Elastic Properties of Crystals

The objective of present day physics is to describe and explain the properties of sohd materials, in particular of crystals, on the basis of their atomic structure. In this chapter we will not pursue this aim. The treatment of matter here will be confined to the continuum model. In this model a body corrsists of a compact continuous set of material points. The boundary of this set of points is called the surface of the body. Frequently it is convenient to consider a body as a material element cut out of the surrounding continuum. The surface then separates the body under consideration from its surroundings. 

---

Generally, the elastic properties of crystals should be described by 36 elasticity constants Cit but usually a proportion of them are equal to zero or are interrelated. It follows that in crystals, the tensors (2.6) and (2.7) are symmetric tensors, owing to which the number of elastic compliance coefficients is reduced, e.g., in the triclinic configuration, from 36 to 21 (Table 2.1). With increasing symmetry, the number of independent co-... 

Our starting point for the analysis of the thermal and elastic properties of crystals is an approximation. We begin with the assertion that the motions of the i atom, will be captured kinematically through a displacement field Uj which is a measure of the deviations from the atomic positions of the perfect lattice. It is presumed that this displacement is small in comparison with the lattice parameter, uj < C qq. Though within the context of the true total energy surface of our crystal (i.e. R2,. .., Rw)) this approximation is unnecessary, we will see... 

Since treatments of the elastic properties of crystals are often very brief, or at best limited to a discussion of cubic crystals, let us begin by giving a short review of this subject as it applies to hexagonal crystals like ice. We shall follow the development given by Nye (1957), to whose book the reader is referred for a fuller treatment. 

For example vibrational spectra and the study of lattice modes, from which information on effective charges may be obtained via a dielectric model, the mechanical and elastic properties of crystals, proton and other NMR which have provided much interesting information on the bonding of metal complexes, and optical effects such as harmonic generation. 

Mashkevich, V.S., Tolpygo, K.B. Electrical, optic and elastic properties of crystals of diamond type. Sov. Phys. - JETP 4, 455-460 (1957)... 

To visualize the elastic properties of crystals it is common to draw a polar diagram with radius vectors of length equal to E (that is the reciprocal of Yj j) in all directions. The endpoints of these radius vectors define the so called elasticity surface of the given crystal. For quartz it is in the form of an oval. The numerical value of Young s modulus for each specified direction may be calculated from Eq. (3.57) and from the transformation equation for s j j. 

Marchenko, V. I. and Parshin, A. Ya. (1980), Elastic properties of crystal surfaces, Soviet Physics JETP 52, 129-131. 

---