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Modeling uniaxial, temperature and strain rate dependent behavior of Al–Mg alloys
Computational Materials Science 49 (2010) 333–339
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Modeling uniaxial, temperature and strain rate dependent behavior of Al–Mg alloys S. Toros, F. Ozturk * Department of Mechanical Engineering, Nigde University, Nigde 51245, Turkey
a r t i c l e
i n f o
Article history: Received 29 March 2010 Received in revised form 2 May 2010 Accepted 7 May 2010
Keywords: Aluminum–magnesium alloy Al–Mg Modeling 5083 5754 Warm forming Flow curve Softening behavior
a b s t r a c t The mechanical properties of 5083-H111 and 5754-O Al–Mg alloys at various testing temperatures (between room temperature (RT) and 300 °C) and strain rates (0.0016–0.16 s1) were determined. A new mathematical model which is based on temperature and strain rate is developed and named as ‘‘softening model”. The model is used to simulate the recorded strain softening behavior of the 5083H111 and 5754-O Al–Mg alloys at warm temperatures and modeling capabilities of softening model are shown. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Aluminum alloys have been widely used in automotive, aircraft, and ship building industries because of their high strength stiffness to weight ratio, good formability, good corrosion resistance, and recycling potential. Moreover, vehicle performance, comfort, and marketability are other advantages of the aluminum alloys. The main goal is to reduce the vehicle weight and fuel consumption about 6–8% for every 10% in weight reduction by using aluminum alloys in vehicles [1–3]. 5XXX (5000 series) Al–Mg alloys are the most common one in aluminum alloys which are particularly used at inner panels of the automotive structures. However, applications of the aluminum and its alloys in this field were far behind steels because of cost and formability issues at room temperature (RT). Therefore, warm forming operations are the most common applications to improve the formability of these materials. Increasing the forming temperature provides increased ductility in the sheet, which can lead to obtain deeper and complex shapes and more stretching to form panels without design modifications to the stamped steel product. In literature, there are many investigations about the deformation behavior of aluminum alloys at various temperatures and strain rates. It is generally observed that the total elongation increases and the flow stress decreases with increasing the temperature level [4–17]. * Corresponding author. Tel.: +90 388 225 2254; fax: +90 388 225 0112. E-mail address: [email protected] (F. Ozturk). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.019
Besides the experimental study, there are some numerical approaches for predicting the flow curves of the materials at elevated temperatures and strain rates [18–20]. In these studies, general approaches are the prediction of variation in the strength coefficient (C), strain and strain rate hardening coefficients with temperature by using extended Nadai model or physically based material model which was described by Bergström [21,22] and adapted by Van Liempt [23] to predict the relationship between stress and strain by considering the physical mechanisms of plastic deformation. Another physically based material model is described by Nes [24] for FCC materials which include the free variation of temperature and strain rate and the effect of dynamic strain ageing due to solutes in microstructure and work hardening. Besides these models, Johnson–Cook phenomenological equation [25] is widely used for high speed metal forming operations. In this approach, the material behavior is modeled as thermo-visco-plastic by considering the plastic strain, the plastic strain rate and temperature variations. Although the physically based material models like Bergström and Nes are able to predict the strain softening behavior of the materials at high temperatures, their complexities limit their use in the commercially used finite element softwares. Therefore, a new easy to use mathematical model which is able to characterize the strain softening (post uniform elongation) is needed. The characterization of 5083-H111 and 5754-O Al–Mg alloys at a wide range of strain rates and temperatures is the first aim of this study. The second objective is to develop a phenomenological constitutive model to describe the mechanical behavior of these Al– Mg alloys at different temperatures and strain rates. The extended
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Nomenclature C
e n e_ e_ ref m k
absolute temperature (K) Ta Tm melting temperature (K) absolute melting temperature (K) T am Th homologous temperature (K) a1, a2, a3, b1, b2, b3, c material parameters for extended Nadai and improved extended Nadai models
strength coefficient (MPa) true strain (mm/mm) strain hardening coefficient strain rate (s1) reference strain rate (s1) strain rate sensitivity fitting parameters
Nadai model was taken as a reference model in order to characterize the variation of the strength coefficient and strain hardening of the materials with temperature in the model development. First, an improved version of the extended Nadai model was developed named as improved Nadai model. Second a new model which is called as softening model was proposed. The proposed model is used to simulate the recorded softening behavior of the 5083H111 and 5754-O Al–Mg alloys at warm temperatures and modeling capabilities of softening model are shown.
3.1. The extended Nadai model A phenomenological model which is referred as Swift equation is actually the classical approach for modeling the material behavior. Macroscopic mechanical test results are fitted to a convenient mathematical function. A good approximation of the stress–strain curve is given in
r ¼ C en
ð3:1Þ
If the material is pre-strained, the relationship changes to 2. Materials and experimental procedure
r ¼ Cðe þ e0 Þn
The uniaxial tensile tests were applied to ASTM E8 standard rectangular dog-bone shape samples which are 3 and 1.82 mm thick at several temperatures ranging from RT to 300 °C in which the material properties like elongation, stress level, were determined and strain rates ranging from 0.0016 to 0.16 s1. Tensile tests were performed on a Shimadzu Autograph 100 kN testing machine with a data acquisition system maintained by a digital interface board utilizing a specialized computer program. Material deformation was measured with a video-extensometer measurement system. Each test was repeated at least three times and their average was used in flow stress strain curves. The elemental compositions of the 5083-H111 and 5754-O Al–Mg alloys were given in Table 1.
with e0 the initial strain. This equation only considers the strain hardening and addition of the strain rate sensitivity to this relation; power law in Eq. (3.3) is used
3. Modeling The optimum temperature distribution and forming speed which affect the manufacturability of the desired shape in a forming operation depend on the blank materials, tool materials, and geometry. In recent years, the usage of finite element simulation programs has been increased remarkably. However, the accuracy of these computational analysis programs is not good enough for all forming conditions such as elevated temperatures and high deformation speeds. To improve the accuracy of the simulation results of a warm forming process, an accurate material model is needed. The hardening property of the aluminum alloys in the plastic deformation region at elevated temperatures is their most important feature, including temperature and strain rate dependency. For numerical analysis of the forming operations, a description of materials’ work hardening features is needed. The classical approach is used to fit the mechanical measurements to a convenient mathematical function.
