Introduction to Functional Analysis.

The properties of linear defects in crystals (dislocation lines) and of planar defects (stacking faults) are important in almost every branch of solid state physics. This article is intended to give a comprehensive introduction to dislocation theory for the physicist who is not a specialist in crystal plasticity. It begins with a survey of the established theory relating to the geometrical and topological properties of dislocations, the elastic theory of dislocations in a continuum, the atomistic or core properties of dislocations, and the dynamics of moving dislocations. General methods for finding the elastic field of an arbitrary dislocation loop are outlined, and the results of some recent calculations using anisotropic elasticity are summarized. These include the prediction, partially confirmed by experiment, that dislocations in certain ranges of orientation may have negative line tension. The current importance of atomistic calculations of core structure and related problems is emphasized, and the methods available for these calculations are discussed. More detailed descriptions of dislocation and stacking fault configurations are given for some of the common crystal structures, and recent work on complex defects resulting from vacancy aggregation in close-packed structures is included. The experimental and theoretical evidence for the recent conclusion that screw dislocations in body-centred cubic metals have an asymmetric core is also reviewed. The rather controversial theory of thermally activated dislocation motion is described in a separate section, and examples are given of the application of this theory to various models of the obstacles encountered by moving dislocations. The final section is concerned with the theory of dislocations in grain boundaries and interphase boundaries, and includes the concept of the surface dislocation tensor.

https://iopscience.iop.org/article/10.1088/0034-4885/33/1/307/pdf

Dislocations and stacking faults

Publisher Summary This chapter examines different aspects of nanomechanics of defects in solids. The methods by which the classical boundary-value problems of continuum mechanics can be imbued with atomistic content are reviewed. Microscopic modeling is founded on the fundamental assertion that beneath the details of observed macroscopic phenomenology, there is a set of microscopic processes which, when understood, rationalize the observed macroscopic behavior to the extent of enabling quantitative predictions. The microscopic simulation of materials is based on the evolution of degrees of freedom that are governed by the Schrodinger equation. It is found that either phenomenologically, or through explicit calculational strategies, the electronic degrees of freedom is implicitly subsumed in the effective pair potential. Once the pair potential has been identified, it is a straightforward matter to evaluate radial derivatives and the corresponding force fields. The energy associated with each distortion may be computed explicitly by recourse to direct atomistics. The contribution due to slip may be extracted by subtracting off the bulk elastic energy. As a result, the exact misfit energy is determined from atomistics. The cohesive-zone theories applied to fracture are also elaborated.

Nanomechanics of Defects in Solids

Distortion of crystals by point defects

Diffraction of waves on square lattice by semi-infinite rigid constraint

The theory of the discrete Fourier transform [1], [2] is applied in solving a system of difference equations describing the positions of atoms in a deformed crystal lattice. The crystal lattice is approximated by the Born-Karman model modified to include the internal energy of the undeformed crystal.

Some applications of the discrete fourier transform to problems of crystal lattice deformation II

Základy numerické matematiky

The paper is concerned with the numerical solution of ordinary differential equations by a new class of methods called overimplicit multistep methods. The effort is devoted to the study of the convergence and $A$-stability of the introduced methods. $A$-stable formulae of arbitrarily high orders are shown to exist in this new class. This implies the efficiency of using these methods for stiff problems.

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Overimplicit multistep methods

Numerické řešení diferenciálních rovnic

Recently, we have developed the necessary and sufficient conditions under which a rational function F(hA) approximates the semigroup of operators exp(tA) generated by an infinitesimal operator A. The present paper extends these results to an inhomogeneous equation u′(t) = Au(t) + f(t).

/pdf/approximate-solution-of-an-inhomogeneous-abstract-3khdbe4hsd.pdf

Emil Vitásek

Papers

Some applications of the discrete fourier transform to problems of crystal lattice deformation II

Základy numerické matematiky

Overimplicit multistep methods

Numerické řešení diferenciálních rovnic

Approximate solution of an inhomogeneous abstract differential equation