Welcome to our today's session.

Our today's second session here.

First of all I was in contact with a colleague responsible for the video.

It should be available right now for June 28th or 24th.

I'm not sure. So all the videos should be available right now.

Okay, so let us quickly recapitulate what we did this morning.

We started to discuss the strong form for the mechanical problem in 2 or 3D.

We considered this system here a body in space with a boundary d omega,

which consists of a Dirichlet and a Neumann part.

We specify a point in this body by its position vector.

This point can have a displacement.

We consider the loadings, the body forces acting in volume and surface tractions acting on the surface of the body.

In order to derive the strong form we have to do 3 basic steps.

The first step we already discussed is kinematics, which is the relation between displacements and strains.

We now continue with the balance equation.

Let me switch here.

Please interrupt me whether there are any issues to be discussed concerning the general problem or the kinematics.

Otherwise we just proceed here.

This is now the balance equation.

And to this end we set up the balance of forces.

Let me just copy this sketch.

Maybe it's better quality from the lecture notes.

So this is our problem.

And the balance equation, as I said, this is a balance of forces.

I just have to find it here.

Balance of forces.

Or one can also say the balance of linear momentum.

So we have two kinds of forces or force like quantities acting on the system.

These are the body forces and the surface tractions.

So all we have to do is the following.

We have to sum up all the body forces.

And summing up this field of body forces means to take the integral over B as a function of x over the entire domain.

Plus the surface tractions acting on the system.

So here we need the surface integral of T of x dA equals zero.

This is a static problem.

In case of a dynamic problem we would have to consider on the right hand side the inertia terms.

But this is not to be considered here.

And we also consider here that the surface traction is nothing else than the stress tensor times the surface normal vector.

This is all we have to do with respect to the balance equation.

And now we can proceed with a constitutive equation.

Which establishes a relation between the stresses and the strains.

So in general we can write sigma.

You remember in the one dimensional case it was just sigma equals Young's modulus times the strain.

Here it is getting a bit more complicated.

Here we have an elasticity tensor of fourth order.

Which is double contracted with a strain tensor.

I will come back to that in a minute.

So C is the elasticity tensor.

And this is a tensor of fourth order.

And we can write in index notation sigma has two indices.