Table 1 Elemental composition of the Al–Mg alloys (in wt.%). Material
Mg
Si
Cu
Fe
Mn
Cr
Al
5083-H111 5754-O
4.09 3.17
0.137 0.112
0.038 0.0096
0.34 0.163
0.54 0.51
0.067 0.0023
Balance Balance
ð3:2Þ
m
e_ r ¼ Cðe þ e0 Þn _ e0
ð3:3Þ
with e_ 0 a reference strain rate. The effect of the temperature on the stress is accounted by assuming C, n, and m as functions of the temperature. The following relations are shown to give good results and referred to as the extended Nadai model [18,19,26].
T 273 CðTÞ ¼ C 0 þ a1 1 exp a2 Tm
ð3:4Þ
T 273 nðTÞ ¼ n0 þ b1 1 exp b2 Tm
ð3:5Þ
T 273 mðTÞ ¼ m0 exp c Tm
ð3:6Þ
In the model, C0, a1, a2, n0, b1, b2, m0, and c are material constants and Tm is a reference temperature. As seen from equations, in the extended Nadai model all basic mechanical properties of the material depend on temperature. However, calculation of the strain rate sensitivity of the Al–Mg alloys at warm temperatures by using extended Nadai model is impossible because it is negative at RT and at 100 °C and then it takes positive values at warm temperature levels. Therefore, the variation of the strength coefficient and strain hardening with temperature of the material was modified and they were defined in the new form (Eqs. (3.7) and (3.8)) which depends on temperature and strain rate. Basically these equations are an improved form of the extended Nadai model. In this study, these improved forms of the equations were used. In the graphs, it is displayed as improved Nadai model.
T 273 e_ þ a3 log 1 þ CðT; e_ Þ ¼ C 0 þ a1 1 exp a2 Tm e_ ref
ð3:7Þ
T 273 e_ þ b3 log 1 þ nðT; e_ Þ ¼ n0 þ b1 1 exp b2 Tm e_ ref
ð3:8Þ
In the improved model, a3 and b3 are fitting parameters.
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drag effects, caused by diffusion of solute atoms around moving dislocations, control the stress at deformation rates and temperatures useful for plastic forming processes and at higher temperature levels, post uniform elongation higher than the uniform elongation. Although the phenomenological model shows good agreement with experimental results, it can just predict the hardening part of the flow curves of the materials. However, Al–Mg alloys generally show superplastic behavior at warm and hot temperatures and the materials exhibit strain softening during the plastic deformation. Based on the experimental observation as shown in Fig. 1, a mathematical model was proposed. If the variations of residuals between extrapo-
3.2. The proposed model Al–Mg alloys may exhibit post uniform elongation with increasing the deformation temperature and decreasing the strain rate. This elongation is also called strain softening behavior and it can be explained with a delay in localized necking after the maximum load. It is believed that this behavior occurs due to the solute drag effect in Al–Mg alloys and after a critical temperature level which depends on the amount of alloying atoms (i.e., Mg atoms for Al–Mg alloys). The movement of alloying atoms is along the grain boundary or lagging behind the boundary exert a drag force on the boundary. Solute
10
140
120 8
Δσ (MPa)
True stress (MPa)
∗
σ −σ
100
80
60
40
4
2 o
20
Exponential fit (y=Aexp(c ε ))
6
204 C
∗
σ -Experimental
3003-H111 0 0.0
0.1
0
Extrapolated after max. stress 0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Strain (mm/mm)
True strain (mm/mm)
Fig. 1. Derivation of the mathematical form of the increasing the stress difference with extrapolated and experimental results.
Table 2 Material parameters for the improved Nadai model (T < 100 °C). 5083-H111 Tm C0 a1
920 K 508.365 MPa 61.104 MPa
5754-O 504.315 MPa 127.931 MPa
a2 n0 m0
5083-H111
5754-O
0.429 0.269 0.002259
1.646 0.31 0.5592
c b1 b2
5083-H111
5754-O
64.713 0.000008 73.107
171.84 0.000013 73.440
Table 3 Fitting parameters for the improved Nadai model (T > 100 °C). 5083-H111 Tm C0 a1
920 K 508.365 MPa 10.155 MPa
5754-O a2 n0 m0
504.315 MPa 1.575 MPa
5083-H111
5754-O
9.557 0.269 0.01483
16.462 0.31 0.04
c b1 b2
5083-H111
5754-O
4.889 0.0928 3.391
3.603 102.198 0.00418
Table 4 Fitting parameters for softening model (T < 100 °C).
