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A physical-based constitutive relation to predict flow stress for Cu–0.4Mg alloy during hot working
Materials Science & Engineering A 615 (2014) 247–254
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
A physical-based constitutive relation to predict flow stress for Cu–0.4Mg alloy during hot working Guoliang Ji, Qiang Li, Lei Li n School of Materials Science and Engineering, Henan Polytechnic University, Jiaozuo 454000, China
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2014 Received in revised form 22 July 2014 Accepted 23 July 2014 Available online 1 August 2014
The hot deformation behaviors of Cu–0.4Mg alloy are studied by the isothermal compression tests on a Gleeble-1500 thermo-mechanical simulator over wide ranges of temperatures and strain rates. Based on the experimental results, a physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy under elevated temperature. Results show that under a given strain rate, the flow stress will decrease with increasing temperature, while under a given temperature the flow stress will increase with increasing strain rate. The true stress–strain curves of Cu–0.4Mg alloy demonstrate the typical characteristics of dynamic recovery and dynamic recrystallization. The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and predicted flow stresses are 0.99747 and 3.3846%, respectively. It indicates that the proposed physical-based constitutive model can accurately characterize the hot deformation behaviors including work-hardening, dynamic recovery and dynamic recrystallization for Cu–0.4Mg alloy. & 2014 Elsevier B.V. All rights reserved.
Keywords: Cu–0.4Mg alloy Deformation behavior Physical-based constitutive model Dynamic recovery and dynamic recrystallization
1. Introduction The constitutive relation of metal materials during the hot deformation process reflects the nonlinear relation between the flow stress and process variables such as strain, strain rate and temperature, and it has complex mathematical forms and profound physical meanings. In the hot working process of metal materials there occur various physical mechanisms such as workhardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX), which are affected by the chemical composition of metals and alloys, deformation temperature, deformation amount and strain rate. The constitutive relation reflects comprehensive effects of various physical mechanisms in the hot working process of metal materials. Constitutive relation can be used to reveal physical mechanisms of deformation, to establish the recrystallization dynamics model, and to develop the processing map by which to uncover the deformation mechanism and to delineate the stability and instability regimes in the hot working window. Furthermore, the constitutive relation is often used to describe the plastic flow properties of metals and alloys, in a form that can be used in the computer code to simulate the thermo-mechanical response of mechanical parts under the prevailing loading conditions [1,2]. Preciseness of the constitutive relation has important significance for achieving accurate simulation results.
n
Corresponding author. Tel.: þ 86 0391 3986939. E-mail address: [email protected] (L. Li).
http://dx.doi.org/10.1016/j.msea.2014.07.082 0921-5093/& 2014 Elsevier B.V. All rights reserved.
The constitutive relations can be divided into three categories, including the phenomenological, physical-based and artificial neural network models [1]. Based on empirical guess and intuitive grasp of effect laws of process parameters on the flow behavior of metal materials (work-hardening, strain rate hardening and temperature softening), the phenomenological constitutive relation does not deeply involve physical mechanisms of materials deformation, and describes flow stress in a simple mathematical form with macroscopic process parameters. The phenomenological approach [3], proposed by Sellars and Tegart, in which the flow stress is expressed by the sine-hyperbolic law in an Arrhenius type of equation, has been extensively used to predict the hot deformation behavior of metal materials. Lin et al. [2] proposed a modified Arrhenius model to characterize the hot deformation behavior of 42CrMo steel by the compensation of strain and strain rate. Based on the classical stress–dislocation relation and the kinetics of dynamic recrystallization, the physical-based constitutive relation was established to describe the flow stress during the work hardening–dynamic recovery and dynamical recrystallization periods for 42CrMo steel [4], nickel-based super-alloy [5], 7050 aluminum alloy [6], and N08028 alloy [7]. An artificial neural network (ANN) is a mathematical model or a computational model that tries to simulate the structure and/or functional aspects of biological neural networks. Since the establishment of the ANN model need not utilize the physical knowledge of deformation mechanisms, it has the powerful ability to predict hot deformation behaviors of metal materials across the whole hot working domain. The ANN model was developed to predict the flow stress
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of austenitic stainless steels [8], Aermet100 [9] and Cu–0.4Mg alloy [10] during hot working with very high accuracy. Cu–0.4Mg alloy contact wires have been successfully used in the high speed railway due to its high strength, high electrical conductivity and high wear resistance. Cu–0.4Mg alloy contact wires manufactured by the traditional production process have coarse grains microstructure resulting from a previous casting process, which damages its mechanical and electrical properties, for example, coarse grain microstructure would cause a great amount of electric sparks when the pantograph slide slides on the contact wires. The new production process of Cu–0.4Mg alloy contact wires is that the Cu–0.4Mg alloy is melted and cast under the air isolated condition in the horizontal contiguous furnace to obtain a rod-shaped billet, then it is hot extruded to crush coarse cast microstructure with a view to obtain a fine recrystallization microstructure, and finally fabricated into contact wires by the cold drawing process. Modeling and prediction of the hot deformation behavior of Cu–0.4Mg alloy is of great importance for controlling the microstructures of Cu–0.4Mg alloy. However, there are few reports about the constitutive model of Cu–0.4Mg alloy. In this study, hot compression tests in a wide range of temperatures and strain rates were carried out to study the hot deformation behavior of Cu–0.4Mg alloy. Based on the stress– dislocation relation and kinetics of dynamic recrystallization, a physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy. Moreover, the prediction ability of the physical-based constitutive model is evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE).
2. Materials and experimental procedure Cu–0.4Mg alloy contact wires in cold drawing were annealed: heating to 600 1C, holding for 3 h and cooling in furnace, then they were machined in cold drawing direction into specimens with a diameter of 8 mm and a length of 12 mm. Compression tests were carried out in the temperature range of 500–800 1C(500, 600, 700, 750, 800 1C) and the strain rate range of 0.005–10 s 1 (0.005, 0.01, 0.1, 1, 5, 10 s 1) on a Gleeble-1500 thermo-mechanical simulator. Cu–0.4Mg alloy specimens were heated to a preset temperature at a rate of 10 1C/s, soaked for 60 s to ensure a uniform temperature in the whole specimen, compressed by 60% in height and then water cooled to the room temperature to obtain the hot deformation microstructure, while true stress–true strain data at different temperatures and strain rates were recorded automatically. In order to decrease the friction effects in the deformation process, a high-temperature lubricant was applied. The hot compression specimens were polished and etched with an etchant (19 g ferric nitrate, 50 ml alcohol and 50 ml water). Optical microstructures in the central region of the section were examined by optical microscopy, and grain sizes were determined by means of the linear intercept method (here ASTM standard E112-12 was used). The microstructure of the annealed Cu–0.4Mg alloy specimens is presented in Fig. 1.
