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Measurement method for stress–strain curve in a super-large strain range
Materials Science & Engineering A 600 (2014) 82–89
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Measurement method for stress–strain curve in a super-large strain range Yasuhiro Yogo a,n, Masatoshi Sawamura a, Masafumi Hosoya b, Michiaki Kamiyama a, Noritoshi Iwata a, Takashi Ishikawa b a b
Toyota Central R&D Labs., Inc., Japan Department of Materials Science and Engineering, Nagoya University, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 9 December 2013 Received in revised form 6 February 2014 Accepted 7 February 2014 Available online 17 February 2014
Flow stress is an essential material property for metal forming. A stress–strain (SS) curve is generally measured by tensile testing or compression testing. In some metal forming processes, the maximum strain exceeds 5.0 (500%); however, standard testing methods can only measure an SS curve up to a strain of 1.0. Therefore, a new method was developed that enables us to measure an SS curve for strains up to 10.0. High-pressure torsion (HPT) was applied for the measurement. Using HPT, it is possible to deform a specimen without fracture. Therefore, larger strains can be introduced into a specimen than with conventional methods. An SS curve measurement was performed for pure aluminum. The measured torque during HPT and the average strain in the specimen were converted into an SS curve. The SS curve measured using the developed method was compared with that obtained using a compression test. It is confirmed that the flow stress was successfully measured up to a strain of 10.0. In addition, the results for the developed method correspond with those of the compression test up to a strain of 1.0. & 2014 Elsevier B.V. All rights reserved.
Keywords: Stress–strain curve Flow stress High pressure torsion Large strain: finite element method
1. Introduction It is well known that bulk material is heavily deformed during industrial metal forming processes. The strain introduced into a material by plastic deformation can be calculated using the finite element method (FEM), i.e., for cold forging [1], hot forging [2], forward [3] and backward extrusion [4], and ring rolling [5]. As shown in those papers, the maximum strain values range from 3.0 to 10.0 (300–1000%). In the design stage for manufacturing, the press load is estimated and the die material is determined using the stress–strain (SS) curve of the material. The SS curve is also used as input data for calculations when using FEM to predict material flow. Tensile testing is a general method used to measure an SS curve. Stress and strain are expanded uniformly within the gauge length of a specimen. Therefore, the experimentally measured load and displacement are easily converted into an SS curve. The measurable strain range is relatively small. The maximum strain measurable in tensile testing depends on the material and is normally less than 0.5. With recent developments, it is possible to measure an SS curve with a strain over 1.0 by using an optical strain measurement system [6].
n
Corresponding author.
http://dx.doi.org/10.1016/j.msea.2014.02.026 0921-5093 & 2014 Elsevier B.V. All rights reserved.
Compression testing is also applicable to bulk materials, and its measurable strain range is larger than that of a tension test. The strain distribution in a material during a compression test is not uniform, making it difficult to convert the measured load and displacement into an SS curve. In order to deal with this, a uniform compression test was developed [7]. In this test, a cylindrical specimen is compressed between smooth flat planes using a Teflon sheet and grease as lubricants. The lubricant is renewed at every 3–5% reduction in height. At every 20–30% reduction in height, the specimens are machined into cylindrical shapes. The machining was performed at a very slow speed to avoid heating the specimen. The Rastegaev test is another uniform compression test. In this test, shallow hollows are machined into both the top and bottom surfaces of the specimen to hold lubricant. During the compression test, the lubricant flows outward through the interface between the specimen and the dies. This reduces the friction force which causes non-uniform compression [8]. On the other hand, a method that allows the strain distribution from a compression test to be used to determine the SS curve was also developed [9]. The average strain and the restraint factor are first calculated with FEM, and then the measured load and displacement curve are converted into an SS curve. Using this compression test, an SS curve can be determined for a strain around 1.0. Using the torsion test [10], a measurable strain range of 1.0–3.0 can be reached, depending on the material. The strain distribution
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
from the torsion test is also not uniform. When the measured rotation angle and torque are converted into an SS curve, the nonuniform strain distribution and nonlinearity of the stress evolution must be considered. There are some reports of measuring an SS curve using torsion tests [11,12]. However, the effects of such strain and stress distributions have not been well studied. Although a large strain is introduced by metal forming, there is no method to measure an SS curve over such a large strain range. When FEM calculations are conducted, an extrapolated SS curve based on the measured SS curve is sometimes used to cover the immeasurable strain range. In the previous study, one of the authors has studied the influence of the difference between the measured and extrapolated SS curves on material flow in FEM simulations. It was confirmed that the SS curve strongly affects material flow even if the difference between the measured and extrapolated SS curves is small, especially when predicting fracturing during the stamping process [13,14]. Therefore, an SS curve should be measured to cover the strain range expected throughout the actual forming process. In this study, a method of measuring the SS curve for strains up to 10.0 is developed. There are two requirements for this method. The first is that it enables us to deform a material up to a strain of 10.0. The second is that the strain is uniform in the material or that the strain distribution is handled with a certain procedure to obtain the SS curve, as was mentioned for the compression test [9]. It is difficult to introduce strains exceeding 1.0 with general material testing methods. On the other hand, for the last 20 years, some material processing methods have been applied to conduct severe plastic deformation (SPD) and to improve material properties. Examples of SPD processing are equal-channel angular pressing (ECAP) [15,16], high-pressure torsion (HPT) [17,18], and accumulative roll bonding (ARB) [19]. With these SPD processing methods, it is possible to continue to deform a material without fracture. In this study, HPT was used because dimensional changes of a specimen while processing is relatively small compared with other SPD processing methods. This is an advantage when experimentally measured data is converted into an SS curve. The strain introduced into a material by HPT testing is not uniform; therefore, a procedure that takes the strain distribution into account must also be developed.