Tm C0 n0
5083-H111
5754-O
920 K 508.315 MPa 0.269
504.354 MPa 0.31
a1 a2 a3
5083-H111
5754-O
125.486 MPa 7.415 2.113
188.193 MPa 2.337 0.746
b1 b2 b3
5083-H111
5754-O
29.297 0.00216 0.0053
127.097 0.00067 0.00486
Table 5 Fitting parameters for softening model (T > 100 °C). 5083-H111 Tm C0 n0
920 K 508.315 MPa 0.269
5754-O 504.354 MPa 0.31
a1 a2 a3
5083-H111
5754-O
42.311 MPa 6.054 76.013
0.264 MPa 22.839 17.317
b1 b2 b3
5083-H111
5754-O
15.491 0.0220 0.0139
0.0090 8.820 0.0085
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(a)
5083-H111
o
100 C
350
(a)
350
200 C 300
o
175 C 250
250
200
150 o
300 C 100
50
0 0,00
100 C
25 C
True stress (MPa)
True stress (MPa)
o
o
300
5754-O
o
25 C
o
Experimental Softening model Improved Nadai model
-1 0.0083 s 0,05
0,10
0,15
0,20
0,25
0,30
o
250 C
200
150
100
50
0,35
0,40
Experimental Softening model Improved Nadai model
-1 0.0016 s
0
0,45
0,1
0,2
True strain (mm/mm)
0,3
0,4
True strain (mm/mm)
(b) 350
(b) 350
5083-H111
o
100 C
o
5754-O
100 C
o
175 C
300
300 o
25 C
o
200 C
250
o
300 C
250
True stress (MPa)
True stress (MPa)
o
25 C 200
150
100
50
-1
250 C
150
100
Experimental Softening model Improved Nadai model
0.042 s
o
200
50
Experimental Softening model Improved Nadai model
-1 0.0083 s
0 0,0
0,1
0,2
0,3
0,4
0 0,00
True strain (mm/mm)
(c)
0,05
0,10
0,15
0,20
0,30
0,35
0,40
350 o
100 C
5083-H111
o
25 C
(c) 350
o
5754-O
100 C
300 o
300 C
300 o
o
250
175 C
200 C 250
True stress (MPa)
True stress (MPa)
0,25
True strain (mm/mm)
200
150
100
50
Experimental Softening model Improved Nadai model
o
25 C
200
150
100
Experimental Softening model Improved Nadai model
-1
0.16 s
50 0 0,0
0,1
0,2
0,3
-1
0.042 s
0,4
0
True strain (mm/mm)
0,05
Fig. 2. True stress vs. true strain curves for 5083-H111 (a) 0.0083 s1, (b) 0.042 s1, (c) 0.016 s1.
lated and experimental results are described with exponential form and then subjected to the temperature, general form of the equation can be described as seen in
rf ðT; e_ Þ ¼ CðT; e_ ÞenðT;e_ Þ expðk logðTÞ T h eÞ
o
250 C
ð3:9Þ
0,10
0,15
0,20
0,25
0,30
0,35
0,40
True strain (mm/mm) Fig. 3. True stress vs. true strain curves for 5754-O (a) 0.0016 s1, (b) 0.0083 s1, (c) 0.042 s1.
In this equation, k is the fitting parameter and it was taken 108 for Al–Mg alloys. Homologous temperature, Th (K) is defined as the absolute temperature Ta (K) divided by the absolute melting temperature T am (K):
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Th ¼
Ta T am
ð3:10Þ
The melting temperature of Al–Mg alloys is around 640 °C. When the temperature is between 25 and 250 °C, the homologous temperature is between 0.33 and 0.57. In this interval, the physics of plastic deformation changes, e.g. the contribution of cross slip increases at higher temperatures [19] and the total elongation of the material is increased while the stress level is decreased. 4. Determination of the material parameters The experimental results are used to determine the material parameters of the models. A least square approximation which is defined in Eq. (4.1) is used to fit the models to the experimental data
f ðxÞ ¼
Z
t2
ðyðx; tÞ /ðtÞÞ2 dt
ð4:1Þ
t1
With t is a scalar, y is the function for the material model which depend on the vector x and / of the experimental data. In the study, plastic deformation region of the materials was used to fit to the material models. After initiation of the necking, the deformation is not uniform any longer. Therefore, this part of the stress–strain curve is not used in determination of the material parameters. In addition to the least square approximation, the simplex minimization technique [27] was used for the minimization of the functions. This algorithm does not rely on the computation or estimation of the derivatives of the loss function. Instead, the function was evaluated for each iteration at m + 1 points in the m dimensional parameter space. For example, in two dimensions (i.e., when there are two parameters to be estimated), the program will evaluate the function at three points around the current optimum. These three points would define a triangle. In more than two dimensions, the ‘‘figure” produced by these points is called a simplex. Intuitively, in two dimensions, three points will allow us to determine ‘‘which way to go,” that is, in which direction in the two dimensional space to proceed in order to minimize the function. The same principle can be applied to the multidimensional parameter space, that is, the simplex will ‘‘move” downhill; when the current step sizes become too ‘‘crude” to detect a clear downhill direction, (i.e., the simplex is too large), the simplex will ‘‘contract” and try again. An additional strength of this method is needed to be found when a minimum appears, the simplex will be expanded again to a larger size to see whether the respective minimum is a local minimum. Thus, in a way, the simplex moves like a smooth single cell organism down to the loss function, contracting and expanding as a local minima or significant ridges are encountered. In the study, the material parameters are determined by dividing of the experimental data in two regions because of the recovery features of the material until the temperature of 100 °C. The extended Nadai model does not able to predict the strength coefficient and strain hardening at this temperature levels if all temperature levels are analyzed at the same time. The results for both materials are tabulated in Tables 2–5 for T < 100 °C and T > 100 °C. 5. Experimental and numerical results Experimental data and simulation results of the improved Nadai model and the proposed softening model at different temperatures and strain rates for 5083-H111 and 5754-O Al–Mg alloys are presented in Figs. 2 and 3, respectively. The temperatures were selected as RT, 100, 200, and 300 °C for 5083-H111 and RT, 100,
