GitLocker: The Coding Marketplace

Description:

FPTE 1.2.1

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 cijkl and sijkl are called the elastic
stiffness coefficients and elastic compliance constants respectively. Here we deal with elastic
stiffness coefficients cijkl, 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 (cij), 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 = σxx ; σ2
= σyy ; σ3 = σzz ; σ4 =
σyz ; σ5 = σzx ; σ6 =
σxy, and the strain components are ε1 = ε
xx ; ε2 = εyy ; ε3 =
εzz ; ε4 = εyz ; ε5
= εzx ; ε6 = εxy. 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:

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.

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

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