Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 102

Fig. 4.15 Two examples of composites formed from an elastic matrix and embedded helicoidal
circular fibers. The two large circular diagrams illustrate the characteristic threefold trigonal
symmetry in the cross-sectional plane. The possible representative volume elements (RVEs) and
their characteristic helical angles are also illustrated. From Fraldi and Cowin ( 2002 )

symmetry, as c 14 passes from positive to negative (or negative to positive) values
through zero, the elastic material is first a trigonal material of a certain chirality, then a
transversely isotropic (or hexagonal) material, and then a trigonal material of an
opposite chirality. A RVE of the composite in Fig. 4.14 may be constructed using a
set of the helicoidal fibers all having identical circular cross-sections and using the
periodicity of the helix (Fig. 4.15 , inset). This construction provides an RVE with a
material neighborhood large enough to adequately average over the microstructure and
small enough to ensure that the structural gradient across it is negligible. An examina-
tion of Fig. 4.15 shows that, in the plane orthogonal to the x 3 axis, the threefold
symmetry characteristic of trigonal symmetry arises naturally.
This example illustrates how chirality is created in a material with a helical
structure. It also demonstrates that the symmetry-breaking chiral elastic constant
^

c 14 in trigonal symmetry is related to the angle of the helical structure of the
material, if the material has a helical structure. Further, it again illustrates how
different levels of RVE's are associated with different types of material symmetry.
In this example the smaller RVE is associated with orthotropic material symmetry
and the larger RVE (obtained by volume averaging over the domain of the smaller
RVE) is associated with monoclinic symmetry. The result demonstrates that a