 and 6, for ER = 1 and 0.9, there is clearly aperiodiclongitudinal variation of maximum σ xx . For ER = 0.8 and lower (until 0.7),the variation is much closer to periodic, as seen in Figs. 7 and 8. A Fouriertransform of the longitudinally varying maximum sectional σ xx (Fig. 9) showsthe presence of a dominant frequency consistent with the nearly periodicbehavior. 10igure 5: Residual stress component σ xx in the extruded billet for ER = 1.0.Figure 6: Maximum of residual stress component σ xx on every section per-pendicular to the medial axis along the LD of the extruded billet for (a) ER= 0.9 and (b) ER = 1.0.Figure 7: Residual stress component σ xx in the extruded billet for ER =0.75. 11igure 8: Maximum residual stress component σ xx on successive sectionsperpendicular to the medial axis along the LD of the extruded billet for (a)ER = 0.7 and (b) ER = 0.8.Figure 9: Fast Fourier Transform (FFT) of the responses shown in (a)Fig. 8(a) and (b) Fig. 8(b).As the ER is reduced further to 0.6, the oscillation in the residual stressfield disappears, as seen in Fig. 10. The through thickness variation of σ xx , asshown in Fig. 11, displays the familiar zigzag pattern developed due to plasticbending and subsequent elastic unloading [Joudaki and Sedighi, 2015].12igure 10: Residual stress component σ xx in the extruded billet for ER =0.6. Figure 11: Residual stress component σ xx in the ND for ER = 0.6.Interestingly, the residual stress distribution along the ND for higher ERsdisplays a sense opposite to that in Fig. 11, as seen in Fig. 12. The transitionof the stress distribution profile begins to occur near ER = 0.7 whereby forthe different sections along the extrusion direction (Fig. 13(a)), the residualstress distribution along the ND starts to change sign (Fig. 13(b)).13igure 12: Residual stress component σ xx in the ND for ER = 0.8 at 2different perpendicular sections along the LD.Figure 13: Through thickness variation of σ xx for ER = 0.7. (a) Sectionsperpendicular to medial axis on which the variation of σ xx along the ND iscompared. (b) Variation of σ xx along the ND. In the previous section, results are presented for different ERs with a fixedhardening rate of 5 MPa as shown in Table 2. In this section, the ER is fixedat 0.75 and the hardening rate is increased from 5 MPa to 560 MPa (Table 3)to analyze its influence on residual stress. The strain hardening behavior formost metals and alloys can be suitably represented by a power law [Hosford14nd Caddell, 2007]. However, in this work a linear fit is considered sinceit provides a reasonable approximation of the stress-strain response beyondthe initial portion of the curve. The linear fits of the power law curves forAluminum alloy, Cobalt alloy, and annealed stainless steel at room tempera-ture are shown in Fig. 14. The parameters of the power law model for thesethree alloys are taken from Callister [2005] and Kalpakjian [2014]. The linearhardening rates are obtained as 70, 900 and 1500 MPa for Aluminum alloy,stainless steel and Cobalt alloy, respectively. Thus, linear hardening rates ofapproximately 70 MPa and higher are representative of room temperatureresponse of metals and alloys while lower rates are more representative ofhardening behavior at elevated temperatures.Figure 14: Linear fits of power law hardening curves of Aluminum and Cobaltalloy, and stainless steel at room temperature. The fits are calculated usingthe stress values obtained from the power law for strain values from 5 to 100%. Consistent with observations made in section 3.1, nearly periodic oscilla-tion along the LD is observed for ER of 0.75 and hardening rate of 5 MPa, anda magnified view of the same is shown in Fig. 15(a). A similar longitudinalvariation of residual stress is obtained from simulations for the other harden-ing rates at ER = 0.75. However, a decrease in the amplitude of oscillationsis observed with increase in hardening rate. To quantify this observation,a variable ‘ D ’ is defined, which is the difference between the maximum and15inimum of the peak sectional σ xx , along the longitudinal direction (LD)and over one cycle, as shown in Fig. 15(a). For a particular hardening rateat ER = 0.75, D remains roughly constant from cycle to cycle. Hence, D can be considered as twice the amplitude of the oscillation. A comparison of D for different hardening rates is shown in Fig. 15(b), and clearly reveals adecrease in oscillation amplitudes with the increase in hardening rate. Thus,hot extrusion can potentially lead to larger oscillations in commonly usedmetals and alloys than cold extrusion. Correspondingly, cold extrusion ofsteel is expected to generate comparatively smaller oscillations in residualstress when compared to aluminium, since the former has a higher hardeningrate than the latter.Figure 15: (a) The variable ’D’, peak to peak amplitude of nearly periodicoscillation of maximum sectional σ xx along the LD. (b) Variation of ’D’ withstrain hardening rate. Our simulations show that details of the contact interaction between thebillet and the exit channel play a strong role in the creation of oscillatoryresidual stresses. During extrusion, the width of the billet actually becomesslightly less than that of the die exit section. Immediately past the bend,contact is steadily maintained between the billet and the lower surface ofthe exit section; however, due to the billet being slightly narrower, contact16s lost at the upper surface. Subsequently, as the billet moves along the exitsection of the die, there is a small upward curvature in the billet which causesloss of contact at the lower die