I'm just reading about the P-K stress tensors, while I've seen Cauchy stress tensor in a course a couple months ago. From a technical point of view, I think it's clear what they represent.
Cauchy stress tensor gives the force on the current configuration with respect to deformed areas on the current configuration. So if $da \hat{n}$ is the infinitesimal area vector in the current configuration, the force would be $df = T (da) \hat{n}$.
First P-K stress tensor gives the force in the current configuration but by means of the undeformed areas in the reference shape, so that $df= P (dA) \hat{N}$ (I denote with capital letters everything in the reference configuration)
Second P-K stress tensor does the same thing of the first P-K tensor, except it gives the pull-back of the force $df$ via the deformation to which the body is subject. So if the deformation is $F$, the second P-K tensor $S$ should be such that $S (dA) \hat{N}= F^{-1} df$
Now, the question is: why do I need them? When do I use the first, when the second, when the third? I'll try to answer myself, but please tell me if I am getting this right.
The first one is pretty clear: you have a deformed body, you measure the deformation and then find the stress forces applied to it. The problem is the other two: when/how to use them?
I have not an engeenering background, so sorry if this is a trivial question but I'd really like some practical, intuitive examples. (Also, if my understanding on the definitions of the three tensors is wrong, please tell me!)