2. Experimental methods 2.1. HPT test The material used for the specimens was pure aluminum (99.9%). The specimen dimensions used are shown in Fig. 1, and the
10
83
Upper die
Ring
Lower die Specimen
Fig. 2. Schematic diagram of the experimental setup for the HPT tests.
experimental setup is shown in Fig. 2. All the surfaces of the specimen were enclosed by the upper and lower dies and the ring. Therefore, the specimen dimensions did not change during HPT testing. In order to suppress sliding between the specimen and the upper and lower dies, small hollows and grooves were machined into the contacting surfaces of the upper and lower dies, as shown in Fig. 3. The specimen was compressed at 1 GPa between the dies and was subsequently torsionally strained by the rotation of the upper die. Due to the high pressure, the specimen can be continuously strained without breaking. The rotation speed of the upper die was 1 rpm. The compression load, rotation angle, and torque of the upper die were measured during the test at room temperature. 2.2. Measurement of the actual rotation angle of a specimen in an HPT test The specimen is torsionally strained by the rotation of the upper die. As reported by Kaveh et al., the rotation angle of the upper die is not always the same as that of the specimen [20]. Therefore, the actual rotation angle of the specimen after HPT testing should be measured to accurately determine the introduced strain. Thin copper (Cu) foil was included in several HPT specimens. The foil thickness was 0.05 mm. A schematic of these specimens is shown in Fig. 4. After testing, each HPT specimen was inserted into a plate, on which radial lines were prescribed on both the front and back surfaces. Using the plate, the torsion angle of the Cu foil was measured from both the front and back views. 2.3. Compression test A compression test was conducted to validate the SS curve measured using the developed method. The initial diameter and height of the specimens were 14 and 21 mm, respectively. For the contacting surface of the upper and lower dies, concentric circles were grooved to prevent the sliding of the specimens. The dimensions of the specimens and the contacting surfaces of the upper and lower dies are shown in Fig. 5. The measured load and displacement were converted into an SS curve based on the procedure proposed by Osakada et al. [9].
1 Fig. 1. Dimensions of the specimen for the HPT tests.
2.4. Torsion test As written previously, it is not possible to measure an SS curve accurately with torsion testing. Therefore, torsion testing [10] was
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Fig. 3. Photograph of the (a) upper die, (b) lower die, and (c) ring, and (d) a magnified view of the die surface to prevent slip between the dies and specimen.
Copper foil (0.05 mm thickness)
14
120 2.4
2.4
120
21
120
0.4
Half-circle specimens (Al) Fig. 4. Schematic diagram of a specimen to measure slippage between a specimen and dies during HPT tests.
only applied to measure the maximum strain obtained by conventional methods. The specimen dimensions are shown in Fig. 6. A specimen was torsionally strained by being gripped at both ends, and the strain was calculated based on the number of revolutions.
3. Calculation method The theoretical equation used to calculate pure torsion strain is described as herein. The shear strain is defined as
γ yz ¼
π D ðθ=360Þ h
ð1Þ
Here, D is the diameter of the specimen, θ is the rotation angle in degrees, and h is the height of the specimen. By applying the von
0.115
0.577
1.2
Fig. 5. Dimensions of a specimen and the contacting surface of the dies for a compression test. (a) Specimen and (b) contacting surface of the die.
Mises equivalent strain, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εeq ¼ 23 ðε2xx þ ε2yy þ ε2zz Þ þ 13 ðγ 2xy þ γ 2yz þ γ 2zx Þ The equivalent strain of pure torsion can then be written as qffiffiffiffiffiffiffiffiffiffi εeq ¼ 13 γ 2yz
ð2Þ
As inferred by the theoretical equation of pure torsion, the strain in an HPT specimen changes along the radius. In addition, it is easy to expect the friction between the specimen and the ring, which affects the strain distribution. Therefore, in order to determine the strain in an HPT specimen, FEM simulations were conducted. The updated Lagrangian formulation is generally applied for metal forming simulations. However, in large-deformation cases, the updated Lagrangian formulation suffers from numerical
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
15
250
4
15
85
True stress / MPa
10
8
20 60
Fig. 6. Dimensions of the specimen for the torsion tests.
Specimen (D10*H1)
Pressure torsion
200 150 n=0.05 n=0.1 100 n=0.2 n=0.3 50 0
Ring
0
0.5
1.0 1.5 True strain / -
2.0
Fig. 8. SS curves defined in the FEM calculations.
Upper die 60
90
CL Upper die
Ring
Specimen
Lower die Fig. 7. Different views of the calculation model. (a) 3D view of the calculation model and (b) schematic view.
Torque / N.m
50 60
40 30
30
20 10 0
Compression load / N
Lower die
0 0 60 120 180 240 300 360 Rotation angle of the upper die / degree
Fig. 9. Torque and compression load measured in an HPT test.
difficulties due to severe mesh distortion. In order to overcome the difficulties caused by such mesh distortion, the arbitrary Lagrangian–Eulerian (ALE) formulation has been proposed [21–23]. ALE combines the respective advantages of the Lagrangian and Eulerian formulations. In the ALE formulation, the mesh motion is taken arbitrarily from the material deformation to keep the element shapes optimal. In addition, ALE is suited to compute deformation in a fixed space. Therefore, ALE was applied to the calculation of strain during HPT. Due to the limitation of the software, a small hole (0.1 mm in radius) was made at the specimen's center. Because the volume ratio of the hole is 0.4% and the strain by HPT approaches zero at the specimen's center, it is clear that the computation accuracy is not adversely affected by the hole. The calculation model is shown in Fig. 7. These FEM simulations were conducted with the commercial FEM software FORGE (Transvalor S.A., France). The effects of the SS curve and the friction between the specimen and the ring on the strain distribution in an HPT specimen were examined. SS curves with different workhardening exponents, n, were defined. The SS curves defined are shown in Fig. 8. Elasto-plastic material properties were defined for each specimen. The friction coefficient, μ, between the specimen and the ring was defined by the Coulomb friction law. The sticking condition between the specimen and the upper and lower dies was assumed to prevent sliding. A pressure of 1 GPa was applied to the specimen by the upper die. The upper and lower dies and the ring were both rigid bodies. The ring is a fixed body; therefore, it does not follow the motion of the specimen.
