Hi Felix,
very interesting thread. I am looking at the paper you mentioned, but for the time being I had the following remarks:
- the implementation in CalculiX with NLGEOM is a linearization of the multiplicative decomposition, i.e. alphaDelta(T) is subtracted from F before building E and it is not subtracted from E. Thus, the approximation is rather F/(1+alpha Delta(T)) approx F(1-alpha Delta(T)) approx F-alpha Delta (T) I, which of course is only valid for alpha Delta (T) small
- for usual applications (e.g. steel with Delta (T) = 1000) alpha Delta(T) is 1.2e-2, which is small. The application here is an exception.
- I was not aware that also S had to be modified, this may have been the essential point.
- also the tangent has to be modified, however, this should only influence the convergence rate (or, in the worst case, lead to divergence).
- If I am not wrong, not doing the linearization of the multiplicative decomposition leads to an expansion coefficient which is logarithmic, i.e. it is based on the actual length and not the initial length while measuring the expansion coefficient. The usual way to measure alpha, though, is linear, i.e. based on the original length. This problem actually occurs in Abaqus: if NLGEOM is off alpha has to be entered as linear, if it is on, as logarithmic. There is a formula to switch, however, if e.g. the user switches on NLGEOM in some step, this causes problems.
- I will look into that in the next days/weeks can get back. Thanks for digging into that!
Guido