Carl Tape's website - elasticity

Elastic symmetry

Walter Tape and Carl Tape

Elastic symmetry with beachball pictures, Geophysical Journal International, 2021

Two complementary methods of inferring elastic symmetry, Journal of Elasticity, 2022

A reformulation of the Browaeys and Chevrot decomposition of elastic maps, Journal of Elasticity, 2024

Suggested approach to the material:

1. check out these google slides

2. check out the 2022 paper

3. check out (and run) the Mathematica notebook here

4. check out the 2024 paper

5. check out these google slides for the 2021 paper

6. check out the 2021 paper, including the supplement on Voigt notation (Section S5)

7. explore the mathematica notebooks that reproduce many of the equations and figures

Reference shapes for each symmetry class, from left to right: monoclinic, trigonal, orthorhombic, tetragonal, cubic, transverse isotropic, isotropic. The 8th symmetry class, trivial (or triclinic), can be represented by any irregular shape.

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. [Tape and Tape, 2021, Figure 2]

Using the Tbb basis, the reference matrices are simpler than the prevailing Voigt 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. [Tape and Tape, 2021, Table 4]

The zero contour (blue) of αmono on the unit sphere for various elastic maps T. Each T here has as symmetry group one of either "reference" groups Utriv, ..., Uiso as indicated. The points of the zero contour αmono are where the axes of the 2-fold symmetries of T intersect the unit sphere. Since the 2-fold rotations in any elastic symmetry group generate the group, the zero contour of αmono determines the symmetry group of T. For any elastic map T, the zero contour of αmono is one of the eight shown here, though probably reoriented. [Tape and Tape, 2022, Figure 1]

Lattice of elastic maps K_Sigma that are closest to the given map T, at bottom, for each symmetry class Sigma (TRIV, MONO, ORTH, etc). The number β = β_Sigma is the angle between T (at bottom) and K_Sigma. The 2-fold points of K_Sigma are shown in green. [Tape and Tape, 2024, Figure 13]

This research was supported in part by awards from the U.S. National Science Foundation (EAR 1829447, EAR 2342129).