4. Experimental results 4.1. HPT results The compression load and torque are shown in Fig. 9. The measured torque shows a drastic increase at the beginning
followed by an almost constant value. The compression force could be maintained at a constant value during the entire HPT test. 4.2. Slippage between the specimen and the upper die The torsion angle of the specimen was measured using an inserted Cu foil. An HPT specimen was observed from both the front and back views. Photographs of the 451 and 3601 torsionally deformed specimens taken from both the front and back views are shown in Fig. 10. The relationship between the rotation angle of the upper die and the torsion angle of the specimen is shown in Fig. 11. From 01 to 451 of rotation of the upper die, no slippage occurred between the specimen and the upper die. Beyond that, a small slippage of about 201 occurred. In Fig. 12, the measured torque is shown. The measured torque of the specimens with Cu foils follows that of the specimen without a Cu foil. This means that the insertion of a Cu foil into the specimen does not affect deformation. 4.3. SS curve from the compression test The SS curve measured by compression testing is shown in Fig. 16. By compressing the specimen to a 50% reduction in height, an SS curve could be measured up to a strain of 0.8. This result is discussed in Section 6.1 to compare it with the SS curve measured using the developed method. 4.4. Maximum strain obtained by the torsion test The torsion test was conducted two times. The rotation angles at fracture were 38401 and 36601. The strain introduced by pure torsion can be calculated using Eq. (2). Therefore, the maximum strain, which is at the circumference, is 4.72 at fracture. The average strain, which is calculated by the integration of Eq. (2) through the radius, is 3.71. This shows that it is not possible to introduce strains up to 10.0 with the torsion test.
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Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
Front view Rotation
Back view Fig. 10. Photographs showing the front and back views and the rotation angles of the upper die after the HPT tests.
50 Theoretical value (without slip)
270
40
180 Experiment
90 0
Torque / N•m
Torsion angle of the specimen / degrees
360
30 20 10 0
0 90 180 270 360 Rotation angle of the upper die / degrees
Fig. 11. Relationship between the rotation angle of the upper die and the torsion angle of a specimen.
5. Calculation results
Rotation angle of the upper die 45 90 180 (With Cu foil) 270 360 360 (Without Cu foil) 360 0 90 180 270 Rotation angle of the upper die / degrees
Fig. 12. The effect of the copper foil on torque.
edge when using a material with a higher work-hardening exponent. In the case where the friction coefficient is higher, the stress concentration at a corner edge is also higher.
5.1. Strain distribution The influence of the friction coefficient, μ, between the specimen and the ring and the work-hardening exponent, n, on the strain distribution is shown in Fig. 13. When a material has a lower defined work-hardening exponent, the strain increases significantly. On the other hand, strain tends to concentrate on a corner
6. Determination procedure for SS curve In this section, the procedure to measure strain using the developed method is first discussed. Then, a procedure to measure stress is discussed. Finally, the SS curve is shown.
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
n=0.1
µ=0.0
n=0.2
µ=0.1
n=0.3
µ=0.2
87
µ=0.3 Center Line
1
2 3 Radius / mm
4
5
Center Line
1
3 2 Radius / mm
4
Equivalent strain 40 36 32 28 24 20 16 12 8 4 5 0
12
350 µ n 0.2 0.0 0.2 0.1 0.2 0.2 0.2 0.3 0.1 0.1 0.3 0.1 Theoretical
10 8 6 4 2 0
Equivalent stress / MPa
Average equivalent strain / -
Fig. 13. Strain distribution in a specimen after 1 revolution. (a) Influence of the work hardening exponent, n (μ¼0.1) and (b) influence of the friction coefficient, μ (n¼ 0.2).
0
60
120
180
240
300
360
Fig. 14. Average strain in a specimen.
Equivalent strain / -
10
Theoretical value (without slip)
6 2 0
With consideration of slippage between the upper die and a specimen
0 90 180 270 360 Rotation angle of the upper die / degrees
100 Compression test
50 0
2
4 6 8 10 Equivalent strain / -
12
6.2. Stress
Fig. 15. Relationship between the rotation angle of the upper die and the average strain in a specimen.
6.1. Strain The way to convert the rotation angle of the upper die into strain is discussed in this section. In Section 5.1, it has been shown that the friction coefficient between the specimen and the ring, μ, and the work-hardening exponent of the material, n, affect the strain distribution. In order to investigate the relationship between the rotation angle of the upper die and the average strain throughout the specimen, the calculation results shown in Fig. 13 were averaged using Eq. (3): ∑V ε εave ¼ i i i ∑i V i
150
value, and the friction coefficient and the work-hardening exponent have no influence on the average strain. As shown in Fig. 11, a small amount of slippage occurred during HPT. It is possible to obtain the relationship between the rotation angle of the upper die and the actual average strain of a specimen by combining Figs. 11 and 14. The relationship obtained is shown in Fig. 15.