175, and 250 °C for 5754-O Al–Mg alloys. The strain rates were 0.0083, 0.042, and 0.16 s1 for 5083-H111 and 0.0016, 0.0083, and 0.042 s1 for 5754-O Al–Mg alloys. The figures show that the flow stresses of the materials are decreasing with increasing the temperature except for 100 °C while the ductility is increasing for both materials. In contrast, the flow stress is increasing with strain rate while the ductility is decreasing particularly for 200 °C and 300 °C. However, the flow stress does not show a significant difference with strain rate at RT and 100 °C. In Al–Mg alloys, one of the most common problems is the stretcher–strain (st–st) marks which appear frequently during the metal forming operations. These problems are not desired, since it decreases the quality of the final products. Warm forming operations can overcome these problems. However, these problems are more dominant at low strain rates (0.0016 s1) and RT for 5754-O Al–Mg alloy which is not hardened. It is observed that the hardening operation can also decrease these undesired problems. Warm forming operations can also increase the elongation and decrease the flow stresses of the materials. Particularly for Al–Mg alloys, the post uniform elongation increases with increasing the temperature and decreasing the strain rate. In general, Al–Mg alloys show solute-drag creep mechanism which affects the tensile ductility of the material above this temperature level and low strain rate. Magnesium is the most effective solute addition for the formation of solute-drag creep in Al and Al–Mg alloys exhibit higher tensile ductility than other nonsuperplastic aluminum alloys [28]. Therefore, the modeling of these mechanisms of the materials is very complicated. The improved Nadai model shows good agreement with experimental results at RT. However, when the forming temperature of materials increases, this agreement shows deterioration because of the variation in the hardening features of the materials at elevated temperatures. It has been seen that only hardening behavior can be simulated with the improved Nadai model. The most important advantage of the softening model is to have capability on predicting of the post uniform elongation of the materials at warm and hot temperatures if the strength coefficient and strain hardening are defined. In addition to worked materials above, the flow curve of the 3003-H111 Al–Mn alloy was also simulated by using derived equation. This material exhibits more post uniform elongation than the other Al–Mg alloys at warm temperatures. Although this material does not belong to Al–Mg alloy group, a case study was performed to determine the ability of the softening model in predicting the strain softening behavior of the materials. The material was tested
Table 6 Chemical composition of AA3003-H111 (wt.%) [24]. Al
Mg
Cu
Mn
Fe
Si
Cr
Zn
Ti
Ni
Bal
0.02
0.07
1.10
0.5
0.21
0.005
0.01
0.02
0.005
Table 7 Mechanical properties of as received AA3003-H111 [24]. Thickness (mm)
Yield strength (MPa)
UTS (MPa)
Total elongation (%)
0.96
63
114
32
17.4 MPa 6.45 77.88
b1 b2 b3
Table 8 Material parameters for softening model. 3003-H111 Tm C0 n0
920 K 199.82 MPa 0.215
a1 a2 a3
0.00274 10.619 0.134
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by a tensile test at RT, 66, 121, 177, 204, and 260 °C at a constant strain rate of 0.0083 s1 by Abedrabbo et al. [29]. Chemical composition and initial mechanical properties of the AA3003-H111 are
(a) 140
shown in Tables 6 and 7, respectively [29]. The main alloying element for this series is manganese. AA3003 aluminum alloy is very soft after the annealing. Therefore, one cold roll pass (H111) was
(d) 140
130 120
120
100
True stress (MPa)
True stress (MPa)
110
90 80 70 60 50
30
Experimental Improved Nadai model Softening model
20
0 0,00
80
60
40
40
10
100
o
3003-H111 0,05
25 C 0,10
0,15
0,20
0,25
0,30
Experimental Improved Nadai model Softening model
20 o
3003-H111 177 C 0 0,0
0,35
0,1
0,2
0,3
0,4
True strain (mm/mm)
True strain (mm/mm)
(b) 140
(e) 140
130 120
120
100
True stress (MPa)
True stress (MPa)
110
90 80 70 60 50 40
Experimental Improved Nadai model Softening model
20
0 0,00
80
60
40
30
10
100
o
66 C
3003-H111 0,05
0,10
0,15
0,20
0,25
0,30
Experimental Improved Nadai model Softening model
20
3003-H111 0 0,0
0,35
o
204 C
0,1
True strain (mm/mm)
0,2
0,3
0,4
0,5
True strain (mm/mm)
(c) 140
(f)
140
130 120
120
100
True stress (MPa)
True stress (mPa)
110
90 80 70 60 50
30
Experimental Improved Nadai Model Softening model
20
0 0,00
80
60
40
40
10
100
3003-H111 0,05
o
121 C
0,10
0,15
0,20
0,25
True strain (mm/mm)
0,30
0,35
0,40
Experimental Improved Nadai model Softening model
20
3003-H111 0 0,0
0,1
o
232 C 0,2
0,3
True strain (mm/mm)
Fig. 4. True stress vs. true strain curves for 3003-H111 at various temperatures (a–f).
0,4
0,5
S. Toros, F. Ozturk / Computational Materials Science 49 (2010) 333–339
applied to these sheets in order to provide some rigidity during handling. This material does not show recovery features with increasing the temperature like 5XXX series. Therefore, experimental data did not divide in two regions and material parameters are determined by applying aforementioned approach. The results obtained from softening model were tabulated in Table 8. As shown in Fig. 4, the flow curves obtained above 177 °C, the material exhibits the softening behavior with increasing the strain at mentioned temperature levels. As revealed in figures, the prediction ability of the improved Nadai model is not as good as the softening model for particularly the post uniform elongation region. Results indicate that the softening model predicts softening behavior in accord with the experiment. 6. Conclusions In this study, a new mathematical equation which is named as softening model is developed. The proposed model can predict softening behavior of the materials at warm temperatures and various strain rates. The predictions from the improved Nadai model and the softening model for 5083-H111 and 5754-O Al–Mg alloys shows acceptable agreement with experimental results at warm temperatures. However, the prediction with softening model is more accurate than the improved Nadai model for the material which has a softening tendency at elevated temperatures. In the case study, 3003-H111 Al–Mn alloy which shows softening behavior and post uniform elongation is more dominant than the uniform elongation. The prediction capability of the softening model is well-observed. Acknowledgements This work is supported by The Scientific and Technological Re_ search Council of Turkey (TÜBITAK). Project Number: 106M058, Title: ‘‘Experimental and Theoretical Investigations of The Effects of _ Temperature and Deformation Speed on Formability”. TÜBITAK support is profoundly acknowledged.
