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Determination of strain-hardening exponent using double compression test
Materials Science and Engineering A 518 (2009) 56–60
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Determination of strain-hardening exponent using double compression test R. Ebrahimi ∗ , N. Pardis Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 9 February 2009 Received in revised form 5 April 2009 Accepted 24 April 2009 Keywords: Double compression test Strain-hardening exponent Pre-strain Finite element analysis
a b s t r a c t In this investigation a new method for determination of strain-hardening exponent (n) is introduced. The presented method is named “Double Compression Test” in which two specimens with the same composition and geometrical dimensions but different processing background are compressed simultaneously. One of the two specimens is in annealed condition while the other has experienced a predetermined amount of pre-strain. This difference leads to different final length of the specimens which can be used in the theoretical relation presented to calculate the strain-hardening exponent. The major advantage of this method is its independency to the stress–strain data which is essential in the conventional method for determination of strain-hardening exponent. The test was performed experimentally and the results were compared with those obtained by the conventional method. Finally, the test was simulated using the commercial finite element code, ABAQUS/Explicit, to investigate it in more details. Experimental and simulation results showed that this test is capable of determining strain-hardening exponent with good accuracy. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The value of the strain-hardening exponent (n) is of major importance in forming operations since it controls the amount of uniform plastic strain the material can undergo during a tensile test before strain localization, or necking, sets in leading to failure. It is easily shown that the maximum amount of uniform plastic deformation in tensile straining is given by the strain-hardening exponent (n) which is known as the Considère criterion. As a result, a high coefficient facilitates complex-forming operations without premature failure [1]. Therefore, strain-hardening exponent is an important parameter reflecting a material’s hardening property and its determination is of great importance. A standard method to perform this task is based on using stress–strain data obtained from uniaxial tensile test. Stress–strain curves are usually represented by the Hollomon equation. Therefore, by plotting stress–strain data on logarithmic coordinates, it can be shown that the slope of the line in the fully plastic region defines the strain-hardening exponent (n) [2,3]. Shinohara [4] investigated relationship between strainhardening exponent and load dependence of Vickers hardness in copper and showed that the slope of the load dependence of the hardness was a good measure for correlating with strain-hardening exponent (n). By using the instrumented spherical-indentation technique, Nayebi et al. [5] presented a relationship between
∗ Corresponding author. Tel.: +98 711 2307293; fax: +98 711 2307293. E-mail address: [email protected] (R. Ebrahimi). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.04.050
applied loads, indenter displacement, flow stress and strainhardening exponent of steels. The model they presented yields to the steel mechanical parameters, y and n from the indentation displacement–load curve. Kim et al. [6] evaluated plastic flow properties by characterizing indentation size effect using a sharp indenter and found a linear relationship between the strain-hardening exponent and the log of the indentation size effect characteristic length for Ni and structural steel samples with different plastic pre-strain values. Antoine et al. [7] showed that there is a linear relationship between the value of the strain-hardening exponent and the yield strength, and presented a model giving the value of the strain-hardening exponent for Ti-IF steel. Nagarjuna et al. [8] investigated the relationship between strain-hardening exponent (n) and grain size of Cu–26Ni–17Zn alloy by analysis of constants in Hollomon equation as a function of grain size. They found that strain-hardening exponent (n) is independent of grain size in the range of 15–120 m. Narayanasamya et al. [9] performed a study on the instantaneous strain-hardening behavior of an aluminum powder metallurgy composite with various percent of iron contents and for the various stress state conditions with two different aspect ratios. They calculated the instantaneous strain-hardening exponent (ni ) and the strength coefficient (ki ) using mathematical expressions. Zhang et al. [10] used two equations correlating the strain-hardening exponent and the strength coefficient with the yield stress–strain behavior, and also the fracture strength with the fracture ductility and presented a simple theoretical method of calculating the strain-hardening exponent and the strength coefficient of metallic materials.
R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60 Table 1 The initial dimensions of the aluminum specimens.
