How to obtain elastic constants using first-principles calculation - 1. Displacement

1. Displacement

On behalf of mechanical properties, elastic constants (or elastic modulus or elastic moduli) consisting of 6 by 6 matrix are definitely important.

In addition, understanding underlying theory of elasticity is indispensable to obtain by atomic scale simulation such as first-principle calculation.

There are significant papers (Classic ones) as follows:

BARRON, T. H. K., and KLEIN. M. L., 1963, Proc. Phys. Soc., 82, 161.
"On the Strain Dependence of the Vibrational Frequency Distributions of Simple Lattices."

BIRCH, F., 1947, Phys. Rev., 71, 809.
"Finite Elastic Strain of Cubic Crystal."

HEARMON, R. F. S., 1946, Rev. Mod. Phys., 18, 409.
"The Elastic Constants of Anisotropic Materials."

HUANG, K., 1950, Proc. Roy. Soc. A, 203, 178.
"On the Atomic Theory of Elasticity."


Among above the reported papers, Hearmon's paper is a great review (I think), which covers isotropic materials, anisotropic materials, temperature, and even the applications.

As fundamental concepts,

The points preserve its identity in all changes of internal configuration, the regions possess a mass, and there is a potential energy related to the changes of internal configuration.

To designate the position of the point,  is defined.
Therefore, the displacement can be defined as .
And it can be denoted by the index notation in the Cartesian coordinates.


Thanks

JF