Page 317 - Introduction to Continuum Mechanics
P. 317
Constitutive Equations for an Orthotropic Linearly Elastic Solid. 301
The coefficients in the stiffness matrix C must satisfy the conditions [See Sect. 5.22] that each
diagonal element C// >0 (no sum on/) for/ =1,2....6 and the determinant of every submatrix
whose diagonal elements are diagonal elements of the matrix C is positive definite [See
Example 5.22.1].
5.25 Constitutive Equations for an Orthotropic Linearly Elastic Solid.
If a linearly elastic solid has two mutually perpendicular planes of symmetry, say Si plane
with unit normal e^ and 82 plane with unit normal 62, then automatically, the $3 plane with a
normal in the direction of 63, is also a plane of material symmetry [see Example 5.25.1 below].
The material is called an Orthotropic material.
For this solid, the coefficient C,y^/ now must be invariant with respect to the transformation
given by Eq. (5.24.1) above as well as the following transformation
Thus, all those C^/which appear in Eq. (5.24.5) and which have an odd number of the subscript
2 must also be zero. For example
That is, in addition to Eqs. (5.24.3), we also have
Therefore, there are now only 9 independent coefficients and the constitutive equations
become:
