Page 240 - Introduction to Continuum Mechanics

Linear Isotropic Elastic Solid 225

from which,

In the following, we first show that if the material is isotropic, then the number of inde-
pendent coefficients reduces to only 2. Later, in Part B, the constitutive equations for
anisotropic elastic solid involving 13 coefficients (monoclinic elastic solid ) , 9 coefficients
(orthotropic elastic solid) and 5 coefficients (transversely isotropic solid), will be discussed.

PART A Linear Isotropic Elastic Solid

5.3 Linear Isotropic Elastic Solid
A material is said to be isotropic if its mechanical properties can be described without
reference to direction. When this is not true, the material is said to be anisotropic. Many
structural metals such as steel and aluminum can be regarded as isotropic without appreciable
error.
We had, for a linear elastic solid, with respect to the e, basis,

and with respect to the e/' basis,

If the material is isotropic, then the components of the elasticity tensor must remain the same
regardless of how the rectangular basis are rotated and reflected. That is

under all orthogonal transformation of basis. A tensor having the same components with
respect to every orthonormal basis is known as an isotropic tensor. For example, the identity
tensor I is obviously an isotropic tensor since its components <5,y are the same for any
Cartesian basis. Indeed, it can be proved (see Prob. 5.1) that except for a scalar multiple, the
identity tensor is the only isotropic second tensor. From d,y, we can form the following three
independent isotropic fourth-order tensors