transform the Slant crack model to straight crack model 1

[Wazy01]

Hi
for plate with slant crack loaded uni-axially, If we want to simplified the case by transforming the axis of loading along the axis of the crack and re-modelling the case as plate with a straight crack under tension(remote Stress*sing(angle)^2), and shear (remote stress*sing(angle)*cos(angle) just like defined in fracture mechanics book of Anderson (i.e see

https://en.wikipedia.org/wiki/Stress_intensity_factor)is

well understood. But is it possible to simulate the plate with slant crack by a plate with straight crack but under equivalent bi-axial loading.

Can we simplified the case to the case of straight crack in bi-axial loading. Are they equivalent? is the shear component equivalent to compression Sx.

Thanks

 

[rb1957]

won't the slant crack be growing across the +ve principal stress ?

You can account for the 2nd principal stress (along the crack, not across it) with a simple SIF.

another day in paradise, or is paradise one day closer ?

 

[Wazy01]

Thanks rb1957 for your response

To clarify my question, please see the attached shows different simulate

First, it is Known that we can simulate/approximate case A in Case B. But is Case C (bi-axial loading +straight crack = uni-axial load+inclined crack)an approximate to case A??
is the shear stress component SIF in slant crack equivalent to the transverse load applied along the crack such in bi-axial load.

You may ask what is the objective of all of this?
As we have SIF formula that valid for different a/r ratios in the case of tensile crack (as in Bowie) we can calculate the K0 as the crack grow including the effect of (free surface, finite width, etc). Then the calculation of crack growth can be done straightforwardly.

How to calculate the KI,KII (DK) for slant crack as the crack grows (a/r) knowing the crack path changing when there is hole or finite width..etc. it well documented for infinite plate. But when there is free surface effect (edge crack, crack from hole) which has effect that changes as the crack grow and the path change is there a direct formula / close solution for K. well the two components are function of Ko (tensile) in terms of the angle. If we can simplified the case of slant crack in terms of the straight crack (including these effects) then may this could be solved.
Any suggestion or guidance will be great help
thanks

 

[rb1957]

IMO, no.

Case A is showing a principal stresses of sigma and zero.
Case B is Mohr's circle, stresses at an angle to the principal direction. btw, you should have the 2nd normal stress (parallel to the crack direction), but also this is negligible so ...
Case C looks like invention ??

another day in paradise, or is paradise one day closer ?

 

[LiftDivergence]

No, I do not think your Case C is equivalent to Case A. The shear stress will have mixed mode cracking effects that are not approximated by the application of a biaxial state as you show in my opinion. To be sure of this, you can just look at the SIF (or beta) solution you intend to use. For example, if the solution you are looking at for the biaxial case is exact (I.E. something like Muskhilishvili's method has been used) very specific boundary conditions will have been applied. You need to look at the solution and the assumptions which have been made.

Instead of trying to simulate your scenario with an approximate model, I feel you should just look for a solution for a slant crack under tension in the first place. They are readily available in handbooks. Isida has covered this I think. Maybe I'm remembering his work for kinked cracks though... I'd have to look. He likes (liked) to use the method of Laurent Series Expansions.

Above, you mention you are interested in this because the presence of edges and holes complicates geometry of your known inclined crack solution. Are you talking about the work of Sih, Paris, and Erdogan from Vol 29 of the Journal of Applied Mechanics in 1962? That is the most common for an inclined crack away from edges I think.

I know of a couple beta solutions for inclined cracks in finite plates and inclined cracks at stress concentrations. Can you draw out the finite geometric configuration you are specifically looking to model?



Keep em' Flying
//Fight Corrosion!

 

[Wazy01]

Thank you rb1957, and LiftDivergence for your valuable comments and generosity with your time to response and help! so much appreciated

rb1957
1- Yes
2- Thanks to bringing this important point I already forgot. it stated somewhere that the 2nd normal stress in this case only has effect on crack path(literature.
3-Case C, it is not my invention , It is my question if it correct or not? It seems not. Where from came my question? well, While I was looking for solution to my problem (which is should be available somewhere), I read from Schijve (Biaxial fatigue of Mtal) that (Simple sheet specimens with oblique/slant through cracks shown in Fig. 19 can be used to obtain basic information about fatigue crack growth under biaxial load sequences)
This sentence seemed to me it might be implying there is an equivalence between the cases A,C.

