Dieser Audiobeitrag wird von der Universität Erlangen-Nürnberg präsentiert.

Ja, herzlich willkommen.

Ladies and Gentlemen, sorry für diese Vertauschung der Termine und dass dann auch noch einer

ausfällt, aber der Herr Friedrich ist dann auch leider noch krank geworden und ich konnte

heute Mittag nicht.

Na ja, gut.

So anyhow, I guess we were talking about the balance equations last time and we discussed

briefly the balance of mass.

That's the first example and just to remind you, there is, the only point I want to make

here is that we have different options to express one and the same statement, in this

case the conservation of mass.

The probably most easiest representation is to say that the material time derivative of

the referential mass density is zero, which has immediate consequence that the referential

mass density is only a function of the position and not of the time.

And then just as a little warm up for whatever follows, we can alternatively write this equation

in two other formats, either as the material time derivative as applied to the spatial

density and then this reads as like this.

And then by changing the material time derivative to the spatial time derivative, the time derivative

at fixed point in space, the end result is essentially that the mass density goes inside

here this divergence.

So alternative to that we could apply here what we have called the spatial time derivative.

I remind you the dot denotes the material time derivative at fixed uppercase x and if

we build in the relations here between these time derivatives, then at the end we can

rewrite this equation here in this format, where the row now goes into the divergence.

And in textbooks you find either of these versions, typically when the background of

the author is more from solid mechanics, you will find everything in terms of the material

time derivative and when the background is more in fluid dynamics, you will find everything

in terms of the spatial time derivative and then of course it matters where you put the

density outside or inside the divergence, as long as the density is a function of the space

coordinates.

If the density is of course homogeneous, constant everywhere, then it doesn't matter, because

then its gradients here would anyhow produce zero.

A homogeneous distribution of density is an assumption that is often made also in fluid

dynamics or sometimes made, but in general of course we have these two different options.

And again, either of these four different expressions are the same statement, namely

mass is conserved.

Just different ways to express the same relation or the same expression or the same statement.

So that was let's say the easiest case.

Now let's go forward to the next balance equation in our list and that is the balance of linear

momentum, the impulse, the normal impulse.

So secondly we are talking about linear momentum.

And for this we should firstly remind ourselves the definition of momentum.

So let's say the linear momentum density, linear momentum you remember is mass times

velocity.

And then the density is of course mass density times velocity.

So here again we can now distinguish between two different versions.

So either we define the linear momentum density per unit volume in the undeformed body, then

we have this mass density times velocity.

Or we define the linear momentum density as density per deformed volume, unit volume.

But in any case it has this flavor mass times velocity per unit volume.