Page 319 - Introduction to Continuum Mechanics

Constitutive Equation for a Transversely Isotropic Linearly Elastic Material 303

Verify that [QjHQz] = -[I] [Q 3 ]-

Solution, (a) With

the equation

becomes

(b)

That is
[QilfQzl =-[i][Q ]
3
From the results of (a) and (b), we see that if thejc-plane and the y-plane are planes of material
symmetry, then the z-plane is also a plane of symmetry.

5.26 Constitutive Equation for a Transversely Isotropic Linearly Elastic Material

If there exists a plane, say £3 plane, such that every plane perpendicular to it, is a plane of
material symmetry, then the material is called a transversely isotropic material. The £3 plane
is called the plane of isotropy and its normal direction 63 is the axis of transverse isotropy.
Clearly, a transversely isotropic material is also orthotropic.
Let Sp represent a plane whose normal ej' is parallel to the £3 plane and which makes an
angle of ft with the ej axis which lies in the £3 plane. Then, for every angle ft, the plane Sp is,
by definition, a plane of symmetry. Thus, if C^i are components of the tensor C with respect
to the basis e/ 'given below:

then, from Eq. (5.24.3), we must have