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Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Modeling uniaxial, temperature and strain rate dependent behavior of Al–Mg alloys S. Toros, F. Ozturk * Department of Mechanical Engineering, Nigde University, Nigde 51245, Turkey
a r t i c l e
i n f o
Article history: Received 29 March 2010 Received in revised form 2 May 2010 Accepted 7 May 2010
Keywords: Aluminum–magnesium alloy Al–Mg Modeling 5083 5754 Warm forming Flow curve Softening behavior
a b s t r a c t The mechanical properties of 5083-H111 and 5754-O Al–Mg alloys at various testing temperatures (between room temperature (RT) and 300 °C) and strain rates (0.0016–0.16 s1) were determined. A new mathematical model which is based on temperature and strain rate is developed and named as ‘‘softening model”. The model is used to simulate the recorded strain softening behavior of the 5083H111 and 5754-O Al–Mg alloys at warm temperatures and modeling capabilities of softening model are shown. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Aluminum alloys have been widely used in automotive, aircraft, and ship building industries because of their high strength stiffness to weight ratio, good formability, good corrosion resistance, and recycling potential. Moreover, vehicle performance, comfort, and marketability are other advantages of the aluminum alloys. The main goal is to reduce the vehicle weight and fuel consumption about 6–8% for every 10% in weight reduction by using aluminum alloys in vehicles [1–3]. 5XXX (5000 series) Al–Mg alloys are the most common one in aluminum alloys which are particularly used at inner panels of the automotive structures. However, applications of the aluminum and its alloys in this field were far behind steels because of cost and formability issues at room temperature (RT). Therefore, warm forming operations are the most common applications to improve the formability of these materials. Increasing the forming temperature provides increased ductility in the sheet, which can lead to obtain deeper and complex shapes and more stretching to form panels without design modifications to the stamped steel product. In literature, there are many investigations about the deformation behavior of aluminum alloys at various temperatures and strain rates. It is generally observed that the total elongation increases and the flow stress decreases with increasing the temperature level [4–17]. * Corresponding author. Tel.: +90 388 225 2254; fax: +90 388 225 0112. E-mail address: [email protected] (F. Ozturk). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.019
Besides the experimental study, there are some numerical approaches for predicting the flow curves of the materials at elevated temperatures and strain rates [18–20]. In these studies, general approaches are the prediction of variation in the strength coefficient (C), strain and strain rate hardening coefficients with temperature by using extended Nadai model or physically based material model which was described by Bergström [21,22] and adapted by Van Liempt [23] to predict the relationship between stress and strain by considering the physical mechanisms of plastic deformation. Another physically based material model is described by Nes [24] for FCC materials which include the free variation of temperature and strain rate and the effect of dynamic strain ageing due to solutes in microstructure and work hardening. Besides these models, Johnson–Cook phenomenological equation [25] is widely used for high speed metal forming operations. In this approach, the material behavior is modeled as thermo-visco-plastic by considering the plastic strain, the plastic strain rate and temperature variations. Although the physically based material models like Bergström and Nes are able to predict the strain softening behavior of the materials at high temperatures, their complexities limit their use in the commercially used finite element softwares. Therefore, a new easy to use mathematical model which is able to characterize the strain softening (post uniform elongation) is needed. The characterization of 5083-H111 and 5754-O Al–Mg alloys at a wide range of strain rates and temperatures is the first aim of this study. The second objective is to develop a phenomenological constitutive model to describe the mechanical behavior of these Al– Mg alloys at different temperatures and strain rates. The extended
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Nomenclature C
e n e_ e_ ref m k
absolute temperature (K) Ta Tm melting temperature (K) absolute melting temperature (K) T am Th homologous temperature (K) a1, a2, a3, b1, b2, b3, c material parameters for extended Nadai and improved extended Nadai models
strength coefficient (MPa) true strain (mm/mm) strain hardening coefficient strain rate (s1) reference strain rate (s1) strain rate sensitivity fitting parameters
Nadai model was taken as a reference model in order to characterize the variation of the strength coefficient and strain hardening of the materials with temperature in the model development. First, an improved version of the extended Nadai model was developed named as improved Nadai model. Second a new model which is called as softening model was proposed. The proposed model is used to simulate the recorded softening behavior of the 5083H111 and 5754-O Al–Mg alloys at warm temperatures and modeling capabilities of softening model are shown.
3.1. The extended Nadai model A phenomenological model which is referred as Swift equation is actually the classical approach for modeling the material behavior. Macroscopic mechanical test results are fitted to a convenient mathematical function. A good approximation of the stress–strain curve is given in
r ¼ C en
ð3:1Þ
If the material is pre-strained, the relationship changes to 2. Materials and experimental procedure
r ¼ Cðe þ e0 Þn
The uniaxial tensile tests were applied to ASTM E8 standard rectangular dog-bone shape samples which are 3 and 1.82 mm thick at several temperatures ranging from RT to 300 °C in which the material properties like elongation, stress level, were determined and strain rates ranging from 0.0016 to 0.16 s1. Tensile tests were performed on a Shimadzu Autograph 100 kN testing machine with a data acquisition system maintained by a digital interface board utilizing a specialized computer program. Material deformation was measured with a video-extensometer measurement system. Each test was repeated at least three times and their average was used in flow stress strain curves. The elemental compositions of the 5083-H111 and 5754-O Al–Mg alloys were given in Table 1.
with e0 the initial strain. This equation only considers the strain hardening and addition of the strain rate sensitivity to this relation; power law in Eq. (3.3) is used
3. Modeling The optimum temperature distribution and forming speed which affect the manufacturability of the desired shape in a forming operation depend on the blank materials, tool materials, and geometry. In recent years, the usage of finite element simulation programs has been increased remarkably. However, the accuracy of these computational analysis programs is not good enough for all forming conditions such as elevated temperatures and high deformation speeds. To improve the accuracy of the simulation results of a warm forming process, an accurate material model is needed. The hardening property of the aluminum alloys in the plastic deformation region at elevated temperatures is their most important feature, including temperature and strain rate dependency. For numerical analysis of the forming operations, a description of materials’ work hardening features is needed. The classical approach is used to fit the mechanical measurements to a convenient mathematical function.