3. Results and discussion 3.1. Stress–strain characteristics of Cu–0.4Mg alloy during hot working The true stress–strain curves for Cu–0.4Mg alloy during hot compression are illustrated in Fig. 2. Under a given temperature (Fig. 2a) the flow stress will increase with increasing strain rate, while under a given strain rate (Fig. 2b) the flow stress will decrease with increasing temperature. General variation of flow
Fig. 1. Microstructures of as annealed Cu–0.4Mg alloy.
stress with temperature and strain rate can be explained quantitatively. The most basic relation [11] among temperature, strainrate and dislocation density can be represented as ΔG n ð1Þ ε_ ¼ ρm A exp kT where ρm ¼ f ρ is the mobile dislocation density, in which f is about 0.1 representing the proportion of the mobile dislocation density in the total dislocation density ρ. A is the materials constant, ΔGn is the heat activity energy and k is the Boltzmann constant. The most basic relation [11] between flow stress and dislocation can be formulated as follows: pffiffiffi σ ¼ σ n þ αμb ρ ð2Þ where σ n is the intrinsic strength of materials representing the crystal lattice frictional resistance, b is the Burgers vector, α is the materials constant and μ is the shear modulus. According to Eqs. (1) and (2), it follows that under a given strain rate, the dislocation density decreases with increasing temperatures, so the total level of the flow stress will decrease with increasing temperatures; similarly under a given temperature, the dislocation density increases with increasing strain rates, so the total level of the flow stress will increase with increasing strain rates. Microstructures of the specimens compressed at strain rates of 0.005 s 1 and 10 s 1 with temperature of 700 1C are presented in Fig. 3a and b, respectively. It can be seen that at a given temperature, with increasing strain rate, the microstructures change from recrystallized equiaxial grains (Fig. 3a) to deformed flat grains (Fig. 3b). At the temperature of 700 1C and the higher strain rate of 10 s 1, dislocation density would increase and interaction between dislocations would become more significant, but due to heat activation of high temperature, larger dislocation density can be annihilated and rearranged by the dislocation climbing, sliding, and cross-slip, which causes polygonization of dislocation and regularization of dislocation cell wall to further form subboundaries, meanwhile dynamic recovery occurs. Microstructures at this deformation condition have typical characteristics of dynamic recovery, as shown in Fig. 3b. At the temperature of 700 1C and the lower strain rate of 0.005 s 1, dislocation density would decrease and interaction between dislocations would become more weak; in addition, due to heat activation of high temperature and having enough time, lower dislocation density can be annihilated and rearranged more easily. Subboundaries formed by polygonization of dislocation and regularization of dislocation cell wall can protrude and grow up from grain boundary with lower storage energy to that with larger storage energy, meanwhile dynamic recrystallization occurs. Microstructures at this deformation
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Fig. 2. True strain–stress curves of Cu–0.4Mg alloys (a) at given temperature of 700 1C and (b) at given strain rate of 10 s 1.
Fig. 3. Microstructures of Cu–0.4Mg alloy deformed at the strain rates of (a) 0.005 s 1 and (b) 10 s 1 with a temperature of 700 1C.
Fig. 4. A schematic of true stress–strain curves characteristic of dynamic recovery and dynamic recrystallization.
condition have typical characteristics of dynamic recrystallization, as shown in Fig. 3a. According to the dynamic softening mechanisms during hot deformation, the true stress–strain curves for Cu–0.4Mg alloy can be divided into two types, i.e., dynamic recovery and dynamic recrystallization. The true stress–strain curves characteristic of dynamic recovery are illustrated in Fig. 4 (marked with ‘a’) [5]. The flow stress rapidly increases due to work-hardening. When a balance between work-hardening and dynamic recovery is reached, a saturation flow stress appears and stays constant; meanwhile grains continue to elongate with increase of strain but the shape and size of subgrains remain unchanged. The true stress–strain curves characteristic of dynamic recrystallization are illustrated in Fig. 4 (marked with ‘b’) [5]. The flow stress rapidly increases to critical stress due to work-hardening, and continues to
slowly increase to the peak stress due to the occurrence of recrystallization; then the flow stress begins to decrease until it reaches a steady stress and stays constant, meanwhile, the shape and size of recrystallized grains remain unchanged. Generally, the true stress–strain curves with dynamic recrystallization characteristics can be divided into three stages [5]: stage I (work-hardening stage and dynamic recovery), stage II (work-hardening, dynamic recovery and dynamic recrystallization) and stage III (steady dynamic recrystallization), as shown in Fig. 4. Stage I is characterized by a sharp increase of flow stress. It is attributed to the fact that the generation and multiplication of dislocation occur rapidly, leading to the high work-hardening rate. Meanwhile, the dynamic recovery caused by the dislocation climbing, sliding, and cross-slip is too weak to overcome the work-hardening effect. In stage II, the dynamic recrystallization takes place at a critical strain (εc ), resulting in decreased workhardening rate with increase of strain. When the dynamic softening (DRV and DRX) rate is in equilibrium with the workhardening rate, the flow stress achieves a peak stress (σ p ), and then gradually decreases to steady stress (σ ss ). When a new balance between dynamic softening and work-hardening is reached, the flow stress is maintained at a fairly constant level regardless of the increased strain (in stage III). 3.2. Constitutive modeling of flow stress The dislocation density theory and the modified Arvami equation can be used to describe the flow behavior of metal materials during hot working, i.e., work-hardening, dynamic recovery and dynamic recrystallization. As shown in Fig. 4, the curve marked ‘a’ represents true work-hardening and DRV behavior of metal materials, or the assumed work-hardening and DRV behavior of metal materials in which work-hardening, DRV and DRX occur simultaneously.
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σ rec represents the true or assumed flow stress of metal materials attributable to work-hardening and DRV. As shown in Fig. 4, the curve marked ‘b’ represents the true work-hardening, DRV and DRX behavior, and its stage II is considered to be the net result of the simultaneous operation of work-hardening and DRV and DRX [7]. σ drx represents the flow stress in stage II, and the difference (Δσ ) between σ drx and σ rec is the net softening directly attributable to DRX [7]. The initial stress, critical strain and stress, peak strain and stress, saturation stress, and steady stress need to be determined first to establish the constitutive relation capable of describing the workhardening, DRV and DRX behaviors. As shown in Fig. 4, for the true stress–strain curves characteristic of dynamic recrystallization (marked ‘b’), the initial stress (σ 0 ), peak strain (εp ) and peak stress (σ p ), and steady stress (σ ss ) can be obtained directly from the true stress–strain curve. However, direct determination of the critical strain (εc ) is difficult, and it can be attained when the value of j dθ=dσ j, where strain hardening rate θ ¼ dσ =dε, reaches the minimum which corresponds to an inflection of dσ =dε versus σ curve [12], as shown in Fig. 5. Then the saturation stress can be determined as the horizontal intercept of the tangent line of θ σ plot through the inflection point. 3.2.1. Modeling the work-hardening and dynamic recovery During the work hardening–dynamic recovery period, the evolution of dislocation density with strain is generally considered to depend on two components: multiplication and annihilation of dislocation, and can be expressed as [5] dρ ¼ U Ωρ dε
ð3Þ
where dρ=dε is the increasing rate of dislocation density with strain; U is a multiplication term which represents the workhardening, and can be regarded as a constant with respect to strain. Ωρ represents the dynamic recovery due to the dislocation annihilation and rearrangement, and Ω is the coefficient of dynamic recovery [13]. When plastic strain ε ¼ 0, dislocation density ρ ¼ ρ0 , where ρ0 is the initial dislocation density, and the corresponding flow stress is the initial yield stress σ 0 . Then the expression of dislocation density can be obtained by integrating Eq. (3) as follows: U U ρ¼ ρ0 e Ωε ð4Þ
Ω
Ω
It is known that the effective stress can be negligible compared with the internal stress at high temperatures [5]. Thus, the applied stress can be directly estimated by the square root of the dislocapffiffiffi tion density, σ ¼ αμb ρ, where α is the material constant, μ is the shear modulus and b is the distance between atoms in the slip direction [14]. Substituting this expression into Eq. (4), the flow stress during the work-hardening and dynamic recovery period can be represented as
σ rec ¼ ½σ 2sat þ ðσ 20 σ 2sat Þe Ωε 0:5
where σ rec represents the flow stress in the work-hardening and DRV period and σ sat is the saturation stress. It can be found in Eq. (5) that three parameters (σ sat , σ 0 and Ω) need to be determined. In general, the combined effects of temperature and strain rate on feature parameters of true stress–strain curves (such as σ sat , σ 0 and Ω) can be characterized by the Zener–Hollomon parameter. The expression of Zener– Hollomon parameter (Z) is given as [15] Z ¼ ε_ expðQ =RTÞ
Fig. 5. The dσ=dε versus σ curve at a temperature 700 1C and a strain rate of 0.1 s 1.
