TRITA-IIP ISSN Han Han. Licentiate Thesis

Transcription

1 TRITA-IIP-0-01 ISSN Determination of Flow Stress and Coefficient of Friction for Extruded Anisotropic Materials under Cold Forming Conditions Han Han Licentiate Thesis Stockholm January 00 Division of Materials Forming Department of Production Engineering Royal Institute of Technology S Stockholm, Sweden

2 Determination of Flow Stress and Coefficient of Friction for Extruded Anisotropic Materials under Cold Forming Conditions by Han Han Licentiate Thesis Division of Materials Forming Department of Production Engineering Royal Institute of Technology S Stockholm, Sweden Stockholm, January 00

3 ABSTRACT The work material in metal working operations always shows some kind of anisotropy. In order to simplify the theoretical analysis, especially considering bulk deformation processes, anisotropy is usually neglected and the material is assumed to be isotropic. On the other hand, the analysis that considered the influence of anisotropy seldom incorporates the influence of friction. For predicting the material flow during plastic deformation and for predicting the final material properties of the product, adequate descriptions of both flow stress curves and coefficients of friction have to be developed. In the present work a number of experimental methods for determining the anisotropy have been utilized and compared: Yield loci, strain ratios (R-values) and establishing flow stress-curves in different directions. The results show that the yield loci measurements are weak in predicting anisotropy when the material strain hardening is different in different directions. It is concluded that also the strain ration (R-value) measurements are unreliable for describing anisotropy. The most trustable and useful results were found from multi-direction determinations of the flow stresses. Three typical cases of ring upsetting conditions were analyzed by theory (3D- FEM) and experiments: 1) An anisotropic ring, oriented 90 0 to the axis of rotational symmetrical anisotropy. The friction coefficient was the same in all directions ) An isotropic ring. The friction coefficient was different in different directions 3) An anisotropic ring oriented 0 0 to the axis of rotational symmetrical anisotropy. The friction coefficient was the same in all directions The cases 1) and ) reveal that the influence of anisotropy on the ring deformation is quite similar to that obtained by changing the frictional condition. The case 3) exposes that if the material flow caused by anisotropy is incorrectly referred to friction, the possible error of the friction coefficient can be as high as 80% for a pronounced anisotropic material. A modified two-specimen method (MTSM) has been established according to an inverse method. Experiments were carried as cylinder upsetting. Here both ordinary cylinders were used as well as so-called Rastegaev specimen. Also plane strain compression tests were utilized. The results show that MTSM is able to evaluate the validity of a selected mathematical model when both the friction coefficient and the flow stress are unknown for a certain process. MTSM can also be used to estimate the friction coefficient and flow stress provided that the selected mathematical model is adequate. Key words: Anisotropy, friction coefficient, flow stress, modified two-specimen method and FE-analysis

4 Acknowledgements During my work on this study many people have helped me in various ways. I am very grateful to Professor Ulf Ståhlberg, my supervisor, for his support, valuable discussions and guidance. Special thanks are put forward to the Royal Institute of Technology, Sweden, for the financial support. The initial part of the work financed by the Brite-Euram project BE is gratefully acknowledged. Sincere thanks go to Professor Jaak Berendson and Professor Bengt Lindberg for their strong moral support. Also, I would like to thank Professor Pavel Huml, my former advisor, for introducing me to the Project of EFFORTS (Enhanced Framework for FOrging design using Reliable 3-Dimensional Simulation), and for his support, discussion and guidance. I am heavily indebted to research engineer, Hans Öberg (Department of Solid Mechanics), for his valuable help during the laboratory work. I would like to thank all my colleagues at the division of Materials Forming for free discussions. Finally, I wish to express my sincere gratitude to my parents who helped me and encouraged me along the way. However, most of all, I am indebted to my wife, Bing Liu, for her continuous encouragement and practical assistance of all kinds.

5 PREFACE The thesis contains a summary and the following papers: A. Han Han, Comparison of Different Methods for Estimating Anisotropy with Rotational Axial Symmetry in Bulk Metal Forming, submitted to the Journal of Materials Processing Technology for publication B. Han Han, Influence of Material Anisotropy and Friction on Ring Deformation, submitted to the Journal of Tribology for publication C. Han Han, The Validity of Mathematical Models Evaluated by Twospecimen Method under the Unknown Coefficient of Friction and Flow Stress, accepted for publication in the Journal of Materials Processing Technology

6 TABLE OF CONTENTS 1. INTRODUCTION RELATIONSHIPS BETWEEN THE PAPERS SUMMARY OF THE PAPERS Paper A: Comparison of Different Methods for Estimating Anisotropy with Rotational Axial Symmetry in Bulk Metal Forming Paper B: Influence of Material Anisotropy and Friction on Ring Deformation Paper C: The Validity of Mathematical Models Evaluated by Two-specimen Method under the Unknown Coefficient of Friction and Flow Stress CONCLUDING REMARKS REFERENCES PAPERS A, B and C

7 1. INTRODUCTION In metal forming processes, the flow stress (material property) and the coefficient of friction (boundary condition) are important variables for the quality control of products. The main reasons are three. First, the flow stress of the workpiece directly influences the prediction of the forces and the energy for carrying out the forming operation, because it is a function of the strain ε, the strain rate ε&, the temperature θ and the microstructure S during plastic deformation. Second, friction increases forming load and power, changes material flow, strain/stress distribution and influences form filling, when the forming pressure is transmitted from the dies to the deforming workpiece. Third, the flow stress and the coefficient of friction are fundamentally formulated in a mathematical model (process model) [1] based on a mechanical analysis for one specimen in a given metal forming process. Therefore, without the accurate flow stress and coefficient of friction, it is impossible to improve the mathematical model for the given process [-4]. In the determination of flow stress and coefficient of friction, off-line methods are usually utilized under the condition that materials remain isotropic. Since metals generally exhibit their lowest formability under tensile stress, upsetting (or compression test) is a common method to obtain the flow stress for high strain values. To reduce the effect of friction, efforts have been made Fig. 1 in the past, such as double-cup specimen method by Riedel [5], narrow-neck specimen method, three specimens method by Rummel [6], concave specimen by Siebel [7], Restegaev specimen method [8] and four specimens method by Cock [9-1]. In spite of these efforts, the influence of friction on forming processes still remains a problem. Thus, this task is directed to tribology. For the quantitative evaluation of friction, ring compression test is a typical method in bulk metal forming. It was first proposed by Kunogi [13] and was later improved by Male and Cockcroft [14]. A number of works [15-33], both experimental and theoretical, showed the usefulness of this method in determining friction at the die/workpiece interface when various friction models (Coulomb s friction, constant friction factor and general friction models) were used for isotropic materials. Unfortunately, anisotropy is an inescapable phenomenon in metalworking [34]. The major cause is crystallographic texture produced in forming processes. That is, with increasing strain, the randomly distributed individual crystals in metals tend to rotate towards a certain preferred crystallographic orientation (texture). The typical modes of forming are rolling, extrusion and drawing. Since Hill 1948 [35, 36] first proposed a quadratic yield criterion for characterizing anisotropy, anisotropy has been recognized as an important factor in sheet metal forming operations, in which the concepts of planar anisotropy and normal anisotropy have been well-established [37-40]. Recently, increased attention has been given to anisotropy in bulk metal forming [41]. In 1998, Pöhlandt et al. [4,43] defined several concepts of anisotropy Fig. and corresponding strain ratios for estimating anisotropy in extruded or drawn metals. Although many studies have been done on anisotropy such as yield criteria 1

8 [44,45] (Hill, 1979, 1993), texture development [46], formability or Forming Limit Diagram [47, 48], no study connects anisotropy with the behavior of friction. The current study will fill in this knowledge gap Fig. 3. In the current work, the focus is on the analysis of influence of anisotropy coupled with the friction on metal flow. The objectives are to identify an appropriate method for predicting anisotropy and its influences on metal flow and to find a suitable method for determining the flow stress and the coefficient of friction for anisotropic materials. Riedel, 1914 Duble-Cup Specimen µ Someone, After 1914 Narrow-neck Specimen Rummel & Meyer, 1919 Three Specimens Method Siebel & Pomp, 197 Concave Specimen and concave tools Rastegaev, 194 Rastegaev Specimen Someone, After 194 Multi-Rastegaev Surface Specimen Cook & Larke, 1945 Graphic Method with Four Specimens Ho/do=1 Ho/do=1.5 Ho/do= Load Tons/Sq. IN Calculated Curve Compression Reduction in height % Cook & Larke, 1945 Extrapolation procedure for Cook and Larke test Alexander, 1963 Fig.1. Methods for reducing friction in upsetting.

9 Yield Surface Yield criteria (or yield function) Anisotropy Sheet metals Planar anisotropy Normal anisotropy Axial anisotropy Strain Ratio Bulk metals (Extruded) Axial-radial anisotropy Axial-tangential anisotropy Radial-tangential anisotropy Fig.. Characterizations of anisotropy. Previous works Yield function Texture Formability Anisotropy? Current work Flow stress Metal flow (Strain) Friction Tensile test Upsetting test Plane strain compression Other methods Isotropy Ring compression test Cigar test Fig. 3. Focus of the current work. The meanings of the lines are: study on isotropic materials, study on anisotropic materials and the current study. 3

10 . RELATIONSHIPS BETWEEN THE PAPERS OF THE THESIS Paper A Anisotropy Methods Influence Isotropy Anisotropy Paper B Anisotropy Friction Ring flow Flow stress Metal flow Friction Paper C Mathematical models Flow stress Friction coefficient Fig. 4. Principle scheme showing the relationship of the three papers of the thesis. The meanings of the lines are: relation between the papers, factors taken into account in the paper A and factors taken into account in the paper B. 4

