Fig。 8。  The relationship of the steady stress with respect to the temperature and the strain rate, respectively: (a) the relationship between lns and lnε_; (b) the relationship between s and lnε_; (c) the relationship between ln½sinhðαsÞ] and lnε_  and (d) the relationship between ln½sinhðαsÞ] and T−1。

combining  the  values  of  n  and  β,   namely   α ¼ β=n ¼   1:24~ 10−2 MPa−1 。

To obtain the value of Q , the natural logarithm of Eq。 (1) results in  the  following equation:

lnε_ ¼ ln A þ nln。sinhðαsÞ。− RT ð6Þ

Based on Eq。 (6), the value of n is modified as the linear proportion factor of lnε_ with respect to ln½sinhðαsÞ]。 The value of n is determined as 4。906 according to the average value of slope of the lines at all the different deformation temperatures as shown in Fig。 8(c)。

For the given strain rates, differentiating T −1 in Eq。 (6) results in

Eq。 (7)

。 ∂ln½sinhðαsÞ 。

Fig。 9。  The relationship between ln½sinhðαsÞ] and lnZ。

Q ¼ nR

∂T −1 ε_

ð7Þ

follows:

7

4:906 3

The  value  of  Q   can  be  calculated  as  101。3 ~ 103  J mol−1 by

ε_ ¼ 1:48 ~ 10  ½sinhð0:0124sÞ]

~ expð−101:3 ~ 10  =RT Þ ð10Þ

combining the values of n and R with the average value of slope of all the lines at the different strain rates as shown in Fig。 8(d)。

In general, the Zener–Hollomon parameter Z can be used to describe the comprehensive influence of the strain rate and the temperature on the flow stress of the metal materials during hot

The substitution of Eq。 (10) into Eq。 (8) leads    to

Z ¼ ε_expðQ =RT Þ¼ 1:48 ~ 107 ½sinhð0:0124sÞ]4:906 ð11Þ

According to Eq。 (8), the following formulation can be obtained:

。Z。1=n

deformation [29]。

n

sinhðαsÞ¼ A

ð12Þ

Z ¼ ε_expðQ =RT Þ¼ A½sinhðαsÞ]

The natural logarithm of Eq。 (8) results  in

According to the definition of the hyperbolic sinhðαsÞ, Eq。 (12) can  be  transformed  into the  following expression:

ln Z ¼ ln A þ nln½sinhðαsÞ] ð9Þ

According to Eq。 (9), the value of lnA is the intercept of the

expðαsÞ−1 ¼ 0 ð13Þ

fitting line of lnZ with respect to ln½sinhðαsÞ] in the lnZ coordinate axis as shown in Fig。 9, so the value of A is further determined as

Eq。 (13) can be further expressed as follows:

1:48 ~ 107 s−1 。

1    2。Z。1=n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3

。Z。

By substituting the values of A, α, nand Q into Eq。 (1), the constitutive  equation  of  7A09  aluminum  alloy  is  expressed    as