Evaluation of low strain rate constitutive equation of 7075 aluminium ...

Evaluation of low strain rate constitutive equation of 7075 aluminium alloy at high temperature A. Jenab and A. Karimi Taheri* The 7075 aluminium alloy is one of the most important engineering alloys utilised extensively in aircraft and transportation industries due to its high specific strength. In the present research, the flow behaviour of this alloy has been investigated using hot compression test at strain rates of 0?001, 0?01, 0?1 and 1 s21 and temperatures of 350, 400 and 450uC. The results reveal that dynamic softening occurred in these temperatures and strain rates. The activation energy, strain rate sensitivity and two constitutive equations (hyperbolic sine law and the power law) are derived from the results. It is shown that the hyperbolic sine law has a better agreement with the experimental results. Keywords: 7075 aluminium alloy, Hot deformation, Strain rate sensitivity, Constitutive equation

Introduction The research on new materials of low densities and coefficients of thermal expansion but high strength, stiffness, fracture toughness and thermal conductivity for the applications in aerospace, automobile and electrical industries has been of tremendous interest to materials scientists.1,2 Among such materials, the use of aluminium alloys in automotive industry has greatly increased in recent years. This has been attributed not only to the issues of fuel economy but also to those of safety, resource conservation and environment compatibility.1 The 7075 aluminium alloy is one of the most important engineering alloys utilised extensively in aircraft and transportation industries due to its high specific strength, weldability and heat treatability conditions.2–4 A deep knowledge about the specific material behaviour, especially the flow curves and their dependencies on temperature, strain and strain rate, is needed in order to successfully control their hot or cold forming processes. In addition, the effects such as worksoftening have to be taken into account when theoretically describing the forming behaviour of materials for simulation purposes. Moreover, the knowledge of the behaviour of material during deformation is essential for the prediction of its final properties. Considering that materials exhibit a better workability in the hot deformation region, it is common to deform them in this region.5 Therefore, determination of dependence of flow stress on processing parameters in hot region is essential. In industry, the deformation of metals and alloys is carried out at rather high strain Department of Materials Science and Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran *Corresponding author, email [email protected]

ß 2011 Institute of Materials, Minerals and Mining Published by Maney on behalf of the Institute Received 18 November 2009; accepted 5 February 2010 DOI 10.1179/026708310X12683158443206

rates, and the low strain rates are relevant to deformation of casting and breakdown rolling. However, the experimental data of the hot deformation of an alloy in wrought condition and laboratory strain rates are useful for the study of the controlled rolling or extrusion processes of the alloy. Referring to the literature, the constitutive equation of 7075 aluminium alloy has been determined at high strain rates, and there is not any report for the low strain rates.2,6 Thus, the lack of research on low strain rate deformation of 7075 aluminium alloy is the motivation of the present work.

Constitutive equation During hot deformation, it is well accepted that the relationship among the steady state flow stress s, strain : rate e and temperature T is generally expressed in the form of an Arrhenius type equation as7–9 {Q : n (1) e~As exp RT Although equation (1), termed also as the power law relation, is among the most common relationships used to correlate the stress, strain rate and activation energy at constant temperature, its correlation of data has been limited to low stresses. It has been shown by Sellars and Tegart10 in 1972 that the exponent parameter n in this relation is dependent on temperature. On the other hand, a hyperbolic sine law relation has been further proposed by Sellars and Tegart10 in modelling the hot working processes. It has been shown that the model predictions using hyperbolic sine law relation are satisfactory in correlating the steady state flow behaviour over large ranges in strain rate and temperature. However, it has been reported that the power law fits the data quite well at low stresses and fails at high stresses.11 Therefore, the hyperbolic sine law has been used in more generalised constitutive equations.12

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Low strain rate constitutive equation of 7075 aluminium alloy

The steady state strain rate is related to the saturated flow stress by a hyperbolic sine function in the following form {Q : (2) e~A½sinhðasÞn exp RT

reaches a peak at a critical strain and then decreases to a steady state stress. This behaviour demonstrates that the rate of workhardening and softening is balanced at the steady state condition. It is essential to mention that at higher strain rates (1 s21), the adiabatic heating generated from deformation is higher. As a result, one can conclude that the flow stress at higher strain rates is reduced due to adiabatic heating. The noise of the load cell of test machine can be the cause of the serrations of the curves shown in Fig. 1. In other words, in this region of temperatures and strain rates, the dynamic strain aging phenomenon can be ruled out since the appropriate strain rate sensitivity is not negative.15 Comparing the curves plotted in Fig. 1a, it is found that, at a specific strain rate, the flow stress decreases markedly with temperature. According to this figure, reduction in flow stress at temperatures of 400 and 450uC is higher than that at 350 and 400uC. The curves plotted in Fig. 1b demonstrate that the flow stress decreases with decreasing strain rate. Since 7075 aluminium alloy is strengthened by precipitates forming during the decomposition of a supersaturated solid solution, the strength of this material results from lattice strain interactions between precipitates and dislocations and is dependent highly on the type, distribution and size of the precipitates. During high temperature loading, dissolution of the precipitates is involved in a thermally activated process. Increasing the temperature accelerates the precipitate dissolution, the rate of dislocation annihilation and, subsequently, the rate of softening. Thus, these physical and metallurgical reactions result in a reduction in flow stress, and correspondingly, the material becomes relatively softer and weaker.2