8 4
Developed method (N=2)
200
Fig. 16. Comparison of SS curves between the developed method and a compression test.
14 12
250
0
Rotation angle of the upper die / degrees
Developed method (N=1)
300
Osakada proposed a procedure to convert the compression load into an SS curve using compression testing [9]. His theory was modified to measure the SS curve using HPT testing. First, an FEM calculation was conducted using a fictitious material with an SS curve described by Eq. (4):
s ¼ ε0:001
Using the calculation results, the average strain and the restraint factor, f, was calculated using Eqs. (4) and (5), respectively:
save ¼ f¼
ð3Þ
Here, εave is the average strain in the specimen, and V i and εi are the volume and equivalent strain of an element (i), respectively. The calculated relationship is shown in Fig. 14. The average strain for all conditions follows the average theoretical value. This means that the average strain can be calculated using the theoretical
ð4Þ
∑i V i si ∑i V i
T FEM
save
ð5Þ ð6Þ
Here, save is the average stress in the specimen, si is the equivalent stress of an element (i), f is the restraint factor, and T FEM is the calculated torque. By combining the measured torque and the calculated restraint factor, f, an SS curve is obtained using Eq. (7):
s¼
T exp f
ð7Þ
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
Here, T exp is the measured torque. Using the obtained SS curve, this procedure was repeated until the difference between the formerly defined stress and the obtained stress became small enough. The obtained SS curve is shown in Fig. 16. The SS curve obtained by compression testing is also shown in Fig. 16. There is a large discrepancy between the two curves. The flow stress is based on the measured torque in the developed test, and the discrepancy appeared because of this measurement. In order to evaluate flow stress using the developed method, only the torque from the torsional deformation of the specimen should be used. However, as shown in Fig. 17, other factors were included in this measurement. It is very difficult to eliminate the effect of the other factors when measuring torque. Therefore, the difference in yield stress is recognized as the effect of the other factors. Here, it is assumed that the other factors caused by the friction force are at a constant value throughout the measurement. By eliminating the effect of the other factors, as shown in Fig. 18, the flow stress was modified, and the modified SS curve is shown in Fig. 19. It is shown that an SS curve was successfully measured for strains up to 10.0. In addition, the SS curve obtained using the developed method
shows good correlation with that obtained by compression testing up to a strain of 1.0. It is natural that both of the yield stresses are the same. However, the work-hardening behavior of the developed method corresponds with that of the compression test. This result shows the validity of the measured results for the developed method.
300 Equivalent stress / MPa
88
Developed method (N=1)
250 200
Developed method (N=2)
150 100
Compression test
50 0
0
2
4 6 8 10 Equivalent strain / -
12
Fig. 19. Comparison of SS curves between the developed method and a compression test.
Upper die
(3)
(1) Specimen
(2)
Lower die
(4) Ring
Measured torque = (1) + (2) + (3) + (4)
Equivalent stress / MPa
300 Developed method (N=1)
250 200
Developed method (N=2)
150
Extrapolated curve
100
Compression test
50 0
0
12
(1) Measure the torque and rotation angle of the upper die during HPT.
of a specimen.
(2): Torque due to the friction force between a specimen and the ring. (3): Torque due to the friction force between the upper die and the ring. (4): Torque due to the friction force between the lower die and the ring.
(2) Measure the relationship between the rotation angle of the upper die and that of the specimen. (3) Define a temporary SS curve for an FEM calculation. (4) Conduct the FEM calculation. (12) The flow stress in (3) should be replaced by the SS curve in (8).
SS curve derived from the law data Equivalent stress
4 6 8 10 Equivalent strain / -
Fig. 20. Comparison of SS curves obtained via the developed method and by extrapolation.
(1): Torque due to torsional deformation
Fig. 17. Schematic diagram of the influence of the friction forces on the measured torque.
2
(5) Calculate the average strain in the specimen. (6) Calculate the constraint factor. (7) Modify the calculated average strain and the constraint factor using the measured data from (2).
Friction force is eliminated.
(8) Determine the SS curve using the data from (1) and (7).
SS curve without friction force
(9) Delete the effect of friction force from the flow stress value determined in (8).
Effect of friction force No
(10) Compare the SS curve from (3) and the result from (9). Is the difference small enough?
SS curve of the compression test Yes
Equivalent strain Fig. 18. Schematic diagram showing the elimination of torque due to the friction forces from the measured torque.
(11) The determined SS curve is an accurate SS curve for the tested material.
Fig. 21. Flow chart to measure an SS curve using the developed method.
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
In Fig. 20, an SS curve, which is extrapolated with the Swift model [24] based on the results of the compression test, is shown to compare with the SS curves obtained using the developed method. It is clearly shown that there is a large discrepancy between the flow stress measured using the developed method and the extrapolated SS curve regarding the work-hardening behavior for strains over 1.0. Therefore, when an SS curve is required for large strain ranges, it is very useful to use the developed method for the measurement.
7. Summary In this paper, a method is proposed for measuring an SS curve for strains up to 10.0. In the developed method, HPT was applied. The procedure for measuring the SS curve is summarized in Fig. 21. It is shown that an SS curve was successfully measured for strains up to 10.0. In addition, the SS curve obtained using the developed method shows good correlation with that by compression testing up to a strain of 1.0.
Acknowledgments The authors would like to thank Drs. S. Kuramoto and T. Furuta (Toyota Central R&D Labs., Inc.) for their useful and stimulating discussions. The authors also would like to express their gratitude to Mr. F. SIMI (Transvalor S.A.) for his kind support helping us to carry out the FEM calculations.
