Everyday experience indicates that pulling identical bars of steel and rubber
by the same axial force and hence axial stress results in different
elongations of the two bars. In mechanics, this difference in the materials
of the two bars is represented through the relationship between the components
of the stress and the components of the strain. Writing each of these as a
column matrix, i.e,
we have
(6.3) |
A material is said to be transversely isotropic
about an axis
a if
the components of the matrix
C are invariant (i.e. are unchanged)
with respect to
rotations of axes about the vector
a, and the direction of
a is called
the axis of transverse isotropy. For rectangular Cartesian coordinate axes
with x3-axis coinciding with the vector
a, the components of the
matrix
Cwill be same no matter how x and y-axes are chosen. For a
transversely isotropic material with the axis of transverse isotropy along the
x3-axis,
The ratio E1/E3 is a measure of the degree of anisotropy. Here
Thus there are 5 independent material parameters,
C11,C12,C13,C33 and G13. Materials having a laminated
structure are usually modeled as transversely isotropic.
A material (e.g. wood) that has three mutually perpendicular planes of elastic
symmetry is called orthotropic. We choose co-ordinate axes so that the
coordinate planes coincide with the planes of elastic symmetry. The material
properties, i.e., values of components of the matrix
C will be unchanged if
the direction of the co-ordinate axes were reversed one at a time. For an
orthotropic material, the elasticity matrix
C has the following
form.
(6.8) |