Peierls-Nabarro stress

These data show clearly that that the intrinsic behavior in pure metals is visco-elastic with the velocity proportional to the applied stress (Newtonian viscosity). Although there is a large literature that speaks of a quasi-static Peierls-Nabarro stress, this is a fiction, probably resulting from studying of insufficiently pure metals. 

Some cubic materials, e.g., TiC and MgAl204, do have enough independent slip systems, but the Peierls-Nabarro stress is high making dislocations immobile except at high temperature. 

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 width of the dislocation, Eq. (3.23), defines the magnitude of the Peierls-Nabarro stress and is a measure of the degree of distortion that has occurred due to dislocation. It is basically the distance over which the dislocation causes disreg-istry thus, it is the magnitude of the displacement of the atoms from their perfect-crystal positions. As indicated in Sect. 3.3.2, when w is several atomic spacings, a dislocation is considered to be wide if it is on the order of one or two atomic spacings, it is narrow. 

At the level of the above discussion, the critical thickness condition G = 0 is based on elastic continuum concepts only. This overlooks a variety of effects that may bear on the process in some cases. Among these are the glide resistance due to the Peierls-Nabarro stress of the material (Matthews 1975), the role of surface or interface energy (Cammarata and... 

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. 

One of the key features that emerges from this solution, and a crucial hint for the type of analyses we will aim for in the future, is the fact that the stresses implied by the Peierls-Nabarro solution do not suffer from the same singularities that plague the linear elastic solution. A detailed examination of this point may be found in Hirth and Lothe (1992) and is illustrated in their eqn (8-13). By introducing an element of constitutive realism in the form of a nonlinear (and in fact nonconvex) interplanar potential, the solution is seen to be well behaved. 

The Peierls-Nabarro model has been used to determine properties of dislocation cores, the misfit energy and particularly changes with pressure. This is based on the assumption of a planar core which is the most able to ghde. It has direct implications for slip systems. In order to move, a dislocation must overcome an energy barrier under an applied stress. The Peierls-Nabarro model has been used to constrain dislocation core sizes and Peierls stresses in several oxides and sihcates relevant to the Earth s mantle, particularly periclase [439], ohvine [440,441], ringwoodite [80],... 

The magnitude of the shear stress needed to move a dislocation along the plane was first determined by Peierls [162] and Nabarro [163]. For the orthorhombic unit cell of a crystal it varies exponentially with the ratio of both miit cell axes perpendicular to the macromolecular chmu direction. For a (100)... 

Having introduced an energy variation as a function of the dislocation position, which leads to an energy barrier to the motion of the dislocation, we can obtain the stress required to move the dislocation without any thermal activation. This stress can be defined as the maximum slope of the variation in the energy as a function of the translation. Using this definition, Peierls and Nabarro showed that the shear stress for dislocation motion, the so called Peierls stress op, is given by... 

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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