Zener-Hollomon parameter approach

As discussed in section 2.5.1, the constitutive equations relating ^ε• to σ and T are:



 



−

=

•

RT A₁σⁿ¹exp Q

ε (2.16)



 



−

=

•

RT A₂exp(βσ)exp Q

ε (2.17)



 



−

=

•

RT A₃[sinh(ασ)]ⁿexp Q

ε (2.18)

As pointed out earlier, though σ is generally taken as the peak flow stress (σp) [MCQU2002, MEDI1996], in a few instances the steady state flow stress (σs) has also been used. The term α is the stress multiplier. n1, β, n, A1, A2 and, A3 are material constants. The hyperbolic sine law (Eq. 2.18) has been found to be the most suitable form for explaining the high temperature deformation behavior of metallic alloys [CHEN2008, LIAN2008, MCQU2002, MEDI1996, SHAO2008, ZHAN2007]. In the present study, peak stress σp

was used in the place of σ in the above expressions.

0 10 20 30 40 50 60 70 80 90

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) _300°C

350°C 400°C

450°C 500°C

(a)

0 20 40 60 80 100 120 140

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) _300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120 140

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C

400°C 450°C 500°C

(c)

0 20 40 60 80 100 120 140 160 180

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C 500°C

(d)

Fig.4.37. Flow curves of Alloy-A at ^ε• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

0 10 20 30 40 50 60 70

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) _300°C

350°C 400°C 450°C

500°C

(a)

0 20 40 60 80 100 120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C

400°C 450°C

500°C

(c)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C 400°C

450°C 500°C

(d)

Fig.4.38. Flow curves of Alloy-B at ε^• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

0 10 20 30 40 50 60 70 80 90

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C

450°C 500°C

(a)

0 20 40 60 80 100 120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C 500°C

(c)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C 400°C

450°C 500°C

(d)

Fig.4.39. Flow curves of Alloy-C at ε^• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

0 20 40 60 80 100 120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C 400°C 450°C 500°C

(a)

0 20 40 60 80 100 120 140

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) 300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) 300°C

350°C 400°C 450°C 500°C

(c)

0 20 40 60 80 100 120 140 160 180

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) 300°C

350°C 400°C 450°C 500°C

(d)

Fig.4.40. Flow curves of Alloy-D at ^ε• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

0 20 40 60 80 100 120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa)

300°C 350°C

400°C 450°C

500°C

(a)

0 20 40 60 80 100 120 140 160 180

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) 300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C 500°C

(c)

0 20 40 60 80 100 120 140 160 180 200

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C

450°C 500°C

(d)

Fig.4.41. Flow curves of Alloy-E at ^ε• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

0 10 20 30 40 50 60 70 80 90 100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) _300°C

350°C 400°C 450°C 500°C

(a)

0 20 40 60 80 100 120 140

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) 300°C

350°C 400°C 450°C 500°C

(b)

0 20 40 60 80 100 120 140 160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C

500°C

(c)

0 20 40 60 80 100 120 140 160 180 200

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True Strain

True Stress (MPa) ^300°C

350°C 400°C 450°C 500°C

(d)

Fig.4.42. Flow curves of Alloy-F at ε^• of (a) 0.001 s^-1 (b) 0.01 s^-1 (c) 0.1 s^-1, and (d) 1.0 s^-1

Table 4.12. Peak stress values of the Al–Cu–Mg alloys at different deformation conditions

ε_{& (s}^-1₎ Temperature

(°C)

Peak flow stress, σp (MPa)