11 3. SUMMARY OF THE PAPERS 3.1 Paper A: Comparison of Different Methods for Estimating Anisotropy of Rotational Axial Symmetry in Bulk Metal Forming The objectives of the study are to evaluate experimental methods of estimating anisotropy of bulk materials with rotational symmetry and to analyze the influence of anisotropy on metal flow in the contact region. The main focus is on the evolution of anisotropy during large strains under different strain histories. Three measurements (of yield loci, strain ratio and flow stress) have been utilized for determining the anisotropy of the axial-symmetrically extruded round bar, aluminum alloy AA608 (Al-Si1Mg0.9Mn0.1). According to the requirements of the three methods, specimens were manufactured in different directions out of the round bars Figs.5-7, in which the strain histories are formed by pre-upsetting to 0, 15, 5 and 35% after annealing. The experiments were carried out with tensile, upsetting and plane strain compression tests. Yield loci were constructed in π-plane by the first yield points of flow curves at the offset plastic strain ε=0.%. Due to evolution of the anisotropy in further plastic deformation, the yield loci measurement can not fully predict anisotropy when the material strain hardening is direction-dependent Fig.8. Second, strain ratios were calculated from axial anisotropy, axial-radial anisotropy and axial-tangential anisotropy, in which the definitions are given by Pöhlandt et al. (1998) for the anisotropy with rotational symmetry. Attention has been paid to ductility of the material in tension Fig. 9. Results showed that the strain ratio measurement is weak to estimate anisotropy because the values of strain ratios are not only determined by the material plastic anisotropy, but also restricted by the ductility in tension Fig.10, in which the pre- and on going strain path is shown in Tab.1. It is clear that the flow stress measurement in different directions can comprehensively determine the anisotropy of bulk materials Figs With this method, direction-dependent strain hardening and Bauschinger effects can be observed based on the analysis of the strain path change Tab.. Moreover, average stress ratios obtained from the flow stress measurement are applied to Hill s (1948) criterion in characterization of the anisotropy of the aluminum alloy AA608 in the FEM. The loading-displacement curves from simulations are in good agreement with the testing results Fig. 14. In addition, the influence of anisotropy on metal flow in contact region has been analyzed by means of the FEM in the cases of upsetting and forward extrusion for five different materials Tab.3. The elements on the contact surface have been traced and the numerical simulation was performed with ABAQUS/Explicit as a non-linear explicit dynamic analysis. The simulation results show that anisotropic materials are more sensitive to friction and the shear strain on the contact surface of the anisotropic workpiece is significantly higher than that of isotropic materials Figs

12 t r y Extruded bar z z x Test specimen (a) P(3,6) r Rzt Rzr Cross-section of extruded bar P(1,4) Rz r P(,5) (b) Fig. 5. (a) Orientation of small specimens, and (b) Specimen locations for strain ratio and yield loci measurements. z r t t z Fig. 6. Plane-strain compression test example of location of specimen, and loading direction. 6

13 A3 A A1 A1 A3 A z A1-A1 t 45 (b) A-A (c) z A3-A3 (a) r 45 r 45 (d) (e) Fig. 7. Locations of specimens for determination of Hill s parameters. Fig. 8. Yield Loci of Annealed Aluminum AA608 at different prestrains 0, 15, 5 and 35% (Experimental results). 7

14 z ε z (Ten) ε x (Ten) x Fig. 9. Cylinder specimen expanded (tensioned) in the circumferencial direction during upsetting in y. Rz Rzt Rz(pre-0%) Rz(pre-15%) Rz(pre-5%) Rz(pre-35%) εy Rzt(pre-0%) Rzt(pre-15%) Rzt(pre-5%) Rzt(pre-35%) εy Rzr Rzr(pre-0%) Rzr(pre-15%) Rzr(pre-5%) Rzr(pre-35%) εy Fig. 10. Strain ratios of Rz, Rzt and Rzr (Experimental results). 8

15 Tab. 1. Relation between pre- and on-going strains R z ε z ε x ε t ε y ε R Pre-strain C T T On-going strain T T C Strain directions Rev. CT Rev. R zt ε z ε x ε t ε y ε r Pre-strain C T T On-going strain T T C Strain directions Rev. CT Rev. R zr ε z ε x ε r ε y ε t Pre-strain C T T On-going strain T T C Strain directions Rev. CT Rev. Where: C=Compression; T=Tension; CT= Continuous; Rev.= Reverse Fig. 11. Flow curves of the annealed aluminum alloy AA608 in directions of r, t and z. C=Compression; T=Tension (Experimental results). 9

16 Fig. 1. Flow curves of the annealed alloy AA608 in directions of r, t and z for 35% pre-straining. C=Compression; T=Tension (Experimental results). Fig. 13. Flow curves of annealed alloy AA608 in the t-direction after pre-straining 0, 15, 5 and 35%. C= Compression; T= Tension (Experimental results). Tab.. Relations between pre- and on-going strain Curves No. 1 No. No. 3 No. 4 No. 5 Pre-strain C T T C T On-going strain C C C T T Strain directions CT Rev. Rev. Rev. CT Note: C=Compression; T=Tension; CT=Continuous; Rev.= Reverse. 10

17 Fig. 14. Cylinder upsetting. Four load-displacement curves are in good agreement with the results from the simulations in which the material is characterized by the Hill s criterion. Table 3 Materials used in FE- analyses Name Characteristics Model (MPa) AA608 σ r =σ θ =0.93σ z σ=190ε 0.15 Ani-30% σ r =σ θ =0.7σ z σ=190ε 0.15 Iso(608) σ r =σ θ =σ z σ=190ε 0.15 Hardening(06) σ r =σ θ =σ z σ=90+190ε 0.6 Non-hardening σ r =σ θ =σ z σ=90 Fig. 15. Distribution of shear strain on the contact surface of cylinder specimen for different materials (Simulation results from ABAQUS). 11

18 Fig. 16. Distribution of friction shear on the contact surface of deformed cylinder specimens (Simulation results from ABAQUS). 1 3 Fig. 17. Shear strains of different materials on the contact surface in forward extrusion (Simulation results from ABAQUS). 1

19 3. Paper B: Influence of Material Anisotropy and Friction on Ring Deformation The deformation of a ring is a complex issue for material flow in the transverse direction during axial reduction. When materials remain isotropic, ring compression test is a typical method for quantitative evaluation of friction in bulk metal forming. Recently, this test has been recommended by Pöhlandt et al. (1998) to determine anisotropy if materials possess anisotropy. The objective of the study is to clarify the influence of material anisotropy and friction on ring deformation. According to the effects of friction or material anisotropy on rings, ring flow was categorized into four patterns Fig. 18 for guiding simulations and experiments. Thus, three typical cases of rings were designed as (1) an anisotropic material ring oriented 90 o to the axis of rotational-symmetrical anisotropy under uniform coefficient of friction; () an isotropic material ring under anisotropic friction condition; and (3) an anisotropic material ring oriented 0 o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction. Figure 19 illustrates the locations and the coordinates of two anisotropic rings. The analyses were conducted by 3D finite element method (FEM), in which material properties were characterized by von Mises and Hill s (1948) yield criteria. The dimensions used in the analyses are (1) deformed ring shapes, () distribution of both normal pressure and frictional shear stress, and (3) the estimated errors in the coefficient of friction. In the first two cases, ring shapes revealed that the influence of anisotropy on ring deformation is quite similar to that obtained by changing the frictional condition Figs. 0,. In reality, a difference between these two cases exits in the distribution of normal pressure and friction shear stress, shown in Figs. 4, 5. This implies that if an anisotropic material is assumed to be isotropic, the influence of the anisotropy will be mistakenly attributed to friction. This error can be easily made in the third case, because the influence of material anisotropy on ring deformation is in the same direction as friction, and cannot be immediately observed in experiments Fig. 6. Figures 7 and 8 show that the possible estimated error for the coefficient of friction can be as high as 80% for a pronounced anisotropic material. The deformed ring shapes have also been verified in experiments using the extruded annealed aluminum alloy AA608 (Al-Si1Mg0.9Mn0.1). The testing results support those of the FEM analyses, Figs. 1, 3, 7. 13

20 Pattern 1 of friction-ring-flow: Neutral Plane Pattern of friction-ring-flow: Neutral Plane Internal diameter increasing When friction is low. Pattern 1 of material-ring-flow: Internal diameter decreasing When friction is high. Pattern of material-ring-flow: θ θ r r Flow competition occurs in the cross section of a ring When σ rr = f (θ ). Fig. 18. Patterns of ring flow. Non-flow competition occurs in the cross section of a ring When σ rr = Const.. z (ring90) Planar isotropic plane A r (ring90) (a) 90 o orientation z (ring0) r (ring0) (b) 0 o orientation z (material) θ (material) r (material) A A z (ring90) y (material) A θ (material) θ (ring0) x (material) r (material) r (ring0) θ (ring90) Rotation 90 o r (ring90) (c) Rotation of coordinates z (material) z (ring0) Fig. 19. Axis of a ring rotated (a) 90 o to the orientation of anisotropy; (b) 0 o to the orientation of anisotropy and (c) Rotation coordinates between orientations of rings and the material anisotropy. 14

21 90 o 90 o 0 o 0 o σ r distribution (a) AA608 σ r distribution (b) AISI01 Fig. 0. Flow competition occurs in the cross section of anisotropic rings, µ=0.07 (Simulation results from ABAQUS). Pattern 1 of material-ring-flow plus Pattern 1 of friction-ring-flow. Fig. 1. Deformed anisotropic ring shape (90 o orientation) with Teflon lubrication. The black line stands for the axis of the extrusion of the AA608 (Experimental result). 15

22 90 o 90 o 0 o 0 o σ r distribution (a) µ 1 /µ =, µ =0.07 σ r distribution (b) µ 1 /µ =3, µ =0.07 Fig.. Under frictional anisotropy conditions, the isotropic ring is formed into an ellipse (Simulation results from ABAQUS). Ring flow is Pattern of friction-ring-flow in 0 o direction, while it is Pattern 1 of friction-ring-flow in 90 o direction. Fig. 3. Influence of frictional anisotropy (µ 1 =dry condition and µ =Teflon) on ring deformation (Experimental result). 16

23 Fig. 4. The magnitude of the frictional shear stress and the normal pressure in 90 o -direction is lower than that in 0 o -direction for anisotropic ring (Simulation results from ABAQUS). 17

24 Fig. 5. Distribution of (a) normal pressure and (b) frictional shear stress in 90 o - and 0 o -direction under the frictional anisotropy condition (Simulation results from ABAQUS). (b) AISI01 (a) AA608 Fig. 6. Final shapes of anisotropic rings with 0 o orientation to the axis of different rotational symmetrical anisotropic materials: (a) aluminum AA608 and (b) steel AISI01 (Simulation results from ABAQUS). 18

25 Fig. 7. Changes in internal diameter versus the reduction in height for different materials under friction conditions: (a) µ=0.07, (b) µ=0.06, (c) µ=0.1 and (d) µ=0. (Results of the simulations and experiments). Pattern of material-ring-flow and two patterns of friction-ring-flow. 19

26 3 y-coordinate (mm) Internal surface (c) Surfaces External surface x-coordinate (mm) Fig. 8. Distribution of (a) normal pressure, (b) frictional shear stress, and (c) profiles of rings internal and external surfaces for materials AISI01 at µ = 0. 1 and isotropy(aisi01) at µ = 0.1, 0.18 (Simulation results from ABAQUS). 0