Table 1 Elemental composition of the Al–Mg alloys (in wt.%). Material
Mg
Si
Cu
Fe
Mn
Cr
Al
5083-H111 5754-O
4.09 3.17
0.137 0.112
0.038 0.0096
0.34 0.163
0.54 0.51
0.067 0.0023
Balance Balance
ð3:2Þ
m
e_ r ¼ Cðe þ e0 Þn _ e0
ð3:3Þ
with e_ 0 a reference strain rate. The effect of the temperature on the stress is accounted by assuming C, n, and m as functions of the temperature. The following relations are shown to give good results and referred to as the extended Nadai model [18,19,26].
T 273 CðTÞ ¼ C 0 þ a1 1 exp a2 Tm
ð3:4Þ
T 273 nðTÞ ¼ n0 þ b1 1 exp b2 Tm
ð3:5Þ
T 273 mðTÞ ¼ m0 exp c Tm
ð3:6Þ
In the model, C0, a1, a2, n0, b1, b2, m0, and c are material constants and Tm is a reference temperature. As seen from equations, in the extended Nadai model all basic mechanical properties of the material depend on temperature. However, calculation of the strain rate sensitivity of the Al–Mg alloys at warm temperatures by using extended Nadai model is impossible because it is negative at RT and at 100 °C and then it takes positive values at warm temperature levels. Therefore, the variation of the strength coefficient and strain hardening with temperature of the material was modified and they were defined in the new form (Eqs. (3.7) and (3.8)) which depends on temperature and strain rate. Basically these equations are an improved form of the extended Nadai model. In this study, these improved forms of the equations were used. In the graphs, it is displayed as improved Nadai model.
T 273 e_ þ a3 log 1 þ CðT; e_ Þ ¼ C 0 þ a1 1 exp a2 Tm e_ ref
ð3:7Þ
T 273 e_ þ b3 log 1 þ nðT; e_ Þ ¼ n0 þ b1 1 exp b2 Tm e_ ref
ð3:8Þ
In the improved model, a3 and b3 are fitting parameters.
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drag effects, caused by diffusion of solute atoms around moving dislocations, control the stress at deformation rates and temperatures useful for plastic forming processes and at higher temperature levels, post uniform elongation higher than the uniform elongation. Although the phenomenological model shows good agreement with experimental results, it can just predict the hardening part of the flow curves of the materials. However, Al–Mg alloys generally show superplastic behavior at warm and hot temperatures and the materials exhibit strain softening during the plastic deformation. Based on the experimental observation as shown in Fig. 1, a mathematical model was proposed. If the variations of residuals between extrapo-
3.2. The proposed model Al–Mg alloys may exhibit post uniform elongation with increasing the deformation temperature and decreasing the strain rate. This elongation is also called strain softening behavior and it can be explained with a delay in localized necking after the maximum load. It is believed that this behavior occurs due to the solute drag effect in Al–Mg alloys and after a critical temperature level which depends on the amount of alloying atoms (i.e., Mg atoms for Al–Mg alloys). The movement of alloying atoms is along the grain boundary or lagging behind the boundary exert a drag force on the boundary. Solute
10
140
120 8
Δσ (MPa)
True stress (MPa)
∗
σ −σ
100
80
60
40
4
2 o
20
Exponential fit (y=Aexp(c ε ))
6
204 C
∗
σ -Experimental
3003-H111 0 0.0
0.1
0
Extrapolated after max. stress 0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Strain (mm/mm)
True strain (mm/mm)
Fig. 1. Derivation of the mathematical form of the increasing the stress difference with extrapolated and experimental results.
Table 2 Material parameters for the improved Nadai model (T < 100 °C). 5083-H111 Tm C0 a1
920 K 508.365 MPa 61.104 MPa
5754-O 504.315 MPa 127.931 MPa
a2 n0 m0
5083-H111
5754-O
0.429 0.269 0.002259
1.646 0.31 0.5592
c b1 b2
5083-H111
5754-O
64.713 0.000008 73.107
171.84 0.000013 73.440
Table 3 Fitting parameters for the improved Nadai model (T > 100 °C). 5083-H111 Tm C0 a1
920 K 508.365 MPa 10.155 MPa
5754-O a2 n0 m0
504.315 MPa 1.575 MPa
5083-H111
5754-O
9.557 0.269 0.01483
16.462 0.31 0.04
c b1 b2
5083-H111
5754-O
4.889 0.0928 3.391
3.603 102.198 0.00418
Table 4 Fitting parameters for softening model (T < 100 °C).
