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TRITA-IIP-02-01 ISSN 1650-1888 Determination of Flow Stress and Coefficient of Friction for Extruded Anisotropic Materials under Cold Forming Conditions Han Han Stockholm January 2002 Licentiate Thesis Division of Materials Forming Department of Production Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden

Determination of Flow Stress and Coefficient of Friction ...7399/FULLTEXT01.pdfIn metal forming processes, the flow stress (material property) and the coefficient of friction (boundary

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Page 1: Determination of Flow Stress and Coefficient of Friction ...7399/FULLTEXT01.pdfIn metal forming processes, the flow stress (material property) and the coefficient of friction (boundary

TRITA-IIP-02-01 ISSN 1650-1888

Determination of Flow Stress and Coefficient of Friction for Extruded Anisotropic Materials

under Cold Forming Conditions

Han Han

Stockholm January 2002

Licentiate Thesis Division of Materials Forming Department of Production Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden

Page 2: Determination of Flow Stress and Coefficient of Friction ...7399/FULLTEXT01.pdfIn metal forming processes, the flow stress (material property) and the coefficient of friction (boundary

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-100 44 Stockholm, Sweden

Stockholm, January 2002

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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 900 to the axis of rotational symmetrical anisotropy. The friction coefficient was the same in all directions 2) An isotropic ring. The friction coefficient was different in different

directions 3) An anisotropic ring oriented 00 to the axis of rotational symmetrical anisotropy. The friction coefficient was the same in all directions

The cases 1) and 2) 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

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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 BE96-3340 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.

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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 Two-

specimen Method under the Unknown Coefficient of Friction and Flow Stress, accepted for publication in the Journal of Materials Processing Technology

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TABLE OF CONTENTS 1. INTRODUCTION………………………………….…………………...1 2. RELATIONSHIPS BETWEEN THE PAPERS………..…….…….…4 3. SUMMARY OF THE PAPERS……………………..…..……………..5 3.1 Paper A: Comparison of Different Methods for Estimating

Anisotropy with Rotational Axial Symmetry in Bulk Metal Forming.…….……………………………………………….5

3.2 Paper B: Influence of Material Anisotropy and Friction on Ring Deformation ………………...….…….…………...13 3.3 Paper C: The Validity of Mathematical Models Evaluated

by Two-specimen Method under the Unknown Coefficient of Friction and Flow Stress .…….…………….21

4. CONCLUDING REMARKS……………………….………..………..28 5. REFERENCES…………………………………………………..…….29 6. PAPERS A, B and C

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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 [2-4]. 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 [9-12]. 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. [42,43] defined several concepts of anisotropy Fig. 2 and corresponding strain ratios for estimating anisotropy in extruded or drawn metals. Although many studies have been done on anisotropy such as yield criteria

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[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, 1914Duble-Cup Specimen

Rummel & Meyer, 1919Three Specimens Method

Someone, After 1914Narrow-neck Specimen

Siebel & Pomp, 1927Concave Specimenand concave tools

Rastegaev, 1942Rastegaev Specimen

Someone, After 1942Multi-Rastegaev Surface

Specimen

µ

1 2 3 4Cook & Larke, 1945

Graphic Method withFour Specimens

Cook & Larke, 1945

Calculated Curve

Reduction in height %

Ton

s/Sq

. IN

Alexander, 1963Extrapolation procedure for Cook and Larke test

Compression

Ho/do=1Ho/do=1.5

Ho/do=2

Loa

d

Fig.1. Methods for reducing friction in upsetting.

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Yield Surface

Strain Ratio

Anisotropy

Bulk metals(Extruded)

Axial-radial anisotropy

Radial-tangential anisotropy

Sheet metals

Yield criteria (or yield function)

Axial-tangential anisotropy

Normal anisotropy

Axial anisotropy

Planar anisotropy

Fig. 2. Characterizations of anisotropy.

Current work

Flow stress

Anisotropy

FrictionMetal flow(Strain)

Isotropy•Ring compression test•Cigar test

•Tensile test•Upsetting test•Plane straincompression•Other methods

Previous works•Yield function•Texture•Formability

?

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.

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2. RELATIONSHIPS BETWEEN THE PAPERS OF THE THESIS

Isotropy

Flow stress

Anisotropy

FrictionMetal flow

•Anisotropy•Methods•Influence

•Anisotropy•Friction•Ring flow

•Mathematical models•Flow stress•Friction coefficient

Paper C

Paper BPaper A

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.

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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 AA6082 (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, 25 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=0.2%. 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. 11-13. With this method, direction-dependent strain hardening and Bauschinger effects can be observed based on the analysis of the strain path change Tab. 2. 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 AA6082 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.15 - 17.

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z z x

yrt

Extruded bar

Test specimen

(a)

r

r

P(3,6)

P(2,5)P(1,4) Rz

Rzr

Rzt

Cross-section of extruded bar

(b)

Fig. 5. (a) Orientation of small specimens, and (b) Specimen locations for strain ratio and yield loci measurements.

z

z

tt

r

Fig. 6. Plane-strain compression test example of location of specimen, and loading direction.

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A2-A2

45°

r

A3-A3

45°

r

z

A1-A1

45°

t

z

A2 A2

A3 A3

A1

A1

(b)

(a)

(d) (e)

(c)

Fig. 7. Locations of specimens for determination of Hill’s parameters.

Fig. 8. Yield Loci of Annealed Aluminum AA6082 at different pre-strains 0, 15, 25 and 35% (Experimental results).

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x

zεz(Ten)

εx(Ten)

Fig. 9. Cylinder specimen expanded (tensioned) in the circumferencial direction during upsetting in y.

0.81

1.21.41.61.82

2.2

0 0.4 0.8 1.2εy

Rz

Rz(pre-0%)Rz(pre-15%)Rz(pre-25%)Rz(pre-35%)

0.81

1.21.41.61.82

2.2

0 0.4 0.8 1.2εy

Rzt

Rzt(pre-0%)Rzt(pre-15%)Rzt(pre-25%)Rzt(pre-35%)

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1 1.2εy

Rzr

Rzr(pre-0%)Rzr(pre-15%)Rzr(pre-25%)Rzr(pre-35%)

Fig. 10. Strain ratios of Rz, Rzt and Rzr (Experimental results).

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Tab. 1. Relation between pre- and on-going strains

Rz εz εx ≈ εt εy ≈ εR Pre-strain C T T On-going strain T T C Strain directions Rev. CT Rev.

Rzt εz εx ≈ εt εy ≈ εr Pre-strain C T T On-going strain T T C Strain directions Rev. CT Rev.

Rzr εz εx ≈ εr εy ≈εt 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 AA6082 in directions of r, t and z. C=Compression; T=Tension (Experimental results).

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Fig. 12. Flow curves of the annealed alloy AA6082 in directions of r, t and z for 35% pre-straining. C=Compression; T=Tension (Experimental results).

Fig. 13. Flow curves of annealed alloy AA6082 in the t-direction after pre-straining 0, 15, 25 and 35%. C= Compression; T= Tension (Experimental results).

Tab. 2. Relations between pre- and on-going strain Curves No. 1 No. 2 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.

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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) AA6082 σr =σθ =0.93σz σ=190ε0.15 Ani-30% σr =σθ =0.7σz σ=190ε0.15 Iso(6082) σr =σθ =σz σ=190ε0.15 Hardening(06) σr =σθ =σz σ=90+190ε0.6 Non-hardening σr =σθ =σz σ=90

Fig. 15. Distribution of shear strain on the contact surface of cylinder specimen for different materials (Simulation results from ABAQUS).

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Fig. 16. Distribution of friction shear on the contact surface of deformed cylinder specimens (Simulation results from ABAQUS).

1 2 3

Fig. 17. Shear strains of different materials on the contact surface in forward extrusion (Simulation results from ABAQUS).

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3.2 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 90o to the axis of rotational-symmetrical anisotropy under uniform coefficient of friction; (2) an isotropic material ring under anisotropic friction condition; and (3) an anisotropic material ring oriented 0o 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, (2) 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. 20, 22. In reality, a difference between these two cases exits in the distribution of normal pressure and friction shear stress, shown in Figs. 24, 25. 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. 26. Figures 27 and 28 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 AA6082 (Al-Si1Mg0.9Mn0.1). The testing results support those of the FEM analyses, Figs. 21, 23, 27.

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Pattern 1 of friction-ring-flow:

Neutral Plane

Internal diameter increasing When friction is low.

Pattern 2 of friction-ring-flow:

Neutral Plane

Internal diameter decreasing When friction is high.

Pattern 1 of material-ring-flow:

θ

r

Flow competition occursin the cross section of a ring

When )(θσ frr = .

Pattern 2 of material-ring-flow:

θ

r

Non-flow competition occursin the cross section of a ring

When .Constrr =σ . Fig. 18. Patterns of ring flow.

r(ring0)

z(ring90)

z(ring0)

r(ring90)r(material)

z(material)

(a) 90o orientation (b) 0o orientation

θ(material)A A

Planarisotropicplane

r(ring90)

r(ring0)

r(material)

z(material)

z(ring90)

x(material)y(material)

θ(ring90)

θ(ring0)

θ(material)

z(ring0)

Rotation 90o

AA

(c) Rotation of coordinates

Fig. 19. Axis of a ring rotated (a) 90o to the orientation of anisotropy; (b) 0o to the orientation of anisotropy and (c) Rotation coordinates between orientations of rings and the material anisotropy.

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90o

0o

σr distribution (a) AA6082

90o

0o

σr distribution (b) AISI201

Fig. 20. Flow competition occurs in the cross section of anisotropic rings, µ=0.027 (Simulation results from ABAQUS). Pattern 1 of material-ring-flow plus Pattern 1 of friction-ring-flow.

Fig. 21. Deformed anisotropic ring shape (90o orientation) with Teflon lubrication. The black line stands for the axis of the extrusion of the AA6082 (Experimental result).

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90o

0o

σr distribution

(a) µ1/µ2=2, µ2=0.027

90o

0o

σr distribution

(b) µ1/µ2=3, µ2=0.027

Fig. 22. Under frictional anisotropy conditions, the isotropic ring is formed into an ellipse (Simulation results from ABAQUS). Ring flow is Pattern 2 of “friction-ring-flow” in 0o direction, while it is Pattern 1 of “friction-ring-flow” in 90o direction.

Fig. 23. Influence of frictional anisotropy (µ1=dry condition and µ2=Teflon) on ring deformation (Experimental result).

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Fig. 24. The magnitude of the frictional shear stress and the normal pressure in 90o-direction is lower than that in 0o-direction for anisotropic ring (Simulation results from ABAQUS).

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Fig. 25. Distribution of (a) normal pressure and (b) frictional shear stress in 90o- and 0o-direction under the frictional anisotropy condition (Simulation results from ABAQUS).

(a) AA6082

(b) AISI201

Fig. 26. Final shapes of anisotropic rings with 0o orientation to the axis of different rotational symmetrical anisotropic materials: (a) aluminum AA6082 and (b) steel AISI201 (Simulation results from ABAQUS).