LiftDivergence
1- I'm actually looking for a SIF & crack propagation solution for slant-crack emanating from hole/circular-or-ellipses
2-Thank you so much to highlight the work of Isida (it discussed the slant crack form the point with max.stress on the hole), see attached please to what Im looking for(in case A, the crack emanating from a (1) hole or (2)ellipse (its longer axis on the same crack line). to predict the life,path, the SIF formula (KI,KII) needed for wide range of (a/r) covering the plate width, or longer than the effect of the hole.
3- i found the paper but it shows only central slant line crack without concentrator

Comments and suggestion are appreciated
Thanks

 

[rb1957]

I think Schijve is saying that an inclined crack in a generally loaded panel (normal and shear) is like a normal crack growing against the max principal stress (or bi-axial stress).

another day in paradise, or is paradise one day closer ?

 

[LiftDivergence]

Here are some excerpt beta solutions for the configuration you are trying to model based on your sketch, I think (see link - the ones with holes are a few pages in). Note for the ellipse you can treat a/b as 1.

These are from Murakami but the authors / methods / accuracy are listed. Since the solutions at holes are not for finite width panels you could try compounding / superimposing with the finite with plate solutions in there.

Hopefully these help you get closer to what you are trying to do.

https://drive.google.com/open?id=1eejSmAGIFp5PI07Uv20hrgDppM_1LIdu

Keep em' Flying
//Fight Corrosion!

 

[Wazy01]

Thanks LiftDivergence for your response
The case in my attached is not there. if you see that the case is exactly (hole-with-symmetric cracks) rotated in different angles. so the the slant crack axis is passing through the centre of hole. it is different form slant crack at the tip of hole.

Thanks

 

[rb1957]

you could transform your cracked hole to an equivalent tip-to-tip crack and then analysis an inclined crack in a panel.

why does the crack want to grow on an incline ? (or rather why do you want to analyze a crack at a hole growing on an incline ?)

An inclined crack at an ellipse is harder to analyze. If this is really what you need to solve, I'd suggest FEA.

another day in paradise, or is paradise one day closer ?

 

[LiftDivergence]

I think what rb1957 is suggesting and what I alluded to above, is I think with the solutions I provided and possibly some other handbooks you should be able to do SIF compounding or superposition to get an approximate solution you are after. You have a solution for a slant crack in a finite width panel and also for a slant crack at a hole in a finite width panel. There is some great work by DP Rooke on methods of compounding. I have these papers but I'm not sure about the legality of sharing them as they are journal articles. One good way to get them is through University Libraries which keep engineering periodicals. I've got hundreds of these papers that way. On compounding:

- "An Improved Compounding Method for Calculating Stress Intensity Factors", D.P. Rooke, Engineering Fracture Mechanics Vol. 23 No. 5, pp. 783-792, 1986 (This has an example similar to what you are trying, but for non-inclined cracks)

- "Simple Methods of Determining Stress Intensity Factors", D.P. Rooke, F.I. Baratta, and D.J. Cartwright, Engineering Fracture Mechanics Vol 14, pp 397-426, 1981

- "Compounded Stress Intensity Factors for Cracks at Fastener Holes", D.P. Rooke, Engineering Fracture Mechanics, Vol 19, No. 2, pp 359-374, 1984

Overall, I think you have three options:
1. As mentioned above, try a simple methodology for SIF compounding. It is hard for me to help more than this without knowing all the specifics of what you're trying to do.
2. Obtain your own beta solution from scratch using analytical methods, or as rb1957, possibly FEM
3. Dig into some research to see if the solution has already been found.