ð5Þ
ð6Þ
where ε_ is the strain rate (s 1), R is the universal gas constant (8.314 J mol 1 K 1), T is the absolute temperature (K), and Q is the activation energy (J mol 1). In order to establish the expression of feature parameters (such as σ sat , σ 0 and Ω), the activation energy (Q ) should be determined first. In the authors' previous publication [16], the calculation procedure of the activation energy Q of Cu–0.4Mg alloy has been elaborated, and it is estimated as Q ¼ 2:138 105 J mol 1. For the true stress–strain curves characteristic of dynamic recovery, the saturation stress (σ sat ) can be directly obtained, as shown in Fig. 4; while for the true stress–strain curves characteristic of dynamic recrystallization it can be determined as the horizontal intercept of the tangent line of θ σ plot through the inflection point, as shown in Fig. 5. Fig. 6a illustrates there is
Fig. 6. Relationship between the saturation stress (σ sat ) and the Zener–Hollomon parameter: (a) linear fitting and (b) polynomial fitting.
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linear relation between the saturation stress (σ sat ) and ln Z. Then, the saturation stress (σ sat ) can be expressed as
σ sat ¼ 15:9998 ln Z 268:186
ð7Þ
Results presented at the end of this article show that saturation stress (σ sat ) has considerable effects on the prediction accuracy of the physical-based constitutive relation of Cu–0.4Mg alloy during hot working. So the correlation coefficient (R) and the average absolute relative error (AARE) between the predicted and measured saturation stresses are employed to evaluate prediction capability of the expression for the saturation stress (σ sat ). The correlation coefficient (R) and the average absolute relative error (AARE) are 0.98894 and 7.6964%, respectively. To further improve prediction accuracy for the saturation stress (σ sat ), a fifth order polynomial is used to fit the measured saturation stress (σ sat ), as shown in Fig. 6b, and can be expressed as follows:
Fig. 8. Relationship between the coefficient of dynamic recovery (Ω) and the Zener–Hollomon parameter.
σ sat ¼ 0:000442ðln ZÞ5 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 94:7969ðln ZÞ2 þ 1233:7397ðln ZÞ 6266:3317
ð8Þ
The correlation coefficient (R) and the average absolute relative error (AARE) between the fifth order polynomial predicted and measured saturation stresses are 0.99639 and 3.4595%, respectively. It can be found that the expression for the saturation stress in the form of the fifth order polynomial greatly improves its prediction accuracy. The yield stress (σ 0 ) at various forming temperatures and strain rates can be directly obtained from the true stress–strain curves, and in this study it is taken as the stress corresponding to a strain of 0.02. Fig. 7 illustrates that a good linear relation exists between the yield stress (σ 0 ) and ln Z. Then, the yield stress (σ 0 ) can be expressed as
σ 0 ¼ 2:9959 ln Z 19:5834
ð9Þ
According to Eq. (5), the coefficient of dynamic recovery (Ω) can be calculated by 2 σ σ2 Ωε ¼ ln 2sat 20 ð10Þ σ sat σ rec Using the true stress–strain data before the saturation strain (corresponding to the saturation stress) for the true stress–strain behavior characteristic of dynamic recovery and using the true stress–strain data before the critical strain for the flow behavior characteristic of dynamic recrystallization, the values of Ω can be determined for all the test conditions. Fig. 8 shows that a linear relation exists between ln Ω and ln Z. It can be found that the dynamic recovery coefficient increases with the decrease of the
Zener–Hollomon parameter, and Ω can be expressed as a function of the Zener–Hollomon parameter.
Ω ¼ 103:9339 Z 0:0891
ð11Þ
Therefore, the physical-based constitutive relation of Cu–0.4Mg alloy during the work hardening–dynamic recovery period can be summarized as 8 σ rec ¼ ½σ 2sat þ ðσ 20 σ 2sat Þe Ωε 0:5 > > > > > > σ sat ¼ 0:000442ðln ZÞ5 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 > > > < 94:7969ðln ZÞ2 þ 1233:7397ðln ZÞ 6266:3317 ð12Þ > σ 0 ¼ 2:9959 ln Z 19:5834 > > > > 0:0891 > Ω ¼ 103:9339 Z > > > : Z ¼ ε_ expð2:138 105 =RTÞ Eq. (12) can be used to predict the flow stress of Cu–0.4Mg alloy during the work hardening–dynamic recovery period, such as at strain rates of 0.1 s 1, 1 s 1, 5 s 1, and 10 s 1 with temperature 500 1C, at strain rates of 1 s 1, 5 s 1, and 10 s 1 with temperature 700 1C, and so on. And it can also be used to predict the assumed flow stress of Cu–0.4Mg alloy due to work-hardening and DRV during the work-hardening, DRV and DRX period. 3.2.2. Modeling the dynamic recrystallization The DRX evolution of metals and alloys mainly depends on the free energy stored in the form of dislocation density during hot deformation. With increase of strain, dislocation continually increases and accumulates to such an extent that at a critical strain, DRX nuclei would form and grow up near grain boundaries, twin boundaries and deformation bands. Especially, under high temperatures and low strain rates, occurrence of dynamic recrystallization becomes more and more obvious. Generally, the volume fraction of dynamic recrystallization (X DRX ) can be expressed as [5,6] ε εc n d ðε Z εc Þ ð13Þ X DRX ¼ 1 exp kd
εp
where εc and εp are the critical strain and the peak strain, respectively. kd and nd are the material constants. Meanwhile, the volume fraction of dynamic recrystallization (X DRX ) can also be represented as [5,6] X DRX ¼
Fig. 7. Relationship between the yield stress (σ0) and the Zener–Hollomon parameter.
σ rec σ drx ðε Z εc Þ σ sat σ ss
ð14Þ
where σ rec is the assumed flow stress of metal materials attributable to work-hardening and DRV during the dynamic recrystallization period, and it can be calculated by Eq. (5). σ drx is the flow stress during the dynamic recrystallization period, σ sat is the saturation
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stress, and σ ss is the steady stress. Substituting Eq. (14) into Eq. (13), the flow stress during the dynamic recrystallization period can be obtained by ε εc n d σ drx ¼ σ rec ðσ sat σ ss Þ 1 exp kd ðε Z εc Þ ð15Þ
εp
In Eq. (15), there are five unknown parameters including εc , εp , σ ss , kd and nd . The steady stress can be easily determined by the true stress–strain curves. Fig. 9 shows that a linear relation exists between the steady stress σ ss and ln Z. So, σ ss can be represented as
σ ss ¼ 9:2155 ln Z 131:1021
ð16Þ
Fig. 10 shows that a linear relation exists between ln(εc ) and ln Z. So, the peak strain εc can be represented as a function of the Zener–Hollomon parameter (Z)
εc ¼ 0:007801 Z 0:1231
Fig. 11. Relationship between the peak strain (εp ) and the Zener–Hollomon parameter.
