Nabarro–Herring creep

Nabarro-Herring creep is a mode of deformation of crystalline materials that occurs at low stresses and held at elevated temperatures in fine-grained materials. In Nabarro-Herring creep, atoms diffuse through the crystals, and the creep rate varies inversely with the square of the grain size so fine-grained materials creep faster than coarser-grained ones. This type of creep results from the diffusion of vacancies from regions of high chemical potential at grain boundaries subjected to normal tensile stresses to regions of lower chemical potential where the average tensile stresses across the grain boundaries are zero. Self-diffusion within the grains of a polycrystalline solid can cause the solid to yield to an applied shearing stress, the yielding being caused by a diffusional flow of matter within each crystal grain away from boundaries where there is a normal pressure and toward those where there is a normal tension. Atoms migrating in the opposite direction accounting for the creep strain. When volume diffusion controls the tensile creep rate ${\dot {\epsilon }}$ is given by:

${\dot {\epsilon }}={A_{n}}{\frac {Db^{3}\sigma }{d^{2}kT}}$

For a general expression of creep rate, the comparison between Nabarro-Herring and Coble creep can be presented as follows:

${\dot {\epsilon }}={\frac {ADGb}{kT}}{({\frac {\sigma }{G}})}^{n}{({\frac {b}{d}})}^{p}$

Mechanism	favorable conditions	Description	A	n	p
Nabarro-Herring Creep	High temperature, low stress and small grain size	Vacancy diffusion through the crystal lattice	10-15	1	2
Coble Creep	low stress, fine grain sizes and temperature less than those for which NH creep dominates	Vacancy diffusion along grain boundaries	30-50	1	3

Where $d$ is the mean grain diameter and $k$ the Boltzmann's constant, G the Shear Modulus and $T$ the absolute temperature. The diffusivity is obtained form the tracer diffusivity, $D^{*}$. The dimensionless constant $A_{n}$ depends intensively on the geometry of grains. The parameters $A$, $n$ and $p$ are dependent on creep mechanisms. Nabbaro-Herring creep does not involve the motion of dislocations. It predominates over high temperature dislocation-dependent mechanisms only at low stresses, and then only for fine-grained materials. Nabarro-Herring creep is characterized by creep rates that increase linearly with the stress and inversely with the square of grain diameter. In contrast, in Coble creep atoms diffuse along grain boundaries and the creep rate varies inversely with the cube of the grain size. Lower temperatures favor Cobol creep and higher temperatures favor Herring-Nabarro creep.