Metadata-Version: 2.1
Name: FPTE
Version: 1.1.0a2
Summary: The FPTE package is a collection of tools for finite pressure temperature elastic constants calculation. Features include, but are not limited to stress-strain method for getting second order elastic tensors using DFT package VASP as well as, ab initio molecular dynamic method for temperature dependent elastic constatns. The package is free and ...
Home-page: https://github.com/MahdiDavari/FPTE
Author: Mahdi Davari
Author-email: Mahdi.Davari@icloud.com
License: UNKNOWN
Project-URL: Bug Reports, https://github.com/MahdiDavari/FPTE/issues
Project-URL: Source, https://github.com/MahdiDavari/FPTE/src
Platform: UNKNOWN
Classifier: Development Status :: 3 - Alpha
Classifier: Programming Language :: Python :: 2
Classifier: Programming Language :: Python :: 3
Classifier: License :: OSI Approved :: MIT License
Classifier: Operating System :: OS Independent
Description-Content-Type: text/markdown
Requires-Dist: matplotlib
Requires-Dist: numpy
Requires-Dist: pandas

# Finite Pressure Temperature Elasticity (FPTE) package. 
<a href="https://ibb.co/gJpS7Js"><img src="https://i.ibb.co/VTZgNTX/Stress-Strain.jpg" alt="Stress-Strain" border="0" /></a>

**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<sub>ijkl</sub> and s<sub>ijkl</sub> are called the elastic stiffness coefficients and elastic compliance constants respectively. Here we deal with elastic stiffness coefficients c<sub>ijkl</sub>, 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<sub>ij</sub>), 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

$$ $$ 
<br>
 and 

 $$ $$

 where the stress components are  &sigma;<sub>1</sub> =  &sigma;<sub>xx</sub> ;  &sigma;<sub>2</sub> =  &sigma;<sub>yy</sub> ;  &sigma;<sub>3</sub> =  &sigma;<sub>zz</sub> ;  &sigma;<sub>4</sub> =  &sigma;<sub>yz</sub> ;  &sigma;<sub>5</sub> =  &sigma;<sub>zx</sub> ;  &sigma;<sub>6</sub> =  &sigma;<sub>xy</sub> , and the strain components are &epsilon;<sub>1</sub> = &epsilon;<sub>xx</sub> ; &epsilon;<sub>2</sub> = &epsilon;<sub>yy</sub> ; &epsilon;<sub>3</sub> = &epsilon;<sub>zz</sub> ; &epsilon;<sub>4</sub> = &epsilon;<sub>yz</sub> ; &epsilon;<sub>5</sub> = &epsilon;<sub>zx</sub> ; &epsilon;<sub>6</sub> = &epsilon;<sub>xy</sub> . When a crystal lattice is deformed with strain (&epsilon;), new lattice vectors a are related to old vectors **a**<sub>0</sub> by **a** = (I + &epsilon;) **a**<sub>0</sub> , 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<sub>ij</sub> . A cubic crystal having highest symmetry is characterized by the lowest number (only three) of independent elastic constants, c<sub>11</sub>, c<sub>12</sub> and c<sub>44</sub>, 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: <br>
 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.

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

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