ð17Þ
Fig. 11 shows that a linear relation exists between ln(εp ) and ln Z. So, the peak strain εp can be represented as a function of the Zener–Hollomon parameter (Z)
εp ¼ 0:037879 Z 0:0776
ð18Þ
Then, substituting the true stress–strain data (after critical strain) into Eq. (14), the parameters kd and nd can be easily obtained for all the tested conditions by the regression analysis. The calculated values of nd are in the range of 1.3337–2.2186, and the average value is evaluated as 1.5693. Fig. 12 shows that a linear
Fig. 12. Relationship between materials constants (kd ) and the Zener–Hollomon parameter.
relation exists between material constants ln(kd ) and ln(Z). So, material constants kd can be represented as a function of the Zener–Hollomon parameter (Z) kd ¼ 0:062777 Z 0:0837
ð19Þ
Therefore, the physical-based constitutive relation of Cu–0.4Mg alloy during the dynamic recrystallization period can be expressed as
Fig. 9. Relationship between the steady stress (σ ss ) and the Zener–Hollomon parameter.
Fig. 10. Relationship between the critical strain (εc ) and the Zener–Hollomon parameter.
n n o 8 ε εc d > ðε Z εc Þ > > σ drx ¼ σ rec ðσ sat σ ss Þ 1 exp kd εp > > > 0:5 > 2 2 2 Ωε > σ rec ¼ ½σ sat þ ðσ 0 σ sat Þe ðε Z εc Þ > > > > 5 > > σ ¼ 0:000442ðln ZÞ 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 sat > > > 2 > > 94:7969ðln ZÞ þ 1233:7397ðln ZÞ 6266:3317 > > > > > σ 0 ¼ 2:9959 ln Z 19:5834 > > < Ω ¼ 103:9339 Z 0:0891 > > σ ss ¼ 9:2155 ln Z 131:1021 > > > > > εc ¼ 0:007801 Z 0:1231 > > > > > εp ¼ 0:037879 Z 0:0776 > > > > > > kd ¼ 0:062777 Z 0:0837 > > > > > nd ¼ 1:5693 > > > : Z ¼ ε_ expð2:138 105 =RTÞ
ð20Þ
Eqs. (12) and (20) together can be used to predict the flow stress of Cu–0.4Mg alloy with dynamic recrystallization behavior, such as at strain rates 0.005 s 1 and 0.01 s 1 with temperature 500 1C, at strain rates 0.005 s 1, 0.01 s 1 and 0.1 s 1 with temperature 700 1C, and so on. Specifically, Eq. (12) is used to predict the first stage (ε o εc ) of flow stress in Cu–0.4Mg alloy with
G. Ji et al. / Materials Science & Engineering A 615 (2014) 247–254
dynamic recrystallization behavior, and Eq. (20) to predict the second stage (ε Z εc ) of flow stress in that. 3.2.3. Verification of the developed constitutive model Eqs. (12) and (20) were used to predict the flow stress of Cu–0.4Mg alloy under all the test conditions. And comparison between the predicted and measured flow stresses under all the test conditions is shown in Fig. 13, which indicates that the predicted flow stresses well agree with the measured ones. Furthermore, the correlation coefficient (R) and the average absolute relative error (AARE) are used to evaluate the prediction capability of the physical-based constitutive relation of Cu–0.4Mg alloy. The AARE is calculated through a term by term comparison of the relative error and therefore is an unbiased statistical parameter for determining the predictability of the constitutive relation. The correlation coefficient (R) provides information on
253
the strength of linear relation between the experimental and predicted values. They can be expressed, respectively, as [9] ∑N i ¼ 1 ðEi EÞðP i PÞ R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ∑i ¼ 1 ðEi EÞ2 ∑N i ¼ 1 ðP i PÞ AARE ¼
1 N P i Ei ∑ 100% N i ¼ 1 Ei
ð21Þ
ð22Þ
where Ei and P i are the measured and the predicted flow stresses, E and P are the mean values of Ei and P i respectively, and N is the total number of data sets used in the study. As shown in Fig. 6, the relationship between the saturation stress and the Zener–Hollomon parameter is fitted by linear and polynomial methods, and the comparison of their predication performances indicates the polynomial fitting method can predict
Fig. 13. Comparisons between the predicted and measured flow stresses of Cu–0.4Mg alloy at temperatures of (a) 500 1C, (b) 600 1C, (c) 700 1C, (d) 750 1C, and (e) 800 1C.
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Fig. 14. Correlation between the measured and predicted flow stresses: (a) the polynomial fitting method predicted flow stress and (b) the linear fitting method predicted flow stress.
the saturation stress more accurately than the linear fitting method. In Eqs. (12) and (20) the saturation stress is expressed by the fifth order polynomial (i.e., Eq. (8)). The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and the predicted flow stresses are 0.99747 and 3.3846%, respectively, as shown in Fig. 14a. It indicates the proposed physical-based constitutive relation can well predict the deformation behaviors including work-hardening, DRV and DRX over a wide range of temperatures and strain rates. But, if the expression for the saturation stress in Eqs. (12) and (20) was replaced by the linear function of ln Z (i.e., Eq. (7)), the correlation coefficient (R) and the average absolute relative error (AARE) between the measured and the predicted flow stresses would be 0.99078 and 6.268%, respectively, as shown in Fig. 14b. So in this study the saturation stress expressed by the polynomial fitting method, rather than by the linear fitting method, can remarkably improve the prediction performance of the physical-based constitutive relation of Cu–0.4Mg alloy during hot working.
during the work hardening–dynamic recovery and dynamic recrystallization period. (4) The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and predicted flow stresses are 0.99747 and 3.3846%, respectively. It indicates that the proposed physical-based constitutive model can accurately characterize the hot deformation behaviors including work-hardening, dynamic recovery and dynamic recrystallization for Cu–0.4Mg alloy over a wide range of temperatures and strain rates.
Acknowledgment The authors appreciate the financial support received from the Materials Processing Engineering Development Funds of Henan Polytechnic University (No. 60621), and from the Science and Technology Research Funds of the Education Department of Henan Province, China (12A430008).
4. Conclusions References In this study, hot compression tests in a wide range of temperatures (500, 600, 700, 750, 800 1C) and strain rates (0.005, 0.01, 0.1, 1, 5, 10 s 1) were carried out to study the hot deformation behavior of Cu–0.4Mg alloy. Important conclusions are presented below. (1) Under a given strain rate the flow stress will decrease with increasing temperature, while under a given temperature the flow stress will increase with increasing strain rate. (2) Dynamic recrystallization occurs at low strain rates and high temperatures, and dynamic recovery does at high strain rates and low temperatures. (3) Based on the stress–dislocation relation and kinetics of dynamic recrystallization, the physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy. The proposed model can well describe the flow stress during the work hardening–dynamic recovery period and
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Y.C. Lin, X.M. Chen, Mater. Des. 32 (2011) 1733–1759. Y.C. Lin, M.S. Chen, J. Zhong, Comput. Mater. Sci. 42 (2008) 470–477. C. Sellars, W.M. Tegart, Acta Metall. 14 (1966) 1136–1138. Y.C. Lin, M.S. Chen, J. Zhong, Mech. Res. Commun. 35 (2008) 142–150. Y.C. Lin, X.M. Chen, D.X. Wen, M.S. Chen, Comput. Mater. Sci. 83 (2014) 282–289. B. Wu, M.Q. Li, D.W. Ma, Mater. Sci. Eng. A 542 (2012) 79–87. L. Wang, F. Liu, Q. Zuo, C.F. Chen, Mater. Des. 47 (2013) 737–745. Sumantra Mandal, P.V. Sivaprasad, S. Venugopal, Model. Simul. Mater. Sci. 14 (2006) 1053–1070. G.L. Ji, F.G. Li, Q.H. Li, H.Q. Li, Z. Li, Comput. Mater. Sci. 48 (2010) 626–632. G.L. Ji, G. Yang, L. Li, Q. Li, J. Mater. Eng. Perform. 23 (2014) 1770–1779. D.K. Shi, W.D. Zhu, Materials Physics, first ed., China Machine Press, Beijing (2006) 55–86. E.I. Poliak, J.J. Jonas, Acta Mater. 44 (1996) 127–136. A. Laasraoui, J.J. Jonas, Metall. Trans. A 22 (1991) 1545–1558. E.P. Busso, F.A. McClintock, Int. J. Plast. 12 (1996) 1–28. C. Zener, J.H. Hollomon, J. Appl. Phys. 15 (1944) 22–32. G.L. Ji, Q. Li, L. Li, Mater. Sci. Eng. A 586 (2013) 197–203.