surface. The degree of upward curvature isfound to vary with time, leading to a sequence of fresh contact points beingestablished between the billet and the upper die surface. Each successivecontact point, thus formed, subsequently moves downstream along with thebillet. When a contact point has moved sufficiently far downstream, thecurvature in the billet increases again, and a new contact point is formedbetween the billet and the upper die surface. This mechanism of space- andtime-varying contact force distribution causes the appearance of oscillatoryresidual stresses. The rest of this section is devoted to a detailed discussionof the same.We begin by comparing the temporal variation of the state of contact andthe spatial variation of the residual stress field along the LD for different cases(Table 2). To this end, we fix a spatial region and examine nodal values of σ yy on the top and bottom surfaces of the portion of the billet that lies insidethis region. Nodes are taken to be not in contact if their σ yy = 0. Conversely,nonzero values of σ yy indicate the presence of normal contact tractions. Theresults obtained are depicted graphically for several discrete instants of time,which shows the evolution of the contact state. From section 3.1 it is observed that the variation of residual stress alongthe LD shows nearly periodic oscillation for case 5 (Table 2). For this case,nodal σ yy on the top surface of the billet inside the region of observationare shown at some instants of time in Fig. 16. As seen from the plots,there is significant variation in the contact state over time. In particular,a region of high contact traction moves to the right with the billet, until anew contact is established; subsequently, both contacts move downstream;as each contact moves downstream, its strength initially increases and thendecreases to zero. Comparing figures 16(a) and 16(f), we see that a nearlyperiodic variation is obtained. This nearly periodic behavior persisted untilthe billet was completely extruded past the bend. For the case shown in thefigure, the time period obtained is 7.68 sec (approximately). The residualstress field in the extruded billet for this case is shown in Fig. 7 which showsnearly periodic oscillation along the LD.17igure 16: Stress component σ yy on the top billet surface in the exit channelfor case 5 (Table 2), at time instants (a) t (an arbitrary starting time), (b)t + 4.41 s, (c) t + 4.48 s, (d) t + 5.35 s, (e) t + 7.05 s, (f) t + 7.68 s.In notable contrast to the traction distributions on the top surface, thecontact state on the bottom surface of the billet remains essentially un-changed over the same time duration, as shown in Fig. 17.18igure 17: Stress component σ yy at the bottom billet surface in the exitchannel for case 5 (Table 2) at different instants of time. We have observed from simulations that aperiodic temporal variation of con-tact results in aperiodic oscillation of residual stress in the billet along theLD. Some details are shown below for case 1 of Table 2. The temporal vari-ation of contact tractions is shown in Fig. 18, where no periodic pattern isseen. Thus, periodicity or aperiodicity in contact conditions between the bil-let and the upper section of the exit channel govern periodicity or aperiodicityof residual stress in the final billet. The residual stress field in the extrudedbillet for this case is shown in Fig. 5 which shows aperiodic oscillation alongthe LD. 19igure 18: Stress component σ yy at time instants (a) t (an arbitrary startingtime) (b) t + 1.89 s (c) t + 11.02 s, on the top billet surface in the exitchannel for case 1. The above mechanism also applies to situations where near zero oscillations inresidual stress along the LD are observed. The temporal variation of contactcondition for case 6 (Table 2) is shown in Fig. 19 as an example. In the figure,the location of the contact region remains nearly the same over time. Minordifferences in traction distribution are observed, but they remain localizedand have a small effect overall. The result is a longitudinally steady residualstress field as evident from Fig. 10. 20igure 19: Stress component σ yy at time instants (a) t (an arbitrary startingtime) (b) t + 2.66 s (c) t + 5.66 s, on the top billet surface in the exitchannel for ER = 0.6. Angular channel extrusion is a continuous forming process that leads to grainrefinement and significant strengthening of the product. However, residualstresses can be developed in the formed products that may degrade theirperformance. Hence, a detailed quantification of the residual stress field ofthe extruded product needs to be obtained. Experimental measurement ofresidual stress is typically performed at few locations to obtain the variationalong ND. Inferences of the developed residual stress field are then madebased on these measurements and with the assumption that variations alongLD are absent. However, the assumption need not be valid in general, andcan depend on the material and process conditions.In the present work, a numerical study of the angular channel extru-sion process has been performed to explore the possibility of development oflongitudinally varying residual stress. Plane strain large deformation elasto-plastic FEM simulations have been performed for different strain hardeningrates and ER. The analyses show that at low strain hardening rate and highER, significant oscillations of residual stress along LD can occur. A closerinvestigation reveals that a small separation between the billet with the dieat the bend into the exit channel occurs due to the intense deformation