A Wolfram Language package for symbolic calculations in anisotropic elasticity. The package provides tools for the manipulation of elastic tensors of arbitrary symmetry, the calculation of Christoffel tensors, phase velocities, polarization vectors, and reflection and transmission coefficients. The package is designed to be used in the context of seismology, geophysics, and materials science.
Mathematica >13.1
<< "https://raw.githubusercontent.com/gpap-gpap/SymbolicAnisotropy/main/SymbolicAnisotropy.wl"
saCreateElasticityTensor[c , "Symmetry" -> sym]
where c
is the tensor label and sym
is the symmetry of the tensor. The following symmetries are supported:
Triclinic
:
Monoclinic
:
Orthotropic
:
Transversely Isotropic
:
Isotropic
:
For example, tilt an orthorhombic tensor by an angle
Block[{\[Psi], c},
With[{
tensor = saCreateElasticityTensor[c, "Symmetry" -> "Orthotropic"],
transform = saRotationTransformation[\[Psi], {1, 0, 0}]},
(saContract[transform, tensor] // saConvert[c, #] &) /. saHumanReadable[c]
]
]
For example, what is the Christoffel tensor for a transversely isotropic medium with symmetry axis along the x-axis?
Block[{c},
With[{
tensor = saCreateElasticityTensor[c, "Symmetry" -> "TransverselyIsotropic"],
transform = saRotationTransformation[\[Pi]/2, {0, 1, 0}]},
saChristoffel[tensor, transform] // saConvert[c, #] &
]
]
For further examples check out the (constantly updated) examples notebook
- Group Velocities
- Ray tracing
- NMO analysis
- Eshelby inclusions
- implement Cowin, S. C., & Mehrabadi, M. M.
gpap+github@mantis-geophysics.com
- Musgrave, M. J. P. (1970). Crystal acoustics: Introduction to the study of elastic waves and vibrations in crystals. Holden-Day, San Francisco
- Nye, J. F. (1985). Physical properties of crystals: their representation by tensors and matrices. Oxford university press.
- Tsvankin, I. (2012). Seismic signatures and analysis of reflection data in anisotropic media. Society of Exploration Geophysicists.
- Structural mechanics MIT lecture notes, https://web.mit.edu/16.20/homepage/
- Helbig, K. (2015). Foundations of Anisotropy for Exploration Seismics: Section I. Seismic Exploration. Elsevier.
- Cowin, S. C., & Mehrabadi, M. M. (1987). On the identification of material symmetry for anisotropic elastic materials. The Quarterly Journal of Mechanics and Applied Mathematics, 40(4), 451-476.