Given:
For an isotropic material
Verify:
where .
From the second equation that has been given
or,
Therefore,
or,
or,
To find out whether the plus or the minus sign should be placed before in the above equation, we put everything in terms of and . Thus,
Plugging these into the equation for , multiplying both sides by and dividing by , we get
The limiting value of is 0.5. Plugging this value into the above equation, we get,
The above can be true only if the sign is positive. Therefore, the
correct relation is
Plugging this relation into the first of the given equations, we have,