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Fig. 27. Changes in internal diameter versus the reduction in height for different materials under friction conditions: (a) µ=0.027, (b) µ=0.06, (c) µ=0.1 and (d) µ=0.2 (Results of the simulations and experiments). Pattern 2 of “material-ring-flow” and two patterns of “friction-ring-flow”.

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0

0.5

1

1.5

2

2.5

3

5 5.5 6 6.5 7 7.517.5 18 18.5 19 19.5 20

x-coordinate (mm)

External surfaceInternal surface

y-co

ordi

nate

(mm

)

(c) Surfaces

Fig. 28. Distribution of (a) normal pressure, (b) frictional shear stress, and (c) profiles of rings’ internal and external surfaces for materials AISI201 at 1.0=µ and isotropy(AISI201) at 0.18 , 1.0=µ (Simulation results from ABAQUS).

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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 (2), minimum differential equation (3), and process conditional equation (4).

)),(,,(),,,( ,,,,,, ikokikokikik GGGfSP εµεεθσ &= (1)

∑= →

=

−=

n

i ioio

ioioii GGGf

GGGfPPQ

1 minlim

2

,1,1,1,1

,2,2,2,2,1,2 )),(,,(

)),(,,(δ

εµεµ (2)

0=

∂∂µQ (3)

),(),( ,2,2,2,1,1,1 ioiioi GGGG εε = (4)

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, ni ,...,1= , Subscript k : specimen number, nk ,...,1= , and Subscript o : original value. Its principle is that the flow stress for a given material is specimen geometry-independent, while the flow resistance depends on it, shown in Fig. 29. The assumptions of the method are, 1. The material used is incompressible in its plastic deformation region. 2. 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-

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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(µ,G2,o , ε2,i(G2,o ,G2,i))

Flow resistancedata domain P1,i

Flow resistancedata domain P2,i

Calculated flow resistancedata domain P2,i

f−1(µ,G1,o , ε1,i(G1,o ,G1,i))

Inverse computationat the same time

Assumed flow stressmodel domain σ

Fig. 29. Explanation of objective function of MTSM.

1

do=10 mm

ho=15 mm2

do=10 mm

ho=10 mm3

do=10 mm

ho=5 mm

4

do=14.14 mm

ho=10 mm

CG1 or RG1: No.1 & No.2CG2 or RG2: No.1 & No.3CG3 or RG3: No.2 & No.3CG4 or RG4: No.2 & No.4

Fig. 30. Cylinder and Rastegaev specimens in upsetting. CG: Cylinder Group; RG: Rastegaev Group.

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u t

Fig. 31. Rastegaev specimen profile.

h = 2,5mm

W = 50 mm

B=5mm

1h = 1,28

mm

W = 50 mm

B=5mm

2

h = 1,28mm

W = 50 mm

B=2,5mm

3

PG1: 1 and 2PG2: 2 and 3

Fig. 32. Plane strain specimens and tools.

z

z

θ

Fig. 33. Location of the plane strain specimen manufactured from a raw material bar.

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Tab. 4. µ calculated from the solver of MTSM Group

No.

Lubricant Friction coeff. µ

CG1 Teflon 0.03055 CG2 Teflon 0.02665 CG3 Teflon 0.02322

Cylinder Upsetting

CG4 Teflon 0.02631 RG1 Molykote 0.03811 RG2 Molykote 0.03818 RG3 Molykote 0.04114

Rastegaev

Specimen upsetting RG4 Molykote 0.04228

PG1 Teflon 0.02955 Plane strain compression test PG2 Teflon 0.02885

Tab. 5. Average value of µ and flow stress σ modeling Cylinder Rastegaev

Specimen Plane strain

Lubricant Teflon Molykote Teflon Average µ 0.0267 0.03991 0.0292

σ 1523.026.190 εσ = 1576.027.190 εσ = 1153.062.178 εσ =

Tab. 6. Coefficient of friction for Teflon Ring Cylinder Plane strain µ 0.028 0.0267 0.0292

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Fig. 34. Loading forces from simulations (ABAQUS) and experiments: (a) cylinder upsetting; and (b) plane strain compression test.

Different equations in cylinder upsetting:

)2

1)2

(exp()2

(2 2

i

i

i

i

i

ii h

rh

rr

hP

⋅−−

⋅⋅

=µµ

µσ (5)

)3

1(hdp µσ += (6)

Tab. 7 µ influenced by models in cylinder upsetting Eq. (5) Eq. (6)

Average µ 0.026683 0.02664

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Different equations in plane strain compression tests:

−′= 1)exp()(

hB

BhP µµ

σ (7)

−+′= 1)exp())(

21(

hB

Bh

Bhp µ

µσ (8)

Fig. 35. Flow stresses influenced by models, Eqs. (7) and (8) in plane strain compression test.

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Tab. 8. µ influenced by models in plane strain. Eq. (7) Eq. (8)

Average µ 0.0292 0.063105 RSQ better than 0.99 worse than 0.91

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4. CONCLUDING REMARKS In this thesis, three experimental methods/measurements (yield loci, strain ratios/R-value 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. 2. 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.

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5. REFERENCE

[1] Roberts, W.L., 1978, Cold Rolling of Steel, New York: Marcel Decker, Inc. [2] Lee, C.H., and Altan, T., 1972, Influence of Flow Stress and Friction upon

Metal Flow in Upset Forging of Ring and Cylinders, ASME, J. Eng. Ind., Vol. 94, pp.775-782.

[3] Pietrzyk, M., Lenard, J.G., 1993, A study of the Plane Strain Compression Test, Annals of the CIRP, 42/1, pp.331-334

[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.1565-1570.

[5] Rieldel, F., 1914, Stahl u. Eisen, Vol. 34, pp.19. [6] Rummel, K., 1919, Stahl u. Eisen, Vol.39, pp.237 [7] Siebel, E., and Pomp, A., 1927, The determination of the deformation

Resistance of Metals by Means of the Compression Test, Mitt. Kaiser Wilhelm Inst. Eisenforsch, Vol. 9, pp.157-171.

[8] Rastegaev, M.V., 1940, Neue Methode Der Homogenen Stauchen, Zavodskaja Laboratoria, Vol. 3, pp.354-355.

[9] Cook, M., Larke, E.C., 1945, Resistance of Copper and Copper Alloys to Homogeneous Deformation in Compression, J. Inst. Metals, Vol.71, pp.371-390.

[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.833-1834.

[12] Becker, N., Pöhlandt, K., 1989, Improvement of the plane-strain compression test for determining flow curves, Annals of the CIRP, 38/1, pp.227-230.

[13] Kunogi, M., 1956, A New Method of Cold Extrusion, J. Sci. Res. Inst., pp.215. [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.121. [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. 92, pp. 389.

[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.21, pp.699.

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[20] Abdul, N.A., 1981, Friction Determination during Bulk Plastic Deformation of Metals, Annals of the CIRP, Vol. 31/1, pp.143.

[21] 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. 431.

[22] 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.

[23] Luong, L.H.S., and Heijkoop, T., 1981, The Influence of Scale on Friction in Hot Metal Working, Wear, Vol. 71, pp. 93.

[24] 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.

[25] Kobayashi, S., 1970, Deformation Characteristics and Ductile Fracture of 1040 Steel in Simple Upsetting of Solid Cylinders and Rings, J. Eng. Ind., pp. 391-399.

[26] 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. 293.

[27] 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. 177.

[28] Kudo, H., 1960, Some Analytical and Experimental Studies of Axi-symmetric Cold Forging and Extrusion, Int. J. Mech. Sci., Vol. 2, pp. 102.

[29] Avitzur, B., 1964, Forging of Hollow Discs, Israel J. Tech., Vol. 3, pp. 295. [30] Liu, J.Y., 1971, J. Eng. Ind., Trans. ASME, Vol. 93, pp. 1134 [31] Lee, C.H. and Altan, T., 1972, Influence of Flow Stress and Friction upon

Metal Flow in Upset Forging of Rings and Cylinders, J. Eng. Ind., Trans ASME, Vol. 94, pp. 775.

[32] Hartley, P., Sturgess, C.E.N. and Rowe, G.W., 1979, Friction in Finite-Element Analyses of Metalforming Processes, Int. J. Mech. Sci., Vol. 21, pp. 301.

[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. 42/1, pp. 335.

[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. 281-297.

[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.

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[38] Wagoner, R.H., Chenot, J.L., 1996, Fundamentals of Metal Forming, New York, John Wiley & Sons. Tnc.

[39] ASTM E 517-96a, 1996, Standard Test Method for Plastic Strain Ratio for Sheet Metal.

[40] Ståhlberg, U., 2001, Materialens Processteknologi, Plastisk Bearnetning, Stockholm.

[41] Montmitonnet, P. and Chenot, J.L., 1995, Introduction of Anisotropy in Viscoplastic 2D and 3D Finite-Element Simulations of Hot Forging, J. Mat. Proc. Tech., Vol. 53, pp. 662-683.

[42] 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. 179.

[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.111-125.

[47] Brunet, M. and Morestin, F., 2001, Experiemntal and Analytical Necking Studies of Anisotropic Sheet Metals, J. Mat. Proc. Tech., Vol. 112, pp. 214-226.

[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.913-925.

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Paper A

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1

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

100 44 Stockholm, Sweden ABSTRACT To determine anisotropy of the extruded aluminum alloy AA6082 (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, 25 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”

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2

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 (1992, 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. 2. ANISOTROPY & EXPERIMENTAL METHODS 2.1. 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

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3

,,, ;)()(

)( ztrkiRli

lklik ==

εεεε

ε (1)

where

)ln( );)(ln()( );)(ln()(o

lo

lili

o

lklk h

hr

rr

r=== ε

εεε

εεε (2)

In Eqs. (1) and (2), it is assumed that a small cylinder specimen is deformed to the principal strain lε where kil ,≠ and that radial strains, )( lk εε and )( li εε , in the cross section of the cylinder are located in the two perpendicular principal axes, in which ki ≠ . 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

1≠ikR . If ikR 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 strain lε . When inverse conditions )()( 1

lkilik RR εε −= are satisfied, only three strain ratios (R-values) are independent, and are defined as )( tzrR ε axial-radial anisotropy,

)( zrtR ε radial-tangential anisotropy and )( rztR ε axial-tangential anisotropy. If a specimen is situated in the center, the difference between )( tzrR ε and )( rztR ε in locations disappears. This strain ratio )( rzR ε is defined as axial anisotropy. In the current work, the strain ratios of )( tzrR ε , )( rztR ε and )( zzR ε are utilized, shown in Fig. 1 (b).

z z x

yrt

(a)

r

r

P(3,6)

P(2,5)P(1,4) Rz

Rzr

Rzt

(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; 1,2,3 represent the strains in compression, while 4,5,6 indicate the strains in tension.