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Measurement method for stress–strain curve in a super-large strain range Yasuhiro Yogo a,n, Masatoshi Sawamura a, Masafumi Hosoya b, Michiaki Kamiyama a, Noritoshi Iwata a, Takashi Ishikawa b a b
Toyota Central R&D Labs., Inc., Japan Department of Materials Science and Engineering, Nagoya University, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 9 December 2013 Received in revised form 6 February 2014 Accepted 7 February 2014 Available online 17 February 2014
Flow stress is an essential material property for metal forming. A stress–strain (SS) curve is generally measured by tensile testing or compression testing. In some metal forming processes, the maximum strain exceeds 5.0 (500%); however, standard testing methods can only measure an SS curve up to a strain of 1.0. Therefore, a new method was developed that enables us to measure an SS curve for strains up to 10.0. High-pressure torsion (HPT) was applied for the measurement. Using HPT, it is possible to deform a specimen without fracture. Therefore, larger strains can be introduced into a specimen than with conventional methods. An SS curve measurement was performed for pure aluminum. The measured torque during HPT and the average strain in the specimen were converted into an SS curve. The SS curve measured using the developed method was compared with that obtained using a compression test. It is confirmed that the flow stress was successfully measured up to a strain of 10.0. In addition, the results for the developed method correspond with those of the compression test up to a strain of 1.0. & 2014 Elsevier B.V. All rights reserved.
Keywords: Stress–strain curve Flow stress High pressure torsion Large strain: finite element method
1. Introduction It is well known that bulk material is heavily deformed during industrial metal forming processes. The strain introduced into a material by plastic deformation can be calculated using the finite element method (FEM), i.e., for cold forging [1], hot forging [2], forward [3] and backward extrusion [4], and ring rolling [5]. As shown in those papers, the maximum strain values range from 3.0 to 10.0 (300–1000%). In the design stage for manufacturing, the press load is estimated and the die material is determined using the stress–strain (SS) curve of the material. The SS curve is also used as input data for calculations when using FEM to predict material flow. Tensile testing is a general method used to measure an SS curve. Stress and strain are expanded uniformly within the gauge length of a specimen. Therefore, the experimentally measured load and displacement are easily converted into an SS curve. The measurable strain range is relatively small. The maximum strain measurable in tensile testing depends on the material and is normally less than 0.5. With recent developments, it is possible to measure an SS curve with a strain over 1.0 by using an optical strain measurement system [6].
n
Corresponding author.
http://dx.doi.org/10.1016/j.msea.2014.02.026 0921-5093 & 2014 Elsevier B.V. All rights reserved.
Compression testing is also applicable to bulk materials, and its measurable strain range is larger than that of a tension test. The strain distribution in a material during a compression test is not uniform, making it difficult to convert the measured load and displacement into an SS curve. In order to deal with this, a uniform compression test was developed [7]. In this test, a cylindrical specimen is compressed between smooth flat planes using a Teflon sheet and grease as lubricants. The lubricant is renewed at every 3–5% reduction in height. At every 20–30% reduction in height, the specimens are machined into cylindrical shapes. The machining was performed at a very slow speed to avoid heating the specimen. The Rastegaev test is another uniform compression test. In this test, shallow hollows are machined into both the top and bottom surfaces of the specimen to hold lubricant. During the compression test, the lubricant flows outward through the interface between the specimen and the dies. This reduces the friction force which causes non-uniform compression [8]. On the other hand, a method that allows the strain distribution from a compression test to be used to determine the SS curve was also developed [9]. The average strain and the restraint factor are first calculated with FEM, and then the measured load and displacement curve are converted into an SS curve. Using this compression test, an SS curve can be determined for a strain around 1.0. Using the torsion test [10], a measurable strain range of 1.0–3.0 can be reached, depending on the material. The strain distribution
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
from the torsion test is also not uniform. When the measured rotation angle and torque are converted into an SS curve, the nonuniform strain distribution and nonlinearity of the stress evolution must be considered. There are some reports of measuring an SS curve using torsion tests [11,12]. However, the effects of such strain and stress distributions have not been well studied. Although a large strain is introduced by metal forming, there is no method to measure an SS curve over such a large strain range. When FEM calculations are conducted, an extrapolated SS curve based on the measured SS curve is sometimes used to cover the immeasurable strain range. In the previous study, one of the authors has studied the influence of the difference between the measured and extrapolated SS curves on material flow in FEM simulations. It was confirmed that the SS curve strongly affects material flow even if the difference between the measured and extrapolated SS curves is small, especially when predicting fracturing during the stamping process [13,14]. Therefore, an SS curve should be measured to cover the strain range expected throughout the actual forming process. In this study, a method of measuring the SS curve for strains up to 10.0 is developed. There are two requirements for this method. The first is that it enables us to deform a material up to a strain of 10.0. The second is that the strain is uniform in the material or that the strain distribution is handled with a certain procedure to obtain the SS curve, as was mentioned for the compression test [9]. It is difficult to introduce strains exceeding 1.0 with general material testing methods. On the other hand, for the last 20 years, some material processing methods have been applied to conduct severe plastic deformation (SPD) and to improve material properties. Examples of SPD processing are equal-channel angular pressing (ECAP) [15,16], high-pressure torsion (HPT) [17,18], and accumulative roll bonding (ARB) [19]. With these SPD processing methods, it is possible to continue to deform a material without fracture. In this study, HPT was used because dimensional changes of a specimen while processing is relatively small compared with other SPD processing methods. This is an advantage when experimentally measured data is converted into an SS curve. The strain introduced into a material by HPT testing is not uniform; therefore, a procedure that takes the strain distribution into account must also be developed.