in21he bend. Subsequently, contact is restored as the billet travels along the exitchannel. The contact location itself can move in space, either periodicallyor aperiodically. The distance of the contact location from the bend canbe thought of as providing a moment arm for the contact force. This mo-ment due to the contact forces influences the deformation process going onwithin the bend. The variation of contact location changes the applied mo-ment which modifies the deformation behavior in the bend. These variabledeformation histories cause sectionwise variable residual stresses, which areswept down the exit channel without further change and eventually emergeas spatially oscillating residual stresses in the billet.The extent of variation of contact location in the exit channel is stronglydependent on the ER. A lower ER reduces the lengths of the regions in whichloss and reestablishment of contact of the billet with the die at the exit chan-nel can occur. Thus, variation of the moment on the deforming billet in thebend due to contact forces at the exit channel is reduced. As this momentinfluences the deformation behavior in the bend, a reduction in its variationlessens the changes in the history of deformation resulting in decreased varia-tion of residual stress along LD. An increase in strain hardening rate reducesthe sensitivity in the deformation behavior of the billet in the bend on themoment due to contact forces at the exit channel. This leads to a reduc-tion in the variation of residual stress in the billet along LD for increasingstrain hardening rate. Thus, for a given ER, if we increase the hardening rate then we expect the oscillatory residual stresses to disappear; and for agiven hardening rate, if we reduce the ER, we expect the residual stresses todisappear as well. Both of these effects have been convincingly observed inthe numerical simulations in this paper.From this study it can be concluded that longitudinal oscillation of resid-ual stress is possible in angular channel extrusion depending on the geometryparameters of the die and material parameters of the billet. This observa-tion clearly indicates that enough sampling points should be considered whilemeasuring residual stresses in formed products for the purpose of materialcharacterization, process design and heat treatment. The observations can beextended to other continuous forming processes where variations on contactis possible. 22 eferences
Callister, W. D. (2005).
Fundamentals of Material Science and Engineering .Wiley.Dassault Systemes (2016). Abaqus FEA.DeLo, D. P. and Semiatin, S. L. (1999). Finite-element modeling of non-isothermal equal-channel angular extrusion.
Metallurgical and MaterialsTransactions A , 30(5):1391–1402.Hosford, W. F. and Caddell, R. M. (2007).
Metal Forming Mechanics andMetallurgy . Cambridge University Press, Cambridge, UK.Iwahashi, Y., Wang, J., Horita, Z., Nemoto, M., and Langdon, T. (1996).Principle of equal-channel angular pressing for the processing of ultra-finegrained materials.
Scripta Materialia , 35(2):143–146.Joudaki, J. and Sedighi, M. (2015). Effect of material’s behavior on residualstress distribution in elastic–plastic beam bending: An analytical solution.
Journal of Materials: Design and Applications , 231(4):361–372.Kalpakjian, S. (2014).
Manufacturing engineering and technology . Addison-Wesley Pub. Co.Khanlari1, H. and Honarpisheh1, M. (2020). Investigation of Microstructure,Mechanical Properties and Residual Stress in Non-equal-Channel AngularPressing of 6061 Aluminum Alloy.
Transactions of the Indian Institute ofMetals , 73(5):1109–1121.Kim, H. S. (2001). Finite element analysis of equal channel angular pressingusing a round corner die.
Materials Science and Engineering:A , 315(1-2):122–128.Kim, H. S., Seo, M. H., and Hong, S. I. (2000). On the die corner gapformation in equal channel angular pressing.
Materials Science and Engi-neering:A , 291(1-2):86–90.Kim, H. S., Seo, M. H., and Hong, S. I. (2001). Plastic deformation analysisof metals during equal channel angular pressing.
Journal of MaterialsProcessing Technology , 113(1-3):622–626.23ee, H. H., Gangwar, K. D., Park, K., Woo, W., and Kim, H. S. (2017).Neutron diffraction and finite element analysis of the residual stress distri-bution of copper processed by equal-channel angular pressing.
MaterialsScience and Engineering:A , 682:691–697.Li, S., Bourke, M. A. M., Beyerlein, I. J., Alexander, D. J., and Clausen,B. (2004). Finite element analysis of the plastic deformation zone andworking load in equal channel angular extrusion.
Materials Science andEngineering:A , 382(1-2):217–236.Nagasekhar, A. V., Yoon, S. C., Tick-Hon, Y., and Kim, H. S. (2009). Anexperimental verification of the finite element modelling of equal channelangular pressing.
Computational Materials Science , 46:347–351.Prangnell, P. B., Harris, C., and Roberts, S. M. (1997). Finite element mod-elling of equal channel angular extrusion.
Scripta Materialia , 37(7):983–989.Romero-Resendiz, L., Flores-Rivera, A., Figueroa, I., Braham, C., Reyes-Ruiz, C., Alfonso, I., and Gonz´alez, G. (2020). Effect of the initial ecappasses on crystal texture and residual stresses of 5083 aluminum alloy.
International Journal of Minerals , Metallurgy and Materials , 27(6):801.Schajer, G. S. (2013).
Practical Residual Stress Measurement Methods . Wiley,West Sussex, UK.Segal, V. M. (1995). Materials processing by simple shear.
Materials Scienceand Engineering:A , 197(2):157–164.Segal, V. M., Reznikov, V. I., Dobryshevshiy, A. E., and Kopylov, V. I.(1981). Plastic working of metals by simple shear.