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Stress concentration at a corner of varying angles, mode I and mode II

I was looking at the stress concentration at a corner for both a mode I and mode II loading situation.

For reference, see: Carpenter, 1984. Mode I and mode II stress intensities for plates with cracks of finite opening. International Journal of Fracture. V. 26. 201-214. 1984.

The problem that is being addresses is the stress concentration found at a corner in an infine body as seen in the figure below. β=0...π.

In general the stress at the corner is a function of r(λ-1) and displacement a function of rλ. We know that stress should go to inf at r = 0 and displacement should be bounded. As such we know that  0 < λ < 1.

When the body is loaded with a remote stress of σyy = σ∞ with all other σ = 0, we have λ as a function of β as shown in the figure below as 1st Eigenvalue (Case I). When we have τxy = τ ∞ with all other σ = 0, we have  λ as a function of β as shown in the figure below as 2nd Eigenvalue (Case II).


The paper presents the work that shows, mathmatically, why these values of λ are what they are, but I am interested in a physical reasoning for these values.

For case I when β = 0 we end up with the same equations for stress and displacement as we have when we have a crack. When β = π, we have the equations that describe the situation of a body with no crack at all. So for case I, I can back the math up with a physical reasoning.

For case II I am able to relate to the β = 0 as a crack with mode II loading. The equations work out the same. What stumps me is why the stress at the corner (~ r(λ-1)) no longer goes to ∞ at  β ≈ π/2 rad. When β = π, we would have no corner and I undestand why we wouldn't have infinite stress. But why does this transition from an infinite stress to a non infinite stress happen at the angle that it happens at?

I can follow the mathmeatical reasoning as to why, but physically what is an explanation for this? Any thoughts?

If anything is unclear just let me know and I'll do my best to clarify.

Dear Justin,

Seems like you have raised a good question.

I also wonder why it went without being answered. This is unfortunate. If an equation is written and manipulation thereof sought (or a mere English restatement thereof is what is being expected), any number of sauve replies could be had. But the moment one seeks a physical explanation, the flow of answers dries up, is it not? ... LOL!

I think I would be able to answer the question. (And, yes, I also think that the query is genuine.) But I have no access to eJournals. Please post an ePrint of the paper (by Carpenter that you refer to) here, or if this is not possible (out of copyright issues), please see if you can send it to my email:

Also, after that, it would also help if I get to know your background and preparation: are you a graduate student? In what kind of research program or department? Please clarify.

-- Ajit


Ajit, I have emailed you a copy of the paper, so please check there for it.

About me, I am a 2nd year graduate student pursuing my masters in Engineering Mechanics at The University of Texas at Austin. I have a mechanical engineering undergraduate degree. I am currently taking a course on fracture mechanics and we solved this problem for homework which is why I found the paper.

I just can't seem to figgure out a physical reason as to why this stress concentration drops to a non infinite value after a ceartain corner. I mean, obviously the stress doesn't go to infinity in actuality; plastic deformation occurs. But that isn't the issue. The issue is why does the stress concentration drop to a very low value before the angle of the corner is such that we just have a flat edged plate.

dear Mister I have a study of simulation of the piping elbow;whith calculated of J-integral; which is the negative significance of J-integarl mercy

j'ai preparer mon etudes sur les coudes d'un pipe (pipe elbow) avec l'utilisation de code Warp3d pour calculer l'integral j pour une fissure demi elliptique a traver l'epaisseur de paroi, le probleme c'est que l'integral soit trouver négative ou le code ne peu pas le calculer pour les tailles de fissure défferentes! alors que ! comment ou qu'est ce qu'en fait pour le permettre du calculer?

Ij' have to prepare my studies on the elbows d' a pipe (pipe elbow) with l' use of Warp3d code to calculate l' integral J for an elliptic crack half has traver l' thickness of wall, the problem c' is that l' integral is to find negative or the code not to little calculate it for the deferent sizes of crack! whereas! how or qu' is this qu' in fact to allow it to calculate?


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