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For convenient description, the local coordinates of three specimens for )( rztR ε , )( tzrR ε and )( rzR ε 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,

)()(

)()(

)()(

)()(

)()(

)()(

)()(

tz

tr

rz

rt

yz

yxz

tz

tr

yz

yxzr

rz

rt

yz

yxzt

R

R

R

εεεε

εεεε

εεεε

εεεε

εε

εε

εεεε

εεεε

=≈=

≈=

≈=

(3)

2.2 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 30o to each tension or compression direction (Lode 1926), shown in Fig. 12. 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. 2.

σ σ

τ τ

σh

σ

τ

Plane strain Pure shear Hydrostatic Component

Fig. 2. 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=0.002. 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

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can be referred to Watts (1952-1953). An example of the plane strain compression tests is illustrated in Fig. 3.

z

z

tt

r

(a) (b)

Fig. 3 Example of plane-strain compression tests: (a) Location of specimen, and (b) Loading direction.

2.3. Yield criteria and parameters In the current work, the anisotropy of the material is characterized by Hill’s criterion (1948) in the cylindrical coordinate ),,( ztr . The yield function is given by

2222222)(2)(2)()( 2

rtNrzMtzLttrrHrrzzGzzttFijf τττσσσσσσσ +++−+−+−= (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 ijξ ,

222

222

222

222

32 ;32 ;32

)1()1()1(2

)1()1()1(2

)1()1()1(2

rtrztz

zzttrr

ttzzrr

rrzztt

NML

H

G

F

ξξξ

ξξξ

ξξξ

ξξξ

===

−+=

−+=

−+=

(5)

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where:

o

tztz

o

rzrz

o

rtrt

o

zzzz

o

tttt

o

rrrr

ττξ

ττξ

ττξ

σσξ

σσξ

σσξ

===

===

;;

;; (6)

The σo and τo 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.

A2-A2

45°

r

A3-A3

45°

r

z

A1-A1

45°

t

z

A2 A2

A3 A3

A1

A1

(b)

(a)

(d) (e)

(c)

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 45o to each principal direction, shown in Fig. 4. 3. 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 AA6082 was constructed in several steps. First, the aluminum round bars (∅51) were heated from room temperature to 410oC, at which it was kept for 2~3 hours, then cooled to 250oC at rate of 30oC/hour, and finally air-

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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%, 25% 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 2 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% 25% 35%

Fig. 5. Pre-straining up to 0, 15, 25 and 35%. 3.2 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 AA6082 possesses planar anisotropy.

)()()( CompCompComp trz σσσ ≈> (7) )()()()( CompTenCompTen ttzz σσσσ ≈>≈ (8)

Fig. 6. Flow curves of the annealed aluminum alloy AA6082 in directions of r, t and z. C=Compression; T=Tension.

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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).

)()()( CompCompComp trz σσσ ≈> (9) )()( TenTen tz σσ < (10)

)()( CompTen zz σσ < (11) 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. 2, 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. 2 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).

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Fig. 7. Flow curves of the annealed alloy AA6082 in directions of r, t and z for 35% pre-straining. C=Compression; T=Tension.

Fig. 8. Flow curves of annealed alloy AA6082 in the t-direction after pre-straining 0, 15, 25 and 35%. C= Compression; T= Tension.

3.3 Anisotropy estimated by strain ratios According to the concepts of )( rztR ε , )( tzrR ε , )( rzR ε in Section 2.1 and the material properties obtained in Section 3.2, 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

)()( TenTen xz σσ > , causing )()( TenTen zx εε > . Meanwhile, with increasing the local

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strain yε (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 )(Tenxε and )(Tenzε in values tends to be smaller and then the strain ratios, )( rztR ε , )( tzrR ε and )( rzR ε decrease with the local straining yε .

Fig. 9. Strain Ratios of Rz, Rzt and Rzr.

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x

zεz(Ten)

εx(Ten)

Fig. 10. Cylinder specimen expanded (tensioned) in the circumferencial direction during upsetting in y.

2. 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. 2. 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 2 in Tab. 2) and a Continuou straining in x (Column 3 in Tab. 2) occur during on-going strain. With increasing pre-strain up to 15, 25, 35%, the difference between )(Tenzσ and )(Tenxσ 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. 2. Relation between pre- and on-going strain Rz εz εx ≈ εt εy ≈ εr

Pre-strain C T T On-going strain T T C Strain direction Rev. CT Rev.

Rzt εz εx ≈ εt εy ≈ εr Pre-strain C T T On-going strain T T C Strain direction Rev. CT Rev.

Rzr εz εx ≈ εr εy ≈εt 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 Rzt is the strongest among the others because of its location, Fig. 1 (b). Thus, the strongest Bauschinger effect can be observed. The evidences are

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)()()( TenTenTen txz σσσ ≈> for 0% pre-strain, Fig. 6; and )()()( TenTenTen txz σσσ ≈< for 35% pre-strain, Fig. 7. This can be called extreme

phenomenon, indicating that the region of the values of )( rztR ε has been changed from 1)( ≥>∞ rztR ε to 1)(0 ≤< rztR ε , shown in Fig. 9.

Fig. 11. Two types of strain ratios.

Fig. 12. Yield Loci of Annealed Aluminum AA6082 at different pre-strain 0, 15 25 and 35%.

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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) 1ratiostrain ≥>∞ , when )()( TenTen xz σσ ≥ ; (b) 1ratiostrain 0 ≤< , when )()( TenTen xz σσ ≤ ; shown in Fig. 11. If

)()( CompTen zz σσ ≠ and )()( CompTen xx σσ ≠ , 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 12 shows that the aluminum alloy AA6082 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 AA6082 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, 2, 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 AA6082 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, 0.2,…, 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

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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 AA6082 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 1.0=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

rrξ ttξ zzξ rtξ rzξ tzξ Isotropy 1 1 1 1 1 1 AA6082 0.93 0.91 1 0.91 0.95 0.96 Ani-30% 0.7 0.7 1 0.65 0.8 0.8

Tab. 4. Cylinder specimens for Fig. 13. No. 1 No. 2 No. 3 No. 4 Height (mm) 15 10 5 14.14 Diameter (mm) 10 10 10 10

Fig.13. Loading-displacement curves of measurement and Hill’s criterion in FEM.

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Tab. 5 Materials used in finite element analyses Name Characteristics Model (MPa) AA6082 σr =σt =0.93σz σ=190ε0.15 Ani-30% σr =σt =0.7σz σ=190ε0.15 Iso(6082) σr =σt =σz σ=190ε0.15 Hardening(06) σr =σt =σz σ=90+190ε0.6 Non-hardening σr =σt =σz 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 AA6082 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.

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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.

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4.2 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 1.0=µ and the extrusion speed is 250mm/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 2 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 2 and 3. Therefore, the result obtained from upsetting case has been further examined.

1 2 3

Fig. 17. Shear strains of different materials in forward extrusion.

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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 strain-hardening 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 )()( CompTen zz σσ ≠ and )()( CompTen xx σσ ≠ , 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 1ztR ≈ 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 BE96-3340 is acknowledged. Also, special thanks go to Professor Pavel Huml for the discussion on experiments. References ASTM E 517-96a, 1996, Standard Test Method for Plastic Strain Ratio r for Sheet Metal.

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Balasubramanian, S. and Anand, L., 1998, Polycrystalline plasticity: Application to earing in cup drawing of A12008-T4 sheet. ASME J. Appl. Mech., Vol. 65, pp. 268-271. Ettouney, O.M. and Stelson, K.A., 1990, J. Eng. Ind., Vol. 112, pp.267. Hill, R., 1948, Proc. Roy. Soc., vol. A193, pp. 281-297 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.2397. Lode, W., 1926, Zeitsch. Phys., Vol. 36, pp. 913. Montmitonnet, P. and Chenot, J.L., 1995, Introduction of anisotropy in viscoplastic 2D and 3 D finite-element simulations of hot forging, J. Mat. Proc. Tech., Vol. 53, pp. 662-683. Pöhlandt, K. and Oberländer, T., 1992. 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., 1952-1953, Proc. Inst. Mech. Eng, 1B, pp. 448-453.

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Paper B

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Influence of Material Anisotropy and Friction on Ring Deformation

Han Han

Materials Forming, Dept. Production Engineering, Royal Institute of Technology

100 44 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 90o to the axis of rotational symmetrical anisotropy under uniform coefficient of friction; (2) an isotropic ring under frictional anisotropy condition; and (3) an anisotropic ring oriented 0o 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 AA6082 (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 [2] 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 [3-21], 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

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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 [22-24]. In conjunction with this, Garmong et al. [22] has investigated the influence of strain-rate-sensitive-material; Danckert [23] has simulated the influence of materials with different strain hardening; Tan et al. [24] 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 [25] because of their mechanic or crystallographic properties [26-28]. In ring experiments, Bhattacharya [26] showed that if the material possesses anisotropy, the deformed ring shapes (such as elliptic or tapering) are influenced by specimen orientation. Pöhlandt [27] 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 90o to the axis of rotational symmetrical anisotropy under uniform coefficients of friction, Fig. 4 (a); (2) an isotropic ring under frictional anisotropy condition; and (3) an anisotropic ring oriented 0o 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 AA6082 (Al-Si1Mg0.9Mn0.1).

2. Ring Flow, Material Constitutive Modeling and FEM 2.1 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 )( ijf σ is usually formulated in Prager’s form,

σε

∂=

)( ijpl fdd

σλ (1)

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Pattern 1 of friction-ring-flow:

Neutral Plane

Internal diameter increasing When friction is low.

Pattern 2 of friction-ring-flow:

Neutral Plane

Internal diameter decreasing When friction is high.

Pattern 1 of material-ring-flow:

θ

r

Flow competition occursin the cross section of a ring

When )(θσ frr = .

Pattern 2 of material-ring-flow:

θ

r

Non-flow competition occursin the cross section of a ring

When .Constrr =σ .