Nomenclature k n H0 Hp Ha H D0 Dp Da A0 Ap Aa ε0
57
strength coefficient strain-hardening exponent initial height of cylinder height of pre-strained cylinder after deformation height of annealed cylinder after deformation total reduction in height initial diameter of cylinder diameter of pre-strained cylinder after deformation diameter of annealed cylinder after deformation initial cross section area of cylinder cross section area of pre-strained cylinder after deformation cross section area of annealed cylinder after deformation amount of pre-strain
Specimen
H0 (mm)
D0 (mm)
Aspect ratio
I II
45 15
30 10
1.5 1.5
where, subscripts “a” and “p” apply to annealed and pre-strained conditions respectively. Thus:
ε0 + ε¯ p ε¯ a
n
=
Aa Ap
(2)
where A0 , Ap and Aa are related to one another according to the following equations: Aa = A0 · exp(¯εa )
(3)
Ap = A0 · exp(¯εp )
(4)
The strain-hardening exponent (n) is calculated using equation: In the present study, a new approach is introduced which is capable of determining strain-hardening exponent without any need to know the stress–strain data. This method just deals with geometrical dimensions of the work piece before and after the test.
2. Theory
n=
ε¯ a − ε¯ p
ln
ε0 +¯εp ε¯ a
(5)
By this method, strain-hardening exponent can be simply calculated by just using geometrical measurements, without needing to know the load values. 3. Experimental procedure
Double compression test is based on simultaneous compression of two specimens having the same composition and geometrical dimensions, one in the annealed condition, while the other undergone a predetermined amount of pre-strain. The scheme of the test is illustrated in Fig. 1. In the absence of internal effects, the same axial force (F) is transmitted through both cylindrical work pieces. Considering a frictionless condition and using the Hollomon equation, this can be expressed by the following equation: k(ε0 + ε¯ p )n · Ap = k(¯εa )n · Aa
Fig. 1. A scheme of double compression test.
(1)
Two sets of cylindrical specimens with aspect ratio 1.5 and dimensions mentioned in Table 1 were machined from an aluminum alloy rod with unknown mechanical properties. These specimens were then annealed at 430 ◦ C for 2 h. The specimens in set (I) were compressed to strain values of 0.2, 0.4, 0.6 and 0.8 as illustrated in Fig. 2. Compression tests were carried out using a screw press with the compression rate of 0.1 mm/s at room temperature. Then, new specimens were machined out of these compressed cylinders. The final dimensions of these specimens were exactly the same as those in set (II). A pair of specimens including an annealed and a strained specimen with pre-strain value ε0 = 0.2, were compressed simultaneously to the amount of H = 10 mm. The test set-up is illustrated in Fig. 3. The same procedure was repeated for specimens with pre-strain values ε0 = 0.4, ε0 = 0.6 and ε0 = 0.8. The specimens’ new heights were measured and are listed in Table 2. Using these dimensions and Eq. (5), it is possible to obtain the value of strain-hardening exponent without considering any load parameter.
Fig. 2. Aluminum specimens compressed to strain values of 0.2, 0.4, 0.6 and 0.8 from left to right.
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R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
Table 2 The experimental results. Pre-strain (ε0 )
H0 (mm)
Hp (mm)
Ha (mm)
n (double compression method)
n (standard method)
% error
0.2 0.4 0.6 0.8
15 15 15 15
10.68 10.78 11.22 11.46
9.86 9.36 9.25 9.02
0.317 0.323 0.316 0.322
0.318
0.31 1.57 0.63 1.26
Meanwhile, the conventional method of determining strainhardening exponent was performed on annealed specimens and the results were compared with those determined previously. In addition, the commercial finite element code, ABAQUS/ Explicit, was used to simulate this new test in order to investigate the process variables in more details. The strength coefficient and strain-hardening exponent used to perform the simulation were determined by standard method. Dimensions of specimens were chosen to be the same as those used in the experiment as mentioned in set (II), Table 1. Simulation was performed for different values of pre-strain and H = 10 mm to investigate the validity of the theoretical results. The effect of friction on the test accuracy was also considered. The friction factor in the experiment was determined experimentally by barrel compression test [11], and in order to use this value in the simulation, it was converted to the friction coefficient using Eq. (6) [12]. =
m 27(1 − m2 )
(6)
Fig. 4. A plot of strain-hardening exponent vs. final specimens’ height ratio for different pre-strain values.
