Dieser Audiobeitrag wird von der Universität Erlangen-Nürnberg präsentiert.

So, welcome to the week of the quarterfinals.

For this lecture I think it's already the semi-finals.

Okay, good. Also, last week we discussed some constraints that we have to put to the free

energy density in order to capture aspects like objectivity or material symmetry.

And today we want to highlight that for the example of elasticity.

And to be more specific we are going to talk today about isothermal.

So, it's the same temperature, isothermal, hyperelasticity.

Okay, so there are two keywords here, isothermal on the one hand and hyperelasticity on the

other hand. And isothermal refers to the case where the free energy density that usually

depends on the strain, in our case the deformation gradient and the temperature, theta, is simplified

insofar as that the temperature is constant and a reference temperature.

It's a room temperature, maybe. Reference temperature.

So indeed temperature is not an argument anymore, it's rather a parameter.

So maybe I indicate that here by the semicolon.

Like let's say elasticity modulus and all these parameters, they depend of course on

temperature but temperature is fixed.

It's not unknown of our problem.

So for isothermal elasticity we can reduce the representation of the free energy simply

to a dependence on the deformation gradient.

We don't have to concern ourselves with the temperature here.

So that is the case of isothermal and hyperelasticity refers to the case where we have zero dissipation.

Remember the dissipation inequality.

Okay for isothermal we can write like this.

So we had the stress power, p int I guess was our notation for that, per unit volume undeformed

configuration let's say, minus the change in free energy whereby this is the material

time derivative.

For general material behavior this expression was positive and for the particular case of

hyperelasticity the dissipation is exactly zero.

And maybe we can add here temperature is only a parameter.

So not an unknown.

Okay.

So we are now considering free energies that depend on the deformation gradient and their

rate if we take the material time derivative should coincide or does coincide with the power

that the stresses perform at the rate of the strains.

So let us recall maybe the representations for the stress power density.

So this internal stress power density in its most elementary format it was just pure stress

times the rate of the deformation gradient.

So if the energy has this format psi not as a function of f then of course its rate is

then its derivative with respect to f times f dot and then from this of course we can

contrast that with the stress power.

So in this case the dissipation is then p minus the derivative of psi with respect to

f multiplied with f dot and for hyperelasticity this is requested to be zero.

So and in order to make this expression zero for arbitrary f dots for arbitrary strain

rates the expression in the brackets has to vanish.

So the conclusion from all this at the end is that this bracket vanishes if p is computed

as a derivative of psi with respect to f.

This is also of course valid in a more general case not only for elasticity.

But now we are going to investigate different expressions for the stress power on the one

hand and for the free energy on the other hand.