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
A physical-based constitutive relation to predict flow stress for Cu–0.4Mg alloy during hot working Guoliang Ji, Qiang Li, Lei Li n School of Materials Science and Engineering, Henan Polytechnic University, Jiaozuo 454000, China
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2014 Received in revised form 22 July 2014 Accepted 23 July 2014 Available online 1 August 2014
The hot deformation behaviors of Cu–0.4Mg alloy are studied by the isothermal compression tests on a Gleeble-1500 thermo-mechanical simulator over wide ranges of temperatures and strain rates. Based on the experimental results, a physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy under elevated temperature. Results show that under a given strain rate, the flow stress will decrease with increasing temperature, while under a given temperature the flow stress will increase with increasing strain rate. The true stress–strain curves of Cu–0.4Mg alloy demonstrate the typical characteristics of dynamic recovery and dynamic recrystallization. The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and predicted flow stresses are 0.99747 and 3.3846%, respectively. It indicates that the proposed physical-based constitutive model can accurately characterize the hot deformation behaviors including work-hardening, dynamic recovery and dynamic recrystallization for Cu–0.4Mg alloy. & 2014 Elsevier B.V. All rights reserved.
Keywords: Cu–0.4Mg alloy Deformation behavior Physical-based constitutive model Dynamic recovery and dynamic recrystallization
1. Introduction The constitutive relation of metal materials during the hot deformation process reflects the nonlinear relation between the flow stress and process variables such as strain, strain rate and temperature, and it has complex mathematical forms and profound physical meanings. In the hot working process of metal materials there occur various physical mechanisms such as workhardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX), which are affected by the chemical composition of metals and alloys, deformation temperature, deformation amount and strain rate. The constitutive relation reflects comprehensive effects of various physical mechanisms in the hot working process of metal materials. Constitutive relation can be used to reveal physical mechanisms of deformation, to establish the recrystallization dynamics model, and to develop the processing map by which to uncover the deformation mechanism and to delineate the stability and instability regimes in the hot working window. Furthermore, the constitutive relation is often used to describe the plastic flow properties of metals and alloys, in a form that can be used in the computer code to simulate the thermo-mechanical response of mechanical parts under the prevailing loading conditions [1,2]. Preciseness of the constitutive relation has important significance for achieving accurate simulation results.
n
Corresponding author. Tel.: þ 86 0391 3986939. E-mail address: [email protected] (L. Li).
http://dx.doi.org/10.1016/j.msea.2014.07.082 0921-5093/& 2014 Elsevier B.V. All rights reserved.
The constitutive relations can be divided into three categories, including the phenomenological, physical-based and artificial neural network models [1]. Based on empirical guess and intuitive grasp of effect laws of process parameters on the flow behavior of metal materials (work-hardening, strain rate hardening and temperature softening), the phenomenological constitutive relation does not deeply involve physical mechanisms of materials deformation, and describes flow stress in a simple mathematical form with macroscopic process parameters. The phenomenological approach [3], proposed by Sellars and Tegart, in which the flow stress is expressed by the sine-hyperbolic law in an Arrhenius type of equation, has been extensively used to predict the hot deformation behavior of metal materials. Lin et al. [2] proposed a modified Arrhenius model to characterize the hot deformation behavior of 42CrMo steel by the compensation of strain and strain rate. Based on the classical stress–dislocation relation and the kinetics of dynamic recrystallization, the physical-based constitutive relation was established to describe the flow stress during the work hardening–dynamic recovery and dynamical recrystallization periods for 42CrMo steel [4], nickel-based super-alloy [5], 7050 aluminum alloy [6], and N08028 alloy [7]. An artificial neural network (ANN) is a mathematical model or a computational model that tries to simulate the structure and/or functional aspects of biological neural networks. Since the establishment of the ANN model need not utilize the physical knowledge of deformation mechanisms, it has the powerful ability to predict hot deformation behaviors of metal materials across the whole hot working domain. The ANN model was developed to predict the flow stress
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of austenitic stainless steels [8], Aermet100 [9] and Cu–0.4Mg alloy [10] during hot working with very high accuracy. Cu–0.4Mg alloy contact wires have been successfully used in the high speed railway due to its high strength, high electrical conductivity and high wear resistance. Cu–0.4Mg alloy contact wires manufactured by the traditional production process have coarse grains microstructure resulting from a previous casting process, which damages its mechanical and electrical properties, for example, coarse grain microstructure would cause a great amount of electric sparks when the pantograph slide slides on the contact wires. The new production process of Cu–0.4Mg alloy contact wires is that the Cu–0.4Mg alloy is melted and cast under the air isolated condition in the horizontal contiguous furnace to obtain a rod-shaped billet, then it is hot extruded to crush coarse cast microstructure with a view to obtain a fine recrystallization microstructure, and finally fabricated into contact wires by the cold drawing process. Modeling and prediction of the hot deformation behavior of Cu–0.4Mg alloy is of great importance for controlling the microstructures of Cu–0.4Mg alloy. However, there are few reports about the constitutive model of Cu–0.4Mg alloy. In this study, hot compression tests in a wide range of temperatures and strain rates were carried out to study the hot deformation behavior of Cu–0.4Mg alloy. Based on the stress– dislocation relation and kinetics of dynamic recrystallization, a physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy. Moreover, the prediction ability of the physical-based constitutive model is evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE).
2. Materials and experimental procedure Cu–0.4Mg alloy contact wires in cold drawing were annealed: heating to 600 1C, holding for 3 h and cooling in furnace, then they were machined in cold drawing direction into specimens with a diameter of 8 mm and a length of 12 mm. Compression tests were carried out in the temperature range of 500–800 1C(500, 600, 700, 750, 800 1C) and the strain rate range of 0.005–10 s 1 (0.005, 0.01, 0.1, 1, 5, 10 s 1) on a Gleeble-1500 thermo-mechanical simulator. Cu–0.4Mg alloy specimens were heated to a preset temperature at a rate of 10 1C/s, soaked for 60 s to ensure a uniform temperature in the whole specimen, compressed by 60% in height and then water cooled to the room temperature to obtain the hot deformation microstructure, while true stress–true strain data at different temperatures and strain rates were recorded automatically. In order to decrease the friction effects in the deformation process, a high-temperature lubricant was applied. The hot compression specimens were polished and etched with an etchant (19 g ferric nitrate, 50 ml alcohol and 50 ml water). Optical microstructures in the central region of the section were examined by optical microscopy, and grain sizes were determined by means of the linear intercept method (here ASTM standard E112-12 was used). The microstructure of the annealed Cu–0.4Mg alloy specimens is presented in Fig. 1.
