Finite Pressure Temperature Elasticity (FPTE) package

Installation

Dependencies

FPTE requires:

• Python (>= 3.7)
• NumPy (>= 1.16.5)
• Pandas (>= 0.25.3)
• Matplotlib (>= 2.2.4)
• joblib (>= 0.11)

FPTE 1.2.0 and later require Python 3.7 or newer. FPTE 1.1.0 and later require Python 3.4 or newer.

FPTE plotting capabilities (i.e., functions start with plot_ and classes end with "Display") require Matplotlib (>= 2.2.4).

User installation

If you already have a working installation of numpy and scipy, the easiest way to install FPTE is using pip:

pip install -U FPTE

or install from source:

git clone https://github.com/MahdiDavari/FPTE
cd FPTE
python setup.py install

In order to check your installation you can use:

python -m pip show FPTE  # to see which version and where FPTE is installed
python -m pip freeze  # to see all packages installed in the active virtualenv
python -c "import FPTE; print(FPTE.__version__)"

Note that in order to avoid potential conflicts with other packages it is strongly recommended to use a virtual environment (venv).

Theory

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 ** a**₀ 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

$$ $$

Crystal System	Space Group Number	No. of Elastic Constants
Cubic	195-230	3
Hexagonal	168-194	5
Trigonal	143-167	6-7
Tetragonal	75-142	6-7
Orthorhombic	16-74	9
Monoclinic	3-15	13
Triclinic	1 and 2	21

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.