In both equations (1) and (2), A is a constant, n is the stress exponent (n5m21, where m is the strain rate sensitivity), a is a material constant, Q is the activation energy for hot working and R is the gas constant.7,12 It has been shown that on the basis of the stress exponent and activation energy values, the deformation mechanisms for microstructure development during deformation can be identified.7 Constitutive equations, including a power law term, have been commonly applied to many alloys with the objective of calculating hot rolling and forging forces.13,14 The power law looses the linearity at high and low stresses. Therefore, the function relating the stress and strain rate has generally been chosen in the form of hyperbolic sine relation.14 To derive the constitutive equation of 7075 aluminium alloy in this research, isothermal hot compression test at different initial strain rates and temperatures is carried out. Then, two constitutive equations (power law and hyperbolic sine) are derived and compared with each other using the test results.

Experimental The chemical composition of the 7075 aluminium alloy used in the present research is listed in Table 1. Isothermal hot compression test at temperatures of 350, 400 and 450uC and strain rates of 0?001, 0?01, 0?1 and 1 s21 was performed using an Instron machine model 8502 to derive the constitutive equations of the alloy. The temperature was controlled with an accuracy of ¡5uC using two thermocouples installed in the furnace and close to the samples. Cylindrical compression specimens of 10 mm diameter and 15 mm height with tolerance of 0?02 mm were machined from an extruded billet. The specimens were heat treated at 345uC for 3 h and air cooled in order to reduce the residual stresses produced by machining on their surfaces. In order to reduce friction, the upper and lower surfaces of the specimens were polished with sand paper no. 3000 before the test. The specimens were deformed to a total strain of 0?7 in each test, and the load stroke data were converted into true stress–true strain data.

Determination of constitutive equations In order to establish the constitutive equations of the 7075 aluminium alloy tested in the present work, the steady state flow stress s data at different temperatures : T and strain rates e at a true strain of 0?6 were used. The true strain of 0?6 was selected because most of the flow stress curves plotted in the present work show a steady state behaviour at this strain. The value of a was chosen as 0?0015, since this value exhibits reliable results as demonstrated in the parallel relation between the ln [sinh(as)] and ln (strain rate) as exhibited in Fig. 2. Figure 3 demonstrates the dependence of s (at 0?6 strain) at strain rates of 0?001, 0?01, 0?1 and 1 s21 on temperature. Using this figure and the MATLAB software, the values of stress at different strain rates and temperatures were derived. The variation of stress with different strain rates at different test temperatures are shown in Fig. 4. According to equations (1) and (2), equations (3) and (4) can be defined respectively. Consequently, the value of n is the inverse of the slope of the line of (ln s versus : : ln e) and {ln [sinh(as)] versus ln e} respectively. The latter is illustrated in Fig. 4.

Results and discussion Flow behaviour The stress–strain curves of the alloy at different temperatures using strain rate of 0?1 s21 are shown in Fig. 1a. Similarly, the stress–strain curves of the alloy at 400uC using different strain rates are exhibited in Fig. 1b. These figures indicate that the alloy shows a flow softening phenomenon, in which the flow stress

Table 1 chemical composition of 7075 aluminum alloy used in this research

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Element

Ca, Bi

V

Ti, Sn

Ni, Ga, Co

Pb

Mn

Cr

Si

Fe

Cu

Mg

Zn

Al

wt-%

0.01

0.013

0.014

0.02

0.05

0.08

0.17

0.2

0.36

1.61

2.18

5.26

Rem.


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a at strain rate of 0?1 s21; b at temperature of 400uC 1 True stress–true strain curves obtained in compression of AA7075

n~

: d ln e d ln s

(3)

n~

: d ln e d ln sinh(as)

(4)

It is observed that in a wide range of temperature and strain rate, the value of n is dependent on strain rate and temperature.15 By neglecting this dependence at these temperature and strain rate regions, the average of n estimated from equations (3) and (4) is presented in Table 2. In order to determine the activation energy, according to equation (5), the slope of ln [sinh(as)] versus 1000 T 21 at different strain rates was calculated for both the power law and the hyperbolic sine constitutive equations. For the latter, the plot is presented in Fig. 5 Q~nR

L ln½sinh(as) 1 L T

(5)

2 Dependence of ln (strain rate) on ln [sinh(as)] at different temperatures, a50?0015

The activation energy values determined from the two constitutive equations are presented in Table 2. Based on these results, the estimated value of average apparent activation energy is ,131?1 kJ mol21 k21 for both constitutive equations. Although the difference between the values of n and Q calculated from equations (1) and (2) is negligible, it will be shown below that the difference between the mean squared errors (MSE) of the predictions of the two equations is noticeable. The activation energy and n value were used to calculate the Zener–Hollomon parameter as below,7 which is plotted versus the flow stress in Fig. 6 using equations (1) and (2) respectively. : Z~e expðQ=RT Þ