Fig. 1. Patterns of ring flow. Based on the normality rule, the incremental plastic strain occurs in the direction normal to the yield surface )( ijf σ . 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 [29]. 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, rrσ 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, rrσ is independent of θ ; in this case,

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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”. 2.2 Material constitutive modeling 2.2.1 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 quasi-static conditions. This means that any influence due by temperature and strain rate will be ignored. Therefore, the strain decomposition is

plel ddd εεε += . (2) The relation of stress-strain in the elastic part can be expressed by

elelel εDσ := . (3) Here, elD is a matrix determined by both Young’s modulus E and Poisson’s ratio ν . The incremental plastic strain pldε in Eq. (2) represents flow rules defined by stress potential criteria as follows. 2.2.2. 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 [28]. Its plastic property commonly has rotational symmetry or cylindrical orthotropy [27]. 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 ),,( zr θ , it is written by

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2222222)(

2)(2)()( 2

θττθτθθσσ

σσσθθσσ

rNrzMzLrrH

rrzzGzzFijf

+++−+

−+−= (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 ijξ ,

222

222

222

222

32 ;32 ;32

)1()1()1(2

)1()1()1(2

)1()1()1(2

θθ

θθ

θθ

θθ

ξξξ

ξξξ

ξξξ

ξξξ

rrzz

zzrr

zzrr

rrzz

NML

H

G

F

===

−+=

−+=

−+=

(5)

where:

o

zz

o

rzrz

o

rr

o

zzzz

oo

rrrr

ττξ

ττξ

ττξ

σσξ

σσξ

σσξ

θθ

θθ

θθθθ

===

===

;;

;; (6)

Substituting Hill’s criterion into Eq. (1), the flow rule associated with Hill’s criterion is written by,

=∂

∂=

−+−−

−−−

−+−−

zL

rzM

rN

rrzzG

zzF

rrH

zzF

rrH

rrzzG

ij

ijpl

fdf

dd

θσ

σ

θσ

σσσθθ

σ

θθσσσ

θθσ

θθσσσσ

σλσ

λ

2

2

2

)()(

)()(

)()(

)()(

σε

(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:

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Hollomon’s model: nkεσ = (8) Ludwik’s model: nkk εσ 21 += (9) 2.3. 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 [32], was used. The upper part of the ring was meshed with 9792 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, )(ringσ and )(materialσ , were utilized. Taking the ring oriented 90o to the axis of the material for example, Fig. 4, the anisotropy is defined by stress ratios in )(materialσ and the corresponding yielding properties in the ring are given by

21)(12)90( ℜℜℜℜ= Tmaterial

Tring σσ (10)

where 1ℜ is a rotation matrix [33] from the cylindrical coordinate of the material to the Cartesian coordinate of the material and 2ℜ 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 [32]. 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 AA6082 round bar was heated from room temperature to 410oC, at which it was kept for 2~3 hours, then cooled to 250oC at rate of 30oC/hour, and finally air-cooled at room temperature.

Six cylinder specimens (∅5x5 mm) were machined from the annealed aluminum alloy AA6082 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. 2.

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

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were 0.01mm/s, chosen to ensure constant temperature and low strain rate ( )/0028.0 s≈ε& .

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.

A2-A2

45°

r

A3-A3

45°

r

z

A1-A1

45°

θ

z

A2 A2

A3 A3

A1

A1

(b)

(a)

(d) (e)

(c)rrσ

θθσzzσ

zθτfor

rzτfor θτ rfor

Fig. 2. Specimens for obtaining stress ratios in Hill’s criterion.

020406080

100120140160180200

0 0.2 0.4 0.6 0.8 1Effective Strain

Effe

ctiv

e St

ress

(MPa

)

1

23

4

51:Compression in z2:Compression in θ3:Compression in r4:Tension in z5:Tension in θ

Fig. 3. Effective stress-strain curves of the extruded annealed aluminum alloy AA6082 in different directions.

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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 zσ and rσ in compression is very small )MPa2( ≈∆σ at initial yield point ( %2.0=plε ), but at 2.0=plε , the difference between them increases to MPa13≈∆σ . Thus, in the current work, the average stress ratio in each direction, for example 7/))7.0(...)1.0(( =++== rrrrrrrrrr εξεξξ , was utilized for Hill’s criterion, in which zzσ was selected for the reference stress oσ . Young’s modulus, E , of the aluminum alloy AA6082 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 AISI201 [35] was also adopted. The values of stress ratios of materials (AA6082, AISI201, 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. 2. It shows that a pronounced anisotropy exists in the material AISI201.

Tab. 1. Stress Ratios for Hill’s criterion (axes: r,θ,z). Stress Ratios

rrξ θθξ zzξ θξ r rzξ zθξ Isotropy 1 1 1 1 1 1 AA6082 0.93 0.91 1 0.91 0.95 0.96 AISI201 0.7 0.7 1 0.65 0.8 0.8

Tab. 2. Reference σ in Eq. (6) and elastic parameters σ (MPa) E (GPa) ν

AA6082 1523.026.190 ε 72 0.3 AISI201 83.01850310 ε+ 200 0.3

4. 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 (AA6082 & AISI201) possess planar isotropy and the characteristics of stresses can be summarized by

θθσσσ )()()( materialrrmaterialzzmaterial ≈> (11)

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Here, we define the plane on which rrmaterial )(σ and θθσ )(material 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 do:di:h = 6:3:2 = 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 90o 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.

r(ring0)

z(ring90)

z(ring0)

r(ring90)r(material)

z(material)

(a) 90o orientation (b) 0o orientation

θ(material)A A

Planarisotropicplane

r(ring90)

r(ring0)

r(material)

z(material)

z(ring90)

x(material)y(material)

θ(ring90)

θ(ring0)

θ(material)

z(ring0)

Rotation 90o

AA

(c) Rotation of coordinates

Fig. 4. Axis of a ring rotated (a) 90o to the orientation of anisotropy; (b) 0o 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 90o to the z-

axis of rotational symmetrical anisotropy; and Fig. 4 (c) presents the rotation of coordinates between the ring (90o) 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

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zzmaterialo

ringrrring

rrmaterialo

ringrrring

)()90()90(

)()90()90(

)0(

)90(

σθσ

σθσ

==

≈= (12)

Since the condition rrmaterialzzmaterial )()( σσ > is maintained, it leads to

)90()0(

)90()0(

)90()90()90()90(

)90()90()90()90(

oringrrring

oringrrring

oringrrring

oringrrring

kk =>=

=>=

θθ

θσθσ (13)

Therefore, rrring )90(σ is a function of )90(ringθ 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 2.3. The values of coefficient of friction ranging from 0,027 to 0,2 were selected on the basis of the experimental conditions. Case 2: Isotropic ring under frictional anisotropy Case 2 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

zzmaterialringrrringzzring )()()()( σσσσ θθ === (14) 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,

ppoo 2

21

1 )90()0( τθµ

τθµ ==>== (15)

In simulations, the ratios of µ1/µ2 ranged from 2 to 3, Fig. 10, and the flow rule associated with von Mises’ criterion was used. Case 3: Anisotropic ring oriented 0o 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.

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Figure 4 (b) shows the anisotropic ring oriented 0o to the z-axis of rotational symmetrical anisotropy. The coordinate of the ring (0o) is shown in Fig. 4 (c). In this case, the yielding properties in the ring is given by

)()0( materialring σσ = (16) Under this condition, rrring )0(σ is independent of )0(ringθ , while the high stress,

zzmaterialzzring )()0( σσ = , 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). 4.2. Results and discussions 4.2.1. Results of Case 1 Simulation results When the anisotropic ring is oriented 90o to the z-axis of rotational symmetrical anisotropy, under the low coefficient of friction µ=0.027, 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

ring 90)90( =θ ) than in the direction of higher yield strength ( oring 0)90( =θ ).

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. 5.

90o

0o

σr distribution (a) AA6082

90o

0o

σr distribution (b) AISI201

Fig. 5. Flow competition occurs in the cross section of anisotropic rings, µ=0.027. (Pattern 1 of material-ring-flow plus Pattern 1 of friction-ring-flow).

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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

)90( 0 ,90oring =θ ).

Fig. 6. The magnitude of the frictional shear stress and the normal pressure in 90o-direction is lower than that in 0o-direction.

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90o

0o

σr distribution (a) AA6082

90o

0o

σr distribution (b) AISI201

Fig. 7. Internal surface of anisotropic rings flows to rings’ center, at µ=0.2. (Pattern 1 of material-ring-flow plus Pattern 2 of friction-ring-flow).

Fig. 8. The influence of material anisotropy still exists since the distribution of normal pressure and frictional shear stress in 90o-direction on the ring surface is not equal to that in 0o-direction, at µ=0.2.

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When the coefficient of friction increased to µ=0.2, the deformed anisotropic ring became a circle for the material with slight anisotropy (AA6082); in other words, the degree of ellipticity decreases if the material possesses higher anisotropy (material AISI201), 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 2 of “friction-ring-flow”. Experimental results Two rings, shown in Fig. 4 (a), were manufactured from the extruded annealed aluminum alloy AA6082 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. (12) 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 (90o orientation) under the friction conditions: (a) Teflon (µ=0.027); and (b) µ=dry condition. The black lines stand for the axis of the original extruded round bar for the aluminum alloy AA6082.

4.2.2. Results of Case 2 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 increases in the lower friction direction ( o

ring 90)( =θ ), and decreases in the higher

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friction direction ( oring 0)( =θ ), (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 µ1/µ2 = 2 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 0o- and 90o-directions are calculated by

100)90ave_value(

)90ave_value()0ave_value(variation o

oo

×−

= (17)

Table 3 Variation of p and τ in two directions Normal Shear

AA6082 µ=0.027

4.7% 4.7%

AISI201 µ=0.027

15% 15%

µ1/µ2=2 1.4% 102% µ1/µ2=3 3% 260%

90o

0o

σr distribution

(a) µ1/µ2=2, µ2=0.027

90o

0o

σr distribution

(b) µ1/µ2=3, µ2=0.027

Fig. 10. Under frictional anisotropy conditions, the isotropic ring is formed into an ellipse. Ring flow is Pattern 2 of “friction-ring-flow” in 0o direction, while it is Pattern 1 of “friction-ring-flow” in 90o direction.

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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 2 (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 90o- and 0o-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 90o on the ring surface, i.e. 1. Dry: µ1(-45o<θ<45o & 135o<θ<225o) ≈ 0,15;

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2. Teflon: µ2(45o<θ<135o & 225o<θ<315o) ≈ 0,027. To keep isotropy in the cross section of the ring, the ring of 0o orientation (material AA6082) was chosen, Fig. 4 (b). Due to frictional anisotropy, the ring was deformed to an ellipse, Fig. 12. The ratio of µ1/µ2 is approximately 5, which is calculated from the testing results in Fig. 14 (a) and (c).

Fig. 12. Influence of frictional anisotropy (µ1=dry condition and µ2=Teflon) on ring deformation. Ring flow is Pattern 2 of “friction-ring-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. 12,

similar elliptic ring shapes were obtained, even though the anisotropic friction conditions between the FEM and the experiment were not exactly the same. 4.2.3. 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 (AA6082 & AISI201) 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.

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(a) AA6082

(b) AISI201

Fig. 13. Final shapes of anisotropic rings with 0o orientation to the axis of different rotational symmetrical anisotropic materials: (a) aluminum AA6082 and (b) steel AISI201.

In fact, different shear strengths between planar isotropic material and isotropic material still exist in the shear direction (in the plane of planar isotropy or in the r-direction of the z-plane). Based on Hill’s criterion, the shear stress ratio can be given by

)()(

isotropykanisotropyk

o

zrzr =ξ (18)

This leads to a difference in the internal diameters of deformed rings between planar isotropic material and isotropic material, in turn causing the friction coefficient to be estimated incorrectly if the conventional calibration curves are used. The reason is that the sensitivity of anisotropic materials to a given friction is different from that of isotropic materials; this can be derived from

zr

ia

zro

zr

oizra

mm

isotropykanisotropyk

isotropykmanisotropykm

ξ

ξ

τ

=

=

==

)()(

)()( (19)

where ma and mi are friction factors for anisotropic and isotropic material. When

1<zrξ , anisotropic materials will be more sensitive to friction than isotropic materials.

