Version 1, last update 4.8.2022
Author
Ali Aminzadeh (aminzadeh@gli.cas.cz), Václav Vavryčuk (vv@ig.cas.cz)
Short description
SYMMETRY is a Matlab code for transformation of the stiffness tensor from an arbitrary coordinate system into the material principal coordinate system. The Matlab code needs just the excel file with the element of the stiffness tensor. The transformation was originally proposed by Cowin and Mehrabadi (1987) and later developed by Ting (1996) from the theoretical point of view.
Details about the method and its accuracy for experimental results are presented in Aminzadeh et al. (2022).
Primary reference
Aminzadeh, A., et al. 2022. Identification of higher symmetry in triclinic stiffness tensor: application to high pressure dependence of elastic anisotropy in deep underground structures, International Journal of Rock Mechanics and Mining Sciences, in press.
Other references
Cowin SC, Mehrabadi MM. On the identification of material symmetry for anisotropic elastic materials. Q J Mech Appl Math. 1987;40(4):451-476. doi:10.1093/qjmam/40.4.451.
Arts RJ. A Study of General Anisotropic Elasticity in Rocks by Wave Propagation: Theoretical and Experimental Aspects. Published online 1993.
Ting T. Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York; 1996.
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User guide
Run the code
The transformation process is performed by a code called SYMMETRY.m. After running the code, you will be asked to enter the experimentally obtained stiffness matrix in the excel file C.xlsx (cells A2:F7) located at the working directory. Then, close the excel file and answer the question “Have you entered the initial matrix?” with y and press enter.
Output
The code forms the 3*3 matrices U (eq. 4a) and V (eq. 4b) and then calculates the average of their eigenvectors.
Output 1: The stiffness matrix is transformed to the principal coordinate system (eqs. 7 and 8).
Output 2: The inverse of this matrix will be the compliance matrix.
Output 3: The eigenvectors for determining the principal coordinate systems on sphere are calculated.
All these results are also written in C.xlsx file and the elastic constants are also calculated in excel.
Each eigenvector must be evaluated whether it satisfies the condition for being normal to a symmetry plane based on eq. 5. If the residual of eq. 5 for any eigenvector is less than 5% of the trace of matrix, the residual is assumed to be approximately zero. This can be changed by the user according to the sensitivity needed.
The type of the symmetry is also defined. If two eigenvalues have less than 5% difference, they are assumed to be equal. This can also be changed due to the sensitivity requirements by users.
Example
As an example, we provide the input file with the stiffness matrix of Grimsel granite under atmospheric pressure.
Copyright
The code can be freely used for research purposes. In the case of publishing the results obtained by this code, please, refer to the paper of Aminzadeh et al (2022). If you intend to use the code for commercial purposes, you should contact the author (vv@ig.cas.cz or aminzadeh@gli.cas.cz) for providing with the commercial license. The use of the code for commercial purposes with no commercial license is prohibited.