So, good morning.

This is just a reminder what we did two weeks ago.

I hope you remember this decomposition of the strain in two different parts.

The volumetric strain and the deviatoric strain.

With the corresponding properties that the trace of the volumetric part is the volume

strain and the trace of the deviatoric part is zero.

In 2D this would be a typical volumetric, purely volumetric deformation and this would

be a purely deviatoric deformation.

Then we invented some fourth order tensors that do the following job for us.

They project the total strain into its volumetric or deviatoric part.

In symbolic notation these fourth order tensors look like that and in particular the volumetric

projection tensor as we call it has this structure here based on the second order unit tensor,

this bold face I here and the deviatoric projection tensor is then given in this format whereby

this symmetric fourth order tensor has this particular combinations of Kronika deltas.

Okay, so this is just to remind you why we call this projection tensors.

They are idempotent so you can take arbitrary powers and you always obtain the same result.

You know that from the number one and the same property has the second order unit tensor.

You can multiply it with itself and again you obtain the second order unit tensor and

this fourth order projection tensor they also have the same property.

However here you have to take this double dot remembering us that we have to sum over

two indices here K and L in this case.

Finally it turns out that these volumetric quantities and the deviatoric quantities are

orthogonal in a certain sense.

This terminology derives from the orthogonality of vectors so if you multiply them you obtain

zero if they are orthogonal and in this case if you multiply here this volumetric and the

deviatoric projection tensors with the double dot contraction you also obtain zero.

Thus we can consider the volumetric strains and the deviatoric strains as being orthogonal

somehow separated from each other.

Then we discussed finally the so called eigenvalue problem.

This is simply a more abstract representation of the topics of the second semester that

you already know namely the determination of the principal strains and the corresponding

principal strain directions.

The main distortion or main stretching if you like and the corresponding main stretching

directions.

Or equivalently we have discussed that for the case of stresses the principal stresses

and the corresponding directions.

The theory behind that is simply that we are looking for a rotated coordinate system n1

n2 n3 so that the corresponding coefficients of the strains in this new coordinate system

only have entries at the main diagonal and all the corresponding shear terms vanish.

You remember from second semester Morse circle and all these discussions if you are in the

coordinate system in the principal axis system then the shear terms vanish.

There are no shear stresses no shear strains if you have found the principal coordinate

system.

And that is exactly this condition here and if we express this more mathematically at

the end we end up with this so called eigenvalue problem.

So lambda here are the eigenvalues respectively the principal strains the Hauptverzerrungen

n are the directions for these unit vectors of this rotated basis system.

So this would be the principal directions of strain in this case and we have to seek

for solutions of this system.

It looks like a linear equation system with a zero on the right hand side so if I copy