Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 87

Fig. 4.10 The hierarchical organization of the eight material symmetries of linear elasticity. The
figure is organized such that the lower symmetries are at the upper left and as one moves down and
across the table to the right one encounters crystal systems with greater and greater symmetry.
From Chadwick et al. ( 2001 )

Table 4.1 The distinct
symmetries of linear
anisotropic elasticity

Triclinic 0 18 (21)

Monoclinic 1 0 12 (13)

Orthotropic or orthorhombic 3 0 9

Tetragonal 5 0 6 (7)

Cubic 9 0 3

Trigonal 3 0 6 (7)

Hexagonal 7 0 5

Transverse isotropy 1 þ1 1 5

2 2

Isotropy 1 1 2

planes of symmetry. The seventh plane of symmetry is the plane containing the
normals to the other six planes of symmetry. The illustration for hexagonal sym-
metry is similar to that for tetragonal symmetry shown in Fig. 4.11 ; the difference is
that there are six rather than four planes with normals all lying in the same plane and
that those normals make angles of 30 rather than 45 with each other. Cubic
symmetry has the nine planes of symmetry illustrated in Fig. 4.12 . The positive
octant at the front of Fig. 4.12 is bounded by three of the symmetry planes with
normals

a 1 , a 2 , and a 3 and contains traces of the six other planes of symmetry.