Tm C0 n0
5083-H111
5754-O
920 K 508.315 MPa 0.269
504.354 MPa 0.31
a1 a2 a3
5083-H111
5754-O
125.486 MPa 7.415 2.113
188.193 MPa 2.337 0.746
b1 b2 b3
5083-H111
5754-O
29.297 0.00216 0.0053
127.097 0.00067 0.00486
Table 5 Fitting parameters for softening model (T > 100 °C). 5083-H111 Tm C0 n0
920 K 508.315 MPa 0.269
5754-O 504.354 MPa 0.31
a1 a2 a3
5083-H111
5754-O
42.311 MPa 6.054 76.013
0.264 MPa 22.839 17.317
b1 b2 b3
5083-H111
5754-O
15.491 0.0220 0.0139
0.0090 8.820 0.0085
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(a)
5083-H111
o
100 C
350
(a)
350
200 C 300
o
175 C 250
250
200
150 o
300 C 100
50
0 0,00
100 C
25 C
True stress (MPa)
True stress (MPa)
o
o
300
5754-O
o
25 C
o
Experimental Softening model Improved Nadai model
-1 0.0083 s 0,05
0,10
0,15
0,20
0,25
0,30
o
250 C
200
150
100
50
0,35
0,40
Experimental Softening model Improved Nadai model
-1 0.0016 s
0
0,45
0,1
0,2
True strain (mm/mm)
0,3
0,4
True strain (mm/mm)
(b) 350
(b) 350
5083-H111
o
100 C
o
5754-O
100 C
o
175 C
300
300 o
25 C
o
200 C
250
o
300 C
250
True stress (MPa)
True stress (MPa)
o
25 C 200
150
100
50
-1
250 C
150
100
Experimental Softening model Improved Nadai model
0.042 s
o
200
50
Experimental Softening model Improved Nadai model
-1 0.0083 s
0 0,0
0,1
0,2
0,3
0,4
0 0,00
True strain (mm/mm)
(c)
0,05
0,10
0,15
0,20
0,30
0,35
0,40
350 o
100 C
5083-H111
o
25 C
(c) 350
o
5754-O
100 C
300 o
300 C
300 o
o
250
175 C
200 C 250
True stress (MPa)
True stress (MPa)
0,25
True strain (mm/mm)
200
150
100
50
Experimental Softening model Improved Nadai model
o
25 C
200
150
100
Experimental Softening model Improved Nadai model
-1
0.16 s
50 0 0,0
0,1
0,2
0,3
-1
0.042 s
0,4
0
True strain (mm/mm)
0,05
Fig. 2. True stress vs. true strain curves for 5083-H111 (a) 0.0083 s1, (b) 0.042 s1, (c) 0.016 s1.
lated and experimental results are described with exponential form and then subjected to the temperature, general form of the equation can be described as seen in
rf ðT; e_ Þ ¼ CðT; e_ ÞenðT;e_ Þ expðk logðTÞ T h eÞ
o
250 C
ð3:9Þ
0,10
0,15
0,20
0,25
0,30
0,35
0,40
True strain (mm/mm) Fig. 3. True stress vs. true strain curves for 5754-O (a) 0.0016 s1, (b) 0.0083 s1, (c) 0.042 s1.
In this equation, k is the fitting parameter and it was taken 108 for Al–Mg alloys. Homologous temperature, Th (K) is defined as the absolute temperature Ta (K) divided by the absolute melting temperature T am (K):
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Th ¼
Ta T am
ð3:10Þ
The melting temperature of Al–Mg alloys is around 640 °C. When the temperature is between 25 and 250 °C, the homologous temperature is between 0.33 and 0.57. In this interval, the physics of plastic deformation changes, e.g. the contribution of cross slip increases at higher temperatures [19] and the total elongation of the material is increased while the stress level is decreased. 4. Determination of the material parameters The experimental results are used to determine the material parameters of the models. A least square approximation which is defined in Eq. (4.1) is used to fit the models to the experimental data
f ðxÞ ¼
Z
t2
ðyðx; tÞ /ðtÞÞ2 dt
ð4:1Þ
t1
With t is a scalar, y is the function for the material model which depend on the vector x and / of the experimental data. In the study, plastic deformation region of the materials was used to fit to the material models. After initiation of the necking, the deformation is not uniform any longer. Therefore, this part of the stress–strain curve is not used in determination of the material parameters. In addition to the least square approximation, the simplex minimization technique [27] was used for the minimization of the functions. This algorithm does not rely on the computation or estimation of the derivatives of the loss function. Instead, the function was evaluated for each iteration at m + 1 points in the m dimensional parameter space. For example, in two dimensions (i.e., when there are two parameters to be estimated), the program will evaluate the function at three points around the current optimum. These three points would define a triangle. In more than two dimensions, the ‘‘figure” produced by these points is called a simplex. Intuitively, in two dimensions, three points will allow us to determine ‘‘which way to go,” that is, in which direction in the two dimensional space to proceed in order to minimize the function. The same principle can be applied to the multidimensional parameter space, that is, the simplex will ‘‘move” downhill; when the current step sizes become too ‘‘crude” to detect a clear downhill direction, (i.e., the simplex is too large), the simplex will ‘‘contract” and try again. An additional strength of this method is needed to be found when a minimum appears, the simplex will be expanded again to a larger size to see whether the respective minimum is a local minimum. Thus, in a way, the simplex moves like a smooth single cell organism down to the loss function, contracting and expanding as a local minima or significant ridges are encountered. In the study, the material parameters are determined by dividing of the experimental data in two regions because of the recovery features of the material until the temperature of 100 °C. The extended Nadai model does not able to predict the strength coefficient and strain hardening at this temperature levels if all temperature levels are analyzed at the same time. The results for both materials are tabulated in Tables 2–5 for T < 100 °C and T > 100 °C. 5. Experimental and numerical results Experimental data and simulation results of the improved Nadai model and the proposed softening model at different temperatures and strain rates for 5083-H111 and 5754-O Al–Mg alloys are presented in Figs. 2 and 3, respectively. The temperatures were selected as RT, 100, 200, and 300 °C for 5083-H111 and RT, 100,
