Continuous Distributions of Dislocations
As shown in Sect. 17, each single dislocation may be considered as a state of self-stress in the classical elasticity theory. Sometimes, however, we are more interested to know the mean values of the strains and stresses produced in a crystal by a large n
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IV
CONTINUOUS DISTRIBUTIONS OF DISLOCATIONS
17. Elastostatics of continuous distributions of dislocations As shown in Sect. 17, each single dislocation may be considered as a state of selfstress in the classical elasticity theory. Sometimes, however, we are more interested to know the mean values of the strains and stresses produced in a crystal by a large number of dislocations. It is then convenient to consider the limiting case of a continuous distribution of dislocations, for which the number of dislocations tends to infinity, while the Burgers vector of each tends to zero, in such a way that the product remains finite in any bounded region. In performing this limiting process it is natural to assume that the product of the number of dislocations and of their individual core volumes tends to zero. In order that the theory of elasticity be applicable to continuous distributions of dislocations it is necessary that the mean strain produced by dislocations be macroscopically continuous, and this may happen only when each macroscopic volume element contains a large number of dislocations. A rough evaluation shows that this is really the case for many situations of practical interest. Indeed, a cube with the side of I mm may usually be considered as sufficiently small with respect to the size of the body and the characteristic wave lengths of its elastic state, and thus may be taken as macroscopic volume element. On the other hand, the total length of the dislocation lines amounts, even in good annealed metals, to 103-104 mm/mm3 and increases during deformation to 108 _1010 mm/mm3. This shows that the real dislocation density is generally sufficiently high to assure a continuous variation of the mean value of the deformation produced by dislocations from one volume element to another. It should be mentioned that the theory of continuous distributions of dislocations may be also used to describe other continuously distributed non-mechanical sources of self-stress, e.g. inhomogeneous thermal or magnetic fields 1, which are sometimes called quasi-dislocations. Furthermore, by using distributions associated with lines or surfaces, it is also possible to use the concepts and methods of the theory of continuous distributions of dislocations to study surface distributions of dislocations or even single dislocations (see, e.g. Kroner [190], Kunin [204], Teodosiu [335]). 1
See Rieder [465], Kroner [192], Anthony [5].
C. Teodosiu, Elastic Models of Crystal Defects © Springer-Verlag Berlin Heidelberg 1982
IV. Continuous distributions of dislocations
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In what follows we shall briefly present the elastostatics of the continuous distributions of dislocations. Consider an elastic body fJI which occupies a region "Y in a self-stressed state produced by a continuous distribution of dislocations; let (k) be the configuration of the body in this state. In this case there exists no global natural configuration, i.e. a stress-free configuration of the whole body. Let N(X) denote a material neighbourhood of a material point X.
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