2. Experimental methods 2.1. HPT test The material used for the specimens was pure aluminum (99.9%). The specimen dimensions used are shown in Fig. 1, and the
10
83
Upper die
Ring
Lower die Specimen
Fig. 2. Schematic diagram of the experimental setup for the HPT tests.
experimental setup is shown in Fig. 2. All the surfaces of the specimen were enclosed by the upper and lower dies and the ring. Therefore, the specimen dimensions did not change during HPT testing. In order to suppress sliding between the specimen and the upper and lower dies, small hollows and grooves were machined into the contacting surfaces of the upper and lower dies, as shown in Fig. 3. The specimen was compressed at 1 GPa between the dies and was subsequently torsionally strained by the rotation of the upper die. Due to the high pressure, the specimen can be continuously strained without breaking. The rotation speed of the upper die was 1 rpm. The compression load, rotation angle, and torque of the upper die were measured during the test at room temperature. 2.2. Measurement of the actual rotation angle of a specimen in an HPT test The specimen is torsionally strained by the rotation of the upper die. As reported by Kaveh et al., the rotation angle of the upper die is not always the same as that of the specimen [20]. Therefore, the actual rotation angle of the specimen after HPT testing should be measured to accurately determine the introduced strain. Thin copper (Cu) foil was included in several HPT specimens. The foil thickness was 0.05 mm. A schematic of these specimens is shown in Fig. 4. After testing, each HPT specimen was inserted into a plate, on which radial lines were prescribed on both the front and back surfaces. Using the plate, the torsion angle of the Cu foil was measured from both the front and back views. 2.3. Compression test A compression test was conducted to validate the SS curve measured using the developed method. The initial diameter and height of the specimens were 14 and 21 mm, respectively. For the contacting surface of the upper and lower dies, concentric circles were grooved to prevent the sliding of the specimens. The dimensions of the specimens and the contacting surfaces of the upper and lower dies are shown in Fig. 5. The measured load and displacement were converted into an SS curve based on the procedure proposed by Osakada et al. [9].
1 Fig. 1. Dimensions of the specimen for the HPT tests.
2.4. Torsion test As written previously, it is not possible to measure an SS curve accurately with torsion testing. Therefore, torsion testing [10] was
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Fig. 3. Photograph of the (a) upper die, (b) lower die, and (c) ring, and (d) a magnified view of the die surface to prevent slip between the dies and specimen.
Copper foil (0.05 mm thickness)
14
120 2.4
2.4
120
21
120
0.4
Half-circle specimens (Al) Fig. 4. Schematic diagram of a specimen to measure slippage between a specimen and dies during HPT tests.
only applied to measure the maximum strain obtained by conventional methods. The specimen dimensions are shown in Fig. 6. A specimen was torsionally strained by being gripped at both ends, and the strain was calculated based on the number of revolutions.
3. Calculation method The theoretical equation used to calculate pure torsion strain is described as herein. The shear strain is defined as
γ yz ¼
π D ðθ=360Þ h
ð1Þ
Here, D is the diameter of the specimen, θ is the rotation angle in degrees, and h is the height of the specimen. By applying the von
0.115
0.577
1.2
Fig. 5. Dimensions of a specimen and the contacting surface of the dies for a compression test. (a) Specimen and (b) contacting surface of the die.
Mises equivalent strain, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εeq ¼ 23 ðε2xx þ ε2yy þ ε2zz Þ þ 13 ðγ 2xy þ γ 2yz þ γ 2zx Þ The equivalent strain of pure torsion can then be written as qffiffiffiffiffiffiffiffiffiffi εeq ¼ 13 γ 2yz
ð2Þ
As inferred by the theoretical equation of pure torsion, the strain in an HPT specimen changes along the radius. In addition, it is easy to expect the friction between the specimen and the ring, which affects the strain distribution. Therefore, in order to determine the strain in an HPT specimen, FEM simulations were conducted. The updated Lagrangian formulation is generally applied for metal forming simulations. However, in large-deformation cases, the updated Lagrangian formulation suffers from numerical
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
15
250
4
15
85
True stress / MPa
10
8
20 60
Fig. 6. Dimensions of the specimen for the torsion tests.
Specimen (D10*H1)
Pressure torsion
200 150 n=0.05 n=0.1 100 n=0.2 n=0.3 50 0
Ring
0
0.5
1.0 1.5 True strain / -
2.0
Fig. 8. SS curves defined in the FEM calculations.
Upper die 60
90
CL Upper die
Ring
Specimen
Lower die Fig. 7. Different views of the calculation model. (a) 3D view of the calculation model and (b) schematic view.
Torque / N.m
50 60
40 30
30
20 10 0
Compression load / N
Lower die
0 0 60 120 180 240 300 360 Rotation angle of the upper die / degree
Fig. 9. Torque and compression load measured in an HPT test.
difficulties due to severe mesh distortion. In order to overcome the difficulties caused by such mesh distortion, the arbitrary Lagrangian–Eulerian (ALE) formulation has been proposed [21–23]. ALE combines the respective advantages of the Lagrangian and Eulerian formulations. In the ALE formulation, the mesh motion is taken arbitrarily from the material deformation to keep the element shapes optimal. In addition, ALE is suited to compute deformation in a fixed space. Therefore, ALE was applied to the calculation of strain during HPT. Due to the limitation of the software, a small hole (0.1 mm in radius) was made at the specimen's center. Because the volume ratio of the hole is 0.4% and the strain by HPT approaches zero at the specimen's center, it is clear that the computation accuracy is not adversely affected by the hole. The calculation model is shown in Fig. 7. These FEM simulations were conducted with the commercial FEM software FORGE (Transvalor S.A., France). The effects of the SS curve and the friction between the specimen and the ring on the strain distribution in an HPT specimen were examined. SS curves with different workhardening exponents, n, were defined. The SS curves defined are shown in Fig. 8. Elasto-plastic material properties were defined for each specimen. The friction coefficient, μ, between the specimen and the ring was defined by the Coulomb friction law. The sticking condition between the specimen and the upper and lower dies was assumed to prevent sliding. A pressure of 1 GPa was applied to the specimen by the upper die. The upper and lower dies and the ring were both rigid bodies. The ring is a fixed body; therefore, it does not follow the motion of the specimen.