27 3.3 Paper C: The Validity of Mathematical Models Evaluated by Two-specimen Method under the Unknown Coefficient of Friction and Flow Stress The purpose of this study is to develop a two-specimen method that would be able to evaluate a mathematical model so as to determine flow stress and coefficient of friction. The focus of the analysis is on the evaluation of the coefficient of friction. A modified two-specimen method (MTSM) has been derived from an objective function of two specimens according to the inverse method. It consists of four equations, which are general equation (1), objective function (), minimum differential equation (3), and process conditional equation (4). P = σ θ, ε, & ε, S) f ( µ, G, ε ( G, G )) (1) k, i k, i ( k, o k, i k, o k, i Q = n f (, G, o, ε, i ( G, o, G, i )), i P1, i = δ = 1 f ( µ, G o i G o G lim min 1,, ε1, ( 1,, 1, i )) P i µ () Q = 0 (3) µ ε G, G ) = ε ( G, G ) (4) 1, i ( 1, o 1, i, i, o, i where G is specimen geometry size, P average flow resistance, S material microstructure factors, σ flow stress, ε average strain, & ε average strain rate, µ coefficient of friction, θ temperature, Subscript i : instantaneous value, i = 1,..., n, Subscript k : specimen number, k = 1,..., n, and Subscript o : original value. Its principle is that the flow stress for a given material is specimen geometryindependent, while the flow resistance depends on it, shown in Fig. 9. The assumptions of the method are, 1. The material used is incompressible in its plastic deformation region.. Two different specimen/tool geometrical sizes have to be designed, as different geometrical ratios lead to different flow resistance due to friction. 3. Two specimens should be tested under the same conditions in terms of boundary condition, process, temperature and strain rate. 4. The coefficient of friction is considered to be an average value on the contact surface. The method has been verified by experiments of cylinder, Rastegaev specimen upsetting and plane strain compression test with the specimens shown in Figs. 30-1

28 33. The flow stress and the coefficient of friction obtained from different tests are in good agreement to each other Tabs. 4 and 5. Also, the values obtained have been examined by both FEM and the ring compression test. Good agreements are found among the results obtained from MTSM, simulations and experiments Fig. 34 and Tab. 6. This method not only can evaluate the validity of a selected mathematical model when both the coefficient of friction and the flow stress are unknown for a given process, Fig. 35 and Tabs. 7, 8; but also it can estimate the on-line coefficient of friction and flow stress when the mathematical model selected is valid. f 1 (µ,g 1,o, ε 1,i (G 1,o,G 1,i )) Flow resistance data domain P 1,i Inverse computation at the same time Flow resistance data domain P,i Assumed flow stress model domain σ f(µ,g,o, ε,i (G,o,G,i )) Calculated flow resistance data domain P,i Fig. 9. Explanation of objective function of MTSM. d o =10 mm d o =10 mm 1 h o =15 mm h o =10 mm d o =10 mm 3 h o =5 mm CG1 or RG1: No.1 & No. CG or RG: No.1 & No.3 CG3 or RG3: No. & No.3 CG4 or RG4: No. & No.4 d o =14.14 mm 4 h o =10 mm Fig. 30. Cylinder and Rastegaev specimens in upsetting. CG: Cylinder Group; RG: Rastegaev Group.

29 u t Fig. 31. Rastegaev specimen profile. W = 50 mm PG1: 1 and PG: and 3 W = 50 mm h =,5 mm B=5mm W = 50 mm h = 1,8 mm B=,5 mm 1 h = 1,8 mm B=5mm 3 Fig. 3. Plane strain specimens and tools. z θ r θ z Fig. 33. Location of the plane strain specimen manufactured from a raw material bar. 3

30 Tab. 4. µ calculated from the solver of MTSM Group No. Lubricant Cylinder Upsetting Rastegaev Specimen upsetting Friction coeff. µ CG1 Teflon CG Teflon CG3 Teflon 0.03 CG4 Teflon RG1 Molykote RG Molykote RG3 Molykote RG4 Molykote Plane strain PG1 Teflon compression test PG Teflon Tab. 5. Average value of µ and flow stress σ modeling Cylinder Rastegaev Plane strain Specimen Lubricant Teflon Molykote Teflon Average µ σ σ = 190.6ε σ = 190.7ε σ = 178.6ε Tab. 6. Coefficient of friction for Teflon Ring Cylinder Plane strain µ

31 Fig. 34. Loading forces from simulations (ABAQUS) and experiments: (a) cylinder upsetting; and (b) plane strain compression test. Different equations in cylinder upsetting: hi µ ri µ ri Pi = σ ( ) (exp( ) 1 ) µ ri hi hi (5) µ d p = σ (1 + ) 3h (6) Tab. 7 µ influenced by models in cylinder upsetting Eq. (5) Eq. (6) Average µ

32 Different equations in plane strain compression tests: h µ B P = σ ( ) exp( ) 1 µ B h (7) h h µ B p = σ (1 + )( ) exp( ) 1 B µ B h (8) Fig. 35. Flow stresses influenced by models, Eqs. (7) and (8) in plane strain compression test. 6

33 Tab. 8. µ influenced by models in plane strain. Eq. (7) Eq. (8) Average µ RSQ better than 0.99 worse than

34 4. CONCLUDING REMARKS In this thesis, three experimental methods/measurements (yield loci, strain ratios/rvalue and flow stresses in multi-directions) for determining material anisotropy have been evaluated. Second, three typical cases of ring deformation have been analyzed using 3D finite element method (FEM) and verified by experiments. Third, a modified two-specimen method (MTSM) has been established according to inverse method and experiments were carried out with cylinder upsetting, Rastegaev specimen upsetting and plane strain compression test. The following conclusions can be drawn from the thesis: 1. Flow stresses in multi-directions are found reliable to estimate material anisotropy for bulk material, compared with yield loci and strain ratio methods.. The influence of anisotropy on ring deformation is quite similar to that obtained by changing the frictional condition. If the anisotropic behavior is mistakenly attributed to friction, the possible estimated error for the coefficient of friction can be as high as 80% for a pronounced anisotropic material. 3. The modified two-specimen method (MTSM) can evaluate the validity of a selected mathematical model when both the coefficient of friction and the flow stress are unknown for a given process, but also it can estimate the coefficient of friction and flow stress when the mathematical model selected is valid. 8

35 5. REFERENCE [1] Roberts, W.L., 1978, Cold Rolling of Steel, New York: Marcel Decker, Inc. [] Lee, C.H., and Altan, T., 197, Influence of Flow Stress and Friction upon Metal Flow in Upset Forging of Ring and Cylinders, ASME, J. Eng. Ind., Vol. 94, pp [3] Pietrzyk, M., Lenard, J.G., 1993, A study of the Plane Strain Compression Test, Annals of the CIRP, 4/1, pp [4] Lee, W.H., Kwak, J.H., Park, C.L., 1996, Proceeding of the Japan-USA symposium on flexible automation, ASME, New York, USA, xviii, pp [5] Rieldel, F., 1914, Stahl u. Eisen, Vol. 34, pp.19. [6] Rummel, K., 1919, Stahl u. Eisen, Vol.39, pp.37 [7] Siebel, E., and Pomp, A., 197, The determination of the deformation Resistance of Metals by Means of the Compression Test, Mitt. Kaiser Wilhelm Inst. Eisenforsch, Vol. 9, pp [8] Rastegaev, M.V., 1940, Neue Methode Der Homogenen Stauchen, Zavodskaja Laboratoria, Vol. 3, pp [9] Cook, M., Larke, E.C., 1945, Resistance of Copper and Copper Alloys to Homogeneous Deformation in Compression, J. Inst. Metals, Vol.71, pp [10] Alexander, J.M., et al., 1963, Manufacturing Properties of Materials, London, Van Nostrand. [11] Woodward, R.L., 1977, A note on the determination of accurate flow properties from simple compression tests, Metallurgical Transactions A, pp [1] Becker, N., Pöhlandt, K., 1989, Improvement of the plane-strain compression test for determining flow curves, Annals of the CIRP, 38/1, pp [13] Kunogi, M., 1956, A New Method of Cold Extrusion, J. Sci. Res. Inst., pp.15. [14] Male, A.T. and Cockcroft, M.G., 1964, Coefficient of Friction under Condition of Bulk Plastic Deformation, J. Inst. Metals, Vol.93, pp.38. [15] Schey, J.A., 1983, Tribology in Metalworking, American Society for Metals. [16] Male, A.T., 1966, Variations in Friction Coefficients of Metals during Compressive Deformation, J. Inst. Metals, Vol. 94, pp.11. [17] Male, A.T. and DePierre, V., 1970, The Validity of Mathematical Solutions for Determining Friction from the Ring Compression Test, J. Lub. Tech., Trans. ASME, Vol. 9, pp [18] Hawkyard, J.B. and Johnson, W., 1967, An Analysis of the Changes in Geometry of A short Hollow Cylinder during Axial Compression, Int. J. Mech. Sci., Vol. 9, pp.163. [19] Janardhana, M.N. and Biswas, S.K., 1979, Modes of Deformation in Aluminum Rings Subjected to Static Compression, Int. J. Mech. Sci., Vol.1, pp

36 [0] Abdul, N.A., 1981, Friction Determination during Bulk Plastic Deformation of Metals, Annals of the CIRP, Vol. 31/1, pp.143. [1] Nagpal, V., Lahoti, G.D., and Altan, T., 1978, A Numerical Method for Simultaneous Prediction of Metal Flow and Temperatures in Upset Forging of Rings, J. Eng. Ind., Trans. ASME, Vol. 100, pp [] Petersen, S.B., Martins, P.A.F. and Bay, N., 1998, An alternative ring-test geometry for the evaluation of friction under low normal pressure, J. Mat. Proc. Tech., Vol. 79, pp.14. [3] Luong, L.H.S., and Heijkoop, T., 1981, The Influence of Scale on Friction in Hot Metal Working, Wear, Vol. 71, pp. 93. [4] Male, A.T., 1964, The Effect of Temperature on the Frictional Behavior of Various Metals during Mechanical Working, J. Inst. Metals, Vol. 93, pp.489. [5] Kobayashi, S., 1970, Deformation Characteristics and Ductile Fracture of 1040 Steel in Simple Upsetting of Solid Cylinders and Rings, J. Eng. Ind., pp [6] Saul, G., Male, A.T. and DePierre, V., 1971, Metal Forming; Interrelation between Theory and Practice, ed. A.L. Hoffmanner, Plenum Press, New York, pp. 93. [7] Male, A.T., DePierre, V. and Saul, G., 1973, The Relative Validity of the Concepts of Coefficient of Friction and Interface Friction Shear Factor for Use in Metal Deformation Studies, Trans. ASME, Vol. 3, pp [8] Kudo, H., 1960, Some Analytical and Experimental Studies of Axi-symmetric Cold Forging and Extrusion, Int. J. Mech. Sci., Vol., pp. 10. [9] Avitzur, B., 1964, Forging of Hollow Discs, Israel J. Tech., Vol. 3, pp. 95. [30] Liu, J.Y., 1971, J. Eng. Ind., Trans. ASME, Vol. 93, pp [31] Lee, C.H. and Altan, T., 197, Influence of Flow Stress and Friction upon Metal Flow in Upset Forging of Rings and Cylinders, J. Eng. Ind., Trans ASME, Vol. 94, pp [3] Hartley, P., Sturgess, C.E.N. and Rowe, G.W., 1979, Friction in Finite- Element Analyses of Metalforming Processes, Int. J. Mech. Sci., Vol. 1, pp [33] Bugini, A., Maccarini, G. and Giardini, C., 1993, The Evaluation of Flow Stress and Friction in Upsetting of Rings and Cylinders, Annals of the CIRP, Vol. 4/1, pp [34] Honeycombe, R.W.K., 1984, The Plastic Deformation of metals, London, Edward Arnold Ltd. [35] Hill, R., 1948, A Theory of the Yielding and Plastic Flow of Anisotropic Metals, Proc. Roy. Soc., Vol. A193, pp [36] Hill, R., 1985, The mathematical theory of plasticity, Oxford University Press, New York. [37] George E. Dieter, 1984, Workability Testing Techniques, American Society for Metals, Ohio. 30