With regard to option 3, here are some papers I found at a cursory glance:

The mixed mode problems for the cracks emanating from a circular hole in a finite plate
Engineering Fracture Mechanics, Volume 32, Issue 2, 1989, Pages 279-288
C.W. Woo, Y.H. Wang, Y.K. Cheung

Analysis of cracks emanating from a circular hole in an orthotropic plate under mixed mode deformationOriginal Research Article
Engineering Fracture Mechanics, Volume 31, Issue 2, 1988, Pages 237-248
S.K. Cheong, C.S. Hong

I can't promise these will give you exactly what you want, but the first one sounds pretty promising based on the abstract:

"The stress intensity factors of two cracks emanating from a hole in a finite plate is analysed by the Muskhelishvili formulation and boundary collocation method. For Mode I case, the present results compared very favorably with the existing solutions. For the mixed mode problems of inclined cracks, the KI and KII values have been obtained for varying crack-length to plate-width ratios, a/b, and different crack angles. It has been demonstrated that the convergence of this method is satisfactory. The proposed stress functions and the calculation procedure may be extended to more complex geometrical and loading cases."

They are available online for ~36.00 USD I think. If you can wait until the weekend I can get them from my library and tell you for sure if they have the solution you're looking for.


Keep em' Flying
//Fight Corrosion!

 

[LiftDivergence]

Right now I'm not getting on option to edit my last post, so I will make a follow up here:

I have obtained both of the papers I mentioned above, they are both very useful for your question. The first paper in particular "The mixed mode problems..." contains a tabular solution for beta values (referred to as "Y" in the paper) for two inclined cracks on either side of a hole in a finite with plate for an a/b of 0.27 - 0.90, and angles from 0-90 degrees. Both KI and KII.

The journal distributor is Elsevier - from their website "Permission of the Publisher is required for resale or distribution outside the institution and for all other derivative works". So I can't send it to you.

Check your local university or commit ~36.00 USD if you think it would be worth helping you with your task.

https://www.sciencedirect.com/science/article/pii/0013794489903007

Keep em' Flying
//Fight Corrosion!

 

[Wazy01]

Thank you so much LiftDivergence for your help and the paper.
It is really interesting and very helpful paper.

Unfortunately the presented results were limited for the case presented, i.e.; the hole is 0.25 of the plate width. similarly the range of the crack length/hole radius was less than 2.5.
I am looking for the case when the hole very small compare to the plate width in the range of 5% maybe. It seems possible to derive the SIF-values for different cases using the complex function in the paper but I do not think I can derive it this way as it may need advanced mathematics skills for finding the stress functions.

Kind regards

 

[Wazy01]

Hi LiftDivergence
I am working to extend the procedure adopted (complex stress function) in the paper of Woo et al 1989 to the case of a plate with smaller hole R to plate width (R/w=0.1) Then the driven K value will cover a wider range like a/W from 0.1 to 0.9.
The series equations are so complicated to apply even the procedure clearly presented. Do you have any idea if any routine published in this regards, please?


Thanks

 

[rb1957]

Is there something that justifies this search for Truth, or "Truth" ? How certain are you of the loading/spectrum ? (I'd expect there's so much uncertainty with the loads so that precision with the crack tip geometry isn't warranted.) How certain are the material properties ? (What material are we talking about, Al, metal, composite ?) Have you considered a probabilistic approach ??

If you've got a small hole and a bi-axial stress field, isn't a crack at a hole under tension (max principal), or bi-axial if you must, accurate enough ?

If you want a particular direction, then you can get the stress distribution around a hole easily enough. Or FEA.

another day in paradise, or is paradise one day closer ?

 

[Andries]

Not an answer to your query, but to obtain information on fracture mechanics problems arising in engineering practice in general I recommend the book "The Practical Use of Fracture Mechanics" by David Broek. He deals with real-life problems in FM, topics not always touched upon in more academic books. New editions are quite expensive but one should be able to buy a used copy.

Andries

 

[LiftDivergence]

Wazy,

Apologies for dropping this - I don't lurk on this forum nearly as much as the aerospace related ones.

Which angle are you specifically interested in, if you are still working this issue?

Keep em' Flying
//Fight Corrosion!