175, and 250 °C for 5754-O Al–Mg alloys. The strain rates were 0.0083, 0.042, and 0.16 s1 for 5083-H111 and 0.0016, 0.0083, and 0.042 s1 for 5754-O Al–Mg alloys. The figures show that the flow stresses of the materials are decreasing with increasing the temperature except for 100 °C while the ductility is increasing for both materials. In contrast, the flow stress is increasing with strain rate while the ductility is decreasing particularly for 200 °C and 300 °C. However, the flow stress does not show a significant difference with strain rate at RT and 100 °C. In Al–Mg alloys, one of the most common problems is the stretcher–strain (st–st) marks which appear frequently during the metal forming operations. These problems are not desired, since it decreases the quality of the final products. Warm forming operations can overcome these problems. However, these problems are more dominant at low strain rates (0.0016 s1) and RT for 5754-O Al–Mg alloy which is not hardened. It is observed that the hardening operation can also decrease these undesired problems. Warm forming operations can also increase the elongation and decrease the flow stresses of the materials. Particularly for Al–Mg alloys, the post uniform elongation increases with increasing the temperature and decreasing the strain rate. In general, Al–Mg alloys show solute-drag creep mechanism which affects the tensile ductility of the material above this temperature level and low strain rate. Magnesium is the most effective solute addition for the formation of solute-drag creep in Al and Al–Mg alloys exhibit higher tensile ductility than other nonsuperplastic aluminum alloys [28]. Therefore, the modeling of these mechanisms of the materials is very complicated. The improved Nadai model shows good agreement with experimental results at RT. However, when the forming temperature of materials increases, this agreement shows deterioration because of the variation in the hardening features of the materials at elevated temperatures. It has been seen that only hardening behavior can be simulated with the improved Nadai model. The most important advantage of the softening model is to have capability on predicting of the post uniform elongation of the materials at warm and hot temperatures if the strength coefficient and strain hardening are defined. In addition to worked materials above, the flow curve of the 3003-H111 Al–Mn alloy was also simulated by using derived equation. This material exhibits more post uniform elongation than the other Al–Mg alloys at warm temperatures. Although this material does not belong to Al–Mg alloy group, a case study was performed to determine the ability of the softening model in predicting the strain softening behavior of the materials. The material was tested
Table 6 Chemical composition of AA3003-H111 (wt.%) [24]. Al
Mg
Cu
Mn
Fe
Si
Cr
Zn
Ti
Ni
Bal
0.02
0.07
1.10
0.5
0.21
0.005
0.01
0.02
0.005
Table 7 Mechanical properties of as received AA3003-H111 [24]. Thickness (mm)
Yield strength (MPa)
UTS (MPa)
Total elongation (%)
0.96
63
114
32
17.4 MPa 6.45 77.88
b1 b2 b3
Table 8 Material parameters for softening model. 3003-H111 Tm C0 n0
920 K 199.82 MPa 0.215
a1 a2 a3
0.00274 10.619 0.134
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by a tensile test at RT, 66, 121, 177, 204, and 260 °C at a constant strain rate of 0.0083 s1 by Abedrabbo et al. [29]. Chemical composition and initial mechanical properties of the AA3003-H111 are
(a) 140
shown in Tables 6 and 7, respectively [29]. The main alloying element for this series is manganese. AA3003 aluminum alloy is very soft after the annealing. Therefore, one cold roll pass (H111) was
(d) 140
130 120
120
100
True stress (MPa)
True stress (MPa)
110
90 80 70 60 50
30
Experimental Improved Nadai model Softening model
20
0 0,00
80
60
40
40
10
100
o
3003-H111 0,05
25 C 0,10
0,15
0,20
0,25
0,30
Experimental Improved Nadai model Softening model
20 o
3003-H111 177 C 0 0,0
0,35
0,1
0,2
0,3
0,4
True strain (mm/mm)
True strain (mm/mm)
(b) 140
(e) 140
130 120
120
100
True stress (MPa)
True stress (MPa)
110
90 80 70 60 50 40
Experimental Improved Nadai model Softening model
20
0 0,00
80
60
40
30
10
100
o
66 C
3003-H111 0,05
0,10
0,15
0,20
0,25
0,30
Experimental Improved Nadai model Softening model
20
3003-H111 0 0,0
0,35
o
204 C
0,1
True strain (mm/mm)
0,2
0,3
0,4
0,5
True strain (mm/mm)
(c) 140
(f)
140
130 120
120
100
True stress (MPa)
True stress (mPa)
110
90 80 70 60 50
30
Experimental Improved Nadai Model Softening model
20
0 0,00
80
60
40
40
10
100
3003-H111 0,05
o
121 C
0,10
0,15
0,20
0,25
True strain (mm/mm)
0,30
0,35
0,40
Experimental Improved Nadai model Softening model
20
3003-H111 0 0,0
0,1
o
232 C 0,2
0,3
True strain (mm/mm)
Fig. 4. True stress vs. true strain curves for 3003-H111 at various temperatures (a–f).
0,4
0,5
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applied to these sheets in order to provide some rigidity during handling. This material does not show recovery features with increasing the temperature like 5XXX series. Therefore, experimental data did not divide in two regions and material parameters are determined by applying aforementioned approach. The results obtained from softening model were tabulated in Table 8. As shown in Fig. 4, the flow curves obtained above 177 °C, the material exhibits the softening behavior with increasing the strain at mentioned temperature levels. As revealed in figures, the prediction ability of the improved Nadai model is not as good as the softening model for particularly the post uniform elongation region. Results indicate that the softening model predicts softening behavior in accord with the experiment. 6. Conclusions In this study, a new mathematical equation which is named as softening model is developed. The proposed model can predict softening behavior of the materials at warm temperatures and various strain rates. The predictions from the improved Nadai model and the softening model for 5083-H111 and 5754-O Al–Mg alloys shows acceptable agreement with experimental results at warm temperatures. However, the prediction with softening model is more accurate than the improved Nadai model for the material which has a softening tendency at elevated temperatures. In the case study, 3003-H111 Al–Mn alloy which shows softening behavior and post uniform elongation is more dominant than the uniform elongation. The prediction capability of the softening model is well-observed. Acknowledgements This work is supported by The Scientific and Technological Re_ search Council of Turkey (TÜBITAK). Project Number: 106M058, Title: ‘‘Experimental and Theoretical Investigations of The Effects of _ Temperature and Deformation Speed on Formability”. TÜBITAK support is profoundly acknowledged.
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