The values of zrξ for various materials (isotropy, AA6082 & AISI201) are listed in Tab. 4. To clearly present a trend of sensitivities of different anisotropic materials for a given friction, an additional anisotropic material named Ani-15% ( θσσσ 15.115.1 == rz ) was assumed.

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Table 4. Shear stress ratio zrξ

Isotropy AA6082 AISI201

zrξ 1 0.95 0.8

In literature, various calibration curves for ring-compression test exist, and there

is a large difference between them due to different methods. Thus, the estimation of coefficient of friction depends on the selection of calibration curves [36]. To avoid this problem in the current work, the calibration procedure of coefficients of friction was carried out with the finite element method, and two assumed isotropic materials (isotropy(AA6082) & isotropy(AISI201)) with different strain hardening exponents (n=0.1523 & n=0.83, Tab. 2), were utilized for corresponding anisotropic rings.

Figure 14 (a)-(d) show the change of internal diameters of two assumed isotropic and three different anisotropic rings under a given friction condition (µ=0.027, 0.6, 0.1 0.2). It can be seen that the isotropic ring with higher strain hardening exponent (isotropy(AISI201) and n=0.83) lowers only the position of friction calibration curves slightly, compared to the isotropic material with lower strain hardening exponent (isotropy(AA6082) and n=0.1523) at 2.0≥µ ; this result confirms Ettouney’s [36] analysis. But the influence of anisotropy on the change of the internal diameter of the ring is relatively large, and the trend is that the higher the anisotropy, the smaller the internal diameter of the ring will be under the same friction conditions. This phenomenon is quite similar to an isotropic ring under high friction condition, e.g. material AISI201, shown in Fig. 15 (a)-(c). The observations are as follows. First, at µ=0.1, the magnitude of frictional shear stress on the anisotropic ring is distributed at the same level as that on the isotropic ring, but the internal diameter of the anisotropic ring decreases much more than does the isotropic ring. Meanwhile, the foldover of the anisotropic ring is also stronger than that of the isotropic ring at both internal and external edges. Second, with increasing µ to 0.18 on the isotropic ring, the internal and external surfaces of the isotropic ring can be identical to that of the anisotropic one at µ=0.1. Accordingly, the magnitudes of both frictional shear and normal pressure on the isotropic ring are also increased. These evidences confirm that anisotropic material is more sensitive to friction, as Eq. 19 proposes.

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Fig. 14. Changes in internal diameter versus the reduction in height for different materials under friction conditions: (a) µ=0.027, (b) µ=0.06, (c) µ=0.1 and (d) µ=0.2. (Pattern 2 of “material-ring-flow” and two Patterns of “friction-ring-flow”).

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0

0.5

1

1.5

2

2.5

3

5 5.5 6 6.5 7 7.517.5 18 18.5 19 19.5 20

x-coordinate (mm)

External surfaceInternal surface

y-co

ordi

nate

(mm

)

(c) Surfaces

Fig. 15. Distribution of (a) normal pressure and (b) frictional shear stress; and (c) profiles of rings’ internal and external surfaces for materials AISI201 at 1.0=µ and isotropy(AISI201) at 0.18 , 1.0=µ .

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Therefore, if the anisotropic ring is assumed to be isotropic, then errors, calculated using Eq. (20), that range from 8% to 80% corresponding to materials AA6028 and AISI201, can be made when determining the coefficient of friction in ring-compression. This trend indicates that the error increases faster than does the degree of anisotropy. In the same way, errors can also be made by using the foldover method suggested by Ettouney [36] for estimation of friction.

100)(

)()(% ×−

=isotropy

isotropyanisotropyerrorµ

µµ (20)

Experimental results In experiments, the ring compressions were carried out in a 500 kN press for AA6082 under two lubrication (Teflon and dry condition) conditions. Figures 14 (a) and (c) present the testing results, which are marked by the symbol “ * ”. These show that the results from the FEM are reasonable. The coefficients of friction calibrated are approximately 0.03 and 0.15 at the end points under Teflon and dry surface conditions. 5. Summary and Conclusions Two typical patterns of ring flow in anisotropic materials have been summarized. The influence of material anisotropy on ring deformation can be easily observed in Pattern 1 of “material-ring-flow” under low friction conditions. Here, ring flow is mainly influenced by anisotropy rather than by friction. Although a similar elliptic ring shape can be obtained through friction anisotropy, the distribution of normal pressure in two directions (90o and 0o) for the two cases varies because normal pressure is a response of the material properties, shown in Tab. 3. If an anisotropic material is assumed to be isotropic, the influence of anisotropy will be mistakenly attributed to friction. This error can be easily made in Pattern 2 of “material-ring-flow” when the conventional calibration curves are used, because the influence of material anisotropy on ring deformation is in the same direction as friction, and cannot be immediately observed in experiments.

As Hill [31] argued, anisotropy is not a rare phenomenon. It is difficult to avoid anisotropy in metalworking and it is invariably developed by any severe strain. The theories of plastic flow for isotropic metals are sufficiently accurate enough for many purposes, but there are also many phenomena that these theories fail to explain. Based on the results as mentioned above, the following conclusions can be drawn:

1 The material flow is a function of material properties and friction conditions. 2 The phenomenon of material flow influenced by material anisotropy in a

transverse direction is quite similar to that by friction.

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3 In the ring test, the coefficient of friction can be overestimated under certain conditions if the material possesses anisotropy.

4 When the ring test is applied to the stock material, two rings are recommended as shown in Fig. 4 (a) and (b). The ring with 90o orientation is used for investigation of material anisotropy. The ring with 0o orientation can be used as long as the deformation of the ring with 90o orientation is close to a circle (isotropy); otherwise, the result has to be questioned.

5 The issue of friction is very important in metal forming processes, and investigations of material properties are desirable.

Nomenclature F,G,H,L,M,N: Hill’s anisotropic parameters kr: Material yield strength in Shear k, k1, k2: Material constant n : Strain hardening exponent p : Normal pressure

ijξ : Stress ratio components

oσ : Reference stress

ijσ : Stress components

iiring )(σ , iiring )90(σ , iiring )0(σ : Principal stresses in rings

iimaterial )(σ : Principal stresses in )(materialσ

)90(ringσ , )0(ringσ : Stress matrix for rings, see Fig. 4

)(materialσ : Stress matrix for the material elσ : Elastic stress matrix

ijτ : Shear stress components

oτ : Reference shear stress elε , plε : Elastic and plastic strain matrix

µ : Coefficient of friction

21 ,ℜℜ : Rotation matrixes, see Section 2.3 λd : Scalar function of stress, strain and strain

history

Acknowledgments Special thanks go to Professor Ulf Stålhberg for reading the manuscript and valuable discussion. Most experiments were carried out at Dept. of Solid Mechanics. The author is indebted to Hans Öberg for his valuable help during the laboratory work.

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The financial support from the Royal Institute of Technology, Sweden and Brite-EUram project BE96-3340 is acknowledged.

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389. [6] Hawkyard, J.B. and Johnson, W., 1967, Int. J. Mech. Sci., Vol. 9, pp. 163. [7] Janardhana, M.N. and Biswas, S.K., 1979, Int. J. Mech. Sci., Vol. 21, pp. 699. [8] Abdul, N.A., 1981, Annals of the CIRP, Vol. 31/1, pp. 143. [9] Nagpal, V., Lahoti, G.D., and Altan, T., 1978, J. Eng. Ind., Trans. ASME, Vol.

100, pp. 431. [10] Petersen, S.B., Martins, P.A.F. and Bay, N., 1998, J. Mat. Proc. Tech., Vol.

79, pp. 14. [11] Luong, L.H.S., and Heijkoop, T., 1981, Wear, Vol. 71, pp. 93. [12] Burgdorf, M., 1967, Industrie-Anzeiger, Vol. 89, pp. 799. [13] DePierre, V. and Gurney, F., 1974, J. Lub. Tech., Trans. ASME, Vol. 96, pp.

682. [14] 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. 293.

[15] Male, A.T., DePierre, V. and Saul, G., 1973, Trans. ASME, Vol. 3, pp. 177. [16] Kudo, H., 1960, Int. J. Mech. Sci., Vol. 2, pp. 102. [17] Avitzur, B., 1964, Israel J. Tech., Vol. 3, pp. 295. [18] Liu, J.Y., 1971, J. Eng. Ind., Trans. ASME, Vol. 93, pp. 1134 [19] Lee, C.H. and Altan, T., 1972, J. Eng. Ind., Trans ASME, Vol. 94, pp. 775. [20] Hartely, P., Sturgess, C.E.N. and Rowe, G.W., 1979, Int. J. Mech. Sci., Vol.

21, pp. 301. [21] Bugini, A., Maccarini, G. and Giardini, C., 1993, Annals of the CIRP, Vol.

42/1, pp. 335. [22] Garmong, G., Paton, N.E., Chesnutt, J.C. and Nevarez, L.F., 1977, Metall.

Trans. A., Vol. 8A, pp.2026. [23] Danckert, J., 1988, Ann. CIRP, Vol. 37 1, pp. 217. [24] Tan, X., Martin, P.A.F., Bay, N. and Zhang, W., 1998, J. Mat. Proc. Tech.,

Vol. Vol. 80-81, pp. 292-297. [25] Edward, M.M., 1991, Metal Working Science and Engineering, McGraw-Hill,

Inc., New York. [26] Bhattacharyya, D., 1981, Annals of the CIRP, Vol. 30/1, pp. 139.

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[27] Pöhlandt, K., Lange, K. and Zucko, M., 1998, Steel Research, Vol. 69, No. 4+5, pp. 171.

[28] Honeycombe, R.W.K., 1984, The Plastic Deformation of metals, Edward Arnold Ltd.

[29] Hill, R., 1985, The mathematical theory of plasticity, Oxford University Press, New York.

[30] Wagoner, R. H. and Chenot, J. L., 1996, Fundamentals of Metal Forming, John Wiley & Sons. Inc., New York.

[31] Hill, R., 1948, Proc. Roy. Soc., Vol. A193, pp. 281-297. [32] Hibbitt, Karlsson and Sorensen, 1997, ABAQUS, Ver. 5.6. [33] Cirsfield, M.A., 1997, Non-linear Finite Element Analysis of Solids and

Structures, John Wiely & Sons, Inc. [34] MNC handbok nr 12, 1983, Aluminium Konstruktions-och materiallära,

Minab/Gotab, Kungälv. [35] Carlsson, B., Doctoral Thesis, 1996, Royal Institute of Technology,

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Paper C

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The Validity of Mathematical Models Evaluated by Two-specimen Method under the Unknown Coefficient of Friction

and Flow Stress

Han Han Materials Forming, Department of Production Engineering, Royal Institute of Technology

100 44 Stockholm, Sweden

Abstract A modified two-specimen method (MTSM) has been derived from an objective function of two-specimen according to the inverse method. Its principle is that the flow stress for a given material is specimen geometry-independent, while the flow resistance depends on it. 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, but also it can estimate the on-line coefficient of friction and flow stress when the mathematical model selected is valid. The method has been verified by experiments of cylinder, Rastegaev specimen upsetting and plane strain compression test with an annealed aluminum alloy AA6082 (Al-Si1Mg0.9Mn0.1). The flow stress and the coefficient of friction obtained from the method have been examined by FEM (finite element method) simulations and the ring test as well.