37 [38] Wagoner, R.H., Chenot, J.L., 1996, Fundamentals of Metal Forming, New York, John Wiley & Sons. Tnc. [39] ASTM E a, 1996, Standard Test Method for Plastic Strain Ratio for Sheet Metal. [40] Ståhlberg, U., 001, Materialens Processteknologi, Plastisk Bearnetning, Stockholm. [41] Montmitonnet, P. and Chenot, J.L., 1995, Introduction of Anisotropy in Viscoplastic D and 3D Finite-Element Simulations of Hot Forging, J. Mat. Proc. Tech., Vol. 53, pp [4] Pöhlandt, K., Lange, K. and Zucko, M., 1998, Concepts and experiments for characterizing plastic anisotropy of round bars, wires and tubes, Steel Research, Vol. 69, No. 4+5, pp. 171 [43] Pöhlandt, K., Lange, K. and Zucko, M., 1999, Effects of anisotropy parameters in cold bulk metal forming, Wire, Vol. 4, pp. 33. [44] Hill, R., 1979, Theoretical plasticity of textured aggregates, Math. Proc. Camb Phi. Soc., Vol.85, pp [45] Hill, R., 1993, A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci., Vol. 35, pp.19. [46] Aukrust, T., et al., 1997, Coupled FEM and Texture Modeling of Plane Strain Extrusion of an Aluminum Alloy, Int. J. Plasticity, Vol. 13, pp [47] Brunet, M. and Morestin, F., 001, Experiemntal and Analytical Necking Studies of Anisotropic Sheet Metals, J. Mat. Proc. Tech., Vol. 11, pp [48] Xu, S. and Weinmann, K.J., 1998, Prediction of Forming Limit Curves of Sheet Metals Using Hill s 1993 User-Friendly Yield Criterion of Anisotropic Materials, Int. J. Mech. Sci., Vol. 40, No. 9, pp

38 Paper A

39 Comparison of Different Methods for Estimating Anisotropy with Rotational Axial Symmetry in Bulk Metal Forming Han Han Materials Forming, Dept. of Production Engineering, Royal Institute of Technology Stockholm, Sweden ABSTRACT To determine anisotropy of the extruded aluminum alloy AA608 (Al- Si1Mg0.9Mn0.1), three methods (of yield loci, strain ratio and flow stress) have been utilized and evaluated. Small specimens were manufactured in different directions out of the round bars that had been pre-strained to 0, 15, 5 and 35%. Experimental results showed that the yield loci measurement could not fully predict anisotropy when the strain hardening of materials is direction-dependent. Second, the strain-ratio measurement is weak to estimate anisotropy during large plastic strains because of instability in tension. Finally, it is clear that the flow stress measurement in different directions can comprehensively determine the anisotropy of bulk materials. And, the average stress ratios calculated from the flow stress measurement are also valid for the anisotropic characterization with Hill s (1948) criterion because the simulation results are in good agreement with the testing results. In addition, the influence of anisotropy on metal flow has been analyzed by means of FEM. In the cases of upsetting and forward extrusion, results from simulations show that anisotropic materials are more sensitive to friction and the shear strain on the contact surface of the anisotropic workpiece is significantly higher than that of isotropic materials. 1. INTRODUCTION Anisotropy is an inescapable phenomenon in metalworking. The major cause of mechanical anisotropy in polycrystalline materials is crystallographic texture. With increasing strain, the randomly distributed individual crystals in metals tend to rotate towards a certain preferred crystallographic orientation (texture). Therefore, the mechanical properties (yield strength or strain hardening) of metals become direction-dependent (anisotropy). The typical modes of forming are rolling, extrusion and drawing. When such products are annealed, the recrystallized grains may also possess a preferred orientation, which in many cases is even stronger than the deformation texture (Honeycombe 1984). The second cause of mechanical anisotropy is related to the development of back stress or internal residual stress 1

40 induced by prior deformation, such as Bauschinger effect. Due to anisotropy, the strength of materials becomes direction-dependent and the flow competition occurs in the workpiece during plastic deformation, in turn, influencing the shapes of final products and formability. One typical example is the earring deformation during cup-deep-drawing from rolled sheets (Balasubramanian 1998). Since Hill (1948) first proposed a yielding criterion to describe the anisotropy of materials, anisotropy has been recognized as an important factor in sheet forming operations, in which concepts of the planar anisotropy and the normal anisotropy are well established. To examine anisotropy of materials, experiments are usually carried out with biaxial stress measurement (Liu 1997) or stress combined with strain measurement (ASTM 1996). These methods are suitable for sheet metal products under the condition of a limited strain that is usually less than 0.1 (Koistinen 1978). Increased attention has been given to anisotropy in bulk metal forming only recently (Montmitonnet 1995). According to the definition of the normal anisotropy in sheet forming, Pöhlandt et al (199, 1998, 1999) defined several anisotropic concepts for extruded or drawn materials and established a strain ratio method for estimating anisotropy. Practically, the metal flow and the strain path change in bulk forming are relatively complex than sheet forming, because the metal can be compressed and tensioned in different directions. Therefore, it is desirable to identify an appropriate method to predict anisotropy and its influence on metal flow of bulk materials. The purpose of this study is to evaluate different methods (yield loci, strain ratio and flow stresses) of determining anisotropy. Results show that the flow stresses measurement in multi-direction gives a better estimation of the anisotropy of bulk materials, compared to the other two methods. Furthermore, to investigate the influence of anisotropy on metal flow in bulk forming process, the cases of upsetting and forward extrusion have been analyzed by means of the finite element method (FEM). Results from simulations demonstrate that the shear strain on the contact surface of an anisotropic workpiece is significantly higher than that of isotropic materials.. ANISOTROPY & EXPERIMENTAL METHODS.1. Radial, tangential and axial anisotropy Usually, the extruded round metallic bars possess orthotropic rotational and symmetrical anisotropy. Like for the R-value (strain ratio) in sheet metal, several definitions of anisotropy were given by Pöhlandt et al. (1998) according to the locations of small cylindrical specimens, shown in Fig. 1 (a)-(b). The general strain ratio is given by

41 R ε k ( ε l ) ( ε l ) = ; i, k r, t, z (1) ε ( ε ) ik = i l where rk ( ε l ) ri ( ε l ) h ε k ( ε l ) = ln( ); ε i ( ε l ) = ln( ); ε l = ln( ) () r r h o o In Eqs. (1) and (), it is assumed that a small cylinder specimen is deformed to the principal strain ε l where l i, k and that radial strains, ε k ( ε l ) and ε i ( ε l ), in the cross section of the cylinder are located in the two perpendicular principal axes, in which i k. Therefore, six strain ratios exist. Due to anisotropy, the shape of the cross section of the deformed cylinder specimen is no longer a circle, so that R ik 1. If R ik cannot be predetermined by the experimental conditions, it implies that the strain ratio is a function of the plastic anisotropy of the material and the 1 strain ε l. When inverse conditions Rik ( ε l ) = Rki ( ε l ) are satisfied, only three strain ratios (R-values) are independent, and are defined as R zr ( ε t ) axial-radial anisotropy, R rt ( ε z ) radial-tangential anisotropy and R zt ( ε r ) axial-tangential anisotropy. If a specimen is situated in the center, the difference between R zr ( ε t ) and R zt ( ε r ) in locations disappears. This strain ratio R z ( ε r ) is defined as axial anisotropy. In the current work, the strain ratios of R zr ( ε t ), R zt ( ε r ) and R z ( ε z ) are utilized, shown in Fig. 1 (b). o z t r z y x P(3,6) r P(1,4) Rz Rzt Rzr r P(,5) (a) (b) Fig. 1. (a) Coordinate systems, and (b) Locations of the specimens for measuring strain ratios and flow stresses. P stands for the strain path; 1,,3 represent the strains in compression, while 4,5,6 indicate the strains in tension. 3

42 For convenient description, the local coordinates of three specimens for R zt ( ε r ), R zr ( ε t ) and R z ( ε r ) are all defined by Cartesian coordinates x, y, z and the directions are shown in Fig. 1 (a). Thus, the strain ratios are written by, R R R zt zr z ε x ( ε y ) ε t ( ε r ) = ε ( ε ) ε ( ε ) ε x ( ε y ) ε r ( ε t ) = ε ( ε ) ε ( ε ) ε x ( ε y ) ε t ( ε r ) ε r ( ε t ) = = ε ( ε ) ε ( ε ) ε ( ε ) z z z y y y z z z r r t z t (3). Yield loci & flow stresses in multi-directions The yield loci measurement is a conventional method to determine the anisotropy of materials. In the cylindrical coordinate, three principal axes in the π-plane stand for the stresses in tension (positive) and compression (negative) directions of r, t and z. Meanwhile, a numbers of six pure shear states are located at 30 o to each tension or compression direction (Lode 196), shown in Fig. 1. These pure shear states can be replaced by plane strain states, because the plane-strain is equivalent to a pure shear plus a hydrostatic component, proved by Mohr s circle in Fig.. τ τ τ σ σ σ h σ Plane strain Pure shear Hydrostatic Component Fig.. Equivalence of the plane-strain. In the current work, flow stresses in different directions were determined using tensile, compression and plane-strain compression tests and yield loci were constructed by the first yield points of all flow curves at the offset plastic strain ε=0.00. Locations of the specimens for tensile and compression tests are shown in Fig. 1 (b). Plane-strain compression tests were conducted using the thin sheets machined out of round bars in different locations and compressed by two parallel rectangular tools in a certain direction. The detailed requirement about the method 4

