Page 313 - Introduction to Continuum Mechanics

Plane of Material Symmetry 297

(c) If every plane perpendicular to the £3 plane is a plane of symmetry, what are the additional
restrictions? In this case, the material is said to be transversely isotropic with respect to
thermal expansion.

Solution, (a) Using the transformation law [See Eq. (2B.13.lc)]

m
we obtain, with Ch fr° Eq. (5.23.Ic)

The requirement that [a ] ' = [a ] results in the restriction that

Thus, only five coefficients are needed to describe the thermo-expansion behavior if there is
one plane of symmetry:

Thus, from Eq. (ii) and (vi)

The requirements that [a ] ' = [a ] results in

Thus, only three coefficients are needed to describe the thermal expansion behavior if there
are two mutually orthogonal planes of symmetry, i.e.,