4. Experimental results 4.1. HPT results The compression load and torque are shown in Fig. 9. The measured torque shows a drastic increase at the beginning
followed by an almost constant value. The compression force could be maintained at a constant value during the entire HPT test. 4.2. Slippage between the specimen and the upper die The torsion angle of the specimen was measured using an inserted Cu foil. An HPT specimen was observed from both the front and back views. Photographs of the 451 and 3601 torsionally deformed specimens taken from both the front and back views are shown in Fig. 10. The relationship between the rotation angle of the upper die and the torsion angle of the specimen is shown in Fig. 11. From 01 to 451 of rotation of the upper die, no slippage occurred between the specimen and the upper die. Beyond that, a small slippage of about 201 occurred. In Fig. 12, the measured torque is shown. The measured torque of the specimens with Cu foils follows that of the specimen without a Cu foil. This means that the insertion of a Cu foil into the specimen does not affect deformation. 4.3. SS curve from the compression test The SS curve measured by compression testing is shown in Fig. 16. By compressing the specimen to a 50% reduction in height, an SS curve could be measured up to a strain of 0.8. This result is discussed in Section 6.1 to compare it with the SS curve measured using the developed method. 4.4. Maximum strain obtained by the torsion test The torsion test was conducted two times. The rotation angles at fracture were 38401 and 36601. The strain introduced by pure torsion can be calculated using Eq. (2). Therefore, the maximum strain, which is at the circumference, is 4.72 at fracture. The average strain, which is calculated by the integration of Eq. (2) through the radius, is 3.71. This shows that it is not possible to introduce strains up to 10.0 with the torsion test.
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Front view Rotation
Back view Fig. 10. Photographs showing the front and back views and the rotation angles of the upper die after the HPT tests.
50 Theoretical value (without slip)
270
40
180 Experiment
90 0
Torque / N•m
Torsion angle of the specimen / degrees
360
30 20 10 0
0 90 180 270 360 Rotation angle of the upper die / degrees
Fig. 11. Relationship between the rotation angle of the upper die and the torsion angle of a specimen.
5. Calculation results
Rotation angle of the upper die 45 90 180 (With Cu foil) 270 360 360 (Without Cu foil) 360 0 90 180 270 Rotation angle of the upper die / degrees
Fig. 12. The effect of the copper foil on torque.
edge when using a material with a higher work-hardening exponent. In the case where the friction coefficient is higher, the stress concentration at a corner edge is also higher.
5.1. Strain distribution The influence of the friction coefficient, μ, between the specimen and the ring and the work-hardening exponent, n, on the strain distribution is shown in Fig. 13. When a material has a lower defined work-hardening exponent, the strain increases significantly. On the other hand, strain tends to concentrate on a corner
6. Determination procedure for SS curve In this section, the procedure to measure strain using the developed method is first discussed. Then, a procedure to measure stress is discussed. Finally, the SS curve is shown.
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
n=0.1
µ=0.0
n=0.2
µ=0.1
n=0.3
µ=0.2
87
µ=0.3 Center Line
1
2 3 Radius / mm
4
5
Center Line
1
3 2 Radius / mm
4
Equivalent strain 40 36 32 28 24 20 16 12 8 4 5 0
12
350 µ n 0.2 0.0 0.2 0.1 0.2 0.2 0.2 0.3 0.1 0.1 0.3 0.1 Theoretical
10 8 6 4 2 0
Equivalent stress / MPa
Average equivalent strain / -
Fig. 13. Strain distribution in a specimen after 1 revolution. (a) Influence of the work hardening exponent, n (μ¼0.1) and (b) influence of the friction coefficient, μ (n¼ 0.2).
0
60
120
180
240
300
360
Fig. 14. Average strain in a specimen.
Equivalent strain / -
10
Theoretical value (without slip)
6 2 0
With consideration of slippage between the upper die and a specimen
0 90 180 270 360 Rotation angle of the upper die / degrees
100 Compression test
50 0
2
4 6 8 10 Equivalent strain / -
12
6.2. Stress
Fig. 15. Relationship between the rotation angle of the upper die and the average strain in a specimen.
6.1. Strain The way to convert the rotation angle of the upper die into strain is discussed in this section. In Section 5.1, it has been shown that the friction coefficient between the specimen and the ring, μ, and the work-hardening exponent of the material, n, affect the strain distribution. In order to investigate the relationship between the rotation angle of the upper die and the average strain throughout the specimen, the calculation results shown in Fig. 13 were averaged using Eq. (3): ∑V ε εave ¼ i i i ∑i V i
150
value, and the friction coefficient and the work-hardening exponent have no influence on the average strain. As shown in Fig. 11, a small amount of slippage occurred during HPT. It is possible to obtain the relationship between the rotation angle of the upper die and the actual average strain of a specimen by combining Figs. 11 and 14. The relationship obtained is shown in Fig. 15.