43 can be referred to Watts ( ). An example of the plane strain compression tests is illustrated in Fig. 3. z r t t z (a) (b) Fig. 3 Example of plane-strain compression tests: (a) Location of specimen, and (b) Loading direction..3. Yield criteria and parameters In the current work, the anisotropy of the material is characterized by Hill s criterion (1948) in the cylindrical coordinate ( r, t, z). The yield function is given by f ( σ ) F( σ σ ) + G( σ σ ) + H ( σ σ ) + Lτ + Mτ + Nτ ij tt zz zz rr rr tt tz rz rt = (4) The σ-values and τ-values are the stress components in the principal and shear directions. If principal axes of the anisotropy are made to coincide with the directions of the reference coordinate, parameters F - N can be expressed by the yield stress ratios ξ ij, F = ( ) + ( ) ( ) ξtt ξ zz ξ rr G = ( ) + ( ) ( ) ξ rr ξ zz ξtt H = ( ) + ( ) ( ) ξ rr ξtt ξ zz L = ; M = ; N = ξ ξ ξ tz rz rt (5) 5

44 where: σ rr σ tt σ zz ξrr = ; ξtt = ; ξzz = σ σ σ o o τ rt τ rz τ tz ξrt = ; ξrz = ; ξtz = τ τ τ o o o o (6) The σ o and τ o stand for the reference yield stress and the shear yield stress. If F=G=H=1 and N=M=L=3, Eq. (4) becomes von Mises criterion, describing the behavior of isotropic materials. A3 A A1 A1 A3 A z A1-A1 t 45 (b) A-A (c) z A3-A3 (a) r 45 r 45 (d) (e) Fig. 4. Locations of specimens for determination of Hill s parameters To obtain stress ratios for Hill s criterion, three specimens were made to coincide with principal directions and another three were manufactured in 45 o to each principal direction, shown in Fig EXPERIMENTAL PROCEDURES AND RESULTS 3.1 Experimental procedures To evaluate different methods in the estimation of the anisotropy, the strain history of the extruded aluminum alloy AA608 was constructed in several steps. First, the aluminum round bars ( 51) were heated from room temperature to 410 o C, at which it was kept for ~3 hours, then cooled to 50 o C at rate of 30 o C/hour, and finally air- 6

45 cooled at room temperature. Second, the annealed aluminum round bars ( 51) were cut into several pieces ( 51x 61mm) and were upset (pre-strained) to 0, 15%, 5% and 35% in the z-direction, shown in Fig. 5. Third, the small specimens were machined out of the pre-strained round bars into the ones with geometrical sizes of 5x5 mm for compression tests and 3x13 mm for tensile tests, also with mm in thickness and 50 mm in width for plane-strain compression tests, shown in Figs. 1,3, 4. In plane strain compression tests, the dimensions of the rectangular compression tools were 5 mm in width and 90 mm in length. Teflon was employed as a lubricant for all tests. 0% 15% 5% 35% Fig. 5. Pre-straining up to 0, 15, 5 and 35%. 3. Anisotropy determined by flow stresses In the case of 0% pre-straining, Fig. 6 shows that the flow curve in the z-direction is 7% higher than those in the r- and t directions, where the values are very close, see Eq. (7). Also, the values of the flow stresses obtained from the tensile and the compression in both z- and t- direction are approximately the same, see Eq. (8). This indicates that the annealed aluminum alloy AA608 possesses planar anisotropy. σ z ( Comp) > σ r ( Comp) σ t ( Comp) (7) σ ( Ten) σ ( Comp) > σ ( Ten) σ ( Comp) (8) z z t t Fig. 6. Flow curves of the annealed aluminum alloy AA608 in directions of r, t and z. C=Compression; T=Tension. 7

46 When the pre-strain increases to 35%, it is observed that, in compression tests, the strength of the flow stress in the z-direction is still higher than those in other directions, see Fig. 8 and Eq. (9). But in tensile tests, the positions of the flow curves No. 4 and No. 5 are reversed, comparing the case of 0% pre-strain, see Eq. (10). Furthermore, the values of the flow stresses between compression and tension tests in the same direction are no longer equal, see Eq. (11). σ z ( Comp) > σ r ( Comp) σ t ( Comp) (9) σ z ( Ten) < σ t ( Ten) (10) σ ( Ten) < σ ( Comp) (11) z z These phenomena can be attributed to the strain path change (strain history). When an aluminum round bar is pre-upset in z, it is naturally expanded (tensioned) in the circumferencial direction which may not be the same as the direction of the on-going strain (testing direction). The strain-path change between pre- and on-going strain in specimens Nos. 1 to 5 (in Fig.7) is shown in Tab. 1. It can be seen that a reverse straining for Specimen Nos., 3, and 4 occurs. Therefore, it is very likely that Bauschinger effect happened, especially to Specimen No. 4 in tension. That is why flow stress No. 4 is lower than No. 5 and also the flow stress in tension is lower than compression, Eq. (11). Similarly, a reverse straining also occurs in the t-direction. Figure 9 shows that with increasing pre-strain, the flow stress in tension is slightly higher than compression. Tab. 1. Relations between pre- and on-going strain Curves No. 1 No. No. 3 No. 4 No. 5 Pre-strain C T T C T On-going strain C C C T T Strain direction CT Rev. Rev. Rev. CT Note: C=Compression; T=Tension; CT= Continuous; Rev.= Reverse In sum, for each pre-strain (0% to 35%), the flow stresses in compression directions demonstrate that the anisotropy of the material is kept approximately at the same level. With increasing pre-strain, the ultimate tensile strain decreases and Bauschinger effect increases gradually (Figs. 6-8). 8

47 Fig. 7. Flow curves of the annealed alloy AA608 in directions of r, t and z for 35% pre-straining. C=Compression; T=Tension. Fig. 8. Flow curves of annealed alloy AA608 in the t-direction after pre-straining 0, 15, 5 and 35%. C= Compression; T= Tension. 3.3 Anisotropy estimated by strain ratios According to the concepts of R zt ( ε r ), R zr ( ε t ), R z ( ε r ) in Section.1 and the material properties obtained in Section 3., the phenomena of anisotropy estimated by the strain ratio method are as follows: 1. Flow competition and instability Figure 9 presents the results. At each level of pre-straining, the material flow in the x-direction is faster than that in the z-direction (flow competition) due to σ ( Ten) > σ ( Ten), causing ε ( Ten) > ε ( Ten). Meanwhile, with increasing the local z x x z 9

48 strain ε y (upsetting), each cylindrical specimen is being expanded (tensioned) along the circumferencial direction, illustrated in Fig. 10. When tensile strains are beyond ultimate strains, the instability in tension occurs, shown in Figs. 6 to 8. Thus, the plastic anisotropy can not be kept in the same level as the beginning. Therefore, the difference between ε x(ten) and ε z(ten) in values tends to be smaller and then the strain ratios, R ε ), R ε ) and R ε ) decrease with the local straining ε y. zt ( r zr ( t z ( r Fig. 9. Strain Ratios of Rz, Rzt and Rzr. 10

49 z ε z (Ten) ε x (Ten) x Fig. 10. Cylinder specimen expanded (tensioned) in the circumferencial direction during upsetting in y.. Flow competition and Bauschinger effects The strain path change between pre- and on-going strains for specimens in Fig. 1 (b) can be summarized in Tab.. It reveals that all specimens are pre-compressed in z and pre-tensioned in local x and y; and that a reverse straining in z (Column in Tab. ) and a Continuou straining in x (Column 3 in Tab. ) occur during on-going strain. With increasing pre-strain up to 15, 5, 35%, the difference between σ z(ten) and σ x(ten) in values becomes gradually smaller due to Bauschinger effect in z. Thus, the flow competition in the cross-section of the specimens is not as strong as the initial stage (pre-strain 0%). All values of the strain ratios are getting close to 1. Tab.. Relation between pre- and on-going strain R z ε z ε x ε t ε y ε r Pre-strain C T T On-going strain T T C Strain direction Rev. CT Rev. R zt ε z ε x ε t ε y ε r Pre-strain C T T On-going strain T T C Strain direction Rev. CT Rev. R zr ε z ε x ε r ε y ε t Pre-strain C T T On-going strain T T C Strain direction Rev. CT Rev. Note: C=Compression, T=Tension, CT= Continuous, Rev.= Reverse 3. Extreme phenomenon When pre-straining up to 35% (see Fig. 5), the degree of pre-tension in the local x on specimen R zt is the strongest among the others because of its location, Fig. 1 (b). Thus, the strongest Bauschinger effect can be observed. The evidences are 11

50 σ ( Ten) > σ ( Ten) σ ( Ten) for 0% pre-strain, Fig. 6; and z x t σ z( Ten) < σ x( Ten) σ t( Ten) for 35% pre-strain, Fig. 7. This can be called extreme phenomenon, indicating that the region of the values of R zt ( ε r ) has been changed from > R ( ε ) 1 to 0 R ( ε ) 1, shown in Fig. 9. zt r < zt r Fig. 11. Two types of strain ratios. Fig. 1. Yield Loci of Annealed Aluminum AA608 at different pre-strain 0, 15 5 and 35%. 1

51 To conclude, the principle of the strain ratio method for estimating anisotropy is based on the flow competition in the cross-section of the cylindrical specimen, in which the flow competition is caused by the different strength of the flow stress in tension along the circumferencial direction, Fig. 10. Thus, the strain ratios are not only determined by the material plastic anisotropy, but also restricted by the instability of the material in tension during large strains. Generally, two patterns of change of the strain ratios exist: (a) > strain ratio 1, when σ z ( Ten) σ x ( Ten) ; (b) 0 < strain ratio 1, when σ z ( Ten) σ x( Ten) ; shown in Fig. 11. If σ z ( Ten) σ z ( Comp) and σ x ( Ten) σ x ( Comp), see Figs. 7, 11 and Eq. (3), the strain ratio method is weak to estimate the anisotropy in the circumferencial compression direction. 3.4 Anisotropy predicted by yield loci Figure 1 shows that the aluminum alloy AA608 possesses the strongest anisotropy after annealing. With increasing pre-strain, a slight kinematic hardening occurs in the z- direction. Meanwhile, it seems that the aluminum AA608 remains isotropy, because the yield surface becomes a circle except the yielding in z-direction at 35% pre-strain. It is likely that yield loci method cannot fully predict anisotropy. For example in Fig. 7, there is no significant difference in the initial yielding among flow curves 1,, 3, but the anisotropy still exists in the material during large strains, due to strain hardening. 4. INFLUENCE OF ANISOTROPY ON SHEAR STRAIN IN CONTACT REGION 4.1 Case of cylinder upsetting First, to verify the flow stress method of determining anisotropy, the finite element method was carried out with ABAQUS, in which the constitutive model of the aluminum AA608 was characterized by Hill s criterion. As Hill s criterion with constant (initial) parameters fails to account for the evolution of the anisotropy during large strains, the average stress ratios calculated from strains (0.1, 0.,, 0.7) were utilized in the current work, shown in Tab. 3. Four loading-displacement curves measured from cylinder upsetting tests have been verified by the FEM. The detailed geometrical sizes of the cylinder specimens are listed in Tab. 4. Figure 13 shows that the average stress ratios obtained are reasonable since the results from the FEM are in good agreement with those from the experiments. Second, in cylinder upsetting, barreling and foldover are common phenomena on the free surface when the friction reaches to a certain level. Ettouney and Stelson (1990) suggested using the measurement of the foldover to estimate friction. This method 13

