Page 242 - Introduction to Continuum Mechanics

Linear Isotropic Elastic Solid 227

Example 5.3.1

Find the components of stress at a point if the strain matrix is

b
and the material is steel with A = 119.2 GPa (17.3 xl() psi) and p = 79.2 GPa
6
(11.5xl0 psi).
Solution. We use Hooke's law 7^- = Ae<5,y + 2 / a£ / y, by first evaluating the dilatation
e - 100 x 10 . The stress components can now be obtained

Example 5.3.2
(a) For an isotropic Hookean material, show that the principal directions of stress and strain
coincide.
(b) Find a relation between the principal values of stress and strain
Solution, (a) Let nj be an eigenvector of the strain tensor E (i.e., Enj = E\ nj). Then, by
Hooke's law we have

Therefore, nj is also an eigenvector of the tensor T.
(b) Let EI, £2, £3 be the eigenvalues of E then e - E\ + E 2 + £3, and from Eq. (5.3.6b),

In a similar fashion,