For N-H creep, creep rate is inversely proportional to d2 , whereas for Coble creep, creep rate is inversely proportional to d3 . What can be the possible explanation for d2 or d3 ?
Hi Yatindra,
Creep at high temperatures is controlled by lattice diffusion; where atoms diffuse through the lattice causing grains to elongate along the stress axis. The resulting flow is known as Nabarro-Herring creep, and its rate is inversely proportional to the square of grain size (1/d2). At lower temperatures, the creep is controlled by grain-boundary diffusion (Coble creep), i.e. atoms diffuse along the grain boundaries and hence have a stronger grain size dependence (1/d3) than the Nabarro-Herring creep.
In both the cases, it was observed that creep rate decreases with increasing grain size because both are diffusion controlled mechanisms and for both cases flow of vacancies from grain boundaries take place. For N-H: Vacancies starts from grain boundaries and flows through lattice and for coble creep vacancies travel along grain boundaries network only. Thus at low temperatures (coble creep), the dependence of creep rate on grain diameter is one order higher because Coble creep is related with grain boundary diffusion only and smaller the grain size, more is grain boundary area per unit volume.
In addition to the two earlier and very helpful replies, the attached notes might be helpful; they give the outline models of the creep mechanisms that show where the grain size dependencies come from. They're from a lecture course I gave about 20 years ago.
Consistent with the mechanistic arguments in the earlier replies one can say:
- Nabarro-Herring creep: If you multiply the grain size by an arbitrary factor p, the creep rate decreases by a factor p because of the increase of the diffusion distance. The creep rate decreases by another factor p because of the decrease of the gradient of the vacancy concentration. The only function of grain size that fulfills this restriction is a power law with the power of -2.
- Coble creep: In addition to the above, the creep rate decreases a third time by a factor of p because of the decrease of the cross section of the diffusion path. The only function of grain size that fulfills this restriction is a power law with the power of -3.
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