Keywords: Mathematical models, coefficient of friction, flow stress and flow resistance

1. Introduction The flow stress of a metallic material and the coefficient of friction at a die/workpiece interface are vital variables almost in all metal forming processes, such as, wire drawing, rolling, extrusion, forging [1] and even in upsetting tests. Fundamentally, the flow stress and the coefficient of friction are formulated in a mathematical model (process model) based on mechanical analysis for one specimen in a given metal forming process. Therefore, without the accurate flow stress and the accurate coefficient of friction, it is impossible to improve the mathematical model for the given process [2-4]. On the other hand, without the accurate mathematical model it is also impossible to obtain the accurate on-line material flow stress, because the flow stress is not a direct measured variable, even if the coefficient of friction can be obtained accurately by a certain method. When the material possesses

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anisotropy, the flow stress determined by off-line methods (tensile test or upsetting test) might not be equal to that estimated by on-line methods, because the flow stress of anisotropic materials is direction-dependent or the evolution of material anisotropy may occurs during large strains [5]. This problem exists not only in practical industry forming processes, but also in a laboratory for obtaining an accurate flow stress during upsetting. To avoid the influence of friction on measurements of the flow stress, Cook [6] in 1945 utilized a method of graphical extrapolation with four specimens and stated that a homogeneous flow curve could be obtained when the diameter of a cylinder

0→d or the height of a cylinder ∞→h . This is because the effect of friction tends to be small when the ratio of hd / decreases. Although Alexander in 1963 [7] described in detail how to use Cook’s method, it is still not convenient for obtaining a stress-strain curve. In 1977, the extrapolating direction of Cook’s method was changed by Woodward [8] into an interpolating one, and four tests were reduced into two. He noted that it is necessary to eliminate the contribution of the frictional work during the two tests so that the flow stress can be obtained from the applied stress. Unfortunately, this method can only be utilized for the simplest mathematical expression of cylinder upsetting. Twelve years later, when applying the two-specimen method to the plane strain compression test for obtaining the flow stress, Becker in 1989 [9] claimed that the contribution of friction has to be eliminated manually based on Woodward’s method. Therefore, the questions that arise here are: first, whether there is a possibility to induce the two-specimen method into a general one that can be applied to many mathematical models; second, whether the flow stress obtained from the two-specimen method is really valid, if the mathematical model selected is not accurate.

The aim of this study is to provide a deeper understanding of the relation among the flow stress, the coefficient of friction and a selected mathematical model through a modified two-specimen method that is derived from an objective function of two specimens according to inverse method. In this method, the variable eliminated is the flow stress instead of the coefficient of friction. The advantage of this change is that the two-specimen method becomes inductive, and it makes the evaluation of the selected mathematical equation possible when both coefficient of friction and flow stress are unknown. Meanwhile, if the selected mathematical model is accurate, the on-line coefficient of friction and the flow stress can be obtained. The method has been used in experiments of cylinder, Rastegaev specimen upsetting and plane strain compression test for determining the flow stress and the coefficient of friction. Good agreement was obtained. And, the values of the flow stress and the coefficient of friction obtained from the method were used as inputs to the FEM and results of simulations for the loading-displacement curves are very similar to those of the experiments. To check the coefficient separately, the standard ring test was used. Also, the result obtained from the MTSM analysis was supported. The workpiece material is an aluminum alloy AA6082 (Al-Si1Mg0.9Mn0.1).

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2. Modified Two-Specimen Method (MTSM) 2.1 Approach of MTSM The mathematical models used in cylinder upsetting, Eqs. (7), (18), strip compressed by the rectangular platens, Eqs. (12), (19), and rolling processes (at least six mathematical models [10-12]), possess one common characteristics. That is, the flow resistance (or loading force) can be expressed as a flow stress multiplied by a function that consists of variables of coefficient of friction and the specimen/tool geometrical size. These mathematical models assume that Tresca or von Mises yield criterion is fulfilled, while the coefficient of friction is from the contact boundary condition where Coulomb’s friction law is considered to be valid. The flow resistance in the models is calculated from the loading force divided by the contact area. If one mathematical model is valid for the corresponding mechanical object or process, the flow stress can be obtained when the value of coefficient of friction is known. In the same way, if the flow stress is known, the coefficient of friction can also be estimated from this model. The modified two-specimen method focuses on one mathematical equation and two unknown variables (flow stress and coefficient of friction). The method consists of its assumed conditions, a general equation, an objective function and a judgement of validity of the mathematical model used. Assumed conditions The two-specimen method tries to avoid analyzing the mechanical behavior for one specimen directly, but it emphasizes identifying the relationship among the flow stress, the coefficient of friction and the selected model. Therefore, the mechanical assumptions mainly rely on the existing mathematical models for one specimen. Here the basic conditions under which the two-specimen method works are listed below. 1. The material used is incompressible in its plastic deformation region. 2. 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. General equation and objective function In a process, if the flow resistance can be expressed as the flow stress multiplied by a function of the coefficient of friction, the initial specimen’s size and its strain, Eq. (1);

)),(,,(),,,( ,,,,,, ikokikokikik GGGfSP εµεεθσ &= (1)

and if two tests satisfy the conditions mentioned above, one objective function (2) exists when the material flow stress is eliminated.

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∑= →

=

−=

n

i ioio

ioioii GGGf

GGGfPPQ

1 minlim

2

,1,1,1,1

,2,2,2,2,1,2 )),(,,(

)),(,,(δ

εµεµ (2)

Meanwhile, the objective function has to satisfy both the minimum differential Eq. (3) and the process conditional Eq. (4). Then, the coefficient of friction can be estimated.

0=∂∂µQ (3)

),(),( ,2,2,2,1,1,1 ioiioi GGGG εε = (4)

f(µ,G2,o , ε2,i(G2,o ,G2,i))

Flow resistancedata domain P1,i

Flow resistancedata domain P2,i

Calculated flow resistancedata domain P2,i

f−1(µ,G1,o , ε1,i(G1,o ,G1,i))

Inverse computationat the same time

Assumed flow stressmodel domain σ

Fig. 1. Explanation of objective function of MTSM The term of objective function usually appears in a fitting problem according to inverse principle [13,14], which tries to find the parameters of the function that can describe the relation between a data domain and a model domain. But the objective function (2) of the two-specimen method in the current work presents a fact that the flow stress for a given material is geometry-independent, while the flow resistance is geometry-dependent. In practice, for a given process, several mathematical models exist as mentioned above. Therefore, the assumed flow stress model domain is defined, shown in Fig.1. This means that when the flow resistance P1,i of the specimen with geometrical size 1 has been transformed from its data domain into the flow stress model domain after removing the effect of friction work in a function

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of )),(,,( ,1,1,1,11

ioio GGGf εµ− , the validity of the flow stress model domain obtained is unknown. In other words, the function of )),(,,( ,1,1,1,1

1ioio GGGf εµ− will be questioned

first. As long as this flow stress obtained in the model domain can be transformed into the second flow resistance iP ,2 data domain in which the specimen size is different from the first one, after considering the effect of friction work, the assumed flow stress model domain remains to be true. Otherwise, the function between the flow stress and the flow resistance has to be modified. The flow stress can be estimated from the general Eq. (1) when the coefficient of friction is obtained from the objective function (2). Judgement of validity of selected mathematical model If a mathematical model for a given process is truly found, and the conditions of the two tests fulfil the assumptions of the two-specimen method mentioned above, then the value of coefficient of friction calculated from the objective function is identified, satisfying both RSQ (R-squared value) = 1 and ii ,2,1 σσ = . Otherwise, the mathematical model obtained can not be fully valid for the given process, leading to deviation of both flow stress and coefficient of friction. The definition of RSQ (R-squared value) is,

2

2222 ))(())((

)()(

−⋅−

−=

∑ ∑∑ ∑

∑ ∑∑

iiii

iiii

YYnXXn

YXYXnRSQ (5)

where:

)),(,,()),(,,(

,1,1,1,1

,2,2,2,2,1

,2

ioio

ioioii

ii

GGGfGGGf

PY

PX

εµεµ

=

= (6)

2.2 Typical Mathematical expression of MTSM in upsetting (or compression test) 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. Its mathematical model contains the flow stress and the coefficient of friction. The typical tests used are the cylinder upsetting and the plane strain compression test.

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2.2.1 Cylinder upsetting In cylinder coordinate, the mathematical model of cylinder upsetting is derived from the Slab method or equation of equilibrium when the deformation of the specimen is considered to be uniform. At this time, the stresses at any arbitrary point in the deformed metal can be assumed to be parallel to principal stresses. Therefore, the flow stress is a result from the Tresca criterion. When the friction at the die/workpiece interface is expressed according to Coulomb’s friction rule, the mathematical model of cylinder upsetting can be given by [15],

)2

1)2

(exp()2

(2 2

i

i

i

i

i

ii h

rh

rr

hP

⋅−−

⋅⋅

=µµ

µσ (7)

During the experiments, the flow resistance is recorded under a certain strain rate, because the loading force depends not only on material properties, but also on the upsetting speed. The effective strain is given by,

)ln(i

oi h

h=ε , thus )exp( ioi hh ε−= (8)

According to the incompressibility law and Eq. (8), the instantaneous radius of the specimen can be written as,

)5.0exp( ioi rr ε= (9) Combining Eqs. (7)-(9), the general Eq. (1) of the two-specimen method for Eq. (7) can be reached. When two tests are carried out under the conditions of the two-specimen method, after removing the flow stress, the objective function for cylinder upsetting is formulated by,

[ ][ ] minlim

2

1 ,1,1

,2,22

,2,1

,1,2,1,2 1)exp(

1)exp(→

=

=

−−

−−

−=∑ δ

µµµµn

i ii

ii

oo

ooii XX

XXrhrh

PPQ (10)

where:

o

ioi

o

ioi

hr

X

hr

X

,2

5.1,2,2

,2

,1

5.1,1,1

,1

))(exp(2

))(exp(2

ε

ε

⋅=

⋅=

(11)

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2.2.2 Plane strain compression test The plane strain compression test is usually more desirable in cold rolling simulations to obtain the off-line flow stress for models of rolling [12]. The specimen is in the form of a thin plate or sheet that is compressed across the width of the strip by flat and parallel tools that are longer than the strip width. Watts and Ford [16] suggested that the plane strain could be obtained when the specimen is W/B>8 for keeping plane strain deformation, and 2≤B/h≤ 4 for both homogenous deformation and remaining the minor effect of friction even in very good lubrication, shown in Fig. 7. Therefore, the mathematical expression of plane strain can also be derived [17] in the same principle way as for cylinder upsetting in Cartesian coordinates under a certain compression speed,

−′= 1)exp()(

hB

BhP µµ

σ (12)

The absolute value of the natural draught in plane strain compression test is defined by

)ln(hho=′ε (13)

The difference between plane strain compression test and cylinder upsetting lies in the criteria employed. Plane strain used von Mises’ yield criterion so that the flow stress and effective strain are given by,

σσ ′=23 (14)

εε ′=3

2 , thus )23exp( ε−= ohh (15)

If the conditions of two tests meet with the assumptions of two-specimen method, the objective function for plane strain is expressed by Eq. (16) (the procedure of derivation is the same for the cylinder upsetting),

minlim

2

1

,1

,1,1

,2

,2,2

,2,1

,1,2,1,2

1)exp(

1)exp(

)(→=

=

−= ∑ δµ

µn

i

o

io

o

io

oo

ooii

hXB

hXB

BhBh

PPQ (16)

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where:

)23exp(

)23exp(

,2,2

,1,1

ii

ii

X

X

ε

ε

=

= (17)

3. MTSM Verified by Experiments 3.1 Upsetting for three types of specimens To verify the modified two-specimen method, the specimens’ geometrical ratios, the surface condition and specimen loading direction have been considered in experiments since the annealed aluminum alloy AA6082 possesses anisotropy in which the flow stress in the z-direction is higher than that in r-and θ-directions as shown in Fig. 2.

