Page 315 - Introduction to Continuum Mechanics

Constitutive Equation for a Monoclinic Anisotropic Linearly Elastic Solid. 299

In obtaining the above equations, we have made use of the fact that e 1? 62, €3 are planes of
a
symmetry so that 0:12 = #21 = a i3 = a 3i = a i2> ~ Ji ~ 0- Now, in addition, since any S@
plane is a plane of symmetry, therefore, [see part (a)]

so that from Eq. (xiiib)

Thus, only two coefficients are needed to describe the thermal expansion behavior of the a
transversely isotropic material.
Finally, if the material is also transversely isotropic with ej as its axis of symmetry, then

so that

and the material is isotropic with respect to thermal expansion with only one coefficient for
its description.

5.24 Constitutive Equation for a Monoclinic Anisotropic Linearly Elastic Solid.
If a linearly elastic solid has one plane of material symmetry, it is called a monoclinic
material. We shall demonstrate that for such a material there are 13 independent elasticity
coefficients.
Let ej be normal to the plane of material symmetry S^ Then by definition, under the change
of basis

the components of the fourth order elasticity tensor remain unchanged, i.e.,

Now, q/ kl = Q mi Q nj Q rk Q sl C mnrs [Sect. 2B14], therefore

where

i.e., Q n = -1, Q 22 = £> 33 = 1, and all other Q tj = 0. Thus,