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2. Strain and Stress In order to describe the changes of displacements, strain is necessary to be defined. The configuration in the infinitesimal neighborhood around points is represented as follows: For the more for this, the Lagrangian strain tensor  can be defined. In case of stress, the Cauchy stress tensor is used to describe the state of stress in the solid. , which is the force acting outwards on a surface region . Thanks JF

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, tempera