52 might be valid for isotropic materials, but not for anisotropy. For a deeper understanding of the influence of anisotropy on metal flow in the contact region during bulk forming, four materials apart from the tested aluminum alloy AA608 were assumed to be as (a) stronger anisotropy, (b) isotropy, (c) isotropy with stronger hardening, and (d) isotropy with non-hardening, see Tabs. 3 and 5. For this reason, the shear strain on the contact surface for different materials was examined by FEM in upsetting case under a friction condition µ = A billet was assumed as 10 mm in diameter and 10 mm in height. The upsetting speed was 10 mm/s. All elements along the contact surface of one quarter of the cylinder were traced. The numerical simulation was performed as a non-linear explicit dynamic analysis. Tab. 3. Stress ratios for Hill s criterion. Stress Ratios ξ rr ξ tt ξ ξ zz rt ξ rz ξ tz Isotropy AA Ani-30% Tab. 4. Cylinder specimens for Fig. 13. No. 1 No. No. 3 No. 4 Height (mm) Diameter (mm) Fig.13. Loading-displacement curves of measurement and Hill s criterion in FEM. 14

53 Tab. 5 Materials used in finite element analyses Name Characteristics Model (MPa) AA608 σ r =σ t =0.93σ z σ=190ε 0.15 Ani-30% σ r =σ t =0.7σ z σ=190ε 0.15 Iso(608) σ r =σ t =σ z σ=190ε 0.15 Hardening(06) σ r =σ t =σ z σ=90+190ε 0.6 Non-hardening σ r =σ t =σ z σ=90 Fig. 14. Distribution of shear strain on the surface of cylinder specimen for different materials. Figure 14 illustrates the distribution of shear strain on the contact surface for five materials. The trend is that the stronger the anisotropy, the higher the shear strain on the surface is. And, the foldover of the free surface was found in the case of the pronounced anisotropic material, showing the lower strength of the shear strain at the edge point due to the fact that the free surface is moving up to the contact area. On the contrary, the strength of the shear strain for isotropic materials remains the same, no matter the isotropic material possesses strain hardening or not. And, the shear strain on the edge for isotropic materials increases in an opposite direction to the middle part. This implies that the shear strain does not easily occur on the surface of isotropic materials. The profiles of foldover of the pronounced anisotropic material and the isotropic material are illustrated in Fig. 15. Furthermore, comparing shear stresses of five materials on the contact surface, anisotropy does not influence the strength of friction shear, because the same values for different anisotropic materials of AA608 and Ani-30% are observed in Fig. 16. In contrast, strain hardening of isotropic materials does. From these phenomena, it is clear that anisotropic materials are more sensitive than isotropic ones for a given friction, and the foldover for anisotropic material occurs easily. 15

54 Foldover (a) Pronounced anisotropic material Foldover (b) Isotropic material Fig. 15. Profiles of foldover of (a) the pronounced anisotropic material, and (b) the isotropic material. Fig. 16. Distribution of friction shear on the contact surface of deformed cylinder specimens. 16

55 4. Case of Forward Extrusion To verify the results obtained from cylinder upsetting, a forward extrusion was employed in the FEM. A billet was assumed to be 100 mm in diameter, 300 mm in length and with 30% reduction. The friction condition is µ = 0. 1 and the extrusion speed is 50mm/s. One element on the surface of the billet was traced from the beginning to the end during extrusion. As the forward extrusion is a process with a forced flow and without free surface, the influence of anisotropy on the shear strain was not found at the beginning of extrusion, region 1 in Fig. 17. When the material was continuously extruded, the shear strain increased in the reduction area, region and large variations of the shear strain were observed in the area between the end of the reduction and the beginning of the exit. The trend is that the stronger the anisotropy, the higher the shear strain is, and these values are kept constantly at the exit, region 3. In contrast, the shear strain of isotropic materials does not increase in the same way as that of anisotropic materials. Instead, it decreases in the area between regions and 3. Therefore, the result obtained from upsetting case has been further examined. 1 3 Fig. 17. Shear strains of different materials in forward extrusion. 17

56 5. SUMMARY AND CONCLUSIONS In the current work, three measurements (of yield locus, strain ratio, flow stress) for estimating anisotropy have been utilized and analyzed in terms of strain path change, strain hardening and instability of materials. Furthermore, the influence of anisotropy on shear strain in the contact region has been studied by the FEM in two cases (upsetting and forward extrusion) with five materials. Conclusions can be drawn as follows: Determination of anisotropy The yield loci method cannot fully predict the anisotropy when the material strainhardening is direction-dependent, because the first yield points of the flow curves may not present the whole anisotropic properties during large strains. Meanwhile, the strain ratio method is weak to estimate anisotropy. One reason is that the strain ratio is not only determined by anisotropy, but also restricted by the instability of the material in tension. Second, if σ z ( Ten) σ z ( Comp) and σ x ( Ten) σ x ( Comp), the strain ratio method cannot estimate the anisotropy in the circumferencial compression direction. Third, during large strain, the strain ratio may indicate that the material is isotropic, for example Rzt 1 at pre-strain 35% in Fig. 9, but anisotropy still exists, shown in Fig. 7. Finally, the flow stress method is the one that can comprehensively estimate the anisotropy of bulk materials. Influence of anisotropy Anisotropic materials are more sensitive to friction and the shear strain on the contact surface of the anisotropic workpiece is significantly higher than that on isotropic one. Acknowledgments The author would like to thank Professor Ulf Ståhlberg for reading the manuscript and valuable discussion. Most experiments were carried out at Dept. of Solid Mechanics. Many thanks go to Hans Öberg for his valuable help during the laboratory work. The financial support from the Royal Institute of Technology, Sweden and Brite-EUram project BE is acknowledged. Also, special thanks go to Professor Pavel Huml for the discussion on experiments. References ASTM E a, 1996, Standard Test Method for Plastic Strain Ratio r for Sheet Metal. 18

57 Balasubramanian, S. and Anand, L., 1998, Polycrystalline plasticity: Application to earing in cup drawing of A1008-T4 sheet. ASME J. Appl. Mech., Vol. 65, pp Ettouney, O.M. and Stelson, K.A., 1990, J. Eng. Ind., Vol. 11, pp.67. Hill, R., 1948, Proc. Roy. Soc., vol. A193, pp Honeycombe, R.W.K, 1984, The Plastic Deformation of Metals, London, Edward Arnold Ltd. Koistinen, D.P. and Wang, N.M., 1978, Mechanics of Sheet Metal Forming, Plenum Press, New York. Liu, C., Huang, Y. and Stout, M.G., 1997, On the Asymmetric Yield Surface of Plastically Orthotropic Materials: A phenomenological study, Acta Mater., Vol. 45, No. 6., pp.397. Lode, W., 196, Zeitsch. Phys., Vol. 36, pp Montmitonnet, P. and Chenot, J.L., 1995, Introduction of anisotropy in viscoplastic D and 3 D finite-element simulations of hot forging, J. Mat. Proc. Tech., Vol. 53, pp Pöhlandt, K. and Oberländer, T., 199. Concepts for the description of plastic anisotropy in cold bulk metal forming, J. Mat. Proc. Tech., Vol. 34, pp.187. Pöhlandt, K., Lange, K. and Zucko, M., 1998, Concepts and experiments for characterizing plastic anisotropy of round bars, wires and tubes, Steel Research, Vol. 69, No. 4+5, p. 171 Pöhlandt, K., Lange, K. and Zucko, M., 1999, Effects of anisotropy parameters in cold bulk metal forming, Wire, Vol. 4, pp. 33. Watts, A. B. and Ford, H., , Proc. Inst. Mech. Eng, 1B, pp

58 Paper B

59 Influence of Material Anisotropy and Friction on Ring Deformation Han Han Materials Forming, Dept. Production Engineering, Royal Institute of Technology Stockholm, Sweden Abstract The influence of material anisotropy and friction on ring deformation has been examined in relation to the distribution of normal pressure and frictional shear stress, deformed ring shapes, and estimated errors in the coefficient of friction. Based on the flow rule associated with von Mises and Hill s yield criteria, the analyses have been carried out with the finite element method (FEM) for three cases, namely, (1) an anisotropic ring oriented 90 o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction; () an isotropic ring under frictional anisotropy condition; and (3) an anisotropic ring oriented 0 o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction. In the first two cases, the results show that the influence of anisotropy on ring deformation is quite similar to that obtained by changing the frictional condition. Therefore, in the third case, if the anisotropic behavior is mistakenly attributed to friction, the possible estimated error for the coefficient of friction can be as high as 80% for a pronounced anisotropic material. Deformed ring shapes have been verified in experiments using the extruded annealed aluminum alloy AA608 (Al-Si1Mg0.9Mn0.1). Keywords: Anisotropy, coefficient of friction and ring deformation 1. Introduction In metal forming processes, friction is one of vital variables. As friction increases forming load and power, changes material flow, strain/stress distribution and influences form filling, it must be determined. The ring-compression test is one of typical methods for the quantitative evaluation of friction in bulk metal forming. It was first proposed by Kunogi [1] for comparing lubricants in cold forging, and was later improved by Male and Cockcroft [] who created the first calibration diagram. With this method, the coefficient of friction can be estimated through the change in the internal diameter of the deformed ring. A number of works [3-1], both experimental and theoretical, showed the usefulness of this method in determining the friction at the die/workpiece interface when various friction models (Coulomb s friction, constant friction factor and general friction models) were used. Approaches of analysis consist of the Slab 1

60 method, the upper bound method and the finite element method (FEM). Recently, it has been noticed that the coefficient of friction estimated by ring-compression is higher or much higher than that by other methods [-4]. In conjunction with this, Garmong et al. [] has investigated the influence of strain-rate-sensitive-material; Danckert [3] has simulated the influence of materials with different strain hardening; Tan et al. [4] has analyzed the change of normal pressure on the ring surface for pre-extruded material. All of these works indicate that the change of a ring s internal diameter is not entirely independent of the material properties. In practice, most materials used in metal forming have varying degrees of anisotropy [5] because of their mechanic or crystallographic properties [6-8]. In ring experiments, Bhattacharya [6] showed that if the material possesses anisotropy, the deformed ring shapes (such as elliptic or tapering) are influenced by specimen orientation. Pöhlandt [7] suggested that ring-compression test can be a method for evaluating material anisotropy according to elliptic shape when the friction on the surface is reduced. From these two experimental works, the questions outstanding are: first, can the influence of anisotropy on ring flow be mistakenly attributed to friction? Second, are there conditions when the influence of anisotropy is present, even though it cannot be observed immediately in a test? The aim of this study is to provide a deeper understanding of the influence of anisotropy and friction on ring deformation. Based on the flow rule associated with von Mises and Hill s yield criteria, the analyses have been carried out with the finite element method (FEM) for three cases: (1) an anisotropic ring oriented 90 o to the axis of rotational symmetrical anisotropy under uniform coefficients of friction, Fig. 4 (a); () an isotropic ring under frictional anisotropy condition; and (3) an anisotropic ring oriented 0 o to the axis of rotational symmetrical anisotropy under uniform coefficients of friction, Fig. 4 (b). In the first two cases, the results show that the influence of anisotropy on ring deformation is quite similar to that obtained by changing the frictional condition. Therefore, in the third case, if the anisotropic behavior is mistakenly attributed to friction, the possible estimated error for the coefficient of friction can be as high as 80% for a pronounced anisotropic material. The deformed ring shapes have been verified in experiments using the extruded annealed aluminum alloy AA608 (Al-Si1Mg0.9Mn0.1).. Ring Flow, Material Constitutive Modeling and FEM.1 Ring flow The deformation of a ring is a particular issue for material flow in the transverse direction during the axial reduction. It is closely related to friction and material properties. In plasticity theory, the flow rule associated with a yield function f ( σ ij ) is usually formulated in Prager s form, dε pl f ( σ ij ) = dλ (1) σ

