Elastic properties of dislocations

In this chapter, elastic constants of single crystals are first summarized, and the anisotropy of elastic properties of single crystals is discussed using Young s modulus, Poisson s ratio, etc. Then, the elastic properties of dislocations are discussed using elastic constants of single crystals, followed by a summary of the elastic moduli of polycrystalline intermetallics. 

Anisotropy of the elastic constants results in a change of elastic properties of dislocations in intermetallics. We have textbooks for calculation of the elastic properties of dislocations using anisotropic elasticity theory (Hirth and Lothe, 1968 Steeds, 1973). The results calculated from the elastic constants of single crystals are summarized. 

The elastic properties of many intermetallic compounds have been investigated extensively since 1967, the year in which the first edition of Intermetallic Compounds, edited by Westbrook, was published. In the present chapter, the elastic constants of single-crystal intermetallic compounds have been summarized, and the elastically anisotropic behaviors of single crystals have been discussed. The elastic properties of dislocations as calculated from the anisotropic elasticity have been shown. The elastic moduli of polycrystalline materials have also been summarized. Finally, the effects of various factors like composition, temperature, etc. on the elastic moduli were described. 

Knowledge of the elastic constants of a single crystal is necessary to calculate or estimate various elastic properties like elastic anisotropy, elastic properties of dislocations, etc., of intermetallic compounds. Such information is also necessary to estimate or check the potential energy between different atoms, which is used for computer simulations like molecular dynamics, etc., although the elastic constants themselves can be calculated using various methods like EAM, F-LAPW, etc. The temperature dependence of the elastic moduli may also be further used for the investigation of various kinds of transformations, since they are sensitive to composition, temperature, etc. 

An epitaxial film is strained in the initial stages of film growth. The strain energy increases with film thickness and may eventually be relaxed by the introduction of misfit dislocations [14.32-14.35], see Fig. 14.7, or by formation of (110) twins in the YBCO [14.36]. The critical thickness at which the misfit dislocations form depends on the lattice mismatch and the elastic properties of the film. The misfit in epitaxial c-axis-oriented YBCO films is accommodated by the formation of twins and edge dislocations with Burgers vectors [100]ybco and [010]ybco [14.37],... 

Elastic properties serve an obvious utility in mechanics of materials, e.g., stress-strain relations and dislocation characteristics (Fisher and Dever, 1967 Fisher and Alfred, 1968). Moreover, elastic properties and their temperature dependencies provide important information and understanding of such physical characteristics as magnetic behavior, polymorphic transformations, and other fundamental lattice phenomena. In this section the elastic properties and their temperature dependencies are presented for all the rare earth metals except promethium, for which there is no data. To the writer s knowledge this is the first one-source compilation of the temperature dependencies of the elastic properties of the rare earth metals. 

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) ... 

Mechanical property measurements of films on substrates are made using the beam deflection techniques discussed under stress measurement except that the beam is loaded with known weights and the deflection is measured with the stress as the known.[ l Measurements can oidy be made as long as the film does not microcrack (tension) or bhster (compression). ] Thin films have been shown to have very high elastic modulus and strength, presumably due to surface pinning of mobile defects (dislocations). An indentation test may be used to determine the elastic properties of coatings. ... 

Dislocations are known to be responsible for die short-term plastic (nonelastic) properties of substances, which represents departure from die elastic behaviour described by Hooke s law. Their concentration determines, in part, not only dris immediate transport of planes of atoms drrough die solid at moderate temperatures, but also plays a decisive role in die behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondaiy creep, die dislocation distribution cannot be considered geometrically fixed widrin a solid because of die applied suess. 

Plastic deformation, unlike elastic deformation, is not accurately predicted from atomic or molecular properties. Rather, plastic deformation is determined by the presence of crystal defects such as dislocations and grain boundaries. While it is not the purpose of this chapter to discuss this in detail, it is important to realize that dislocations and grain boundaries are influenced by things such as the rate of crystallization, particle size, the presence of impurities, and the type of recrystallization solvent used. Processes that influence these can be expected to influence the plastic deformation properties of materials, and hence the processing properties. 

An elastic continuum model, which takes into account the energy of bending, the dislocation energy, and the surface energy, was used as a first approximation to describe the mechanical properties of multilayer cage structures (94). A first-order phase transition from an evenly curved (quasi-spherical) structure into a... 

He and Hutchinson (1989) considered a crack approaching an interface as a continuous distribution of dislocations along a semi-infinite half space. The effect of mismatch in elastic properties on the ratio of the strain energy release rates, Gi/Gj, is related to two non-dimensional parameters, the elastic parameters of Dundurs, a and /f (Dundurs, 1968) ... 

Despite the similarities in brittle and ductile behavior to ceramics and metals, respectively, the elastic and permanent deformation mechanisms in polymers are quite different, owing to the difference in structure and size scale of the entities undergoing movement. Whereas plastic deformation (or lack thereof) could be described in terms of dislocations and slip planes in metals and ceramics, the polymer chains that must be deformed are of a much larger size scale. Before discussing polymer mechanical properties in this context, however, we must first describe a phenomenon that is somewhat unique to polymers—one that imparts some astounding properties to these materials. That property is viscoelasticity, and it can be described in terms of fundamental processes that we have already introduced. 

The essential difference between treatments of chemical processes in the solid state and those in the fluid state is (aside from periodicity and anisotropy) the influence of the unique mechanical properties of a solid (such as elasticity, plasticity, creep, and fracture) on the process kinetics. The key to the understanding of most of these properties is the concept of the dislocation which is defined and extensively discussed in Chapter 3. In addition, other important structural defects such as grain boundaries, which are of still higher dimension, exist and are unknown in the fluid state. 

As a force is applied to the item through the die, the metal first becomes elastically strained and would return to its initial shape if the force were removed at this point. As the force increases, the metal s elastic limit is exceeded and plastic flow occurs via the motion of dislocations. Many of these dislocations become entangled and trapped within the plastically deformed material thus, plastic deformation produces crystals which are less perfect and contain internal stresses. These crystals are designated as cold-worked and have physical properties which differ from those of the undeformed metal. 

The simplest defect in a semiconductor is a substitutional impurity, such as was discussed in Section 6-E. There are also structural defects even in pure materials, such as vacant lattice sites, interstitial atoms, stacking faults (which were introduced at the end of Section 3-A) and dislocations (see, for example, Kittel, 1971, p. 669). They are always in small concentration but can be important in modifying conduction properties (doping is an example of this) or elastic properties (dislocations arc an example of this). 

There is a counterbalancing disadvantage to this simplicity the defects that can occur in metals are less easily detected and followed. Instead of direct observation of the local properties of a defect by ESR, one has to employ such indirect and less sensitive measurements as those of electrical conductivity or modulus of elasticity to get an estimate of the concentrations of vacancies or interstitials. Dislocations can, to be sure, be counted by etching techniques, but these again are inconvenient, not adapted to powder samples, and not highly informative. 

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