Symmetries table for elastic tensor

The table follows Bragger’s (1965) paper.

Brugger, K. (1965). Pure Modes for Elastic Waves in Crystals. Journal of Applied Physics, 36(3), 759–768. https://doi.org/10.1063/1.1714215

Tensor symmetry relationship constraints

Cubic

c11 = c22 = c33
c12 = c13 = c23
c44 = c55 = c66
c14 = c15 = c16 = 0
c24 = c25 = c26 = 0
c34 = c35 = c36 = 0
c45 = c46 = c56 = 0

Orthorhombic

c14 = c24 = c34 = 0
c15 = c25 = c35 = 0
c16 = c26 = c36 = 0
c45 = c46 = c56 = 0

Tetrahedral (6)

c11 = c22
c13 = c23
c44 = c55
c14 = c24 = c34 = 0
c15 = c25 = c35 = 0
c16 = c26 = c36 = 0
c45 = c46 = c56 = 0

Tetrahedral (7)

c11 = c22
c13 = c23
c16 = -c26
c44 = c55
c14 = c24 = c34 = 0
c15 = c25 = c35 = 0
c36 = c45 = c46 = c56 = 0

Hexagonal

c11 = c22
c13 = c23
c44 = c55
c66 = (c11 - c12) / 2
c14 = c24 = c34 = 0
c15 = c25 = c35 = 0
c16 = c26 = c36 = 0
c45 = c46 = c56 = 0

Trigonal (6)

c11 = c22
c13 = c23
c14 = -c24 = c56
c44 = c55
c66 = (c11 - c12) / 2
c16 = c26 = c34 = c35 = c36 = c45 = 0
c15 = -c25 = -c46 = 0

Trigonal (7)

c11 = c22
c13 = c23
c14 = -c24 = c56
c15 = -c25 = -c46
c44 = c55
c66 = (c11 - c12) / 2
c16 = c26 = c34 = c35 = c36 = c45 = 0

Monoclinic

c14 = c16 = 0
c24 = c26 = 0
c34 = c36 = 0
c45 = c56 = 0