Page 312 - Introduction to Continuum Mechanics

296 The Elastic Solid

5.23 Plane of Material Symmetry

Let Si be a plane whose normal is in the direction of ej. The transformation

describes a reflection with respect to the plane 5j. This transformation can be more con-
veniently represented by the tensor Q in the equation

where

If the constitutive relations for a material, written with respect to the {e/} basis, remain the
same under the transformation [Qj_], then we say that the plane Si is a plane of material
symmetry for that material. For a linearly elastic material, material symmetry with respect to
the Si plane requires that the components of C,yjy in the equation

be exactly the same as C/^/ in the equation

under the transformation Eq. (5.23.1). When this is the case, restrictions are imposed on the
components of the elasticity tensor, thereby reducing the number of independent components.
Let us first demonstrate this kind of reduction with a simpler example, relating the thermal
strain with the rise in temperature.

Example 5.23.1
Consider a homogeneous continuum undergoing a uniform temperature change
A0 = Q - Q O% Let the relation between the thermal strain e,y and A0 be given by

where a^ is the thermal expansion coefficient tensor.
(a) If the plane Si defined in Eq. (5.23.1) is a plane of symmetry for the thermal expansion
property of the material, what restrictions must be placed on the components of a^ ?
an
(b) If the planes $2 d 53 whose normals are in the direction of 62 and 63 respectively are also
planes of symmetry, what are the additional restrictions? In this case, the material is said to
be orthotropic with respect to thermal expansion.