Dear Pasha,

 I must correct my answer by replacing "residual vector" by "internal force 
> vector"


Ok, great. So this is a slightly different situation. I was going to say 
that I think that we're cross-talking here, and you saved me from sketching 
out a whole bunch of equations to clarify things :-) 

Yes, we can use internal forces for all situations. 


Hmm... I still don't think so. What about if you have body forces? Then the 
internal stresses balance both the externally applied loads plus the volume 
forces. Do you agree with this? However, one can of course compute the 
internal stress plus the body force loading... 

But then consider a cube with a non-uniform pressure on more than one side. 
How do I compute from this global vector the average or total load applied 
to any *individual* surface? You can't, because the values in the 
degrees-of-freedom shared by any two sides of the geometry contain load 
contributions from two faces. For reaction forces at constrained surfaces - 
no problem of course!

Also, I think that one should consider which vectors one typically 
constructs during assembly. The most efficient way to do assembly (but I 
recognise that this is not universally followed - not even by me) is to 
compute the constrained RHS vector and tangent matrix in one shot during 
assembly. That means that you never need to actually construct the 
individual internal, external and body force vectors in order to solve your 
linear system because, well, they get bundled into the RHS and then further 
manipulated when imposing constraints.

Dependent on what it is you want to achieve, I do concur that what you 
suggest may be perfectly plausible. But in general... I'm still 
unconvinced. To be introspective, what I suggest (and what Hamed has done 
here) might be overkill for plain elasticity alone since you can always 
recompute the traction vector itself as it is you that specified the 
loading at each time-/load-step. In coupled problems, however, external 
tractions are not always so easily defined and its actually sometimes 
easier to simply do this stress-type integration procedure.

However, can we calculate internal force by integrating over boundary 
> surfaces when material model is history dependent and has internal state 
> variable such as plasticity


Yes, you can - its just not very convenient. You could do this in one of 
two ways:

1. Extrapolate the internal variables to the faces and recompute the 
stresses with these values. Its clear that this might be problematic 
because one has to make an assumption about whether there is plastic yield 
at the extrapolated points (as can be attested to by the note in the 
ContinuousQuadratureDataTransfer 
<https://www.dealii.org/developer/doxygen/deal.II/classparallel_1_1distributed_1_1ContinuousQuadratureDataTransfer.html>
 
class - we'd like to extend this at some point and we've already worked out 
a couple of options that may work for plasticity but simply haven't tried 
to implement them yet), so I would recommend option 2...
2. Simply create and track additional quadrature point data (with internal 
data) at the faces at which you'll be doing this post-processing. That way 
you can update the internal variables as necessary and they'll always be in 
sync with the solution field. You can then compute stresses etc. at these 
points whenever necessary. Its annoying if the local problem is expensive 
to compute, but the number of face QP data that you'd create is likely far 
less than those required for assembly etc.

Best,
Jean-Paul

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