Stress tensor characteristics

[toothroot]

Hi experts,

For a fatigue analysis of a part, I need to compute principal stresses from nodal stress tensors. The fatigue analysis will we done by a self-written programme, input comes from FE-result files. The problem is that I have to take a general 3D stress state into account. Thus, the principal stress values can be solved in two ways:

1. by computation of the eigenvalues of the stress tensor S

2. by finding the roots of the polynomial
s^3+I1*s^2+I2*s-I3
where I1, I2, I3 are the three invariants of the stress tensor.

The two approaches are actually identical, because the polinomial in 1. is the characteristic polinomial of the stress tensor and its roots therefore the principal stresses.

My questions would be:

1. Is the stress tensor always a regular matrix? Can it become singular? What would be the physical consequence of a singular stress tensor?

2. Are stress tensors possible with imaginary roots? What is the physical meaning of that?

3. Does anybody have a hint what routine or algorithm is best suited for solving that problem (calculating prinicipal stresses from a general stress tensor)? The algorithm has to be absolutely stable, because I wouldn't want to see it crash after a night of computing because of an arkward combination of numerical values. The model size could be quite large.

Thank you very much in advance,
Daniel

 

[EdR]

For any real structure the stress tensor (3x3) will be symmetric (an equilibrium requirement e.g. tauxy = tauyx) and the roots (principal values) will be real.

If the full 3x3 is used to solve a twod plane stress problem the 3x3 will be singular (sigz=0). In that case just solve as a 2-d problem and set sigz to 0.

Computation by either method you have identified will give what you want but you must check for a number of special cases (such as a 2-d case) during the solution..

I'd also be a bit careful when looking at nodal stresses. Depending on how they were determined they may be fine or they may be garbage...

Ed.R.