Coble creep

Coble creep, a diffusive mechanism, is driven by a vacancy (or mass) concentration gradient. The change in vacancy concentration ${\displaystyle \Delta C}$ from its equilibrium value ${\displaystyle C_{0}}$ is given by

${\displaystyle \Delta C=C_{0}{\frac {\sigma \Omega }{kT}}}$

This can be seen by noting that ${\displaystyle \Delta C\propto C_{0}e^{\sigma \Omega /kT}-C_{0}e^{-\sigma \Omega /kT}}$ and taking a high temperature expansion, where the first term on the right hand side is the vacancy concentration from tensile stress and the second term is the concentration due to compressive stress. This change in concentration occurs perpendicular to the applied stress axis, while parallel to the stress there is no change in vacancy concentration (due to the resolved stress and work being zero).[2]

We continue by assuming a spherical grain, to be consistent with the derivation for Nabarro–Herring creep; however, we will absorb geometric constants into a proportionality constant ${\displaystyle A}$. If we consider the vacancy concentration across the grain under an applied tensile stress, then we note that there is a larger vacancy concentration at the equator (perpendicular to the applied stress) than at the poles (parallel to the applied stress). Therefore, a vacancy flux exists between the poles and equator of the grain. The vacancy flux is given by Fick's first law at the boundary: the diffusion coefficient ${\displaystyle D_{v}}$ times the gradient of vacancy concentration. For the gradient, we take the average value given by ${\displaystyle \Delta C/(\pi R/2)}$ where we've divided the total concentration difference by the arclength between equator and pole then multiply by the boundary width ${\displaystyle \delta '}$ and length ${\displaystyle 2\pi R}$.

${\displaystyle J_{v}=A'D_{v}{\frac {\Delta C}{\pi R/2}}\delta '(2\pi R)}$

where ${\displaystyle A'}$ is a proportionality constant. From here, we note that the volume change ${\displaystyle \pi R^{2}dR/dt}$ due to a flux of vacancies diffusing from a source of area ${\displaystyle \pi R^{2}}$ is the vacancy flux ${\displaystyle J_{v}}$ times atomic volume ${\displaystyle \Omega }$:

${\displaystyle J_{v}\Omega =\pi R^{2}dR/dt=\pi R^{3}{\dot {\epsilon }}}$

Where the second equality follows from the definition of strain rate: ${\displaystyle {\dot {\epsilon }}=(1/R)(dR/dt)}$. From here we can read off the strain rate:

${\displaystyle {\dot {\epsilon }}_{C}={\frac {J_{v}\Omega }{\pi R^{3}}}=AD_{v}\Delta C\Omega {\frac {\delta '}{R^{3}}}=AD_{GB}{\frac {\sigma \Omega }{kT}}{\frac {\delta '}{R^{3}}}}$

Where ${\displaystyle A'\rightarrow A}$ has absorbed constants and the vacancy diffusivity through the grain boundary ${\displaystyle D_{GB}=D_{v}C_{0}\Omega }$.

Since Coble creep involves mass transport along grain boundaries, cracks or voids would form within the material without proper accommodation. Grain boundary sliding is the process by which grains move to prevent separation at grain boundaries.[1] This process typically occurs on timescales significantly faster than that of mass diffusion (an order of magnitude quicker). Because of this, the rate of grain boundary sliding is typically irrelevant to determining material processes. However, certain grain boundaries, such as coherent boundaries or where structural features inhibit grain boundary movement, can slow down the rate of grain boundary sliding to the point where it needs to be taken into consideration. The processes underlying grain boundary sliding are the same as those causing diffusional creep[1]

This mechanism is originally proposed by Ashby and Verrall in 1973 as a grain switching creep.[5] This is competitive with Coble creep; however, grain switching will dominate at large stresses while Coble creep dominates at low stresses.

This model predicts a strain rate with the threshold strain for grain switching ${\displaystyle 0.72\gamma /d}$. [1]

${\displaystyle {\dot {\epsilon }}_{GS}\propto {\frac {\Omega }{kT}}{\frac {\delta 'D_{GB}}{d^{3}}}(\sigma -{\frac {0.72\gamma }{d}})}$

The relation to Coble creep is clear by looking at the first term which is dependent on grain boundary thickness ${\displaystyle \delta '}$ and inverse grain size cubed ${\displaystyle d^{-3}}$.