Herring-Nabarro creep

The energy to create a vacancy under acting stress is given by  

V is the atomic volume (here, it is the volume of a vacancy) and Ep is defined by Eq. (6.35). There is a small concentration difference in the vacancies between the faces of AB and BC in the above figure, where tensile and compressive stresses are acting, respectively. Denoting the vacancy concentrations at the respective 

Clearly, in this relation Ep was replaced by Eq. (6.36). Equations (6.37) and (6.38) represent the local equilibiium concentrations under tension and compression in Fig. 3.50a. Recalling that  

As indicated, there is a flow of atoms from the tensile to the compressed faces and an opposite flow of vacancies. When a concentration gradient exists, diffusion flux will occur. This flux of vacancies may be expressed as  

For small values of stress, and since the nominator is always smaller than the denominator, the quotient is small and sinh(crV/kT) = cV/kT. Substituting this value into Eq. (6.45), one obtains  

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N1 -acylsulfanilamides, 23 508 A21-heterocyclic derivatives, 23 508 Ar -heterocyclic-Ar -acylsulfanilamides, 23 508 A21-heterocyclic sulfanilamides, 23 507—508 2V-(2-aminoethyl)-l,3-propylenediamine physical properties, 5 486t 2V-(2-aminoethyl)-piperazine (AEP), 5 485 N2 oxidation, Birkeland-Eyde process of, 27 291-292, 316. See also Dinitrogen entries Nitrogen entries N3 -P5 phosphoramidates, 27 630-631 Na+, detection in blood, 24 54. See also Sodium entries Nabarro-Herring creep, 5 626 Nacol 18, chain length and linearity, 2 10t Nacreous pigments, 7 836-837 19 412 Nacrite, 6 659... 

Figure 14-3. Nabarro-Herring creep in a grain of a polycrystalline sample which is (inhomogeneously) stressed (ct2

Let us finally mention that in polycrystalline samples, Nabarro-Herring(-Coble) creep occurs as already introduced in Section 14.3.2. The Nabarro-Herring creep rate is inversely proportional to the square of the average grain size, l2, if volume diffusion of point defects prevails. It is inversely proportional to /3 if grain boundary diffusion determines the transport. 

Mass diffusion between grain boundaries in a polycrystal can be driven by an applied shear stress. The result of the mass transfer is a high-temperature permanent (plastic) deformation called diffusional creep. If the mass flux between grain boundaries occurs via the crystalline matrix (as in Section 16.1.3), the process is called Nabarro-Herring creep. If the mass flux is along the grain boundaries themselves via triple and quadjunctions (as in Sections 16.1.1 and 16.1.2), the process is called Coble creep. 

Figure 16.5 Deformation mechanism map for Ag polycrystal a = applied stress, p = shear modulus, grain size = 32 pm, and strain rate = IGF8 s 1. The diffusional creep field is divided into two subfields the Coble creep field and the Nabarro-Herring creep field. From Ashby [20].

Fine-grained materials, when subjected to high temperatures and low applied stresses, deform by mutual accommodation of grains assisted by grain boundary sliding and transport of matter (diffusion). Under conditions where lattice diffusion dominates, the diffusional creep rate is reasonably well characterized by the Nabarro-Herring creep process. (For a review of this and other classical creep mechanisms, see Refs. 5 and 6.) Here the strain rate is expressed as... 

The creep of materials can also occur solely by diffusion, i.e., without the motion of dislocations. Consider a crystal under the action of a combination of tensile and compressive stresses, as shown in Fig. 7.4. The action of these stresses will be to respectively increase and decrease the equilibrium number of vacancies in the vicinity of the boundaries. (The boundaries are acting as sources or sinks for the vacancies.) Thus, if the temperature is high enough to allow significant vacancy diffusion, vacancies will move from boundaries under tension to those under compression. There will, of course, be a counter flow of atoms. As shown in Fig. 7.4, this mass flow gives rise to a permanent strain in the crystal. For lattice diffusion, this mechanism is known as Nabarro-Herring creep. The analysis showed that the creep rate e is given by... 

Derive an expression for the critical grain size below which Coble creep dominates and above which Nabarro Herring creep dominates (at constant temperature). 

For Nabarro-Herring creep the creep rate is given by... 

The following important points apply to Nabarro-Herring creep ... 

Nabarro, Frank Reginald Nunes (1916-2006) is another exception to our rule. He studied at Oxford and Bristol University. During World War II he worked on the explosive effect of shells and was made a member of the Order of the British Empire (OBE). In 1953 he became head of the physics department at the University of the Witwatersrand in South Africa. He is perhaps best known for Nabarro-Herring creep and the Peierls-Nabarro force. 

The densification by plastic deformation and power-law creep is, in principle, independent of particle (grain) size. In the case of diffusion (both lattice and grain boundary), on the other hand, densification depends on not only the effective pressure but also the grain size. The densification by diffusion under an external pressure is similar to diffusional creep Nabarro-Herring creep due to lattice diffusion, and Coble creep due to grain boundary diffusion. The dependency of densification on grain size is the same as that of diffusional creep. 

Ashby, M. F., On interface-reaction control of Nabarro-Herring creep and sintering, Scripta MetalL, 3, 837—42, 1969. 

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