Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 103

material symmetry that permits chirality (i.e., trigonal, monoclinic, or triclinic
symmetry) is obtained by averaging over a domain that is characterized by
a symmetry that does not permit chirality (i.e.,isotropic,cubic,transverseisotro-
pic, tetragonal, and orthotropic). Clearly the result presented depends on the fact
that the (non-chiral) orthotropic material symmetry is helically curvilinear. The
association of the micro-geometric chiral character of c 14 with a helix is not a
unique association. The basic property of c 14 is its symmetry-breaking character,
and it may be associated with structural gradients in the material (Cowin 2002 ;
Fraldi and Cowin 2002 ).
There are many natural and man-made examples of both chiral materials as
structures and as local components in globally non-chiral composites. Chiral
materials that form chiral structures occur in nature (Neville 1993 ). Perhaps the
most famous is the tusk of the narwhal (in the middle ages the tusk of the narwhal
was thought to be the horn of the mythical unicorn). This whale is edentulous
except for the upper lateral incisors. The right incisor normally remains embedded
in the jaw, but in adult males the left tooth forms a tusk, which can in large
specimens reach a length of 2.4 m, and have a diameter of 8 cm at the point of
eruption. Normally the tusk is imprinted with the curvature of the bone socket as it
erupts or extrudes itself from a bone socket. However, if the tusk slowly twists in
the socket as it grows, the imprinted curvature will be neutralized or averaged and
the tusk will grow straight with the spiral structure. A second example of a natural
chiral structure occurs in trees, both hardwoods and softwoods, due to a combina-
tion of genetic and environmental factors. The spiral structure in trees causes a
practical problem with telephone and power poles. Changes in the moisture content
of the wood of the pole cause the pole to twist after it has been employed as part of a
transmission network.
Chiral materials that form chiral and non-chiral structures occur very frequently
in nature (Neville 1993 ). A typical such natural structure is illustrated in Fig. 4.14 .
The structure is a set of concentric coaxial cylinders, each lamina or cylinder
characterized by a different helical angle. The angle of the helices often rotates
regularly from one cylinder to the next. This type of structure is called helicoidal
and described as a cylinder made of "twisted plywood" (see Figs. 10.33 and 10.34).
The helical fibers may or may not be touching, as illustrated in Fig. 4.15 . The
examples of this structure in nature are numerous and include fish scales and plant
stem walls. Man also uses cylinders made of "twisted plywood" to create structures.
4.10 Relevant Literature
A very interesting and perceptive topic on symmetry in general, but including all
the symmetries of interest in the present text, is the topic of Weyl ( 1952 ) with the
title Symmetry . Material symmetry is well explained in the topics of Nye ( 1957 ) and
Fedorov ( 1968 ). The treatment of material symmetry in this chapter does not follow
the standard treatments of material symmetry contained, for example, in the topics