s ¼ α ln4 A

þ A þ 1 5 ð14Þ

In terms of the Zener–Hollomon parameter Z, the constitutive equation of 7A09 aluminum alloy is expressed by

zones of the good workability or the zones of the unstable flow。 Therefore,  it  is  necessary  to  obtain  the  instability  regimes  of   the

s ¼ 1=αlnfðZ=AÞ1=n  þ ½ðZ=A  2=n

1=2

þ  ] g

0:2038

7   0:4076

1=2

plastic flow。 Ziegler [32] firstly put forward the instability criterion of

plastic   flow   by   applying   the   extremum   principle   of  irreversible

¼ 80:645lnf½Z=ð1:48 ~ 107 Þ]

þ f½Z=ð1:48 ~ 10  Þ]

thermodynamics  to the plastic  flow of large  strain。  Consequently,

ð15Þ

The constitutive equation can be used to describe the flow behavior of 7A09 aluminum  alloy  during  hot  deformation based on dynamic recovery and dynamic recrystallization。 The constitu- tive equation of 7A09 aluminum alloy can be determined as the material model during finite element simulation of isothermal precision forging of aluminum alloy rotating    disk。

5。Hot processing map

the stable flow will occur if the differential quotient satisfies the following inequality:

dD      D

dR  4 R ð24Þ

where R is the function of the strain rate, and D is the dissipation function at the given temperature。 According to the dynamic material model, D is equal to J。 Therefore, the flow instability criterion can be obtained  as  follows  [33]:

∂ln。m=ðm þ 1Þ。

5。1。Fundamentals  of  hot  processing map

ξðε_Þ¼ 

∂lnðε_Þ 

þ m o0 ð25Þ

According to dynamic material modeling, the power dissipated during plastic working of the metal material can be described by the following formulation:

P ¼ G þ J ð16Þ

where G is the dissipated content expressed by

Z  ε_

where the parameter ξðε_Þ may be viewed as a function of tempera-

ture and strain rate to obtain an instability processing map。 Accord- ing to the instability processing map, metallurgical instability during plastic flow occurs in the regimes where ξðε_Þ is negative。 If the internal entropy production rate in the material system is less than the strain rate imposed on the  system,  the  flow  becomes  localized and causes flow instability。 The hot processing map can be obtained by   combining   the   power   dissipation   map   and   the     instability

processing map。

and  J  is  the  dissipated co-content expressed by

Z s

J ¼ ε_ds ð18Þ

0

It can be generally accepted that the flow behavior of metal materials is strongly dependent on the strain rate and the temperature。 The general form of the flow behavior at the given strain and temperature is expressed  as