And so, I can put my tensor or my state of stress in a matrix.

Remember, these off diagonal terms are equal by equilibrium.

And here's the matrix notation.

What I'd like you to do now is to go back and review modules 24, 25, and

26 of my advanced course on three dimension dynamic motion.

And in those series of modules,

we looked at something called the inertia matrix, which was also a tensor.

Representing physical quantities with an array of numbers.

And so these are completely mathematically analogous.

And if you recall back and you look back at these earlier modules,

you noticed we found that for a particular coordinate orientation,

the products of inertia would vanish.

And we arrived at what we called the principle moments of inertia with respect

to principle axes.

And that matrix looked like this.

Now, we can do the same thing with our matrix array of stresses.

And so for a given general set of stresses, shown here, there

is a particular coordinate orientation where the sheer stresses vanish, and

we arrive at what are the principle normal stresses acting on the principle planes.

And we call those sigma 1, sigma 2, and sigma 3.

Now, we found sigma 1 and sigma 2 for

two dimensional plane stress using Mohr's circle.

We're going to use this technique to find the the three principle stresses

in this module, which could also be used in the plain stress problem,

but we're going to use something called the eigenvalue property.

And so, for this matrix notation, again, we're going to go to

our principle stresses and this is solved via the eigenvalue problem.

Again, go back to my 3D course to see what the eigenvalue problem's all about.

And the eigenvalues are the principle stresses and

we get a minimum, a maximum, and one in-between and those are on my diagonal.

And the corresponding eigenvectors

are the three sets of directional cosines, which define

the normals to the three principle planes where these principle stresses occur.

And so I can orient my block in such a manner

that I get three principle stresses, no shear stresses on those faces.

And so, now we can see from my first slide in this module,

that you can orient the block in any direction and

completely describe the state of stress at a point for

three dimensions using just a cube with three orthogonal faces.

And so, just another way of looking at