4. Results and discussion The values of strain-hardening exponent obtained by the presented method and standard method are shown in Table 2. As can be seen in this table, strain-hardening exponent can be calculated by this method with very good accuracy. Based on Eq. (5), the relationship between strain-hardening exponent and Hp /Ha for different pre-strain values with fixed amount of total reduction in height (H = 10 mm) is plotted as illustrated in Fig. 4 wherein the experimental values mentioned in Table 2 are depicted as solid squares. It is observed that by increasing the amount of pre-strain, the slope of the curve decreases, indicating that the change in the ratio Hp /Ha has less effect on the predicted value of strain-hardening exponent. As a result, by apply-
Fig. 3. Double compression test set-up.
ing higher pre-strain values, any error in measurement of heights after deformation would have less effect on determination of strainhardening exponent. Such graphs can be used as a calibration curve instead of using the relations mentioned in the theory. As observed in Fig. 5, by increasing the total height reduction (H) of a specified material, the ratio Hp /Ha decreases. Therefore, for small values of pre-strain, it is recommended to choose small values of total displacement, which is also proper for testing materials with low workability. A pair of samples corresponding to a pre-strain value of 0.8 is illustrated in Fig. 6. The difference in heights of these specimens is clearly due to the difference in their initial conditions (annealed and pre-strained condition). In order to investigate the effect of friction on deviation of results from the ideal frictionless condition as considered in the theory, a simulation was performed for a material with strainhardening exponent n = 0.318, H = 10 mm, ε0 = 0.6 and different
Fig. 5. Effect of the total height reduction on final specimens’ height ratio.
R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
59
Fig. 6. The specimens after the test with H = 10 mm. The left specimen has experienced a pre-strain amount of 0.8 and the right one was in the annealed condition. Fig. 9. Changes of the height ratio vs. the total height reduction for different materials and a specified value of pre-strain.
Fig. 7. Effect of friction, on the deviation of results from the frictionless condition.
friction coefficients. The final specimens’ heights obtained from simulation were used in Eq. (5) to determine the strain-hardening exponent in each condition. As illustrated in Fig. 7, by decreasing the friction coefficient, deviation from the actual value decreases resulting in more accurate strain-hardening exponent values. The coefficient of friction was predicted to be 0.03 in the conducted experiments. Referring to Fig. 7, it is concluded that the error due to friction effect is negligible and therefore, the determined strain-hardening exponent values in Table 2 are acceptable. In Fig. 8, strain is plotted versus time for double compression test specimens as obtained by simulation in frictionless condition. Two distinct zones are observed in this figure. At zone (I) the deformation is limited to the annealed specimen and the pre-strained specimen merely transmits the load. When deformation of the annealed specimen reaches a certain value, both specimens deform simultaneously (zone II). The effect of strain-hardening exponent on the relation between ratio Hp /Ha and total reduction in height (H) is presented in Fig. 9. According to this figure, the test would be more sensitive by using lower total height reduction values. Fig. 10 shows that the sensitiv-
Fig. 10. Effect of values of pre-strain on the changes of the height ratio for different n values.
ity of the test for a specified amount of reduction in height increases by increasing the amount of pre-strain. 5. Conclusion Double compression test is introduced as a new method for determination of strain-hardening exponent. By this method, the n value of any material can be predicted according to Eq. (5) without any need to know stress–strain data. The experimental results of this method show good correlation with those obtained by conventional method. The validity of the theory was also studied by conducting finite element simulations of the process using ABAQUS/Explicit. The simulation results also show that the effect of friction on the strain-hardening exponent value obtained by double compression test is negligible. Acknowledgement Financial support by the office of Research Council of Shiraz University through grant numbers 87-GR-ENG-15 is appreciated. References
Fig. 8. Changes of strain with time for annealed and pre-strained specimens during double compression test.
