Creep constitutive behavior is generally described using the minimum creep rate or in most cases the steady-state creep rate. Norton’s law is the equation that is used for describing the dependence of stress on strain rate at a given temperature:

e = A1an,

where Aj is a constant dependent on the test temperature and n is the stress expo­nent (basically it is the reciprocal of strain rate sensitivity m, as given in Eq. (5.23)). It is important not to confuse between strain hardening exponent and stress expo­nent, which have the same symbol, n. The increase of the steady-state creep rate with test temperature (Figure 5.27a) under a given stress follows an Arrhenius equation with a characteristic activation energy for creep (Qc):

e = A2e—Qc/RT (5.36)

where A2 is a constant dependent on the applied stress/load. For creep at high tem­peratures (T> 0.4Tm), the activation energy for creep (Q;) was shown to be equal to that for self-diffusion (Qd). Thus, the temperature and stress variations of creep rate can be combined into one equation:

e = A3e-Qc/RT an, (5.37)

where A3 is a material constant and R is the gas constant (1.987 cal mol-1 K-1). It has been shown that factors affecting self-diffusivity also influenced the creep rates similarly; for example, pressure dependence, effect of C on self-diffusion and creep of у-iron, influence of ferromagnetism and crystal structure (diffusion and creep of a-iron versus у-iron), influence of crystal structure on creep and diffusion in thal­lium, effect of composition (stoichiometry) on diffusion and creep, and so on. Thus, one can state that the steady-state creep rate is proportional to DL (lattice or self-diffusion) so that the creep rate equation becomes

/a n

e = A4Dan = A5D(jJ. (5.38)

Here, A4 and A5 are constants dependent on the material and microstructure such as the grain size. Thus, the steady-state creep rate results plotted as e/D versus a/E or a/G in double-logarithmic scale yield a straight line with a slope of n (stress exponent); this is known as Sherby plot and takes care of the temperature variation of the elastic modulus (E = E0 — E(T — T0)). At very high stresses (>10—3E), however, power-law breakdown is noted with creep rates increasing

more rapidly (Figure 5.28). This creep regime is due to dislocation glide-climb with climb of edge dislocations being the rate controlling process as exhibited by pure metals (as well as ceramics) and some alloys.

However, these relationships do not show the effect of other parameters in a use­ful way. In fact, the strain rate is a function of stress, temperature, and micro­structural factors. The Bird-Mukherjee-Dorn (BMD) equation is commonly used to express the constitutive behavior during creep deformation:

where A is a material constant, d is the grain size, and p is known as the grain size exponent. It should be noted that in the above Dorn equation, the strain rate, stress, and grain size are all normalized to dimensionless parameters. The values for A, n, and p along with appropriate diffusivity {D = D0exp(—Q/RT)} in Eq. (5.39) charac­terize the underlying creep mechanisms. The underlying creep mechanisms have been dealt with in detail in Section 5.1.6.4.