So here we are. Good. So and to summarize what we did last week, let me just recall

you this picture of the bar, the round bar that we pull where we can distinguish

the undeformed surface area A from after pulling in the case of metals that could

be the snacking, so that probably the cross sectional area here changes.

Thank you. And then this is the area after the deformation and the deformation

of course is due to action of forces and with that of course we have the

possibility to introduce tractions and therefore also stresses as force per

area before and after the deformation. That was the basic thing and that

introduced what is called the nominal stress. That is the essentially force

per area undeformed and we could compare that versus the so-called true stress

which is force per deformed area here, the lowercase a, and the nominal

stress for historical reasons is usually denoted by P and it is related to the

true stress sigma by this little formula that involves the so-called cofactor of

the deformation gradient and that is simply a reminder to us that the

Nansen's formula that relates essentially the vectorial area elements

before and after the deformation is involved in these considerations. The P

as I said is called P because it refers also to the name Piola so we also

call that the Piola stress and the true stress in honor of Cauchy because he

introduced that concept of stress is usually called the Cauchy stress. The

Cauchy stress as you know is the stress that acts in the deformed configuration.

This is what we call a spatial tensor. So all the indices are lowercase indices

that live in the deformed configuration and the Piola stress is as the

deformation gradient also an example of a so-called two-point tensor. You can

easily understand that if you consider that that P times the surface normal

in the undeformed configuration gives the contribution to the force in the

spatial configuration. So the left side is the spatial index and the right side

is a material index and in this regard P lives in these two configurations that's

what we call the two-point tensor. Okay there is an important consequence of

this what is called a Piola transformation. Quantities that are

related in this fashion are denoted as being related by a so-called Piola

transformation. So this is an example of a Piola transformation and the

consequence of that of the Piola transformation is that also the

divergence of these quantities has a particular relation and this is

something I just want to remind you. The consequence of the Piola

transformation of the stresses in this case is the following that if you

evaluate the divergence with respect to the coordinates in the undeformed

configuration and maybe let's for the fun of it just add here this is the

undeformed configuration this is the deformed configuration these potatoes

that we had have drawn usually is now this concrete example of the bar. Yeah

yes so if you evaluate the divergence with respect to the coordinates that

describes this undeformed configuration of P that is related to the Jacobian to

the determinant of the deformation gradient times the divergence of I

evaluated in the deformed configuration and applied to Sigma and why is that so

important we will see later in a couple of minutes when we discuss the balance

equations let's say in quotes the equilibrium equation that is expressed

in terms of this divergence of Sigma and then we can likewise express that in

terms of the divergence of P which is in a sense easier to handle because here

the derivatives with respect to the coordinates that we know before the

deformation are of course in a sense easier to evaluate than derivatives

with respect to coordinates that we don't know that are part of the solution