Alloy-A Alloy-B Alloy-C Alloy-D Alloy-E Alloy-F

0.001

300 77.83 62.23 76.35 100.26 109.62 94.98

350 58.86 48.63 57.67 71.92 81.27 72.22

400 32.28 38.04 37.65 50.13 69.88 49.70

450 26.95 26.71 32.45 35.29 47.64 46.03

500 17.61 22.6 22.53 25.20 36.12 29.62

0.01

300 122.9 113.43 108.65 126.06 169.53 119.29

350 88.40 77.89 69.24 87.53 107.31 107.41

400 62.79 56.69 53.51 68.59 82.84 64.68

450 37.02 37.54 40.29 50.77 61.65 62.78

500 22.29 30.10 35.10 37.12 43.78 41.49

0.1

300 131.47 110.04 141.88 146.75 139.29 135.11

350 122.94 102.73 92.03 110.23 111.86 102.20

400 66.63 83.93 68.39 83.46 88.17 94.77

450 49.93 66.13 53.10 61.51 85.25 62.00

500 38.99 50.16 42.66 46.50 60.61 49.62

1.0

300 153.82 146.61 147.79 160.21 190.41 175.22

350 116.93 115.12 116.65 124.97 145.22 133.15

400 96.85 104.65 96.38 100.65 105.83 106.02

450 69.51 78.75 79.38 78.51 94.23 76.80

500 58.89 56.53 60.89 61.56 69.90 71.58

Determination of the value of stress multiplier α is very important for this analysis.

No single solution exists for α and its value for Al alloys found in the literature varies from 0.01 to 0.08. However, α = 0.052 has been widely used by researchers [MCQU2002, SPIG2003, BARD2003]. The value of α can also be defined as α ≈ β / n1 [LIAN2008, CHEN2008], where β and n1 are taken as the average values of the slopes of the ln(^ε

•

) vs. σ plots and ln(ε

•

) vs. ln(σ) plots, respectively, in the range of T studied. Figure 4.43 shows the corresponding plots for Alloy-A and Alloy-D. The values of β and n1 were evaluated for all the investigated alloys by this method. The values of α determined were 0.016, 0.014,

0.016, 0.013, 0.012 and 0.013 for Alloy-A, Alloy-B, Alloy-C, Alloy-D, Alloy-E and Alloy- F, respectively. α merely facilitates mathematical fitting procedure. Appropriate value of α ensures linear and parallel fits for the ln(^ε• ) vs. ln[sinh(ασ)] data [BARD2003, BJØR2001, CAVA2002, CERR2007, GHOL2009, MEDI1996, MCQU2002, QING2009, SPIG2003].

The plots of ln (ε

• ) vs. ln[sinh(ασ)] for all the alloys studied are shown in Figure 4.44 to Figure 4.49. α = 0.01 yielded the best goodness of fit (R²) value for the data. Hence in the present analysis, the optimal value of α = 0.01 was used for all the investigated alloys.

-8 -7 -6 -5 -4 -3 -2 -1 0 1

0 50 100 150 200

Peak flow stress, σ_p (MPa)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-8 -7 -6 -5 -4 -3 -2 -1 0 1

2 3 ln (σ4_p) 5 6

ln ( )

573 K 623 K 673 K 723 K 773 K

(b)

-8 -7 -6 -5 -4 -3 -2 -1 0 1

0 50 100 150 200

Peak flow stress, σ_p (MPa)

ln ( )

573 K 623 K 673 K 723 K 773 K

(c)

-8 -7 -6 -5 -4 -3 -2 -1 0 1

2 3 ln (σ4_p) 5 6

ln ( )

573 K 623 K 673 K 723 K 773 K

(d)

Fig.4.43. Plots of (a) ln ^ε• vs. σp for Alloy-A (b) ln ^ε• vs. ln σp for Alloy-A (c) ln ^ε• vs. σp for Alloy-D, and (d) ln ^ε• vs. ln σp for Alloy-D

The activation energy (Q) for high temperature deformation can be obtained from a plot of ln(ε

• ) vs. 1/T at constant [sinh(ασ)] value [GEOR1988]. The expression for Q therefore becomes:

( )

sinh

ln Q R 1

T

_ασ

ε



•



 ∂ 

 

= −    ∂         

(4.9)

Eqn. 4.9 can also be expressed by the following relationship, which is considered as the general equation for evaluating the activation energy term when the hyperbolic-sine constitutive modeling is being used [BARD2003, BJØR2001, CAVA2002, CERR2007, CHEN2008, GHOL2009, LIAN2008, SPIG2003]:

