For the indirect method, a starting resonant frequency spectrum, f _ n ^ { cal } ( n = 1, 2, & ) is calculated using estimated values for the elastic constants and the known sample dimensions and density.
22.
Once the lattice strains are calculated, the product of these values and the x-ray shear modulus, also known as the diffraction elastic constant GR ( HKL ), provides the stress on different lattice planes, ? ( HKL ).
23.
It has been extended by P . N . Keating to include anharmonic effects ( and calculate third-order elastic constants ), and many other researchers have extended it to include forces between the covalent bonds, and augment it in other ways.
24.
The position of these peaks occurs at the natural frequencies f _ n ( from which the elastic constants are determined ) and the quality factor Q ( a measure of how narrow the resonance is ) provides information about the dissipation of elastic energy.
25.
Interest in elastic properties made its entrance with 17th century philosophers, but the real theory of elasticity, indicating that the elastic constants of a material could be obtained by measuring sound velocities in that material, was summarized only two hundred of years later.
26.
J . S . Koehler theoretically predicted that by using alternate ( nano-) layers of materials with high and low elastic constants, shearing resistance is improved by up to 100 times as the Frank-Read source of dislocations cannot operate in the nanolayers.
27.
Relationships between elastic constants and X for cubic systems depending upon the direction of propagation of the phonon, q, and the eigenvector of the phonon, U, where L = longitudinal and T = transverse acoustic waves ( Hayes & Loudon, 1978 ).
28.
The nature and strength of these forces is important for the scientific understanding of solids since they determine the way the solid responds to these stresses ( elastic constants ), the velocity of sound waves in it, its infra-red absorption, and many other properties.
29.
The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21 C _ { ijkl } = C _ { klij } = C _ { jikl } = C _ { ijlk }.
30.
Ionization produces electron-hole pairs; the holes are trapped in the lattice near the Al atom, the resulting Li and Na atoms are loosely trapped along the Z axis; the change of the lattice near the Al atom and the corresponding elastic constant then causes a corresponding change in frequency.
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