Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 82

Fig. 4.8 Material anisotropy
induced by deformation, from
Hull ( 1981 )

a

b

c

The illustration in this figure might represent the fiber deformation in a fibrous
composite manufacturing process. However, it could also represent deformation of
the collagen fibers in the deformation of a soft tissue.
For noncrystalline materials there are only three material symmetries tradition-
ally considered, orthotropy, transverse isotropy, and isotropy. However, the forms
of C for orthotropy and transverse isotropy are the same as the forms of C for the
rhombic and hexagonal crystal systems, respectively. Hence when the crystalline
and the traditional noncrystalline elastic material symmetries are combined, there
are only eight distinct forms of

C , one for each of the seven crystal systems and

isotropy.

4.4 Planes of Mirror Symmetry

Symmetry elements are operations used in the analysis of symmetry. The principal
symmetry element of interest here is the plane of mirror or reflective symmetry. We
begin with a discussion of congruence and mirror symmetry. Two objects are
geometrically congruent if they can be superposed upon one another so that they
coincide. The two tetrahedra at the top of Fig. 4.9 are congruent. Congruence of two
shapes is a necessary but not sufficient condition for mirror symmetry. A pair of
congruent geometric objects is said to have mirror symmetry with respect to a plane
if for each point of either object there is a point of the other object such that the pair
of points is symmetric with respect to the plane. The two congruent tetrahedra at the
bottom of Fig. 4.9 have the special relationship of mirror symmetry with respect to
the plane whose end view is indicated by an m. Each congruent geometric object is
said to be the reflection of the other. The relationship between the two objects with