Finite Pressure Temperature Elasticity (FPTE) package.

Elastic Stifness Coefficients from Stress-Strain Relations:

According to Hooke's law, the second-rank stress and strain tensors for a slightly deformed crystal are related by

$$ $$

where the fourth rank tensors c_ijkl and s_ijkl are called the elastic stiffness coefficients and elastic compliance constants respectively. Here we deal with elastic stiffness coefficients c_ijkl, which govern the proper stress-strain relations at nite strain. In general, we can write

$$ $$

where X and x are the coordinates before and after the deformation. There are 81 independent stiffness coefficients in general; however, this number is reduced to 21 by the requirement of the complete Voigt symmetry. In Voigt notation (c_ij), the elastic constants form a symmetric 6x6 matrix

$$ $$

In single suffix notation (running from 1 to 6), we can also use the matrix representations for stress and strain

$$ $$
and

$$ $$

where the stress components are σ₁ = σ_xx ; σ₂ = σ_yy ; σ₃ = σ_zz ; σ₄ = σ_yz ; σ₅ = σ_zx ; σ₆ = σ_xy , and the strain components are ε₁ = ε_xx ; ε₂ = ε_yy ; ε₃ = ε_zz ; ε₄ = ε_yz ; ε₅ = ε_zx ; ε₆ = ε_xy . When a crystal lattice is deformed with strain (ε), new lattice vectors a are related to old vectors a₀ by a = (I + ε) a₀ , where I is identity matrix. The stress-strain relations are then simply given by

$$ $$

The presence of the symmetry in the crystal reduces further the number of independent c_ij . A cubic crystal having highest symmetry is characterized by the lowest number (only three) of independent elastic constants, c₁₁, c₁₂ and c₄₄, which in matrix notation is

$$ $$

Note: For more information regarding the second-order elastic constant see reference:

1. Golesorkhtabar, Rostam, et al., “ElaStic: A Tool for Calculating Second-Order Elastic Constants from First Principles.” Computer Physics Communications 184, no. 8 (2013): 1861–73.

2. Karki, Bijaya B. “High-Pressure Structure and Elasticity of the Major Silicate and Oxide Minerals of the Earth’s Lower Mantle,” 1997.

3. Barron, THK, and ML Klein. “Second-Order Elastic Constants of a Solid under Stress.” Proceedings of the Physical Society 85, no. 3 (1965): 523.