(6)

These plots show a good linear correlation between the flow stress and the Z value. The regression coefficient R2 presented in the figures is acceptable and

3 Dependence of flow stress at strain of 0?6 on temperature at different strain rates


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5 Variation of ln [sinh(as)] versus 1000 T 21 at different strain rates 4 Variation of flow stress at strain of 0?6 versus strain rate at different temperatures (calculated from Fig. 3)

shows better results for hyperbolic sine relationship. The slope of the plot gives a stress exponent value close to that presented in Table 2. With the utilisation of these plots, the value of A in equations (1) and (2) was calculated. As a result, two constitutive equations were determined as equations (6) and (7) : {131100 : e~½exp ({7:77)s6 77 exp (6) RT : {131120 : e~½exp (35:2)½sinhð0:0015sÞ6 74 exp RT

(7)

Figure 7 has been plotted using these equations. Many researchers use this plot in order to investigate the accuracy of prediction of their constitutive equations.16–18 It is notable that R2 value is slightly higher using the hyperbolic sine law equation, indicating its better prediction. However, the results show a good agreement between the experimental and predicted stress for both of the constitutive equations. Constitutive equation presented by Lee et al.2 and used by Sheikh-Ahmad et al.6 for high strain rate deformation of 7075 aluminium alloy is in the form of equation (8) : s~s0 enem (1{bDT) (8) where s is the flow stress, e is the strain, n is the : workhardening coefficient, e is the strain rate, m is the strain rate sensitivity index, T is the temperature and s0 and b are the constants. The values of the specific material parameters for 7075 Al alloy are determined as s05510?58 MPa, n56?861022, m50?144 and b51?3561023. In order to demonstrate that equation (8) is specifically suitable for high strain rate deformation and it loses its accuracy for low strain rate deformations, Table 2 Average values of n and Q calculated from equation (1) and (2)

Average value of n Q, kJ mol21 k21

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Equation (1)

Equation (2)

6.77 131.10

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the present research conditions (strain e50?6; strain rates : (s21), 0?001,e,1; and temperature (uC), 350,T,450) were utilised to predict the true flow stress by equation (8). Subsequently, Fig. 8 was plotted. As it can be seen from this figure, the scatter of the predicted flow stress values is higher using equation (8) than those predicted using equations (6) and (7). The MSE has been used in some researches to compare the output of a neural network and the initial data.6,19,20 Mathematically, the residual for a specific predictor value is the difference between the response value y and the predicted response value y^ : Therefore, the MSE can be calculated by equation (9) MSE~1=v

v X

(yi {y^ i )2

(9)

i~1

where v indicates the number of independent pieces of information. Figure 9 shows a comparison between the MSE values calculated for equation (6)–(8). In addition, the accuracy of approximate values for different quantities can be compared with one another using the relative error calculated from equation (10), where a is the approximate value and x is the exact value;21 a can be taken as the predicted flow stress values, and x can be measured from the flow stress data a{x Relative error~ (10) x Utilising equation (10) and calculating the average of relative errors from equations (6)–(8), Fig. 10 can be plotted. Referring to Figs. 9 and 10, equation (8) is not suitable for prediction of flow stress values at low strain rates since the high strain rate deformation leads to an enhancement of strength due to the increase in workhardening and dislocation density. In fact, the true stress values predicted by equation (8) are higher than the measured values at lower strain rates. Furthermore, it is notable from Figs. 9 and 10 that equation (7) is more appropriate than equation (6) since its values of MSE and relative error are both smaller.

Conclusions In the present research, the uniaxial hot compression testing of 7075 aluminium alloy at different temperatures of 350, 400 and 450uC and strain rates of 0?001, 0?01, 0?1

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6 Variation of ln Z versus a ln s and b ln [sinh(as)]

a power law equation; b hyperbolic sine equation 7 Comparison between measured and predicted flow stress values

8 Compassion between measured and predicted flow stress values determined by equations (6)–(8)

9 Comparison of mean squared error among constitutive equations (6)–(8)


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Sharif University of Technology, Tehran, Iran, for the provision of research facilities used in the present work.

References

10 Comparison among averages of relative errors made by constitutive equations (6)–(8)

and 1 s21 has been conducted to evaluate the constitutive equation of the alloy. The following conclusions may be drawn from the results. 1. Activation energy of deformation of 7075 aluminium alloy at temperature and strain rate region of the present study was calculated as 131?1 kJ mol21 k21. 2. Constants of hyperbolic sine and power law type constitutive equations were calculated, and therefore, the equations can be rewritten and demonstrated as {131100 : 6:77 : e~½exp ({7 77)s exp RT {131120 : 6:74 : : e~½expð35 2Þ½sinhð0 0015sÞ exp RT 3. The predictions of hyperbolic sine law show better agreement with the experimental results.

Acknowledgement The authors would like to thank the research board and the Department of Materials Science and Engineering of

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