[1] B. Verlinden, J. Driver, I. Samajdar, R.D. Doherty, Plasticity, in: R.W. Cahn (Ed.), Thermo-Mechanical Processing of Metallic Materials, Pergamon Materials Series, Pergamon, UK, 2007, pp. 33–56. [2] ASTM E 646-78, Standard test method for tensile strain-hardening exponent (nvalue) of metallic sheet materials, Annual Book of ASTM standards, Philadelphia, 03.01, 1989, pp. 639–643. [3] W.F. Hosford, R.M. Caddell, Metal Forming: Mechanics and Metallurgy, 2nd ed., Prentice-Hall, USA, 1983. [4] K. Shinohara, J. Mater. Sci. 28 (1993) 5325–5329. [5] A. Nayebi, R. El Abdi, O. Bartier, G. Mauvoisin, Mech. Mater. 34 (2002) 243–254.
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R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
[6] J.Y. Kim, S.K. Kang, J.R. Greer, D. Kwon, Acta Mater. 56 (2008) 3338–3343. [7] P. Antoine, S. Vandeputte, J.-B. Vogt, Mater. Sci. Eng. A 433 (2006) 55– 63. [8] S. Nagarjuna, B. Gopalakrishna, M. Srinivas, Mater. Sci. Eng. A 429 (2006) 169–172.
[9] R. Narayanasamy, T. Ramesh, K.S. Pandey, Mater. Sci. Eng. A 394 (2005) 149–160. [10] Z. Zhang, Q. Sun, C. Li, W. Zhao, J. Mater. Eng. Perf. 15 (2006) 19–22. [11] R. Ebrahimi, A. Najafizadeh, J. Mater. Process. Technol. 152 (2004) 136–143. [12] N. Bay, J. Mech. Work. Technol. 14 (1987) 203–223.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Determination of strain-hardening exponent using double compression test R. Ebrahimi ∗ , N. Pardis Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 9 February 2009 Received in revised form 5 April 2009 Accepted 24 April 2009 Keywords: Double compression test Strain-hardening exponent Pre-strain Finite element analysis
a b s t r a c t In this investigation a new method for determination of strain-hardening exponent (n) is introduced. The presented method is named “Double Compression Test” in which two specimens with the same composition and geometrical dimensions but different processing background are compressed simultaneously. One of the two specimens is in annealed condition while the other has experienced a predetermined amount of pre-strain. This difference leads to different final length of the specimens which can be used in the theoretical relation presented to calculate the strain-hardening exponent. The major advantage of this method is its independency to the stress–strain data which is essential in the conventional method for determination of strain-hardening exponent. The test was performed experimentally and the results were compared with those obtained by the conventional method. Finally, the test was simulated using the commercial finite element code, ABAQUS/Explicit, to investigate it in more details. Experimental and simulation results showed that this test is capable of determining strain-hardening exponent with good accuracy. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The value of the strain-hardening exponent (n) is of major importance in forming operations since it controls the amount of uniform plastic strain the material can undergo during a tensile test before strain localization, or necking, sets in leading to failure. It is easily shown that the maximum amount of uniform plastic deformation in tensile straining is given by the strain-hardening exponent (n) which is known as the Considère criterion. As a result, a high coefficient facilitates complex-forming operations without premature failure [1]. Therefore, strain-hardening exponent is an important parameter reflecting a material’s hardening property and its determination is of great importance. A standard method to perform this task is based on using stress–strain data obtained from uniaxial tensile test. Stress–strain curves are usually represented by the Hollomon equation. Therefore, by plotting stress–strain data on logarithmic coordinates, it can be shown that the slope of the line in the fully plastic region defines the strain-hardening exponent (n) [2,3]. Shinohara [4] investigated relationship between strainhardening exponent and load dependence of Vickers hardness in copper and showed that the slope of the load dependence of the hardness was a good measure for correlating with strain-hardening exponent (n). By using the instrumented spherical-indentation technique, Nayebi et al. [5] presented a relationship between
