Again, let’s start with :
For what is maximized?
In index notation,
to find local optimums
Simplifying, we get:
This further reduces to: ;
Eigenvalue problem is precisely
This “Eigenvalue problem” was mentioned in Section 1: Trace, Scalar Product, Eigenvalues. An alternative proof of the principal stress eigenvalue problem can be found in [Boresi].
And, at the principal plane, traction is in the direction of the normal –i.e. no shear. (max shear = )
Principal shear stresses can be found in [Holzapfel].
Simple 2D e.x.: Problems like the following are typical in undergrad “mechanics of materials”, where Mohr’s Circle would be used to find the maximum stresses. Tensor methods are faster and can be more easily extended to 3D.
The above values of stress and angle agree with the transformation equations from undergrad, of course.