이것伊ndar

Cool!

So...

At this point we talked about which stresses and which strains form pairs

and we did this based on the combination of stresses and strains that make the same stress power.

If I go back two slides, we were discussing the kinetic energy theory

that expressively changes the kinetic energy due to the difference of the external forces

and the force of the internal forces changing the force.

The internal power is essentially given by this integral here, the polar force,

the gradient of the velocity, which is essentially nothing else than the individual forces of the formation of the waves.

And then we tried to combine or modify this expression here

to combine different stresses and different strains.

So we went through all that and without going into details, we were acting like this,

we were acting with this, let's say, acting with the expression for the stress power

and then using that expression to see that we get combinations of other stresses with other kinds of energy relatives,

and we can do so in other situations.

At the end of the day, I think this is the result that we got two weeks ago

that they were all stress and strains from the Heng A Tso Tzu Tzu Piola stress and the deformation gradients.

And the same power is used by combining, for instance, the Piola stress, which actually means the symmetric stress,

so kind of like symmetric stress as well as dynamic symmetric stresses,

and the Piola stress then performs work with these hand derivatives,

and these hand derivatives, E or C, okay, that's C and C,

so they are essentially the pair S and E or S and E are for these states in conjugation.

And likewise, we consider the so-called push forward of the Piola stress with the C-stress,

then it comes out that the push forward of the C-stress hand derivative of these strain measures

with the C-stress hand derivative of these strain measures, the little E that was a unit strain,

and then the A-strain with the unit strain, that's maybe a little bit strange to think of a unit strain,

but it's a particular time derivative of that time dot that has the same, okay, so the design of pair of

contravariant stresses, the covariant strain, and of course you could also do that for the covariant versus the contravariant strain, also that is a combination that is rarely used, so just here for completeness.

Okay, so this was a little interlude, okay, to play with the expression for the power that is performed by the stresses,

and this brings us to the next balance, the balance of energy.

So we begin roughly with the class that already last semester, so here is only this actual aspect of the kinematics that comes in,

but essentially we find the energy as the addition of the kinetic energy and the internal, so-called internal energy,

and here the density is then the density of the energy per unit volume before deformation, and again, little k and little u are likewise densities of the kinetic energy in the internal energy respectively.

And then for all the quantities for which we wrote the balance, we computed the total quantities for which we contained in the control volume, so this will be the total energy,

and we express that by integrating E0 or Ep over the expected volume. Okay, so the balance of energy tells us that this quantity is changed due to power input if you like,

and this power input can come from various sources, here we will restrict ourselves to power input due to the kinetic power and heat power, which is 50 degrees in temperature, which is the same.

Okay, there could be other inputs here on the right hand side from temperature power or from district or magnetic power, and whatever terms we add here will end this theory further.

Let's say the basic combination of the basic ingredient would be the power that is extended by the total in our body or our control volume will lead to a change of this total energy here as we will add the so-called thermal power input.

The thermal power input consists of two contributions, one is the heat forces, which are 0, which are distributed within the volume,

and the second is the so-called heat flux through the boundary of our control volume, which is here, as you can see, energy per time per area, 0 energy per time per volume,

and again this 0 indicates that these quantities per volume in a certain area, for certain conditions, like for example, heat, 1 sub T, 2 sub T, or 20.5 T per volume and area after the differential.

The external power we have just seen, I go back to record it again.

External power is essentially the power extended by the volume forces, and okay here we still have to bring in the temperature conditions, so we will write it out for you.

So this would be external power, and that is the line S, the total volume, so that is the power performed by the volume forces.

The power is performed by the boundary voltage, or the pressure.

The thermal power is the fraction, followed by the force in theory, as you can see here, and that is the expression that is in here.

So that would be the power input, and this would be the mechanical forces, then the thermal power input, then given that we have seen it already,

so this is the distributed heat sources plus the flux, the heat flux over the boundary.

Of course, if you compare these expressions, then it is particularly convenient to introduce the heat flux in a similar manner as the fraction by a Cauchy-Lac-Theorem.