8 4
Developed method (N=2)
200
Fig. 16. Comparison of SS curves between the developed method and a compression test.
14 12
250
0
Rotation angle of the upper die / degrees
Developed method (N=1)
300
Osakada proposed a procedure to convert the compression load into an SS curve using compression testing [9]. His theory was modified to measure the SS curve using HPT testing. First, an FEM calculation was conducted using a fictitious material with an SS curve described by Eq. (4):
s ¼ ε0:001
Using the calculation results, the average strain and the restraint factor, f, was calculated using Eqs. (4) and (5), respectively:
save ¼ f¼
ð3Þ
Here, εave is the average strain in the specimen, and V i and εi are the volume and equivalent strain of an element (i), respectively. The calculated relationship is shown in Fig. 14. The average strain for all conditions follows the average theoretical value. This means that the average strain can be calculated using the theoretical
ð4Þ
∑i V i si ∑i V i
T FEM
save
ð5Þ ð6Þ
Here, save is the average stress in the specimen, si is the equivalent stress of an element (i), f is the restraint factor, and T FEM is the calculated torque. By combining the measured torque and the calculated restraint factor, f, an SS curve is obtained using Eq. (7):
s¼
T exp f
ð7Þ
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
Here, T exp is the measured torque. Using the obtained SS curve, this procedure was repeated until the difference between the formerly defined stress and the obtained stress became small enough. The obtained SS curve is shown in Fig. 16. The SS curve obtained by compression testing is also shown in Fig. 16. There is a large discrepancy between the two curves. The flow stress is based on the measured torque in the developed test, and the discrepancy appeared because of this measurement. In order to evaluate flow stress using the developed method, only the torque from the torsional deformation of the specimen should be used. However, as shown in Fig. 17, other factors were included in this measurement. It is very difficult to eliminate the effect of the other factors when measuring torque. Therefore, the difference in yield stress is recognized as the effect of the other factors. Here, it is assumed that the other factors caused by the friction force are at a constant value throughout the measurement. By eliminating the effect of the other factors, as shown in Fig. 18, the flow stress was modified, and the modified SS curve is shown in Fig. 19. It is shown that an SS curve was successfully measured for strains up to 10.0. In addition, the SS curve obtained using the developed method
shows good correlation with that obtained by compression testing up to a strain of 1.0. It is natural that both of the yield stresses are the same. However, the work-hardening behavior of the developed method corresponds with that of the compression test. This result shows the validity of the measured results for the developed method.
300 Equivalent stress / MPa
88
Developed method (N=1)
250 200
Developed method (N=2)
150 100
Compression test
50 0
0
2
4 6 8 10 Equivalent strain / -
12
Fig. 19. Comparison of SS curves between the developed method and a compression test.
Upper die
(3)
(1) Specimen
(2)
Lower die
(4) Ring
Measured torque = (1) + (2) + (3) + (4)
Equivalent stress / MPa
300 Developed method (N=1)
250 200
Developed method (N=2)
150
Extrapolated curve
100
Compression test
50 0
0
12
(1) Measure the torque and rotation angle of the upper die during HPT.
of a specimen.
(2): Torque due to the friction force between a specimen and the ring. (3): Torque due to the friction force between the upper die and the ring. (4): Torque due to the friction force between the lower die and the ring.
(2) Measure the relationship between the rotation angle of the upper die and that of the specimen. (3) Define a temporary SS curve for an FEM calculation. (4) Conduct the FEM calculation. (12) The flow stress in (3) should be replaced by the SS curve in (8).
SS curve derived from the law data Equivalent stress
4 6 8 10 Equivalent strain / -
Fig. 20. Comparison of SS curves obtained via the developed method and by extrapolation.
(1): Torque due to torsional deformation
Fig. 17. Schematic diagram of the influence of the friction forces on the measured torque.
2
(5) Calculate the average strain in the specimen. (6) Calculate the constraint factor. (7) Modify the calculated average strain and the constraint factor using the measured data from (2).
Friction force is eliminated.
(8) Determine the SS curve using the data from (1) and (7).
SS curve without friction force
(9) Delete the effect of friction force from the flow stress value determined in (8).
Effect of friction force No
(10) Compare the SS curve from (3) and the result from (9). Is the difference small enough?
SS curve of the compression test Yes
Equivalent strain Fig. 18. Schematic diagram showing the elimination of torque due to the friction forces from the measured torque.
(11) The determined SS curve is an accurate SS curve for the tested material.
Fig. 21. Flow chart to measure an SS curve using the developed method.
Y. Yogo et al. / Materials Science & Engineering A 600 (2014) 82–89
In Fig. 20, an SS curve, which is extrapolated with the Swift model [24] based on the results of the compression test, is shown to compare with the SS curves obtained using the developed method. It is clearly shown that there is a large discrepancy between the flow stress measured using the developed method and the extrapolated SS curve regarding the work-hardening behavior for strains over 1.0. Therefore, when an SS curve is required for large strain ranges, it is very useful to use the developed method for the measurement.
7. Summary In this paper, a method is proposed for measuring an SS curve for strains up to 10.0. In the developed method, HPT was applied. The procedure for measuring the SS curve is summarized in Fig. 21. It is shown that an SS curve was successfully measured for strains up to 10.0. In addition, the SS curve obtained using the developed method shows good correlation with that by compression testing up to a strain of 1.0.
Acknowledgments The authors would like to thank Drs. S. Kuramoto and T. Furuta (Toyota Central R&D Labs., Inc.) for their useful and stimulating discussions. The authors also would like to express their gratitude to Mr. F. SIMI (Transvalor S.A.) for his kind support helping us to carry out the FEM calculations.
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