( )

[ ]

( )

[ ]

•



















 



∂

∂













∂

= ∂

•

ε

ασ ασ

ε

T R

Q

T

1 sinh ln sinh

ln

ln (2.22)

i.e. Q = R n S where n is the mean slope of ln(ε

•

) vs. ln[sinh(ασ)] plots at different T and S the mean slope of the ln[sinh(ασ)] vs. 1/T plots at various ^ε• . These plots for all the investigated Al–Cu–Mg alloys are shown in Figure 4.44 to Figure 4.49. Table 4.13 presents the values of n, S and Q obtained for all the alloys studied. Figure 4.50 plots the variation of Q with Sn wt.% in the investigated alloy system. Q value of 183.4 kJ/mole, obtained for Alloy-A, is in general agreement with the values reported for other Al alloys [BARD2003, CAVA2002, EVAN1990, MCQU2002]. Q increased to 223.3 kJ/mol with the addition of 0.06 wt.% Sn but decreased when the Sn content was increased beyond 0.08 wt.%.

The Zener-Hollomon parameter (Z) can be expressed as

( )

[ ]

ⁿ

RT A

Z ε_exp Q _sinhασ

= 3



 



= 

•

(2.21)

Taking logarithm of Eq. (2.21), one gets

ln(Z) = ln (A3) + n ln[sinh(ασ)] (4.10) from which A3 and n can be obtained as mentioned in section 2.5.1. Figure 4.51 shows the ln(Z) vs. ln[sinh(ασ)] plots, in which the solid lines represent linear fit to the data. The values of ln(Z) at various ^ε• and deformation T are presented in Table 4.14 to Table 4.19 for

all the alloys studied. It is evident that Z increases with increase in ^ε

•

and decrease in T.

Dynamic recrystallization is generally favoured at low ^ε• and high T, i.e, at low Z values [QING2009].

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-2 -1.5 -1 -0.5 0 0.5 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b)

Fig.4.44. Plots of (a) ln (sinh ασp) vs. ln ^ε• and (b) ln (sinh ασp) vs. 1000/T for Alloy-A

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-2 -1.5 -1 -0.5 0 0.5 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b) Fig.4.45. Plots of (a) ln (sinh ασp) vs. ln ^ε

•

and (b) ln (sinh ασp) vs. 1000/T for Alloy-B

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-2 -1.5 -1 -0.5 0 0.5 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b) Fig.4.46. Plots of (a) ln (sinh ασp) vs. ln ^ε

•

and (b) ln (sinh ασp) vs. 1000/T for Alloy-C

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-2 -1.5 -1 -0.5 0 0.5 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b) Fig.4.47. Plots of (a) ln (sinh ασp) vs. ln ^ε

•

and (b) ln (sinh ασp) vs. 1000/T for Alloy-D

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-1.5 -1 -0.5 0 0.5 1 1.5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b) Fig.4.48. Plots of (a) ln (sinh ασp) vs. ln ^ε

•

and (b) ln (sinh ασp) vs. 1000/T for Alloy-E

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-2 -1 0 1 2

ln (Sinh ασ_p)

ln ( )

573 K 623 K 673 K 723 K 773 K

(a)

-1.5 -1 -0.5 0 0.5 1 1.5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

1000/T (K^-1) ln (Sinh ασp)

0.001 s-1 0.01 s-1 0.1 s-1 1.0 s-1

(b)

Fig.4.49. Plots of (a) ln (sinh ασp) vs. ln ^ε• and (b) ln (sinh ασp) vs. 1000/T for Alloy-F

Table 4.13. Values of the constants α, n, S, A3 and Q corresponding to the Al–Cu–Mg alloys Sample ID α (MPa^-1) n S A3 (s^-1) Q (kJ/mole)