61 Pattern 1 of friction-ring-flow: Neutral Plane Pattern of friction-ring-flow: Neutral Plane Internal diameter increasing When friction is low. Pattern 1 of material-ring-flow: Internal diameter decreasing When friction is high. Pattern of material-ring-flow: θ θ r r Flow competition occurs in the cross section of a ring When σ rr = f (θ ). Non-flow competition occurs in the cross section of a ring When σ rr = Const.. Fig. 1. Patterns of ring flow. Based on the normality rule, the incremental plastic strain occurs in the direction normal to the yield surface f ( σ ij ). When ring specimens remain isotropic, the material properties are direction-independent and the plastic incremental strain, Eq. (1), can be described by Levy-Mises flow rule on the basis of von Mises criterion [9]. Thus, only the influence of friction on ring flow is recognized. As the internal diameter of a deformed ring can increase or decrease depending on low or high friction at the die/ring interface, two patterns of ring flow have been categorized by researchers, Fig. 1. When ring materials possess anisotropy, the yield function, however, is no longer direction-independent. Its incremental plastic strain will differ from isotropic cases under the same friction conditions. If the material anisotropy remains rotational symmetric (cylindrical orthotropic), another two typical patterns of ring flow can be identified according to the characteristics of stresses in the cross section of the ring determined by its orientation. In the first pattern, σ rr is a function of θ. Due to stress equilibrium, flow competition occurs in the cross section of the deformed ring. Thus, the internal diameter of the deformed ring is no longer constant with θ. In the second pattern, σ is independent of θ ; in this case, rr 3

62 flow competition cannot be observed and the internal hole of the deformed ring has rotational symmetry, Fig. 1. For convenience of discussion, ring flow patterns influenced by friction and material anisotropy in the current work are labeled as friction-ring-flow and material-ring-flow respectively. In practice, ring flow is a combination of friction-ring-flow and material-ring-flow.. Material constitutive modeling..1 Large plastic strain plus small elastic strain The ring flow of metallic materials can be treated as large plastic strain plus small elastic strain in large strain elasto-plasticity since the elastic strain is quite small compared to its plastic strain. To investigate the influence of anisotropy on ring flow, the deformation of a ring was restricted to room temperature and under quasistatic conditions. This means that any influence due by temperature and strain rate will be ignored. Therefore, the strain decomposition is el pl dε = dε + dε. () The relation of stress-strain in the elastic part can be expressed by el el el σ = D : ε. (3) el Here, D is a matrix determined by both Young s modulus E and Poisson s ratio pl ν. The incremental plastic strain dε in Eq. () represents flow rules defined by stress potential criteria as follows.... Plastic modeling In some cold forging processes, billets are cut from extruded round bars. Anisotropy is caused by deformation due to extrusion. With increasing strain, the randomly distributed individual crystals in metals tend to rotate towards a certain preferred crystallographic orientation (texture), so that the mechanical properties (such as yield strength and strain hardening) become direction-dependent (anisotropy). When such products are annealed, the recrystallized grains may also possess a preferred orientation, which in many cases is even stronger than the deformation texture [8]. Its plastic property commonly has rotational symmetry or cylindrical orthotropy [7]. Wagoner and Chenot [30] pointed out all existing formulations are approximate at best, and few have any connection to the micro-mechanism s response for a given material. Thus, Hill s criterion (1948) [31], Eq. (4), is still widely used for characterization of the macroscopic behavior of anisotropy in three dimensions. In the cylindrical coordinate ( r, θ, z), it is written by 4

63 5 ) ( ) ( ) ( ) ( θ τ τ θ τ θθ σ σ σ σ σ θθ σ σ r N rz M z L rr H rr zz G zz F ij f = (4) If the principal axes of anisotropy are made to coincide with the directions of the reference coordinate, parameters F-N can be expressed by yield stress ratios ij ξ, 3 ; 3 ; 3 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( θ θ θθ θθ θθ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ r rz z zz rr zz rr rr zz N M L H G F = = = + = + = + = (5) where: o z z o rz rz o r r o zz zz o o rr rr τ τ ξ τ τ ξ τ τ ξ σ σ ξ σ σ ξ σ σ ξ θ θ θ θ θθ θθ = = = = = = ; ; ; ; (6) Substituting Hill s criterion into Eq. (1), the flow rule associated with Hill s criterion is written by, = = + + z L rz M r N rr zz G zz F rr H zz F rr H rr zz G ij ij pl f d f d d θ σ σ θ σ σ σ σ θθ σ θθ σ σ σ θθ σ θθ σ σ σ σ σ λ σ λ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( σ ε (7) If F=G=H=1 and N=M=L=3 in Eq. (7), the flow rule is associated with von Mises criterion. In the current work, two criteria (von Mises and Hill s) were adopted, and Hollomon s and Ludwik s models were utilized for the reference flow stress in Eq. (6) when different materials were used:

64 n Hollomon s model: σ = kε (8) n Ludwik s model: = k k ε (9) σ Finite element method (FEM) To analyze the influence of anisotropy and friction on ring deformation, 3D finite element method (FEM), as implemented in the ABAQUS/Standard software [3], was used. The upper part of the ring was meshed with 979 elements in which each element was constructed with eight nodes. The orientations of a ring and its rotational symmetrical anisotropy were defined separately by their z-axes in two cylindrical coordinates. Accordingly, two stress spaces, σ (ring ) and σ (material), were utilized. Taking the ring oriented 90 o to the axis of the material for example, Fig. 4, the anisotropy is defined by stress ratios in σ (material) and the corresponding yielding properties in the ring are given by T T ( ring90) = R R1σ ( material) R1 R σ (10) where R 1 is a rotation matrix [33] from the cylindrical coordinate of the material to the Cartesian coordinate of the material and R is a rotation matrix [33] from the Cartesian coordinate of the material to the cylindrical coordinate of the ring, Fig. 4 (c). Combining Eqs. (5), (6) and (10), the Hill s parameters in the stress space of a ring can be obtained; and the calculation is a straightforward process in ABAQUS [3]. The contact condition at the die/ring interface was assumed as non-separated. The numerical simulation was performed as a non-linear quasi-static analysis. The friction condition was described by Coulomb s law (τ=µp). In the analysis, the constant friction model (τ=mk) was also considered. 3. Experimental Determination of Young s Modulus and Stress Ratios for Hill s criterion The commercial extruded aluminum alloy AA608 round bar was heated from room temperature to 410 o C, at which it was kept for ~3 hours, then cooled to 50 o C at rate of 30 o C/hour, and finally air-cooled at room temperature. Six cylinder specimens ( 5x5 mm) were machined from the annealed aluminum alloy AA608 bar ( 51) for compression, while a further two specimens ( 3x13 mm) were used for tensile tests. The specific locations of the specimens for stress ratios of Hill s criterion Eq. 6 are illustrated in Fig.. The experiments were carried out with a MTS 160 kn dynamic Press controlled by an Instron 8500 Model dynamic system. For testing accuracy, a suitable loading cell was chosen. Each specimen was tested at room temperature; loading speeds 6

65 were 0.01mm/s, chosen to ensure constant temperature and low strain rate ( ε& / s). Figure 3 shows that the value of the stress in the z-direction is 7% higher than in the r- and θ - directions; those values, incidentally, are very close to each other. A3 A A1 A1 A3 A z for τ θz A1-A1 θ 45 σ rr σ zz σ θθ (a) (b) A-A for τ rθ r 45 (c) z A3-A3 for τ rz r 45 (d) (e) Fig.. Specimens for obtaining stress ratios in Hill s criterion. Effective Stress(MPa) :Compression in z :Compression in θ 3:Compression in r 4:Tension in z 5:Tension in θ Effective Strain 1 3 Fig. 3. Effective stress-strain curves of the extruded annealed aluminum alloy AA608 in different directions. 7

66 Hill s theory is a criterion with constant (initial) parameters, and fails to account for the evolution of anisotropy during large strain, meaning that each stress ratio in Eq. 6 may varies with increasing strain. For example, in Fig. 3, it can be seen that the difference between σ z and σ r in compression is very small ( σ MPa) at pl pl initial yield point ( ε = 0.% ), but at ε = 0., the difference between them increases to σ 13MPa. Thus, in the current work, the average stress ratio in each direction, for example ξ rr = ( ξ rr ( ε rr = 0.1) ξ rr ( ε rr = 0.7)) / 7, was utilized for Hill s criterion, in which σ zz was selected for the reference stress σ o. Young s modulus, E, of the aluminum alloy AA608 was obtained from the tensile test in the z-direction and its Poisson s ratio, ν, was chosen from the reference [34]. To further investigate the influence of anisotropy on ring flow in the FEM, the published data for the wire steel AISI01 [35] was also adopted. The values of stress ratios of materials (AA608, AISI01, assumed isotropy) for Hill s and von Mises criteria are presented in Tab. 1, while the reference flow stresses and the elastic parameters are listed in Tab.. It shows that a pronounced anisotropy exists in the material AISI01. Tab. 1. Stress Ratios for Hill s criterion (axes: r,θ,z). Stress Ratios ξ rr ξ θθ ξ ξ zz r θ ξ rz ξ θz Isotropy AA AISI Tab.. Reference σ in Eq. (6) and elastic parameters σ (MPa) E (GPa) ν AA ε AISI ε Influence of Anisotropy and Friction on Ring Deformation 4.1 Three cases of ring compression design Based on Fig. 3 and the values of stress ratios in Tab. 1, materials (AA608 & AISI01) possess planar isotropy and the characteristics of stresses can be summarized by σ ( material ) zz σ ( material ) rr σ ( material ) θθ > (11) 8

