Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 100

4.9 Symmetries that Permit Chirality

Thus far in the consideration of material symmetries the concern has been with the
number and orientation of the planes of material symmetry. In this section the
consideration is of those material symmetries that have planes that are not normals
to planes of reflective symmetry. The triclinic, monoclinic, and trigonal symmetries
are the only three of the eight elastic symmetries that permit directions that are not
normals to planes of reflective symmetry. Every direction in triclinic symmetry is a
direction in which a normal to the plane of material symmetry is not permitted.
Every direction that lies in the single symmetry plane in monoclinic symmetry is a
direction in which a normal to the plane of material symmetry is not permitted. The
only direction in trigonal symmetry in which a normal to the plane of material
symmetry is not permitted is the direction normal to a plane of threefold symmetry.
There are not other such directions. The triclinic, monoclinic, and trigonal
symmetries are also the only three of the eight elastic symmetries that, in their
canonical symmetry coordinate system, retain cross-elastic constants connecting
normal stresses (strains) to shear strains (stresses) and vice versa. In the C matrices
listed in Table 4.4 these cross-elastic constants appear in the lower left and upper
right 3

3 sub-matrices for the triclinic, monoclinic, and trigonal symmetries. In
the A matrices listed in Table 4.3 only monoclinic symmetry has a cross-elastic
constant. The nonzero cross-elastic constants and the directions that are not normals
to planes of reflective symmetry are directly related; such planes disappear when
the cross-elastic constants are zero. It is the existence of such planes and associated
cross-elastic constants that allow structural gradients and handedness (chirality).
Trigonal symmetry, because it is the highest symmetry of the three
symmetries, admits a direction that is not a direction associated with a normal
to a plane of reflective symmetry, nor any projected component of a normal to a
plane of reflective symmetry. An interesting aspect of trigonal symmetry is the
chiral or symmetry-breaking character of the cross-elastic constant c 14 . Note that ^

c 14 is not constrained to be of one sign; the sign restriction on ^ ^ c 14 from the positive

definiteness of strain energy is

r

c 44 ð c 11 c 12 Þ
2

r

c 44 ð c 11 c 12 Þ
2

< ^ c 14 < : (4.20)

c 14 vanishes, the C matrix in Table 4.4 for trigonal symmetry specializes to the C
matrix in Table 4.4 for hexagonal or transversely isotropic symmetry. Hexagonal
symmetry is a sixfold symmetry with seven planes of mirror symmetry. Six of the
normals to these seven planes all lie in the seventh plane and make angles of 30
with one another. A single plane of isotropy characterizes transversely isotropic
symmetry. A plane of isotropy is a plane of mirror symmetry in which every vector
is itself a normal to a plane of mirror symmetry. Since a plane of isotropy is also a
plane of symmetry, there are an infinity plus one planes of symmetry associated
with transverse isotropy.

If ^