∗ Corresponding author. Tel.: +98 711 2307293; fax: +98 711 2307293. E-mail address: [email protected] (R. Ebrahimi). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.04.050
applied loads, indenter displacement, flow stress and strainhardening exponent of steels. The model they presented yields to the steel mechanical parameters, y and n from the indentation displacement–load curve. Kim et al. [6] evaluated plastic flow properties by characterizing indentation size effect using a sharp indenter and found a linear relationship between the strain-hardening exponent and the log of the indentation size effect characteristic length for Ni and structural steel samples with different plastic pre-strain values. Antoine et al. [7] showed that there is a linear relationship between the value of the strain-hardening exponent and the yield strength, and presented a model giving the value of the strain-hardening exponent for Ti-IF steel. Nagarjuna et al. [8] investigated the relationship between strain-hardening exponent (n) and grain size of Cu–26Ni–17Zn alloy by analysis of constants in Hollomon equation as a function of grain size. They found that strain-hardening exponent (n) is independent of grain size in the range of 15–120 m. Narayanasamya et al. [9] performed a study on the instantaneous strain-hardening behavior of an aluminum powder metallurgy composite with various percent of iron contents and for the various stress state conditions with two different aspect ratios. They calculated the instantaneous strain-hardening exponent (ni ) and the strength coefficient (ki ) using mathematical expressions. Zhang et al. [10] used two equations correlating the strain-hardening exponent and the strength coefficient with the yield stress–strain behavior, and also the fracture strength with the fracture ductility and presented a simple theoretical method of calculating the strain-hardening exponent and the strength coefficient of metallic materials.
R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60 Table 1 The initial dimensions of the aluminum specimens.
Nomenclature k n H0 Hp Ha H D0 Dp Da A0 Ap Aa ε0
57
strength coefficient strain-hardening exponent initial height of cylinder height of pre-strained cylinder after deformation height of annealed cylinder after deformation total reduction in height initial diameter of cylinder diameter of pre-strained cylinder after deformation diameter of annealed cylinder after deformation initial cross section area of cylinder cross section area of pre-strained cylinder after deformation cross section area of annealed cylinder after deformation amount of pre-strain
Specimen
H0 (mm)
D0 (mm)
Aspect ratio
I II
45 15
30 10
1.5 1.5
where, subscripts “a” and “p” apply to annealed and pre-strained conditions respectively. Thus:
ε0 + ε¯ p ε¯ a
n
=
Aa Ap
(2)
where A0 , Ap and Aa are related to one another according to the following equations: Aa = A0 · exp(¯εa )
(3)
Ap = A0 · exp(¯εp )
(4)
The strain-hardening exponent (n) is calculated using equation: In the present study, a new approach is introduced which is capable of determining strain-hardening exponent without any need to know the stress–strain data. This method just deals with geometrical dimensions of the work piece before and after the test.
2. Theory
n=
ε¯ a − ε¯ p
ln
ε0 +¯εp ε¯ a
(5)
By this method, strain-hardening exponent can be simply calculated by just using geometrical measurements, without needing to know the load values. 3. Experimental procedure
Double compression test is based on simultaneous compression of two specimens having the same composition and geometrical dimensions, one in the annealed condition, while the other undergone a predetermined amount of pre-strain. The scheme of the test is illustrated in Fig. 1. In the absence of internal effects, the same axial force (F) is transmitted through both cylindrical work pieces. Considering a frictionless condition and using the Hollomon equation, this can be expressed by the following equation: k(ε0 + ε¯ p )n · Ap = k(¯εa )n · Aa
Fig. 1. A scheme of double compression test.
