Elastic symmetry with beachball pictures
Walter Tape and Carl Tape
Geophysical Journal International, 2021
The elastic map, or generalized Hooke's Law, associates stress with strain in an elastic material. A symmetry of the elastic map is a reorientation of the material that does not change the map. We treat the topic of elastic symmetry conceptually and pictorially. The elastic map is assumed to be linear, and we study it using standard notions from linear algebra, not tensor algebra. We depict strain and stress using the "beachballs" familiar to seismologists. The elastic map, whose inputs and outputs are strains and stresses, is in turn depicted using beachballs. We are able to infer the symmetries for most elastic maps, sometimes just by inspection of their beachball depictions. Many of our results will be familiar, but our versions are simpler and more transparent than their counterparts in the literature.
Suggested approach to the material:
1. check out these google slides
2. check out the paper (and supplement)
3. explore the mathematica notebooks that reproduce many of the equations and figures
4. check out the follow-up study, which uses the B-basis
Orthonormal basis of 3 x 3 symmetric matrices used to represent the elastic map Tbb, which inputs a strain matrix and outputs a stress matrix.
Example elastic map Tbb depicted in the following six-beachball figure.
Visualization of the elastic map Tbb as six eigenvectors (beachballs) and eigenvalues. In this example, the third beachball is most informative and reveals the tetragonal symmetry that characterizes Tbb.
Table 4 from TapeTape2021. Using the Tbb basis, the reference matrices are simpler than the prevailing forms in the literature (e.g., Nye, 1957). For example, the matrices for isotropic (ISO) and cubic (CUBE) are diagonal, and transverse isotropic (XISO) and tetragonal (TET) have one off-diagonal entry.