.. _soec_tutorial: ============================================= Second-Order Elastic Moduli Analysis tutorial ============================================= :Last updated: |today| :Author: **Gianfranco Ulian** Preliminary operations ====================== Download the :download:`hydroxylapatite input file<../downloads/hydroxylapatite.dat>`, which contains the second-order elastic tensor of hydroxylapatite in Voigt notation (:math:`6 \times 6` matrix, values in GPa) and the density of the mineral, expressed in :math:`kg\ m^{-3}`. [1]_ Put this file in a folder of your choice and enter in this folder via the command prompt (or console under Linux/Mac OSX). Analysis of the second-order elastic constants ============================================== This analysis is conducted in an automated mode by Quantas, so it is sufficient to type: .. code-block:: console > quantas soec hydroxylapatite.dat to perform it. Quantas reports the initial settings used in this analysis: .. code-block:: console ________ __ \_____ \ __ _______ _____/ |______ ______ / / \ \| | \__ \ / \ __\__ \ / ___/ / \_/. \ | // __ \| | \ | / __ \_\___ \ \_____\ \_/____/(____ /___| /__| (____ /____ > \__> \/ \/ \/ \/ v0.9.0 Authors: Gianfranco Ulian Copyright 2020, University of Bologna Calculator: Second-order elastic constants analysis Measurement units ------------------------------------- - pressure: GPa .. warning:: At the moment, only elastic constants expressed in GPa are supported. If you want to follow this tutorial with elastic constants for a system of your choice, and their value are not in GPa, please, convert them in this units before creating the input file and starting the analysis. Then, the input file is read and relevant properties are printed on screen (and in the output file ``hydroxylapatite_SOEC.txt``): .. code-block:: console Reading input file: hydroxylapatite.dat Elastic analysis of Hydroxylapatite System is hexagonal Density: 3178.0 kg m^-3 Stiffness matrix (values in GPa) 187.2080 65.1930 84.7030 0.0000 0.0000 0.0000 65.1930 187.2080 84.7030 0.0000 0.0000 0.0000 84.7030 84.7030 222.6580 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 39.6870 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 39.6870 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 61.0070 Compliance tensor (values in TPa^-1) 6.758054 -1.437660 -2.023971 0.000000 0.000000 0.000000 -1.437660 6.758054 -2.023971 0.000000 0.000000 0.000000 -2.023971 -2.023971 6.031101 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 25.197168 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 25.197168 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 16.391562 A symmetry analysis on the values of the SOECs matrix (correctly) revealed that the system is hexagonal, and the stiffness and compliance matrices are reported. Polycrystalline (average) properties: .. code-block:: console Average properties Bulk Young's Shear Poisson's modulus modulus modulus ratio (GPa) (GPa) (GPa) Voigt 118.47467 136.63989 52.24120 0.30778 Reuss 116.60441 131.05436 49.91864 0.31268 Hill 117.53954 133.85036 51.07992 0.31021 and the eigenvalues of the stiffness matrix: .. code-block:: console Eigenvalues of the stiffness matrix: lambda_1: 39.68700 lambda_2: 39.68700 lambda_3: 61.00700 lambda_4: 116.82176 lambda_5: 122.01500 lambda_6: 358.23724 are calculated and reported. The eigenvalues are all positive, meaning that the system is mechanically stable. .. note:: If any of the eigenvalues were negative, the analysis would have stopped, detecting the instability of the system. Quantas then proceeds searching for the minimum and maximim values of: - Young's modulus; - linear compressibility; - shear modulus; - Poisson's ratio - seismic waves (*if the density is present in input*) along crystal directions, assuming the system as a monocrystal. The results of this procedure are reported in tabular format for Young's modulus and linear compressibility: .. code-block:: console Variations of the elastic moduli: -------------------------------------------------------------------------------- | Young's modulus | Linear compressibility -----------|----------------------------------|--------------------------------- | E_min E_max | beta_min beta_max Values | 117.6414 165.8072 | 1.9832 3.2964 -----------|----------------------------------|--------------------------------- Anisotropy | 1.4094 | 1.6622 -----------|----------------------------------|--------------------------------- | 0.5213 0.0000 | 0.0000 0.7071 Axis | 0.5213 0.0000 | 0.0000 0.7071 | 0.6757 1.0000 | 1.0000 0.0000 -------------------------------------------------------------------------------- Notes: E min/max values in GPa, beta min/max values in TPa^-1 for shear modulus and Poisson's ratio: .. code-block:: console -------------------------------------------------------------------------------- | Shear modulus | Poisson's ratio -----------|----------------------------------|--------------------------------- | G_min G_max | nu_min nu_max Values | 39.6870 61.0075 | 0.1944 0.4857 -----------|----------------------------------|--------------------------------- Anisotropy | 1.5372 | 2.4987 -----------|----------------------------------|--------------------------------- | 0.5000 -0.6832 | 0.0000 0.7356 1st Axis | 0.8660 0.7302 | -1.0000 -0.0002 | 0.0000 0.0000 | -0.0000 -0.6775 -----------|----------------------------------|--------------------------------- | 0.5000 -0.6832 | 0.0000 0.7356 2nd Axis | 0.8660 0.7302 | -1.0000 -0.0002 | 0.0000 0.0000 | -0.0000 -0.6775 -------------------------------------------------------------------------------- Notes: G min/max values in GPa and for seismic wave velocities: .. code-block:: console Variations of the seismic velocities: ------------------------------------------------------------------------------------- | V_s1 | V_s2 | V_p -----------|------------------------|------------------------|----------------------- | min max | min max | min max Values | 3.5338 4.1768 | 3.5338 4.3814 | 7.5397 8.3703 -----------|------------------------|------------------------|----------------------- Anisotropy | 1.1819 | 1.2398 | 1.1102 -----------|------------------------|------------------------|----------------------- | 0.0000 0.8597 | -0.0000 0.7071 | 0.5987 0.0000 Axis | -0.0000 -0.0000 | -0.0000 0.7071 | 0.5987 0.0000 | -1.0000 0.5109 | 1.0000 0.0000 | -0.5320 -1.0000 ------------------------------------------------------------------------------------- Notes: min/max values in km s^-1 Analysis of elastic properties on :math:`(xy)`, :math:`(xz)` and :math:`(yz)` planes ==================================================================================== By using the ``--polar`` option, the elastic properties are evaluated on the cited planes: .. code-block:: console > quantas soec hydroxylapatite.dat --polar The analysis proceeds calculating the bi-dimensional variations of the cited properties on the :math:`(xy)`, :math:`(xz)` and :math:`(yz)` planes: .. code-block:: console - Calculation of polar (2D) properties: * along (xy) a. Young's modulus b. Linear compressibility c. Shear modulus d. Poisson's ratio e. Wave velocities * along (xz) a. Young's modulus b. Linear compressibility c. Shear modulus d. Poisson's ratio e. Wave velocities * along (yz) a. Young's modulus b. Linear compressibility c. Shear modulus d. Poisson's ratio e. Wave velocities Calculation time: 62.7 sec Some polar plots of the elastic properties can be produced in an automated mode if the command is launched as: .. code-block:: console > quantas soec hydroxylapatite.dat --polar --plot .. note:: To generate publication-ready picture, it is recommended to increase the dot-per-inch (DPI) of the output images by using, for example: .. code-block:: console > quantas soec hydroxylapatite.dat --polar --plot --dpi 300 If plots are requested, the following lines will be printed: .. code-block:: console Plotting results as requested: - figure hydroxylapatite_SOEC_E.png generated - figure hydroxylapatite_SOEC_LC.png generated - figure hydroxylapatite_SOEC_G.png generated - figure hydroxylapatite_SOEC_Nu.png generated - figure hydroxylapatite_SOEC_waves.png generated Calculated data exported to hydroxylapatite_SOEC.hdf5 The produced polar plots should be like the following ones: .. image:: ../_static/hydroxylapatite_SOEC_E.png :align: center :alt: Young's modulus of hydroxylapatite on (xy), (xz) and (yz) planes .. image:: ../_static/hydroxylapatite_SOEC_LC.png :align: center :alt: Linear compressibility of hydroxylapatite on (xy), (xz) and (yz) planes .. image:: ../_static/hydroxylapatite_SOEC_G.png :align: center :alt: Shear modulus of hydroxylapatite on (xy), (xz) and (yz) planes .. image:: ../_static/hydroxylapatite_SOEC_Nu.png :align: center :alt: Young's modulus of hydroxylapatite on (xy), (xz) and (yz) planes .. image:: ../_static/hydroxylapatite_SOEC_waves.png :align: center :alt: Seismic wave velocities of hydroxylapatite on (xy), (xz) and (yz) planes .. note:: The calculated data reported in the ``hydroxylapatite_SOEC.hdf5`` contains the values used to generate the 2D polar plots of the elastic properties of the crystalline material. They can be extracted to generate plots according to the user's preferences via: .. code-block:: console quantas export soec hydroxylapatite_SOEC.hdf5 .. rubric:: References .. [1] Ulian, G., Valdre, G., 2018. Second-order elastic constants of hexagonal hydroxylapatite (P63) from ab initio quantum mechanics: comparison between DFT functionals and basis sets. Int. J. Quantum Chem. 118, e25500