(1)
Two sets of cylindrical specimens with aspect ratio 1.5 and dimensions mentioned in Table 1 were machined from an aluminum alloy rod with unknown mechanical properties. These specimens were then annealed at 430 ◦ C for 2 h. The specimens in set (I) were compressed to strain values of 0.2, 0.4, 0.6 and 0.8 as illustrated in Fig. 2. Compression tests were carried out using a screw press with the compression rate of 0.1 mm/s at room temperature. Then, new specimens were machined out of these compressed cylinders. The final dimensions of these specimens were exactly the same as those in set (II). A pair of specimens including an annealed and a strained specimen with pre-strain value ε0 = 0.2, were compressed simultaneously to the amount of H = 10 mm. The test set-up is illustrated in Fig. 3. The same procedure was repeated for specimens with pre-strain values ε0 = 0.4, ε0 = 0.6 and ε0 = 0.8. The specimens’ new heights were measured and are listed in Table 2. Using these dimensions and Eq. (5), it is possible to obtain the value of strain-hardening exponent without considering any load parameter.
Fig. 2. Aluminum specimens compressed to strain values of 0.2, 0.4, 0.6 and 0.8 from left to right.
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R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
Table 2 The experimental results. Pre-strain (ε0 )
H0 (mm)
Hp (mm)
Ha (mm)
n (double compression method)
n (standard method)
% error
0.2 0.4 0.6 0.8
15 15 15 15
10.68 10.78 11.22 11.46
9.86 9.36 9.25 9.02
0.317 0.323 0.316 0.322
0.318
0.31 1.57 0.63 1.26
Meanwhile, the conventional method of determining strainhardening exponent was performed on annealed specimens and the results were compared with those determined previously. In addition, the commercial finite element code, ABAQUS/ Explicit, was used to simulate this new test in order to investigate the process variables in more details. The strength coefficient and strain-hardening exponent used to perform the simulation were determined by standard method. Dimensions of specimens were chosen to be the same as those used in the experiment as mentioned in set (II), Table 1. Simulation was performed for different values of pre-strain and H = 10 mm to investigate the validity of the theoretical results. The effect of friction on the test accuracy was also considered. The friction factor in the experiment was determined experimentally by barrel compression test [11], and in order to use this value in the simulation, it was converted to the friction coefficient using Eq. (6) [12]. =
m 27(1 − m2 )
(6)
Fig. 4. A plot of strain-hardening exponent vs. final specimens’ height ratio for different pre-strain values.
4. Results and discussion The values of strain-hardening exponent obtained by the presented method and standard method are shown in Table 2. As can be seen in this table, strain-hardening exponent can be calculated by this method with very good accuracy. Based on Eq. (5), the relationship between strain-hardening exponent and Hp /Ha for different pre-strain values with fixed amount of total reduction in height (H = 10 mm) is plotted as illustrated in Fig. 4 wherein the experimental values mentioned in Table 2 are depicted as solid squares. It is observed that by increasing the amount of pre-strain, the slope of the curve decreases, indicating that the change in the ratio Hp /Ha has less effect on the predicted value of strain-hardening exponent. As a result, by apply-
Fig. 3. Double compression test set-up.
ing higher pre-strain values, any error in measurement of heights after deformation would have less effect on determination of strainhardening exponent. Such graphs can be used as a calibration curve instead of using the relations mentioned in the theory. As observed in Fig. 5, by increasing the total height reduction (H) of a specified material, the ratio Hp /Ha decreases. Therefore, for small values of pre-strain, it is recommended to choose small values of total displacement, which is also proper for testing materials with low workability. A pair of samples corresponding to a pre-strain value of 0.8 is illustrated in Fig. 6. The difference in heights of these specimens is clearly due to the difference in their initial conditions (annealed and pre-strained condition). In order to investigate the effect of friction on deviation of results from the ideal frictionless condition as considered in the theory, a simulation was performed for a material with strainhardening exponent n = 0.318, H = 10 mm, ε0 = 0.6 and different
Fig. 5. Effect of the total height reduction on final specimens’ height ratio.
