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strain dependence could be introduced without affecting the derivation that follows, provided it is insensitive to temperature and strain rate. The apparent activation energy associated with Eq. [1] is R01noQ - m -O(-U~ ~=

REFERENCES 1. H. Kwon, C.M. Kim, K.B. Lee, H.R. Yang, and J.H Lee: MetalL Mater. Trans. A, 1996, vol. 27A, pp. 3343-46. 2. H. Kwon: Sertpta Metall., 1989, vol. 23, pp. 1001-04. 3. R.J. Tunney and N. Ridley: Met Sci., 1979, vol. 13, pp. 585-90.

rdlnm

F. MONTHEILLET and J.J. JONAS When the rate controlling glide process is thermally activated, the high-temperature flow stress of materials is both strain rate sensitive and temperature dependent, v] In agreement with numerous observations, the following two assumptions are commonly (although sometimes only implicitly) made: (1) the flow stress o- and strain rate ~ are related by means of a power law, i.e., the rate sensitivity 0 In or I is independent of strain rate, although it m- 01n~, r

Q-

R 01nor ~

m 0-~)

=

- R 01n& [ is independent of temperO(1/T) 1,~

ature, although it may depend on strain rate or flow stress. Here, R is the gas constant and (~, T) and (o-, T) are the two different sets of independent variables associated with the first and second expressions for Q, respectively. It will be shown here that if these two assumptions are valid simultaneously, the flow rule necessarily takes two alternative forms which are derived and discussed subsequently. In the case of metal working, the strain rate ~ and the temperature T are the two independent variables. The power-law viscoplastic relationship (assumption (1)) can be written in the form or = k(T)k',cr,

[1]

where m(T) denotes a temperature-dependent rate sensitivity and k(T) is a viscosity-like parameter that also depends on T. Since only steady-state flow is addressed here, the strain e does not appear in Eq. [1]. However, a type of

F. MONTHEILLET, Directeur de Recherche au CNRS, is with Ecole des Mines (SMS), 42023 Saint-Etienne, France. J.J. JONAS, Professor, is with the Department of Metallurgical Engineering, McGill University, Montreal, PQ, Canada H3A 2A7. Manuscript submitted May 29, t996. 334~-VOLUME 27A, OCTOBER 1996

[2]

-

a

[3a]

/3

[3b]

and 1 dlnk m d(1/T)

-

where a and /3 are constants. These two coefficients are both expected to be positive, since m generally increases and k decreases with temperature. The integration of Eq. [3a] leads to m = mo exp ( - a / T )

[4]

where mo is a constant. Substitution of Eq. [4] into Eq. [3b] and integration then gives

lnk - /3mo[ 1 _

i

can be a function of temperature T; and (2) the temperature dependence of the flow stress takes the form of an Arrhenius law, i.e., the apparent activation energy

1] m_1

Thus, Q is independent of T for any ~ (assumption (2)) if and only if din m d(1/T)

Temperature Dependence of the Rate Sensitivity and Its Effect on the Activation Energy for High-Temperature Flow

dlnk

RLd(-q In k + ~ f ~

exp ( _ T) ]

[5]

where A is an integration constant. Now, instead of using the parameter/3, it