Introduction to Elasticity Theory for Crystal Defects

Abstract

Self-sufficient and user-friendly, this book provides a complete introduction to the anisotropic elasticity theory necessary to model a wide range of crystal defects. Assuming little prior knowledge of the subject, the reader is first walked through the required basic mathematical techniques and methods. This is followed by treatments of point, line, planar and volume type defects such as vacancies, dislocations, grain boundaries, inhomogeneities and inclusions. Included are analyses of their elastic fields, interactions with imposed stresses and image stresses, and interactions with other defects, all employing the basic methods introduced earlier. This step by step approach, aided by numerous exercises with solutions provided, strengthens the reader's understanding of the principles involved, extending it well beyond the immediate scope of the book. As the first comprehensive review of anisotropic elasticity theory for crystal defects, this text is ideal for both graduate students and professional researchers.

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

“…Conclusion: Summarising the estimates for difference operators applied to g (1) , g (2) ,g (3) and choosingg = g (1) + g (2) − g (3) we obtain both of the decay estimates claimed in parts …”

“…This motivates the formulation of a Galerkin-type approximation scheme for (1) (see Sections 2.3 and 3.4 This is a finite dimensional optimisation problem with dim(Ẇ 1,2 N ) ≈ N , and our framework yields a straightforward proof of: supposeū is a strongly stable (cf. (9)) solution to (1) then, for N sufficiently large, there exists a solutionū N to (3) such that…”

“…Note that N is directly related to the computational cost of solving (3). It is interesting to note that the rate N −1/2 is generic; that is, it is independent of any details of the particular defect.…”

“…One can also formulate anti-plane models for pure screw dislocations by restricting A to displacements of the form u = (0, 0, u 3 ) and also computing a predictor of the form u lin = (0, 0, (u lin ) 3 ). Note also that for pure screw dislocations, (33) is ignored.…”

“…Since practical schemes necessarily work in small computational domains (for example, "supercells") they cannot explicitly resolve these fields but must employ artificial boundary conditions (periodic boundary conditions appear to be the most common). To assess the accuracy and in particular the cell size effects of such simulations, numerous formal results, numerical explorations, or results for linearised problems can be found in the literature; see for example [3,8,17,27] and references therein for a small representative sample.…”