3. Results and discussion 3.1. Stress–strain characteristics of Cu–0.4Mg alloy during hot working The true stress–strain curves for Cu–0.4Mg alloy during hot compression are illustrated in Fig. 2. Under a given temperature (Fig. 2a) the flow stress will increase with increasing strain rate, while under a given strain rate (Fig. 2b) the flow stress will decrease with increasing temperature. General variation of flow
Fig. 1. Microstructures of as annealed Cu–0.4Mg alloy.
stress with temperature and strain rate can be explained quantitatively. The most basic relation [11] among temperature, strainrate and dislocation density can be represented as ΔG n ð1Þ ε_ ¼ ρm A exp kT where ρm ¼ f ρ is the mobile dislocation density, in which f is about 0.1 representing the proportion of the mobile dislocation density in the total dislocation density ρ. A is the materials constant, ΔGn is the heat activity energy and k is the Boltzmann constant. The most basic relation [11] between flow stress and dislocation can be formulated as follows: pffiffiffi σ ¼ σ n þ αμb ρ ð2Þ where σ n is the intrinsic strength of materials representing the crystal lattice frictional resistance, b is the Burgers vector, α is the materials constant and μ is the shear modulus. According to Eqs. (1) and (2), it follows that under a given strain rate, the dislocation density decreases with increasing temperatures, so the total level of the flow stress will decrease with increasing temperatures; similarly under a given temperature, the dislocation density increases with increasing strain rates, so the total level of the flow stress will increase with increasing strain rates. Microstructures of the specimens compressed at strain rates of 0.005 s 1 and 10 s 1 with temperature of 700 1C are presented in Fig. 3a and b, respectively. It can be seen that at a given temperature, with increasing strain rate, the microstructures change from recrystallized equiaxial grains (Fig. 3a) to deformed flat grains (Fig. 3b). At the temperature of 700 1C and the higher strain rate of 10 s 1, dislocation density would increase and interaction between dislocations would become more significant, but due to heat activation of high temperature, larger dislocation density can be annihilated and rearranged by the dislocation climbing, sliding, and cross-slip, which causes polygonization of dislocation and regularization of dislocation cell wall to further form subboundaries, meanwhile dynamic recovery occurs. Microstructures at this deformation condition have typical characteristics of dynamic recovery, as shown in Fig. 3b. At the temperature of 700 1C and the lower strain rate of 0.005 s 1, dislocation density would decrease and interaction between dislocations would become more weak; in addition, due to heat activation of high temperature and having enough time, lower dislocation density can be annihilated and rearranged more easily. Subboundaries formed by polygonization of dislocation and regularization of dislocation cell wall can protrude and grow up from grain boundary with lower storage energy to that with larger storage energy, meanwhile dynamic recrystallization occurs. Microstructures at this deformation
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Fig. 2. True strain–stress curves of Cu–0.4Mg alloys (a) at given temperature of 700 1C and (b) at given strain rate of 10 s 1.
Fig. 3. Microstructures of Cu–0.4Mg alloy deformed at the strain rates of (a) 0.005 s 1 and (b) 10 s 1 with a temperature of 700 1C.
Fig. 4. A schematic of true stress–strain curves characteristic of dynamic recovery and dynamic recrystallization.
condition have typical characteristics of dynamic recrystallization, as shown in Fig. 3a. According to the dynamic softening mechanisms during hot deformation, the true stress–strain curves for Cu–0.4Mg alloy can be divided into two types, i.e., dynamic recovery and dynamic recrystallization. The true stress–strain curves characteristic of dynamic recovery are illustrated in Fig. 4 (marked with ‘a’) [5]. The flow stress rapidly increases due to work-hardening. When a balance between work-hardening and dynamic recovery is reached, a saturation flow stress appears and stays constant; meanwhile grains continue to elongate with increase of strain but the shape and size of subgrains remain unchanged. The true stress–strain curves characteristic of dynamic recrystallization are illustrated in Fig. 4 (marked with ‘b’) [5]. The flow stress rapidly increases to critical stress due to work-hardening, and continues to
slowly increase to the peak stress due to the occurrence of recrystallization; then the flow stress begins to decrease until it reaches a steady stress and stays constant, meanwhile, the shape and size of recrystallized grains remain unchanged. Generally, the true stress–strain curves with dynamic recrystallization characteristics can be divided into three stages [5]: stage I (work-hardening stage and dynamic recovery), stage II (work-hardening, dynamic recovery and dynamic recrystallization) and stage III (steady dynamic recrystallization), as shown in Fig. 4. Stage I is characterized by a sharp increase of flow stress. It is attributed to the fact that the generation and multiplication of dislocation occur rapidly, leading to the high work-hardening rate. Meanwhile, the dynamic recovery caused by the dislocation climbing, sliding, and cross-slip is too weak to overcome the work-hardening effect. In stage II, the dynamic recrystallization takes place at a critical strain (εc ), resulting in decreased workhardening rate with increase of strain. When the dynamic softening (DRV and DRX) rate is in equilibrium with the workhardening rate, the flow stress achieves a peak stress (σ p ), and then gradually decreases to steady stress (σ ss ). When a new balance between dynamic softening and work-hardening is reached, the flow stress is maintained at a fairly constant level regardless of the increased strain (in stage III). 3.2. Constitutive modeling of flow stress The dislocation density theory and the modified Arvami equation can be used to describe the flow behavior of metal materials during hot working, i.e., work-hardening, dynamic recovery and dynamic recrystallization. As shown in Fig. 4, the curve marked ‘a’ represents true work-hardening and DRV behavior of metal materials, or the assumed work-hardening and DRV behavior of metal materials in which work-hardening, DRV and DRX occur simultaneously.
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σ rec represents the true or assumed flow stress of metal materials attributable to work-hardening and DRV. As shown in Fig. 4, the curve marked ‘b’ represents the true work-hardening, DRV and DRX behavior, and its stage II is considered to be the net result of the simultaneous operation of work-hardening and DRV and DRX [7]. σ drx represents the flow stress in stage II, and the difference (Δσ ) between σ drx and σ rec is the net softening directly attributable to DRX [7]. The initial stress, critical strain and stress, peak strain and stress, saturation stress, and steady stress need to be determined first to establish the constitutive relation capable of describing the workhardening, DRV and DRX behaviors. As shown in Fig. 4, for the true stress–strain curves characteristic of dynamic recrystallization (marked ‘b’), the initial stress (σ 0 ), peak strain (εp ) and peak stress (σ p ), and steady stress (σ ss ) can be obtained directly from the true stress–strain curve. However, direct determination of the critical strain (εc ) is difficult, and it can be attained when the value of j dθ=dσ j, where strain hardening rate θ ¼ dσ =dε, reaches the minimum which corresponds to an inflection of dσ =dε versus σ curve [12], as shown in Fig. 5. Then the saturation stress can be determined as the horizontal intercept of the tangent line of θ σ plot through the inflection point. 3.2.1. Modeling the work-hardening and dynamic recovery During the work hardening–dynamic recovery period, the evolution of dislocation density with strain is generally considered to depend on two components: multiplication and annihilation of dislocation, and can be expressed as [5] dρ ¼ U Ωρ dε
ð3Þ
where dρ=dε is the increasing rate of dislocation density with strain; U is a multiplication term which represents the workhardening, and can be regarded as a constant with respect to strain. Ωρ represents the dynamic recovery due to the dislocation annihilation and rearrangement, and Ω is the coefficient of dynamic recovery [13]. When plastic strain ε ¼ 0, dislocation density ρ ¼ ρ0 , where ρ0 is the initial dislocation density, and the corresponding flow stress is the initial yield stress σ 0 . Then the expression of dislocation density can be obtained by integrating Eq. (3) as follows: U U ρ¼ ρ0 e Ωε ð4Þ
Ω
Ω
It is known that the effective stress can be negligible compared with the internal stress at high temperatures [5]. Thus, the applied stress can be directly estimated by the square root of the dislocapffiffiffi tion density, σ ¼ αμb ρ, where α is the material constant, μ is the shear modulus and b is the distance between atoms in the slip direction [14]. Substituting this expression into Eq. (4), the flow stress during the work-hardening and dynamic recovery period can be represented as
σ rec ¼ ½σ 2sat þ ðσ 20 σ 2sat Þe Ωε 0:5
where σ rec represents the flow stress in the work-hardening and DRV period and σ sat is the saturation stress. It can be found in Eq. (5) that three parameters (σ sat , σ 0 and Ω) need to be determined. In general, the combined effects of temperature and strain rate on feature parameters of true stress–strain curves (such as σ sat , σ 0 and Ω) can be characterized by the Zener–Hollomon parameter. The expression of Zener– Hollomon parameter (Z) is given as [15] Z ¼ ε_ expðQ =RTÞ
Fig. 5. The dσ=dε versus σ curve at a temperature 700 1C and a strain rate of 0.1 s 1.