67 Here, we define the plane on which σ (material) rr and σ (material) θθ are lying as the plane of planar isotropy, Fig. 4. Two typical orientations of the anisotropic rings and one assumed isotropic ring were utilized in the FEM analyses. For all the simulations and tests, the initial geometrical sizes of rings are d o :d i :h = 6:3: = 30 mm:15 mm: 10 mm [3]. The purpose of the three cases, as well as the yielding properties in the rings and the friction conditions, are as follows: Case 1: Anisotropic ring oriented 90 o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction Case 1 is to examine the influence of anisotropy on ring deformation in relation to the shape, the distribution of normal pressure and frictional shear. z (ring90) Planar isotropic plane A r (ring90) (a) 90 o orientation z (ring0) r (ring0) (b) 0 o orientation z (material) θ (material) r (material) A A y (material) A θ (material) x (material) r (material) θ (ring0) r (ring0) θ (ring90) z (material) r (ring90) z (ring0) z (ring90) Rotation 90 o (c) Rotation of coordinates Fig. 4. Axis of a ring rotated (a) 90 o to the orientation of anisotropy; (b) 0 o to the orientation of anisotropy and (c) Rotation of coordinates between orientations of rings and the material anisotropy. Figure 4 (a) illustrates the location of the anisotropic ring oriented 90 o to the z- axis of rotational symmetrical anisotropy; and Fig. 4 (c) presents the rotation of coordinates between the ring (90 o ) and the material. In this case, the loading direction of the ring is vertical to the axis of the material. The yielding property in the ring s coordinate system is given by Eq. (10) and the typical yielding properties in the radial direction of the ring can be simply described as 9

68 σ σ ( ring90) rr ( ring90) rr ( θ ( θ ( ring90) ( ring90) o o = 90 ) σ = 0 ) = σ ( material) rr ( material) zz (1) Since the condition σ > is maintained, it leads to ( material ) zz σ ( material ) rr σ ( ring90) rr ( θ = 0 ) > σ ( θ = 90 k ( θ = 0 ) > k ( θ = 90 ( ring90) rr ( ring90) ( ring90) o o ( ring90) rr ( ring90) rr ( ring90) ( ring90) o o ) ) (13) Therefore, σ ( ring 90) rr is a function of θ (ring90) and the stress anisotropy exists in the cross section of the ring. In the simulations, the flow rule associated with Hill s criterion was used, as described in Section.3. The values of coefficient of friction ranging from 0,07 to 0, were selected on the basis of the experimental conditions. Case : Isotropic ring under frictional anisotropy Case is to verify the influence of friction anisotropy on isotropic ring deformation with regard to the shape, the distribution of normal pressure and frictional shear, which may lead to a better understanding of the phenomena in Case 3. Thus, the ring material was intentionally assumed as isotropic, in which the yielding properties in three principal directions were given by σ = θθ = (14) ( ring ) zz σ ( ring) rr = σ ( ring) σ ( material) zz while the frictional anisotropy was applied at the die/ring interface. The definition of the frictional anisotropy is that the values of coefficient of friction in two perpendicular directions are physically different under the same normal pressure, and formulated by Coulomb s expression, o τ1 o τ µ 1 ( θ 0 ) = > µ ( θ = 90 ) = p p = (15) In simulations, the ratios of µ 1 /µ ranged from to 3, Fig. 10, and the flow rule associated with von Mises criterion was used. Case 3: Anisotropic ring oriented 0 o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction The third case is to clarify the influence of anisotropy on estimation of the coefficient of friction in ring-compression. 10

69 Figure 4 (b) shows the anisotropic ring oriented 0 o to the z-axis of rotational symmetrical anisotropy. The coordinate of the ring (0 o ) is shown in Fig. 4 (c). In this case, the yielding properties in the ring is given by σ = σ (16) ( ring 0) ( material) Under this condition, ( ring 0) zz σ ( material) zz σ ( ring 0) rr is independent of θ (ring 0), while the high stress, σ =, has been moved to the loading direction of the ring, see Eq. (11) and Fig. 4 (b), meaning that the stress isotropy is only kept in the cross section of the ring. In simulations, the flow rule and the values of coefficient of friction used were the same as in Case 1, but Eq. (10) was replaced by Eq. (16). 4.. Results and discussions Results of Case 1 Simulation results When the anisotropic ring is oriented 90 o to the z-axis of rotational symmetrical anisotropy, under the low coefficient of friction µ=0.07, both internal and external diameters of the ring increase (Pattern 1 of material-ring-flow plus Pattern 1 of friction-ring-flow ). Based on the stress equilibrium and the flow rule associated with Hill s criterion, the material flows more easily along the direction of lower yield strength ( o θ ( ring90) = 90 ) than in the direction of higher yield strength ( o θ ( ring90) = 0 ). The influence of material anisotropy on ring deformation is dominant. Due to flow competition, the final shape of the ring becomes elliptic and the degree of ellipticity depends on the degree of anisotropy (planar isotropy), as shown in Fig o 90 o 0 o 0 o σ r distribution (a) AA608 σ r distribution (b) AISI01 Fig. 5. Flow competition occurs in the cross section of anisotropic rings, µ=0.07. (Pattern 1 of material-ring-flow plus Pattern 1 of friction-ring-flow). 11

70 As the coefficient of friction is not zero, frictional shear stress exists at the die/workpiece interface. Figure 6 shows the influence of anisotropy on the distribution of frictional shear stresses and normal pressures in two directions ( o θ ). ( ring 90) = 90 o, 0 Fig. 6. The magnitude of the frictional shear stress and the normal pressure in 90 o -direction is lower than that in 0 o -direction. 1

71 90 o 90 o 0 o 0 o σ r distribution (a) AA608 σ r distribution (b) AISI01 Fig. 7. Internal surface of anisotropic rings flows to rings center, at µ=0.. (Pattern 1 of material-ring-flow plus Pattern of friction-ring-flow). Fig. 8. The influence of material anisotropy still exists since the distribution of normal pressure and frictional shear stress in 90 o - direction on the ring surface is not equal to that in 0 o -direction, at µ=0.. 13

72 When the coefficient of friction increased to µ=0., the deformed anisotropic ring became a circle for the material with slight anisotropy (AA608); in other words, the degree of ellipticity decreases if the material possesses higher anisotropy (material AISI01), shown in Fig. 7. Thus, the influence of friction on ring deformation becomes dominant, even though the influence of anisotropy still exists; this can be seen in the distribution of normal pressure and frictional shear stress in Fig. 8. The ring flow is a combination of Pattern 1 of material-ring-flow and Pattern of friction-ring-flow. Experimental results Two rings, shown in Fig. 4 (a), were manufactured from the extruded annealed aluminum alloy AA608 round bar. The deformation of rings was examined under two friction conditions (Teflon and dry condition). The tests were performed in a 500 kn Press. The compression speed is 0.01 mm/s. The direction of the original axis of the round bar was marked with black line on the surfaces of the rings Fig. 9. The stress characteristics are given in Eqs. (1) and (13). Comparing results from the FEM, the same patterns were found. That is, the degree of ellipticity decreases when the coefficient of friction increases. (a) Lubricant: Teflon (b) Lubricant: dry Fig. 9. Final shapes of rings (90 o orientation) under the friction conditions: (a) Teflon (µ=0.07); and (b) µ=dry condition. The black lines stand for the axis of the original extruded round bar for the aluminum alloy AA Results of Case Simulation results When frictional anisotropy condition is applied to the surface of the isotropic ring, the ring flow is naturally controlled by friction. The internal diameter of the ring o increases in the lower friction direction ( θ ), and decreases in the higher ( ring ) = 90 14

73 o friction direction ( θ ( ring ) = 0 ), (two Patterns of friction-ring-flow ). Therefore, the final ring s shape is elliptic. The degree of ellipticity depends on the ratio of friction anisotropy, such as µ 1 /µ = or 3, Fig. 10. Compared to the results of Figs. 6 & 8, Figure 11 shows that distribution of normal pressure and frictional shear stress is different from that in Case 1. The detailed variations of the average normal pressure and the average frictional shear in the 0 o - and 90 o -directions are calculated by o o ave_value(0 ) ave_value(90 ) variation = 100 (17) o ave_value(90 ) Table 3 Variation of p and τ in two directions Normal Shear AA % 4.7% µ=0.07 AISI01 15% 15% µ=0.07 µ 1 /µ = 1.4% 10% µ 1 /µ =3 3% 60% 90 o 90 o 0 o 0 o σ r distribution (a) µ 1 /µ =, µ =0.07 σ r distribution (b) µ 1 /µ =3, µ =0.07 Fig. 10. Under frictional anisotropy conditions, the isotropic ring is formed into an ellipse. Ring flow is Pattern of friction-ring-flow in 0 o direction, while it is Pattern 1 of friction-ring-flow in 90 o direction. 15

74 The calculated values are listed in Tab. 3, which indicates a fact that different frictional shear stresses in the two directions occur in all cases, but the reasons for this phenomenon are different. In Case 1 (anisotropic material), the percentage variation in the anisotropic frictional shear stresses corresponds to its variation of the anisotropic normal pressures. In Case (anisotropic friction), the anisotropic shear stresses are mainly determined by the corresponding coefficient of friction, and the variation of normal pressures in the two directions is very small. Fig. 11. Distribution of (a) normal pressure and (b) frictional shear stress in 90 o - and 0 o -direction under the frictional anisotropy condition. Experimental result In the experiment, the friction anisotropy condition was created by alternating between Teflon and dry conditions every 90 o on the ring surface, i.e. 1. Dry: µ 1 (-45 o <θ<45 o & 135 o <θ<5 o ) 0,15; 16

75 . Teflon: µ (45 o <θ<135 o & 5 o <θ<315 o ) 0,07. To keep isotropy in the cross section of the ring, the ring of 0 o orientation (material AA608) was chosen, Fig. 4 (b). Due to frictional anisotropy, the ring was deformed to an ellipse, Fig. 1. The ratio of µ 1 /µ is approximately 5, which is calculated from the testing results in Fig. 14 (a) and (c). Fig. 1. Influence of frictional anisotropy (µ 1 =dry condition and µ =Teflon) on ring deformation. Ring flow is Pattern of frictionring-flow in the area under dry condition, while it is Pattern 1 of friction-ring-flow in the Teflon area. Comparing the FEM results in Fig. 10 with the experimental result in Fig. 1, similar elliptic ring shapes were obtained, even though the anisotropic friction conditions between the FEM and the experiment were not exactly the same Results of Case 3 Simulation results When the orientation of the anisotropic ring is made to coincide with the z-axis of rotational symmetrical anisotropy, the shapes of the deformed anisotropic ring for two planar isotropic materials (AA608 & AISI01) are all circular, shown in Fig 13. The phenomena are quite similar to the behavior of isotropic material. Therefore, an incorrect impression might be obtained. That is, planar isotropy could be mistaken for isotropy because the deformed ring is not elliptic. 17