Peierls-Nabarro force

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. 

Peierls, Sir Rudolf Ernst (1907-1995) helped a colleague (Orowan) solve some simple math. The result is the Peierls valley and the Peierls-Nabarro force. Peierls was also part of the British contingent of the Manhattan Project. 

Our treatment in this section will cover three primary thrusts in modeling dislocation core phenomena. Our first calculations will consider the simplest elastic models of dislocation dissociation. This will be followed by our first foray into mixed atomistic/continuum models in the form of the Peierls-Nabarro cohesive zone model. This hybrid model divides the dislocation into two parts, one of which is treated using linear elasticity and the other of which is considered in light of a continuum model of the atomic-level forces acting across the slip plane of the dislocation. Our analysis will finish with an assessment of the gains made in direct atomistic simulation of dislocation cores. 

Fig. 8.26. Schematic of key elements in Peierls-Nabarro dislocation model (courtesy of R. MiUer). The key idea is the use of a nonlinear constitutive model on the relevant slip plane which is intended to mimic the atomic-level forces resulting from slip.

Our preliminary analysis centers around the geometry shown in fig. 11.18 in which an atomically sharp crack is subjected to mode II loading and the slip distribution is assumed to occur along the prolongation of the crack plane. Borrowing from the Peierls-Nabarro analysis described in section 8.6.2, it is assumed that the atomic-level forces across the slip plane may be characterized in terms of an interplanar potential (5). This description asserts that the tractions which arise on account of the sliding discontinuity are given by r = —d

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The force opposing the motion of a dislocation is balanced by the force encouraging its motion. Thus, in a first approximation, there is no net force on the dislocation and the motive stress required is zero. Nevertheless, the small force actually required to move a dislocation has been explained by Peierls-Nabarro. Peierls-Nabarro stress , as it became known, is given as ... 

The stress necessary to move a single dislocation through an otherwise perfect crystal is known as the Peierls-Nabarro stress. Its exact calculation is difficult and needs a detailed knowledge of the molecular arrangement in the crystal and the intermolecular force law. A simplified discussion will, therefore, be given in terms of dislocations in a simple cubic crystal. First, the molecular arrangement around a dislocation in a simple cubic crystal will be considered qualitatively. 

Elastic constants are fundamentetl physical constants that are measures of the interatomic forces in materials, and are often used for the estimation of an interatomic potential that is applied in a computer simulation. They give information about the stiffness of the material and are used for understanding of mechanical properties. For example, the properties of dislocations like Peierls stress, self-energy, interaction between dislocations, etc., are explained by elastic theory. The Peierls stress rp is given by the following equation (Peieris, 1940 Nabarro, 1947) ... 

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