ð5Þ
ð6Þ
where ε_ is the strain rate (s 1), R is the universal gas constant (8.314 J mol 1 K 1), T is the absolute temperature (K), and Q is the activation energy (J mol 1). In order to establish the expression of feature parameters (such as σ sat , σ 0 and Ω), the activation energy (Q ) should be determined first. In the authors' previous publication [16], the calculation procedure of the activation energy Q of Cu–0.4Mg alloy has been elaborated, and it is estimated as Q ¼ 2:138 105 J mol 1. For the true stress–strain curves characteristic of dynamic recovery, the saturation stress (σ sat ) can be directly obtained, as shown in Fig. 4; while for the true stress–strain curves characteristic of dynamic recrystallization it can be determined as the horizontal intercept of the tangent line of θ σ plot through the inflection point, as shown in Fig. 5. Fig. 6a illustrates there is
Fig. 6. Relationship between the saturation stress (σ sat ) and the Zener–Hollomon parameter: (a) linear fitting and (b) polynomial fitting.
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linear relation between the saturation stress (σ sat ) and ln Z. Then, the saturation stress (σ sat ) can be expressed as
σ sat ¼ 15:9998 ln Z 268:186
ð7Þ
Results presented at the end of this article show that saturation stress (σ sat ) has considerable effects on the prediction accuracy of the physical-based constitutive relation of Cu–0.4Mg alloy during hot working. So the correlation coefficient (R) and the average absolute relative error (AARE) between the predicted and measured saturation stresses are employed to evaluate prediction capability of the expression for the saturation stress (σ sat ). The correlation coefficient (R) and the average absolute relative error (AARE) are 0.98894 and 7.6964%, respectively. To further improve prediction accuracy for the saturation stress (σ sat ), a fifth order polynomial is used to fit the measured saturation stress (σ sat ), as shown in Fig. 6b, and can be expressed as follows:
Fig. 8. Relationship between the coefficient of dynamic recovery (Ω) and the Zener–Hollomon parameter.
σ sat ¼ 0:000442ðln ZÞ5 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 94:7969ðln ZÞ2 þ 1233:7397ðln ZÞ 6266:3317
ð8Þ
The correlation coefficient (R) and the average absolute relative error (AARE) between the fifth order polynomial predicted and measured saturation stresses are 0.99639 and 3.4595%, respectively. It can be found that the expression for the saturation stress in the form of the fifth order polynomial greatly improves its prediction accuracy. The yield stress (σ 0 ) at various forming temperatures and strain rates can be directly obtained from the true stress–strain curves, and in this study it is taken as the stress corresponding to a strain of 0.02. Fig. 7 illustrates that a good linear relation exists between the yield stress (σ 0 ) and ln Z. Then, the yield stress (σ 0 ) can be expressed as
σ 0 ¼ 2:9959 ln Z 19:5834
ð9Þ
According to Eq. (5), the coefficient of dynamic recovery (Ω) can be calculated by 2 σ σ2 Ωε ¼ ln 2sat 20 ð10Þ σ sat σ rec Using the true stress–strain data before the saturation strain (corresponding to the saturation stress) for the true stress–strain behavior characteristic of dynamic recovery and using the true stress–strain data before the critical strain for the flow behavior characteristic of dynamic recrystallization, the values of Ω can be determined for all the test conditions. Fig. 8 shows that a linear relation exists between ln Ω and ln Z. It can be found that the dynamic recovery coefficient increases with the decrease of the
Zener–Hollomon parameter, and Ω can be expressed as a function of the Zener–Hollomon parameter.
Ω ¼ 103:9339 Z 0:0891
ð11Þ
Therefore, the physical-based constitutive relation of Cu–0.4Mg alloy during the work hardening–dynamic recovery period can be summarized as 8 σ rec ¼ ½σ 2sat þ ðσ 20 σ 2sat Þe Ωε 0:5 > > > > > > σ sat ¼ 0:000442ðln ZÞ5 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 > > > < 94:7969ðln ZÞ2 þ 1233:7397ðln ZÞ 6266:3317 ð12Þ > σ 0 ¼ 2:9959 ln Z 19:5834 > > > > 0:0891 > Ω ¼ 103:9339 Z > > > : Z ¼ ε_ expð2:138 105 =RTÞ Eq. (12) can be used to predict the flow stress of Cu–0.4Mg alloy during the work hardening–dynamic recovery period, such as at strain rates of 0.1 s 1, 1 s 1, 5 s 1, and 10 s 1 with temperature 500 1C, at strain rates of 1 s 1, 5 s 1, and 10 s 1 with temperature 700 1C, and so on. And it can also be used to predict the assumed flow stress of Cu–0.4Mg alloy due to work-hardening and DRV during the work-hardening, DRV and DRX period. 3.2.2. Modeling the dynamic recrystallization The DRX evolution of metals and alloys mainly depends on the free energy stored in the form of dislocation density during hot deformation. With increase of strain, dislocation continually increases and accumulates to such an extent that at a critical strain, DRX nuclei would form and grow up near grain boundaries, twin boundaries and deformation bands. Especially, under high temperatures and low strain rates, occurrence of dynamic recrystallization becomes more and more obvious. Generally, the volume fraction of dynamic recrystallization (X DRX ) can be expressed as [5,6] ε εc n d ðε Z εc Þ ð13Þ X DRX ¼ 1 exp kd
εp
where εc and εp are the critical strain and the peak strain, respectively. kd and nd are the material constants. Meanwhile, the volume fraction of dynamic recrystallization (X DRX ) can also be represented as [5,6] X DRX ¼
Fig. 7. Relationship between the yield stress (σ0) and the Zener–Hollomon parameter.
