Publisher Summary This chapter discusses the elastic constants of crystals. The elastic properties of solid matter hold interest for both technology and basic research. In the first field applied elasticity is an important discipline for those fundamental considerations of engineering design which are usually included under the designation “strength of materials.” The treatment of structural materials requires semiempirical methods, because their compositions are complex and prior treatment has a pronounced effect. On the other hand, basic research into elastic properties is usually concerned with work on specimens in the simplest state which can be obtained reproducibly—for example, annealed single crystals. The complete tabulation of elastic constants for such specimens is valuable, not only for itself, but also because the data can be correlated with other physical measurements and thereby provide possible insight into the nature of the atomic forces in solid matter. The research aspect of elastic studies is of primary interest for this chapter. The case of the isotropic medium is important both for the chronological development of the subject of elasticity and also for its applicability to polycrystals and glasses.

The Elastic Constants of Crystals

Work-hardening phenomena are based on the very fundamental principles (i) that at the position of every dislocation axis the respective resolved shear stress cannot exceed the friction stress, including the self-stress of bowing dislocations, and (ii) that always that structure forms which among those accessible by the dislocations minimizes stored energy per unit length of dislocation line. Such dislocation structures have been named LEDSs. The corresponding work-hardening theory, the mesh length theory, is applicable to all materials deforming via gliding dislocations and to all types of deformation. Results previously achieved with the mesh length theory are summarized, and a number of new developments are discussed. Depending on the dislocation structures formed, the work-hardening behavior differs. Easily intersecting glide causes dislocation cell structures with almost dislocation-free cell interiors delineated by dislocation rotation boundaries. Pronounced planar glide causes Taylor lattices characterized by local planar order parallel to the one or perhaps two most highly stressed glide plane(s), no systematic lattice rotations, and overall uniform dislocation density. The most widely observed basic features of work hardening are explained in general terms. Specific applications are indicated for layer-type crystals, h.c.p. single crystals, single-crystal and polycrystalline pure f.c.c. metals and α-brass-type alloys, precipitation-hardened materials and steels. Included are the different stages of work hardening, dynamical effects in low temperature plasticity, the general characteristics of grain boundary strengthening and the Hall-Petch relationship. In addition, proposed explanations for (i) glide system interactions in polyslip resulting in microbands and affecting texture formation, and (ii) creep without stress dependence of dislocation density, are discussed.

Theory of plastic deformation: - properties of low energy dislocation structures

A detailed discussion of data obtained over the past 15 years concerning the damping of mechanical vibrations in the kilocycle and megacycle range is given. The dependence of the decrement and modulus change on the variables of frequency and strain‐amplitude and many other parameters is compared with predictions of the dislocation theory developed in an earlier paper. Although general agreement is obtained, and many interesting quantitative results are found, it is not possible to say that the theory agrees everywhere since not all the necessary parameters are known well enough theoretically. A number of new experiments are suggested which may permit stronger conclusions to be made. This part may be read independently of the earlier paper by the reader who does not wish to follow the development of the theory in detail.

Application of Dislocation Theory to Internal Friction Phenomena at High Frequencies

Non-linear deformation mechanisms during nanoindentation

Dynamic nonlinear elasticity in geomaterials

A quantitative theory of damping and modulus changes due to dislocations is developed. It is found that the model used by Koehler of a pinned dislocation loop oscillating under the influence of an applied stress leads to two kinds of loss, one frequency dependent and the other not. The frequency dependent loss is found to have a maximum in the high megacycle range. The second type of loss is a hysteresis loss which proves to be independent of frequency over a wide frequency range which includes the kilocycle range. This loss has a strain‐amplitude dependence of the type observed in the kilocycle range. The theory provides a quantitative interpretation of this loss.

A. Granato

Papers

Theory of Mechanical Damping Due to Dislocations