On the basis of his studies, Andrade1 proposed that the creep curve could be described by an equation of the form

Є = £0 (1 + , [3.1]

where e is the strain, t is the time and в and к are constants. The transient creep, that is the primary creep stage, is described by в; the constant к rep­resenting the secondary stage describes an extension per unit length that proceeds at a constant rate. Elsewhere, the dependence of creep rate on time was described by a power series5 where ё is the creep rate, a, and n, are functions of both temperature and stress. The primary stage of a normal creep curve, namely curve A, can be described by Equation [3.2] when n attains a value of 2/3. In such a case, the time dependence of creep strain (є) is described by

є- Є0 + Д1/3, [3.3]

where є0 is the instantaneous strain, в isa constant and t is time. Equation [3.3] is in accordance with the time law of creep proposed by Andrade,1 known as Andrade’s в-flow.

For steady-state creep, n = 0. The creep curve is then described by

є — £0 + £st, [3.4]

where es is the steady-state creep rate. The total creep curve consisting of the primary, secondary and tertiary region can be then described by

є — є0 + вШ + est + Yt3, [3.5]

where yt3 describes the tertiary component of the creep curve. There were several other time-creep law equations proposed to describe creep data. A major objection to the Andrade’s P-flow equation is that it predicts infinite creep-rate at the instant of loading (i. e. as t approaches 0) which is consid­ered to be unrealistic.4 Garofalo6 proposed the following equation:

є — єо + є (l- e~rt)+ ёst, [3.6]

where e, is the limit for transient creep and r is the rate of exhaustion of the transient creep that is a function of the ratio of initial creep strain rate (et^0).4 An extension of the Garofalo equation that further includes the tertiary creep regime, would be7:

є — єо + є (l — e~rt) + Gt + єьер{‘(-°,), [3.7]

where eL is a constant equal to the smallest strain deviation from steady state at the onset of tertiary creep, p is a constant and tot is the time for onset of tertiary creep.

In comparison to the normal creep equation, the logarithmic creep behav­ior is usually described by

£ = £0 + aln (+ y), [3.8]

where a and у are constants. This equation indicates that over a long period of time, the strain rate of deformation tends to become zero. Such an equa­tion, as discussed in the previous section, would be useful for describing exhaustion creep.

While Andrade’s equation is an empirical correlation borne out of his experimental observations, a number of researchers have derived similar relations on the basis of physically based mechanisms. For example, the Garofalo equation has been derived by considering sub-structural changes during deformation and by modeling the whole phenomenon as a first order reaction. Webster et at.8 and Amin et at.9 have correlated the stress and tem­perature dependence of eT, r and £s to the rate controlling mechanisms of high temperature creep using first order reaction rate concepts. They found that the Garofalo equation can be derived by assuming that the transient creep follows a first order kinetic reaction rate theory with a rate constant 1/t = K£s that depends on stress and temperature. Here 1/t is the relaxa­tion frequency which is similar to r in Equation [3.6] and т is the relaxation time for rearrangement of dislocations during transient creep controlled by dislocation climb. The physical mechanisms, for example, dislocation climb, that have been suggested to control or govern creep will be discussed in a subsequent section.

During their analyses of the creep results on Zr-based alloys, Murty10 found the following equation better describes the primary creep compared to the Garofalo equation:

Ak-t. £j 1

£ = — , A = L and B =

1 + AB £ st £ s £tr

where A is the ratio of the initial strain rate to steady-state value and B is the inverse of the extent of the primary creep. A was found to be around 10 as reported earlier by Dorn and co-workers49 for many materials that behave like pure metals and class-M alloys.

A distinctly different approach was used by incorporating anelastic11 strain in the description of the primary creep regime and especially in predicting the transients in creep strain due to sudden stress changes. i 2 Accordingly, the total strain at any given time is given by

£ = £e + £a + £ p, [310]

where the subscripts E, a and p correspond to the elastic, anelastic and plastic strains; the plastic strain (ep) is given by Equation [3.9]. The anelas­tic strain (ea) is time dependent, completely recoverable strain in contrast to the permanent plastic strain which is not recoverable and elastic strain which is recoverable but instantaneous (time-independent). The anelastic strain at the loading is given by

[3.11]

In the above equation, v ~ 7 and a* (~2.5 x 1022) are material constants in Hart’s equation of state and p is the anelastic modulus.

In the following section we discuss the importance of the different param­eters, namely stress temperature and microstructure. The strain rate of deformation, є can be expressed as

є = f (a, T, microstructure), [3.12]

where a is the applied stress and T is the test temperature.