As has already been noted above, the theory of planar defects organically includes the mechanics of twinning, grain boundaries, Somigliani dislocations, translational dislocations, disclination, and dispiration. The fundamental propositions of the theory and methods of giving the tensor T are listed in Table 4. The mathematical formalism remains the same throughout, and it is applicable to both discrete objects (it is then necessary to conserve the δ-function apparatus), and to a continuous (then appropriate “smoothing” is needed, which usually reduces to replacement of the multiplication procedure by the normal n or by the direction Τ, to operations of finding the gradient, divergence, and curl of regular expressions, and “discarding” the δ-functional), In particular, the problem of thermoelasticity is formulated successfully by such a method in the terminology of the present theory.
In a broad sence of the word, the development of the theory should be perceived as an extension of the concept of imperfection to “defects” of sufficiently arbitrary origin. A completely developed formalism was worked out earlier for just linear defects; in the symbols used here, for the case b=b0 + Ω X (r − r0) for constant b0,Ω, and r0, and without taking account of processes on the boundary S if the linear defect contained such a feature. Let us emphasize that to describe three-dimensional “defects” occurring because of homogeneous distortion Β = Τδ (V)., it is sufficient to use the apparatus of just the theory of planar defects since the fundamental phenomena are associated with precisely the presence of boundaries and in a formal plane, with the spatial derivatives of Β, they are always expressed in terms of the functional δ (S), while in the case of finite surface gradients in terms of δ(L). The time derivatives of the distortion T, i.e.,\(\dot T\) is written down in the developed representations in terms of the form\(\dot \alpha = n \times T\delta (S), \dot \eta = (\dot \alpha \times \nabla )^s \) with all the resulting consequences.
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Likhachev, V.A. Cooperative plasticity due to the motion of disorientation and phase separation boundaries. Soviet Physics Journal 25, 541–557 (1982). https://doi.org/10.1007/BF00898749
- Regular Expression
- Spatial Derivative
- Present Theory
- Planar Defect
- Develop Representation