R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
59
Fig. 6. The specimens after the test with H = 10 mm. The left specimen has experienced a pre-strain amount of 0.8 and the right one was in the annealed condition. Fig. 9. Changes of the height ratio vs. the total height reduction for different materials and a specified value of pre-strain.
Fig. 7. Effect of friction, on the deviation of results from the frictionless condition.
friction coefficients. The final specimens’ heights obtained from simulation were used in Eq. (5) to determine the strain-hardening exponent in each condition. As illustrated in Fig. 7, by decreasing the friction coefficient, deviation from the actual value decreases resulting in more accurate strain-hardening exponent values. The coefficient of friction was predicted to be 0.03 in the conducted experiments. Referring to Fig. 7, it is concluded that the error due to friction effect is negligible and therefore, the determined strain-hardening exponent values in Table 2 are acceptable. In Fig. 8, strain is plotted versus time for double compression test specimens as obtained by simulation in frictionless condition. Two distinct zones are observed in this figure. At zone (I) the deformation is limited to the annealed specimen and the pre-strained specimen merely transmits the load. When deformation of the annealed specimen reaches a certain value, both specimens deform simultaneously (zone II). The effect of strain-hardening exponent on the relation between ratio Hp /Ha and total reduction in height (H) is presented in Fig. 9. According to this figure, the test would be more sensitive by using lower total height reduction values. Fig. 10 shows that the sensitiv-
Fig. 10. Effect of values of pre-strain on the changes of the height ratio for different n values.
ity of the test for a specified amount of reduction in height increases by increasing the amount of pre-strain. 5. Conclusion Double compression test is introduced as a new method for determination of strain-hardening exponent. By this method, the n value of any material can be predicted according to Eq. (5) without any need to know stress–strain data. The experimental results of this method show good correlation with those obtained by conventional method. The validity of the theory was also studied by conducting finite element simulations of the process using ABAQUS/Explicit. The simulation results also show that the effect of friction on the strain-hardening exponent value obtained by double compression test is negligible. Acknowledgement Financial support by the office of Research Council of Shiraz University through grant numbers 87-GR-ENG-15 is appreciated. References
Fig. 8. Changes of strain with time for annealed and pre-strained specimens during double compression test.
[1] B. Verlinden, J. Driver, I. Samajdar, R.D. Doherty, Plasticity, in: R.W. Cahn (Ed.), Thermo-Mechanical Processing of Metallic Materials, Pergamon Materials Series, Pergamon, UK, 2007, pp. 33–56. [2] ASTM E 646-78, Standard test method for tensile strain-hardening exponent (nvalue) of metallic sheet materials, Annual Book of ASTM standards, Philadelphia, 03.01, 1989, pp. 639–643. [3] W.F. Hosford, R.M. Caddell, Metal Forming: Mechanics and Metallurgy, 2nd ed., Prentice-Hall, USA, 1983. [4] K. Shinohara, J. Mater. Sci. 28 (1993) 5325–5329. [5] A. Nayebi, R. El Abdi, O. Bartier, G. Mauvoisin, Mech. Mater. 34 (2002) 243–254.
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R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60
[6] J.Y. Kim, S.K. Kang, J.R. Greer, D. Kwon, Acta Mater. 56 (2008) 3338–3343. [7] P. Antoine, S. Vandeputte, J.-B. Vogt, Mater. Sci. Eng. A 433 (2006) 55– 63. [8] S. Nagarjuna, B. Gopalakrishna, M. Srinivas, Mater. Sci. Eng. A 429 (2006) 169–172.
[9] R. Narayanasamy, T. Ramesh, K.S. Pandey, Mater. Sci. Eng. A 394 (2005) 149–160. [10] Z. Zhang, Q. Sun, C. Li, W. Zhao, J. Mater. Eng. Perf. 15 (2006) 19–22. [11] R. Ebrahimi, A. Najafizadeh, J. Mater. Process. Technol. 152 (2004) 136–143. [12] N. Bay, J. Mech. Work. Technol. 14 (1987) 203–223.