Alloy-A 0.01 6.24 3.53 7.83x10¹³ 183.4

Alloy-B 0.01 6.77 2.86 1.28x10¹² 161.2

Alloy-C 0.01 6.97 2.87 4.16x10¹² 166.3

Alloy-D 0.01 8.55 3.14 5.37x10¹⁶ 223.3

Alloy-E 0.01 9.09 2.98 1.25x10¹⁶ 225.5

Alloy-F 0.01 8.4 2.8 1.96x10¹⁴ 195.5

100 125 150 175 200 225 250

0 0.02 0.04 0.06 0.08 0.1 0.12

Sn wt. %

Activation Energy, Q (kJ/mole)

Fig.4.50. Variation of activation energy of deformation with Sn content in Al–Cu–Mg alloys

Table 4.14. Values of ln (Z) for Alloy-A at various deformation conditions ε

•

(s^-1) T (K)

573 623 673 723 773

0.001 31.59 28.5 25.87 23.6 21.63

0.01 33.89 30.8 28.17 25.9 23.93

0.1 36.19 33.1 30.47 28.2 26.23

1.0 38.49 35.4 32.77 30.51 28.53

Table 4.15. Values of ln (Z) for Alloy-B at various deformation conditions ε

•

(s^-1)

T (K)

573 623 673 723 773

0.001 26.93 24.21 21.90 19.91 18.17

0.01 29.23 26.51 24.20 22.21 20.47

0.1 31.53 28.82 26.50 24.51 22.78

1.0 33.83 31.12 28.81 26.81 25.08

Table 4.16. Values of ln (Z) for Alloy-C at various deformation conditions ε

•

(s^-1) T (K)

573 623 673 723 773

0.001 28.01 25.20 22.82 20.76 18.97

0.01 30.31 27.51 25.12 23.06 21.28

0.1 32.61 29.81 27.42 25.37 23.58

1.0 34.91 32.11 29.73 27.67 25.88

Table 4.17. Values of ln (Z) for Alloy-D at various deformation conditions ε

•

(s^-1)

T (K)

573 623 673 723 773

0.001 39.97 36.20 33.00 30.24 27.84

0.01 42.27 38.51 35.30 32.54 30.14

0.1 44.57 40.81 37.61 34.85 32.44

1.0 46.87 43.11 39.91 37.15 34.75

Table 4.18. Values of ln (Z) for Alloy-E at various deformation conditions ε

•

(s^-1)

T (K)

573 623 673 723 773

0.001 40.43 36.63 33.40 30.61 28.18

0.01 42.74 38.94 35.70 32.91 30.49

0.1 45.04 41.24 38.00 35.22 32.79

1.0 47.34 43.54 40.31 37.52 35.09

Table 4.19. Values of ln (Z) for Alloy-F at various deformation conditions ε

•

(s^-1) T (K)

573 623 673 723 773

0.001 34.14 30.84 28.04 25.62 23.52

0.01 36.44 33.15 30.34 27.92 25.82

0.1 38.74 35.45 32.64 30.23 28.12

1.0 41.04 37.75 34.95 32.53 30.42

R² = 0.9629

15 20 25 30 35 40 45 50

-2 -1 0 1

ln (sinh ασ_p)

lnZ

(a)

R² = 0.9584

15 20 25 30 35 40 45 50

-2 -1 0 1

ln (sinh ασ_p)

lnZ

(b)

R² = 0.9863

15 20 25 30 35 40 45 50

-2 -1 0 1

ln (sinh ασ_p)

lnZ

(c)

R² = 0.9928

15 20 25 30 35 40 45 50

-2 -1 0 1 2

ln (sinh ασ_p)

lnZ

(d)

R² = 0.9522

15 20 25 30 35 40 45 50

-2 -1 0 1 2

ln (sinh ασ_p)

lnZ

(e)

R² = 0.9661

15 20 25 30 35 40 45 50

-2 -1 0 1 2

ln (sinh ασ_p)

lnZ

(f)

Fig.4.51. Plots of ln (sinh ασp) vs. ln Z for (a) Alloy-A (b) Alloy-B (c) Alloy-C (d) Alloy-D (e) Alloy-E, and (f) Alloy-F