σ rec σ drx ðε Z εc Þ σ sat σ ss
ð14Þ
where σ rec is the assumed flow stress of metal materials attributable to work-hardening and DRV during the dynamic recrystallization period, and it can be calculated by Eq. (5). σ drx is the flow stress during the dynamic recrystallization period, σ sat is the saturation
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stress, and σ ss is the steady stress. Substituting Eq. (14) into Eq. (13), the flow stress during the dynamic recrystallization period can be obtained by ε εc n d σ drx ¼ σ rec ðσ sat σ ss Þ 1 exp kd ðε Z εc Þ ð15Þ
εp
In Eq. (15), there are five unknown parameters including εc , εp , σ ss , kd and nd . The steady stress can be easily determined by the true stress–strain curves. Fig. 9 shows that a linear relation exists between the steady stress σ ss and ln Z. So, σ ss can be represented as
σ ss ¼ 9:2155 ln Z 131:1021
ð16Þ
Fig. 10 shows that a linear relation exists between ln(εc ) and ln Z. So, the peak strain εc can be represented as a function of the Zener–Hollomon parameter (Z)
εc ¼ 0:007801 Z 0:1231
Fig. 11. Relationship between the peak strain (εp ) and the Zener–Hollomon parameter.
ð17Þ
Fig. 11 shows that a linear relation exists between ln(εp ) and ln Z. So, the peak strain εp can be represented as a function of the Zener–Hollomon parameter (Z)
εp ¼ 0:037879 Z 0:0776
ð18Þ
Then, substituting the true stress–strain data (after critical strain) into Eq. (14), the parameters kd and nd can be easily obtained for all the tested conditions by the regression analysis. The calculated values of nd are in the range of 1.3337–2.2186, and the average value is evaluated as 1.5693. Fig. 12 shows that a linear
Fig. 12. Relationship between materials constants (kd ) and the Zener–Hollomon parameter.
relation exists between material constants ln(kd ) and ln(Z). So, material constants kd can be represented as a function of the Zener–Hollomon parameter (Z) kd ¼ 0:062777 Z 0:0837
ð19Þ
Therefore, the physical-based constitutive relation of Cu–0.4Mg alloy during the dynamic recrystallization period can be expressed as
Fig. 9. Relationship between the steady stress (σ ss ) and the Zener–Hollomon parameter.
Fig. 10. Relationship between the critical strain (εc ) and the Zener–Hollomon parameter.
n n o 8 ε εc d > ðε Z εc Þ > > σ drx ¼ σ rec ðσ sat σ ss Þ 1 exp kd εp > > > 0:5 > 2 2 2 Ωε > σ rec ¼ ½σ sat þ ðσ 0 σ sat Þe ðε Z εc Þ > > > > 5 > > σ ¼ 0:000442ðln ZÞ 0:063764ðln ZÞ4 þ 3:5448ðln ZÞ3 sat > > > 2 > > 94:7969ðln ZÞ þ 1233:7397ðln ZÞ 6266:3317 > > > > > σ 0 ¼ 2:9959 ln Z 19:5834 > > < Ω ¼ 103:9339 Z 0:0891 > > σ ss ¼ 9:2155 ln Z 131:1021 > > > > > εc ¼ 0:007801 Z 0:1231 > > > > > εp ¼ 0:037879 Z 0:0776 > > > > > > kd ¼ 0:062777 Z 0:0837 > > > > > nd ¼ 1:5693 > > > : Z ¼ ε_ expð2:138 105 =RTÞ
ð20Þ
Eqs. (12) and (20) together can be used to predict the flow stress of Cu–0.4Mg alloy with dynamic recrystallization behavior, such as at strain rates 0.005 s 1 and 0.01 s 1 with temperature 500 1C, at strain rates 0.005 s 1, 0.01 s 1 and 0.1 s 1 with temperature 700 1C, and so on. Specifically, Eq. (12) is used to predict the first stage (ε o εc ) of flow stress in Cu–0.4Mg alloy with
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dynamic recrystallization behavior, and Eq. (20) to predict the second stage (ε Z εc ) of flow stress in that. 3.2.3. Verification of the developed constitutive model Eqs. (12) and (20) were used to predict the flow stress of Cu–0.4Mg alloy under all the test conditions. And comparison between the predicted and measured flow stresses under all the test conditions is shown in Fig. 13, which indicates that the predicted flow stresses well agree with the measured ones. Furthermore, the correlation coefficient (R) and the average absolute relative error (AARE) are used to evaluate the prediction capability of the physical-based constitutive relation of Cu–0.4Mg alloy. The AARE is calculated through a term by term comparison of the relative error and therefore is an unbiased statistical parameter for determining the predictability of the constitutive relation. The correlation coefficient (R) provides information on
253
the strength of linear relation between the experimental and predicted values. They can be expressed, respectively, as [9] ∑N i ¼ 1 ðEi EÞðP i PÞ R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ∑i ¼ 1 ðEi EÞ2 ∑N i ¼ 1 ðP i PÞ AARE ¼
1 N P i Ei ∑ 100% N i ¼ 1 Ei
ð21Þ
ð22Þ
where Ei and P i are the measured and the predicted flow stresses, E and P are the mean values of Ei and P i respectively, and N is the total number of data sets used in the study. As shown in Fig. 6, the relationship between the saturation stress and the Zener–Hollomon parameter is fitted by linear and polynomial methods, and the comparison of their predication performances indicates the polynomial fitting method can predict
Fig. 13. Comparisons between the predicted and measured flow stresses of Cu–0.4Mg alloy at temperatures of (a) 500 1C, (b) 600 1C, (c) 700 1C, (d) 750 1C, and (e) 800 1C.
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Fig. 14. Correlation between the measured and predicted flow stresses: (a) the polynomial fitting method predicted flow stress and (b) the linear fitting method predicted flow stress.
the saturation stress more accurately than the linear fitting method. In Eqs. (12) and (20) the saturation stress is expressed by the fifth order polynomial (i.e., Eq. (8)). The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and the predicted flow stresses are 0.99747 and 3.3846%, respectively, as shown in Fig. 14a. It indicates the proposed physical-based constitutive relation can well predict the deformation behaviors including work-hardening, DRV and DRX over a wide range of temperatures and strain rates. But, if the expression for the saturation stress in Eqs. (12) and (20) was replaced by the linear function of ln Z (i.e., Eq. (7)), the correlation coefficient (R) and the average absolute relative error (AARE) between the measured and the predicted flow stresses would be 0.99078 and 6.268%, respectively, as shown in Fig. 14b. So in this study the saturation stress expressed by the polynomial fitting method, rather than by the linear fitting method, can remarkably improve the prediction performance of the physical-based constitutive relation of Cu–0.4Mg alloy during hot working.
during the work hardening–dynamic recovery and dynamic recrystallization period. (4) The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and predicted flow stresses are 0.99747 and 3.3846%, respectively. It indicates that the proposed physical-based constitutive model can accurately characterize the hot deformation behaviors including work-hardening, dynamic recovery and dynamic recrystallization for Cu–0.4Mg alloy over a wide range of temperatures and strain rates.
Acknowledgment The authors appreciate the financial support received from the Materials Processing Engineering Development Funds of Henan Polytechnic University (No. 60621), and from the Science and Technology Research Funds of the Education Department of Henan Province, China (12A430008).
4. Conclusions References In this study, hot compression tests in a wide range of temperatures (500, 600, 700, 750, 800 1C) and strain rates (0.005, 0.01, 0.1, 1, 5, 10 s 1) were carried out to study the hot deformation behavior of Cu–0.4Mg alloy. Important conclusions are presented below. (1) Under a given strain rate the flow stress will decrease with increasing temperature, while under a given temperature the flow stress will increase with increasing strain rate. (2) Dynamic recrystallization occurs at low strain rates and high temperatures, and dynamic recovery does at high strain rates and low temperatures. (3) Based on the stress–dislocation relation and kinetics of dynamic recrystallization, the physical-based constitutive model is developed to predict the flow stress of Cu–0.4Mg alloy. The proposed model can well describe the flow stress during the work hardening–dynamic recovery period and
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