What is a stress singularity and why does it disappear at certain angles?

[Xishan]

Hello!
I've found in my FE Analyses (For an 'L' shaped model) that for each value of Poisson's Ratio there exists a certain substanded angle above which the elastic solution has no stress singularity. Can somebody tell me what really a stress singularity means mathematically? and that why does it disappear above/below certain angles. I would also appreciate a detailed mathematical treatment of this problem.

Thanks,
Xishan

 

[Xishan]

Is it that I've posted in a wrong forum?

 

[Clausius2]

Hello!
I've found in my FE Analyses (For an 'L' shaped model) that for each value of Poisson's Ratio there exists a certain substanded angle above which the elastic solution has no stress singularity. Can somebody tell me what really a stress singularity means mathematically? and that why does it disappear above/below certain angles. I would also appreciate a detailed mathematical treatment of this problem.

Thanks,
Xishan

You don't mistake posting it here. I don't know about it very much, but I recall when studying FEM a stress singularity was a point in which elasticity equations give an in infinite stress. It can be found in axisymmetric bodies due to central symmetry or on points in which you are directly applying a load. Maybe a better answer would be provided if you especify the load acting on your body.

 

[PerennialII]

You're familiar with the Williams' series solution ? It's as far as I know a first complete solution for general loading (there've been specific ones like Westergaard's etc. earlier) for elastic stress singularities arising from a corner of a specific angle. When the corner "converges" the case reduces to a crack, which it is typically intented for, but the Airy stress function can be solved for whatever angle (using traction free boundary conditions), as such altering the singularity. For a isotropic linear-elastic medium with a crack the stress singularity is the "familiar" $$\frac{1}{\sqrt{r}}$$, where $$r$$ is the distance of a material point in a polar coordinate system stationed at the singularity.

For the detailed theory & derivation of the solution look e.g. :

http://www.engin.brown.edu/courses/En224/airyexample/airyexample.html
http://ceae.colorado.edu/~saouma/Lecture-Notes/lecfrac.pdf

 

[Xishan]

Well! let me see... the second link that you've mentioned doesn't contain the full text. As far as I can recall from my undergrad course, the Airy's stress function doesn't work here. It doesn't take the geometry as input does it? For example it doesn't consider the crack angle, all it will give is the point where stresses are infinite and that I already know to be the corner... or may be I'm not getting it right! please help

 

[PerennialII]

The stress function developed by Williams is for a boundary layer type of an analysis (i.e. an infinite domain) with generic far-field boundary conditions. The solution of the stress function for different corner angles (360 being a crack) will give you different kinds of singularities depending on the corner angle ... which I think is close to what you're doing in your FEM analysis (with a different geometry for sure, but the corner is most dependent on the angle and so in principle it ougth to work fine).