Good Morning everybody.

Yeah, okay the

So good morning everybody.

Okay.

Yeah, okay.

So as a little warm up, a small quick reminder of what we have been doing last time.

So this again is a picture here of our continuum body and various quantities, strain, stresses,

heat fluxes, temperatures and so on are parameterized in the position X that we shall express in

coordinate form with respect to the Cartesian basis for simplicity.

Okay, maybe a remark here.

The body has a boundary dB and that splits into parts where either the displacements

or the tractions are prescribed.

These are the two parts of the boundary denoted here with the U like displacements and T like

tractions and the intersection of these two parts is empty and both the union of both

sets gives the total boundary.

Okay, then just a little reminder on notation.

We introduced the notation for gradients, derivatives with respect to the coordinates

here in particular the shorthand notation UI, J or in symbolic notation the snubla U.

Okay, we discussed that we have three sets of equations essentially plus solution methods,

kinematics, balance equations, constitutive laws.

That we are going to discuss and as an example we considered the very simple case of statics

in mechanics where we recalled the equilibrium conditions expressed in terms of the stresses

and at the end the equilibrium equations that we developed here by summing all forces in

the various directions can be expressed in this simplified index notation whereby in

particular this derivatives here sigma ij, i denotes the so called divergence of the

stresses and in symbolic notation this notation dh here denotes the divergence.

d are the body forces and we sometimes also call these quantities that show up here in

the divergence we call them fluxes, flusse and the divergence of the fluxes plus this

quantity that we call sources, quellen or senken they add up to zero and we also remember

that divergence sigma as we will see later has to do with all the forces that are applied

at the boundary and b has to do with all the forces that are applied in the body and the

sum of all the forces, the surface forces and the body forces it sums up to zero, simply

equilibrium.

Then as an example for the constitutive law this would be the Hooke's law a linear relation

between the strains, the stress and the stresses.

Here we have a quantity with four indices we call this a fourth order tensor and that

is the elasticity tensor if you count this has three times three times three times three

numbers 81 numbers and we will later see how this reduces to only two different material

parameters in the case of isotropic linear elasticity.

But in general these are 81 numbers that we have in a condensed format in this expression.

In symbolic notation we introduce this double dot here to denote this double summation about

two indices k and l.

You remember repeated indices denotes summation.

Finally the kinematics defines the strains here as the symmetric part of the displacement

gradient in various notations and in particular in symbolic notation the strain tensor epsilon

is computed from the symmetric gradient of the displacement field.

The example for a non-mechanical quantity is temperature and for the case of thermostatics

which is a very special case we also derived the relevant equations from considering let's

say an energy balance where we considered essentially heat power flowing in and out

of an elementary volume and then by equilibrating that with the distributed heat sources at