Fig. 2. Flow stresses of the annealed aluminum alloy AA6082 in directions of r, θ and z.

Figure 3 shows the testing procedures that consist of two sections. In the first section, cylinder upsetting and Rastegaev specimen upsetting [18], Fig. 5, were carried out, to evaluate the validity of the tested material flow stress when the coefficient of friction is changed. In the second section, the cylinder upsetting and the plane strain compression test were grouped to evaluate the validity of the coefficient of friction in different loading directions. The detailed specimens’ designs for the three tests are as follows, Figs. 4, 5, 6 and 7.

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Rastegaev SpecimenUpsetting

CylinderUpsetting

Plane StrainCompression Test

Changed:LubricantFriction Distribution

Same:Loading Direction (Material Property)

Changed: Loading Direction

(Material Property)Same: Lubricant

Obtained: Same σ ? Different µ ?

Obtained: Same µ ? Different σ ?

Result: MTSM is valid ?

Fig. 3. Flow chart of the experiments’ design.

1

do=10 mm

ho=15 mm2

do=10 mm

ho=10 mm3

do=10 mm

ho=5 mm

4

do=14.14 mm

ho=10 mm

CG1 or RG1: No.1 & No.2CG2 or RG2: No.1 & No.3CG3 or RG3: No.2 & No.3CG4 or RG4: No.2 & No.4

Fig. 4. Geometrical size of cylinder and Rastegaev specimens in upsetting. CG: Cylinder Group; RG: Rastegaev Group.

Cylinder specimens Four specimens with ratios ( oo dh / ) of 1.5, 1, 0,7 and 0.5 for original height versus original diameter were machined in the middle of a raw material bar (∅51) along the z-axis. The detailed specimen sizes are shown in Fig. 4. The structures of four groups, CG1 (Specimen No. 1 and No 2), CG2 (Specimen No. 1 and No. 3); CG3 (Specimen No. 2 and No. 3), and CG4 (Specimen No. 2 and No. 4) were made according to the requirements of the two-specimen method.

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The mathematical model selected was Eq. (7), because it provides the coefficient of friction with a wider range of applications than Siebel [19] Eq. (17). Secondly, Woodward’s two-specimen method can not be applied to this model directly. Rastegaev specimens In cylinder upsetting, Pearsall and Backofen [20] pointed out that the film of a lubricant would be broken down like a circumferential ring at both ends of a specimen during compression. Commonly the friction will be increased in this area, and the distribution of friction is different from friction hill. To simulate this effect in testing, Rastegaev specimen [18] was utilized as shown in Fig. 5.

u t

Fig. 5. Rastegaev specimen geometrical profile. The concave of the specimen was filled with a lubricant (Molykote) in experiments. Although the lubrication condition between the area of the lubricant and the edge of circumferential ring differs, the two-specimen method does not concern the problem of distribution of friction, but it emphasizes only the single average value of the coefficient of friction on the surface. Therefore, the sizes of u = 0,2 mm and t = 3 mm were designed. The rest of geometrical sizes of Rastegaev specimens were completely copied from cylinder upsetting mentioned above, Fig. 4, including the mathematical model of Eq. (7). Plane strain specimens The aim of using plane strain is to change the loading direction or strain path since the annealed alloy AA6082 possesses anisotropy. The specimens used in the plane strain compression test were machined in the middle of a raw material bar, and the compression was in the θ direction, Fig. 6. Based on the requirement of specimen’s geometrical size proposed by Watts and Ford [16], three specimens were used, Fig. 7. These specimens were categorized into two groups, which are PG1 and PG2 (Plane Group). PG1 consists of specimens No. 1 & No. 2, while specimen No. 2 & No. 3 are included in PG2, in conformity with two compression plate dies with width in 5 mm and 2.5 mm. The expression is given by Eq. (11).

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z

z

θ

Fig. 6. Location of the plane strain specimen manufactured from a raw material bar.

h = 2,5mm

W = 50 mm

B=5mm

1h = 1,28

mm

W = 50 mm

B=5mm

2

h = 1,28mm

W = 50 mm

B=2,5mm

3

PG1: 1 and 2PG2: 2 and 3

Fig. 7. Plane strain specimens and tools. 3.2 Instruments and testing conditions The experiments were carried out with MTS 160 kN dynamic Press controlled by Instron 8500 Model dynamic system. For testing accuracy, a suitable loading cell has been chosen. Each specimen was tested at room temperature and the loading speed was 0.01mm/s for keeping temperature constant. The compression dies were polished carefully. Teflon with a thickness of 0.075 mm was used as lubrication for cylinder, plane strain compression test, while the lubricant of Molykote was employed for Rastegaev specimen upsetting. 3.3 Results from experiments The results indicate that each flow resistance increases when the specimen thickness decreases, provided the initial contact area is the same. Similarly, each flow resistance also increases with its initial contact area when the size of initial thickness is the same, taking Rastegaev specimen and plane strain compression test as examples, shown in Figs. 8 and 9.

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Table 1 presents the coefficients of friction calculated from the solver of the two-specimen method for each group. Although a little deviation is observed in the values listed, the values of RSQ (R-squared values) for all fitting curves are better than 0.99 (close to 1). These −µ values imply that the mathematical models selected are valid, and the flow stress and the coefficient of friction can be estimated at the same time. Here, CG, RG and PG stand for the groups of cylinder, Rastegaev specimen and plane strain compression test.

Tab. 1. µ calculated from the solver of MTSM Group

No.

Lubricant Friction coeff. µ

CG1 Teflon 0.03055 CG2 Teflon 0.02665 CG3 Teflon 0.02322

Cylinder Upsetting

CG4 Teflon 0.02631 RG1 Molykote 0.03811 RG2 Molykote 0.03818 RG3 Molykote 0.04114

Rastegaev Specimen upsetting

RG4 Molykote 0.04228 PG1 Teflon 0.02955 Plane strain

compression test PG2 Teflon 0.02885 Substituting these coefficients of friction into Eqs. (7) and (11), the flow stresses for cylinder, Rastegaev specimen and plane strain compression test can be obtained. The curve number 5 in Fig. 8 consists of eight flow curves computed from cylinder and Rastegaev specimen upsetting, while the number 4 in Fig. 9 is made up of three flow curves from plane strain compression test. It can be seen that these stress-strain curves are in good agreement with each other. The average coefficient of friction for each type of upsetting and the corresponding flow stresses that are characterized by Hollomon’s model [21] are illustrated in Tab. 2. The trend is that the values of parameters of Hollomon’s model are very close between cylinder and Rastegaev specimen upsetting due to the same loading direction, but the coefficients of friction differ since various lubricants were employed. As the material AA6082 possesses anisotropy, the flow stress is direction-dependent. So the different flow stress was obtained from the cylinder upsetting and the plain strain compression test, while the coefficients of friction between them are rather close because of the same lubricant (Teflon).

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Tab. 2. Average value of µ and flow stress σ modeling Cylinder Rastegaev

Specimen Plane strain

Lubricant Teflon Molykote Teflon Average µ 0.0267 0.03991 0.0292

σ 1523.026.190 εσ = 1576.027.190 εσ = 1153.062.178 εσ =

60

80

100

120

140

160

180

200

0 0,2 0,4 0,6 0,8Strain

Stre

ss (M

Pa)

1: h/D=1.5, flow resistance2: h/D=1, flow resistance3: h/D=0.5, flow resistance4: h/D=0.7, flow resistance5: Eight flow curves from cylinder and Rastegaev specimen upsetting

123 4

5

Fig. 8. The flow resistance of four Rastegaev specimens with different geometrical ratios in upsetting. No.5 consists of eight flow curves from cylinder and Rastegaev specimen upsetting.

0

50

100

150

200

250

0 0,2 0,4 0,6 0,8 1 1,2 1,4Strain

Stre

ss (M

Pa)

13

2

4

1: L:W:h=50:5:2.5, flow resistance2: L:W:h=50:5:1.28, flow resistance3: L:W:h=50:2.5:1.28, flow resistance4: Three flow curves

Fig. 9. The flow resistance of three specimens with different geometrical ratios in plane strain compression test. The No.4 stand for three flow curves calculated from the corresponding flow resistance.

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3.4 Different mathematical models evaluated by MTSM 3.4.1 Cylinder upsetting As early as 1927, an approximate mathematical expression (17) was derived by Siebel [19] from Eq. (7) on the basis of Taylor’s extended equation under the condition 35.0/ ≤hdµ [10]. This condition is suitable for the current work.

)3

1(hdp µσ += (18)

Compared to the values of coefficient of friction obtained from the mathematical models in Eqs. (7) and (18), the results show that there is no significant difference in them, shown in Tab. 3. However, the same pattern can also be found in the flow stresses for the two mathematical models. Richardson (1985) pointed out that the error of Eq. (18) is less than 1% compared with Eq. (7) at the condition 35.0/ ≤hdµ [10]. Here, MTSM confirms this conclusion and shows that both Eqs. (7) and (18) are valid under the current testing condition.

Tab. 3. µ influenced by models in cylinder upsetting Eq. (7) Eq. (18)

Average µ 0.026683 0.02664 3.4.2 Plane strain compression test In plane strain compression test, an expression (19) was mentioned by Becker [9] in 1989, which the non-homogenous deformation has been taken into account, based on elementary plasto-mechanics.

