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# Stress intensity factors for a slanted crack under compression

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Hello everyone,

Here is the problem I have: I'm modeling the geometry of a simple straight edge crack in a 2D elastic medium using Abaqus. I assume plain stress conditions. The crack makes an angle with the horizontal, is small enough to be considered as embedded in an infinite domain (ratio crack length/size of domain < 1/10) and I apply a vertical compressive load on top of my domain. I fixed one point in displacement at the bottom of it and the whole bottom edge is constrained not to move vertically.

Abaqus allows the computation of the J-integral and the stress intensity factors KI, KII by use of the **Contour integral** keyword. Results in tension are pretty accurate, however I seem to get problems when it comes to compressive loads. According to many authors ([Maji91], [Lauterbach98], [Rao03], [Zhu05], [Albrecht06], etc.), this problem gives non-zero shear mode SIF (**KII=σ∞√(πa).cosα.sinα**), which is understandable, but zero mode I SIF, which I explain as the closed character of the crack under such conditions. Abaqus gives rather good values for KII and their path-independence is correct, but also gives very negative values for KI, of the same order as the values of KII. The J-integral is calculated with Abaqus according to the following formulation (as defined by Rice, for instance):

**J=lim_{Γ→0}∫_{Γ}[Wδ1j-σijui,1]njdΓ**

This integral is then transformed into a surface integral (using the well-known ramp function **q**), and an interaction integral method can be used to separate modes I and II stress intensity factors, using the fact that in the framework of linear elastic mechanics in an homogeneous medium, we have:

**J=1/E'.(KI2+****KII2****)**

Of course the compressive character of the fields is not taken into account in the definition of the J-integral presented above, which is valid in case the crack is opened. I have been trying to reconstruct the analytical field around the crack tip in the conditions of my problem (that is a slanted crack subjected to a remote compressive vertical load in an infinite domain), but did not quite manage to do so using simple terms. My questions are:

- Do you think there would be any way to "extract" the compressive part of the stress field and substract it from the whole field to get the singular field around the crack tip -- then use this field to compute the SIF?
- Does the negative character of mode I stress intensity factor physically mean anything? I personally would not think it does, since plugging those negative values of KI into the singular displacement field fomulation around a crack tip (in √r) would impose an interpenetration of the crack lips.
- Is there any closed-form formulation for the displacement/stress fields for such a problem?

Thank you in advance for any remarks/suggestions you may have regarding my issue.

Kind regards,

Julien Jonvaux

Ph.D student at the University of Illinois at Chicago

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## Comments

## Some comments

Julien,

Some comments. If you are not actually imposing contact constraints then finite element simulations will allow the crack faces to interpenetrate. This is why you are not getting zero for KI. The answer to your question 2 is yes and no. As you have realized, if KI < 0 then this implies that the crack faces overlap. It also implies that the stresses ahead of the crack are compressive and square root singular. However, contact will prevent this from happening. The answer to your 3rd question is also yes. If you are interested in a centered through crack in an infinite medium then there is quite a bit that you can do. You should even be able to solve the contact case. Contact me if you are interested in the details.

Chad

P.S. I just re-read your post and noticed it's an edge crack, not a center crack. You still might want to look at the center crack if you are interested in analytic solutions.

## Negative K2

Hi I am modelling a plate with two non aligned cracks

I got negative value for K2 at one of the crack tips from ABAQUS

If I derive K2 from the post processing of nodal forces and displacements from MVCCI technique

Then K2 = (G2 x E) ^ 0.5

If I get negative G2 how do I get K2from it

thank you

M.Surendran

## Re: Some comments

Dear Chad,

Thank you so much for your reply. I actually forgot to mention that my model does impose contact constraints on the crack faces so that they cannot overlap (use of penalty terms). However Abaqus still gives negative values for KI, which I was quite surprised about. The stress ahead of the crack should have shear terms (which is the case, as KII≠0 and the crack faces slip on one another), but as you mentioned, no square root compressive terms (which I apparently get). The J-integral maybe needs to be redefined another way in case of compressive loading, that is what I'm trying to figure out.

Anyhow, I would be very interested in the analytic solutions for the problem of a center crack in an infinite medium, this would probably help me a lot. My email is jjonva2@uic.edu.

Thank you very much, again, I highly appreciate your help.

Julien

## Slanted crack

Julien,

The case where the body is under uniform tension at infinity was solved by V. V. Panasiuk and A. P. Datsyshin (see their paper, On the limiting equilibrium of a half-plane with an arbitrarily oriented crack at its boundary, Soviet Materials Science, vol. 7, No. 6, 1974). To my knowledge, there is no exact solution to this problem available in the literature. You can also look at the Handbook of Stress Intensity Factors by Murakami, to see if there is any exact soultion derived by any one. The Panasiuk-Datshyshin solution is based on a pair of singular integral equations derived by using Muskhelishvili's complex variable approach.

Regarding the case where the solid is under compressive load, you can introduce some kind of contact condition of the crack faces, and then use the same approach as Panasiuk-Datshyshin. Of course, in this case the problem willl be lot more complicated, but from a purely conceptual standpoint, I do not see any signficant problem. You should also look at the works of Dundurs and Comninou, who did a considerable amount of work on crack problems involving contacting crack faces.

Hope this helps!

Mujibur Rahman

## Negative stress intensity factors for crack growth

Hello, I am using a simple Paris law for crack growth, and in calculating Delta K, one of the values I get is negative, should I set it to zero before calculating Deta K? For example. DK=Kfinal-KInitial, and say KFinal=20, and Kinitial=-5, is DK 20 or 25?

im using analytical solutions for tge Ks, based on polynomial distributions, which allow negative values as far as I can see. Physically it doesn't make sense to increase the range if there are compressive stresses at one point in the cycle. Thanks !

## Hello,

Hello,

It is not clear to me what you meant by "analytical solutions for tge Ks, based on polynomial distribution, which allows negative values". But, anyway, I agree with you that you can set Kinitial to zero in cases where they turn out to be negative. As a matter of fact, the crack faces would close even before the complete unloading, because of the plastic deformations at the crack tips left behind by the maximum loading -- a phenomenon discovered by Elber back in 1968. Therefore, the effective DK would be even less than the DK that you would get if you set Kinitial to zero when Kinitial is negative. You may want to read a little bit about the Elber crack closure concept.

Hope this is helpful!

Mujibur Rahman