Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 101

Fig. 4.14 A composite
structure composed from sets
of left- and right-handed
helically wound fibrous
laminae that are in the form of
concentric coaxial cylinders;
the fibers of each lamina or
component cylinder are
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”. From Fraldi and
Cowin ( 2002 )

A simple thought model is possible for the visualization of the symmetry-breaking
character of the elastic constant c 14 . This constant could be described as a chiral
constant, chiral being a word coined by Kelvin (Thompson 1904 )( I call any geometri-
cal figure, or group of points, chiral, and say it has chirality, if its image in a plane
mirror, ideally realized, cannot be brought to coincide with itself. ) and widely used in
describing the structure of molecules. It means that a structure cannot be superposed on
its mirror image, that the structure has a handedness. For example, helical spirals are
chiral; they are either left-handed or right-handed. A composite structure of alternate
left- and right-handed helical spirals is illustrated in Fig. 4.14 . Consider a composite
material constructed of an isotropic matrix material reinforced by only right-handed
spiral helices whose long axes are all parallel. These helical spirals may be either
touching or separated by a matrix material (Fig. 4.15 ). Let the helical angle be

y

(Fig. 4.15 ) and let negative values of y correspond to otherwise similar left-handed

helices; the vanishing of then corresponds to a straight reinforcement fiber. Assume

that when the effective elastic constants for this material are calculated, the sign of

y

^ c 14 is

determined by the sign of is zero. It is then possible to

geometrically visualize the chiral, symmetry-breaking character of

y and vanishes when y

c 14 as it passes
from positive to negative (or negative to positive) values through zero as the vanishing
of a helical angle of one handedness occurs and the initiating of a helical angle of the
opposite handedness commences. At the dividing line between the two types of
handedness,

^

the reinforcing fibers are straight. In terms of the elastic material