)1)()(2

1( −+′= hB

eBh

Bhp

µ

µσ (19)

Tab. 4. µ influenced by models in plane strain Eq. (12) Eq. (19)

Average µ 0.0292 0.063105 RSQ better than 0.99 worse than 0.91

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Fig. 10. Flow stresses influenced by models in plane strain compression test: (a) Eq. (19) for µ=0,063105; (b) Eq. (19) for µ=0,0292.

This equation is derived from the specimen geometrical ratio B/h=1, which does not meet with the better condition of homogeneous deformation 2B/h ≥ . If Eq. (19) is used, Table 4 shows that the average value of coefficient of friction is bigger than 0.06 and the value of RSQ (R-squared value) is smaller than 0.91. Substituting µ=0.063105 into Eq. (19), the flow curves obtained can not be well fitted into each other. Also, the 22% error occurs compared with the results from Eq. (12), shown in Fig. 10 (a). This means that if the coefficient of friction is simply removed from the selected mathematical model without any discussion on its validity, the value of flow stress obtained becomes inaccurate. Moreover, when substituting µ=0.0292 obtained in the section 3.3 into Eq. (19), the value of flow stresses will be deviated largely to each other, Fig. 10, which describes a fact that the accurate value of flow stress also can not be obtained even the accurate coefficient of friction is utilized. Therefore, Eq. (19) is not suitable for the current work since the flow stress of material is geometry-dependent.

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Fig. 11. Loading forces from FEM (ABAQUS) and experiments: (a) cylinder upsetting; and (b) plane strain compression test.

4. Validity of MTSM Examined by the FEM and Ring Test 4.1 Loading force examined by FEM simulation The task of MTSM is to evaluate a given mathematical model and to determine the flow stress and the coefficient of friction in a given process when the material is loaded. Comparatively, the FEM simulation employs an inverse approach to calculate the loading force in the current work. Figures 11(a) and (b) show that the loading force simulated from FEM can be well fitted into the testing results. Here, numbers (1 to 4) stand for specimen’s number, Figs. 4 and 7. From these results, the conclusion can be drawn that the values of the flow stress and the coefficient of friction obtained from the two-specimen method in Tab. 2 are reasonable and suitable for the FEM. 4.2 Coefficient of friction examined by ring-test The ring compression test has been proven to be a simple method for determining friction parameters in bulk metal forming. The geometrical size of ring specimen

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usually is utilized as 6:3:2 [23]. The ring specimens were machined into 30mm (outer-diameter), 15mm (inner-diameter) and 10mm (height). In the ring-compression test, the result of coefficient of friction µ of Teflon was approximately 0.028 according to Male’s [24] calibration curves. Compared to the values obtained from cylinder and plane strain compression test with the two-specimen method, the coefficient of friction of Teflon considered to be valid, Tab. 5.

Tab. 5. Coefficient of friction for Teflon Ring Cylinder Plane strain µ 0.028 0.0267 0.0292

5. Discussion 5.1 Coefficient of friction, flow stress and mathematical models Usually, Rastegaev specimen is used for better lubrication when the geometrical size, u/t =µ, is selected [18, 22]. But in our case, the edge and the concave size were designed as u/t >µ for increasing the effect of friction on the edge of the circle. The results show that the assumption of the average single value of the coefficient of friction is acceptable for the two-specimen method, Fig. 8. In Section 3.4, the results indicate that the main function of the two-specimen method is to evaluate the validity of the selected models rather than to determine the flow curve directly. When a model used is inaccurate, both the values of flow stress and the coefficient of friction are deviated. This finding is not available in Woodward and Becker’s work. 5.2 Testing issues It is found that the two-specimen method needs high accurate loading measurement. The resolution of equipment should be good enough to distinguish the increased loading force caused by friction. In the current work, the resolution of the equipment is better than 0.1%, while the sensitivity of loading force to friction for each specimen ranges from 1.5% to 7.6%, shown in Fig. 12. Regarding testing accuracy, the sensitivity of loading should be at least 10 times higher than the equipment resolution, which could cause 10% measurement error. Therefore, in our cylinder upsetting for the specimen with the h/d geometrical ratio of 1.5, the measurement ability has been reached to its maximum limitation. This means that either decreased coefficient of friction or increased h/d ratio will cause more errors in experiments. It is also found that the measurement results will be influenced by transducer’s location due to elastic property of tools. Taking plane strain compression test for example, the shorter the distance between the transducer and the specimen is, the more closer the value measured is to the true plastic deformation of the specimen, shown in Fig. 13.

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0%1%2%3%4%5%6%7%8%

Resolutio

n

C1: H/D=1

,5

C2: H/D=1

C4: H/D=0

,7

C3: H/D=0

,5

R1: H/D=1

,5

R2: H/D=1

R4: H/D=0

,7

R3: H/D=0

,5

P1:L/W

/h= 50/5

/2,5

P2:L/W

/h= 50/5

/1,28

∆ L

oadi

ng

Fig. 12. Sensitivity of loading to friction in upsetting of cylinder (C), Rastegaev specimen (R) and plane strain compression test (P).

0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

40,00

45,00

0,00 0,50 1,00 1,50 2,00 2,50displacement(mm)

forc

e(kN

)

Position:disp. - force(kN)Plate: disp. - force(kN)Clipgage: disp.- force(kN)Plastic: disp. - force(kN)

1

43

2

4 3 2 1

Fig. 13. Measurement results influenced by transducer’s location.

5.3 Application of the Method As most materials possess anisotropy in a different degree in metal forming processes, the identification of a material property becomes extremely important. Commonly, multi-direction testing is required to investigate the detailed characteristics of the material as well as to identify the constitutive equation [5]. However, in practice the specific flow stress may be only taken into account under a given process or strain path or loading direction. To solve this problem by the two-specimen method, only two tests are required to obtain the flow curve in the given direction if the mathematical model is valid as shown in Figs. 8 and 9.

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The application of the two-specimen method could start with the cold flat rolling processes to evaluate the models (at least six mathematical expressions) as well as to estimate the on-line average flow stress and the on-line coefficient of friction. Although several methods (at least five methods) to predict the on-line average coefficient of friction exist, some of them do not take the flattening of work-roll into account and some methods can be only applied to a single stand rolling process. As an example of creating two specimens in a flat rolling process, the suggestions are given as follows,

ho hfV1o V1fR1

R1VR1

VR1

(a)

ho hfV2o V2f

R2

R2VR2

VR2

(b)

Fig. 14. Application of the two-specimen method in flat rolling. (a) Specimen one: Work-rolls with radius R1 and (b) Specimen two: Work-rolls with radius R2, where R2=2R1.

1. The strips can be rolled separately by two different work-rolls with radii R1 and

R2 for the same draught fo hhh −=∆ , Fig. 14. Here assumes, 12 2RR = . (20) 2. The peripheral velocities of work-rolls,

1RV and 2RV , should meet with Eq. (21),

122 RR VV = , (21)

for keeping the same reduction strain rate in the two processes.

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3. Since the on-line coefficient of friction is a function of the radius R , the peripheral velocity of work-roll RV and draught h∆ , the relation between 1µ for 1R and 2µ for 2R can be derived as ,

21 2µµ ≈ , (22) through Ford’s expression [25]. 4. Taking Stone equation [11] for example,

∆′′= 1)exp(

o

oR h

hRWhF µ

µσ , (23)

the objective function can be expressed as

∑= →

=

−∆′

−∆′

−=n

i

o

oiRiR

hhR

hhR

FFQ1 minlim

2

112

221

,1,2

)1)(exp(

)1)(exp(δ

µµ

µµ (24)

where:

fo hhh −=∆ (25)

∆⋅

−+=′

∆⋅

−+=′

hWEF

RR

hWEF

RR

iR

iR

πν

πν

,22

22

,12

11

)1(161

)1(161

. (26)

After substituting the coefficient of friction µ1 of Eq. (22) into Eq. (24), the on-line coefficients of friction (µ1 for R1 and µ2 for R2) and the on-line average flow stress can be obtained. Repeating this procedure in all the existing models of rolling, the best mathematical model, the coefficient of friction and flow stress corresponding to the model can be found. 6. Conclusions The modified two-specimen method (MTSM) has been verified satisfactorily under our testing conditions with cylinder, Rastegaev specimen and plane strain

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compression test. The validity of the flow stress and the coefficient of friction obtained from the method has been examined by the FEM (finite element method) simulation and ring test. Compared with Woodward’s two-specimen method, the advantages of MTSM are as follows. First, it can be applied to more mathematical models since the variable eliminated is the flow stress rather than the coefficient of friction. Second, with this method, the validity of a mathematical model selected can be identified through inverse computation, under the unknown coefficient of friction and flow stress. The judgement of the validity of a model depends on the RSQ (R-squared value) in which the physical meaning is exhibited in Fig. 1. The findings of the current work have demonstrated that MTSM clarifies the relationship among the mathematical model, the coefficient of friction and the flow stress. This means that the purpose of the two-specimen method is to determine the validity of a selected mathematical model rather than to obtain the flow stress directly. The reason is that both the flow stress and the coefficient of friction obtained from the two-specimen method would be deviated if the mathematical model is not accurate, Fig. 8. Thus, MTSM could be recommended as a way to evaluate the validity of a selected mathematical model for a given process, and to estimate the frictional coefficient and the flow stress of materials. However, the coefficient of friction at the die/workpiece interface is relatively low in the current work and some questions might be still opened. In the near future, our research will focus on the application of the flat rolling process. Nomenclature B: width of plane strain tools d : diameter of cylinder specimen f (…) : a function of … FR : rolling force G : specimen geometry size h : specimen height

h∆ : rolling draught P : average flow resistance

io rr , : original and instantaneous radii of cylinder specimen R : radius of work-roll, mm R’ : radius of flattened work roll RSQ: R-squared value S : material microstructure factors t: see Fig. 5 u: see Fig. 5

10V , 20V : entry velocity

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fV1 , fV2 :exit velocity

1RV : peripheral velocity of work rolls with 1R

2RV : peripheral velocity of work rolls with 2R W : width of strip σ : flow stress

ik ,σ ′ : major stress under plane strain 'σ : average major stress under plane strain

ε : strain or average strain ε : average strain ε& : strain rate ε& : average strain rate

ik ,ε′ : absolute value of the natural draught in plane strain α : geometrical factor µ : coefficient of friction θ : temperature Subscript i : instantaneous value, ni ,...,1= Subscript k : specimen number, nk ,...,1= Subscript o : original value Subscript f : final value

Acknowledgments

Special thanks go to Professor Ulf Ståhlberg for reading the manuscript and valuable discussions. Most experiments were carried out in Dept. of Solid Mechanics, Royal Institute of Technology, Sweden. The author is indebted to Hans Öberg, Dept. of Solid Mechanics, Royal Institute of Technology, Sweden for his valuable help during the laboratory work. The financial support from the Royal Institute of Technology, Sweden and Brite-EUram project BE96-3340 is acknowledged.

References

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[3] M. Pietrzyk, J.G. Lenard, A study of the plane strain compression test, Annals of the CIRP, 42/1 (1993) 331-334

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