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# On the Path-dependence of the J-integral near a Stationary Crack in an Elastic-Plastic Material

Submitted to the Journal of Applied Mechanics on 2/1/2010.

The path-dependence of the *J*-integral is investigated numerically, via the finite element method, for a range of loadings, Poisson's ratios, and hardening exponents within the context of *J*2-flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This feature allows for a dense finite element mesh within the plastic zone and accurate infinite boundary conditions. Features of the crack tip field that have been computed previously by others, including the existence of an elastic sector in Mode I loading, are confirmed. The somewhat unexpected result is that *J* for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for Mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for Mode II. The applications of *T* or *S *stresses, whether applied proportionally with the *K*-field or prior to *K*, have only a modest effect on the path-dependence.

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## Comments

## No one is surprised?

At the risk of looking silly and being the only person that comments on his own post, I thought I might try to stoke some discussion. This work was not actually planned. Maria and I were planning to study crack tip fields in ferroelastic materials and wanted to develop "exact" boundary conditions for the small scale yielding problem. So we did that and the first thing we looked at was standard J2 plasticity. We found that the J-integral was not path-independent. Now, I had expected maybe a few percent deviation, but we were getting much more than this. So we looked for the error. And we looked. And when no errors were to be found any longer, the result persisted. I also contacted 5 colleagues who know a great deal more about non-linear fracture than I do, to see what they thought. I got two responses that expected a few percent deviation, two that expected path-independence, and one that said 20-25% deviation. So we did this study and we are still left with some questions. For example, why does the path-dependence go away in mode II?

## Re: No one is surprised?

Dear Chad:

I put your paper in my reading list as soon as I saw it and now your comment pushed me to read it right away. Regarding the title of your comment, I would say a "theorem" is valid when all its (implicit and explicit) assumptions hold; changing any of the underlying assumptions, the theorem would not be a theorem anymore. So, in this sense it is not surprising to see some path dependency. What is certainly very interesting is to see how path dependent J-integral is when the material is not elastic anymore (and this is what you have done very nicely). Regarding path independence of KII, which is certainly interesting, the only thing that comes to my mind is the following. If I understand it correctly, you have restricted your calculations to circular paths. I would repeat the calculations for non-symmetric paths and see if still J-integral is path independent for KII.

Regards,

Arash

## Other paths

Arash,

Thanks for the suggestion. I will look at some other paths and get back to you. I agree that one should not expect perfect path-independence, but as you said we were surprised by the magnitude. I think that my intuition on this was guided by Rice's results for mode III, were path-independence is rigourously valid in SSY for elastic-power-law hardening materials.

Chad

## Partial answer and re-direct

Arash,

Consider any path that is symmetric about the x-axis and fields obeying pure mode II symmetry. It is strainghtforward to show that the contribution to J from the bottom part of the path is identical to that from the top. Our caclulations suggest path-independence for circular paths. Now consider a path consisting of a semicircle of radius R1 in the lower half-plane connected by a line along the x-axis to another semicircle of radius R2 in the upper half-plane. Since the contribution to J from the line segement along the x-axis must be zero for mode II symmetry, J for this path must be the same as J for all circular paths.

So it appears that the question is not on the symmetry of the path, but on its shape. By that I mean, do rectagular shapes yield path-independent results for J? Unfortunately, due to the geometry of our mesh, answering this question quantitatively would require some retooling that I do not want to undertake at this time. My question for you is, would you expect different results from a rectangular path?

Chad

## Re: Partial answer and re-direct

Dear Chad:

I agree with your argument that knowing path independence for circular paths would imply path independence for any path made of half circles (centered at the crack tip) and joined by line segments (parallel to the crack faces). I don't know what will happen when you consider rectangular paths (or any other path). If there is no proof, one would need a direct calculation to answer this question.

Regards,

Arash

## One new path

Arash,

Again, I agree with what you have said here. As I am sure you already realize, it is impossible for me to perform a direct calculation for all possible integration paths. So, along those lines my hands are tied. However, I did do one other path that is not a circle. It is not really a rectangle either, but in some way it approximates a rectangle and it is not centered on the crack tip and it goes through the plastic zone. Given the coarseness of the path and the fact that I have had to do a path integral instead of a domain integral, the result is in agreement with path-independence for mode II. This path differs from a path in the elastic region by 0.13%. I can give you more details of the path if you would like to see them, and of course I can share the numerical results that are used for the calculation.

Chad

## Re: One new path

Dear Chad:

Of course, no one can check this for all possible paths. Having the extra path I think now you have a stronger conjecture. If I were you I would add all this in the (revised) paper.

Regards,

Arash

## Re: One new path

Arash,

We will make a comment about this in the revision. However, I'm not sure why circular paths would yield some special behavior in the first place. Is there an example you could think of where you have path-independence for circular paths but path-dependence for some other path? Just thinking out loud here.

So another observation is the following. For incompressible pure power-law hardening materials proportional loading and path-independence can be proven (Ilyushin). Furthermore, rigid-perfectly-plastic materials are a limiting case of this constitutive description. Incompressible rigid-perfectly-plastic materials are also a limiting case of the behavior we study in the paper when s0/E -> 0 and v=0.5. However, s0/E has no effect on the results. So under one description there is path-independence but for the other there is significant path-dependence. An interesting case where there is a difference between the two limits.

Chad

## Path-dependent J integral

Dear Chad:

Your Fig. 3 is intriguing indeed. Several thoughts are running through my mind, but I'll ask one question now.

I searched through your paper on my screen, but could not find detailed comments on Fig. 9 of McMeeking JMPS 25, 357 (1977). His Fig. 9 showed that the J integral appraoaches the J_app when the contours are of radius a few times the crack opening displacement.

For all these years, and in drafting my lecture notes, I have had his Fig. 9 in mind.

You must have talked to McMeeking abou your Fig. 3 and his Fig. 9. Do you have any comment?

## RE: Path-dependent J integral

Zhigang,

I have had conversations with Bob about this, but my recollections of the specifics are vague. I would prefer to reserve any comments until perhaps you and I can talk to him together.

After your comment and reading Bob's paper again, I did some more calculations last night to convince myself yet again that everything we did was correct. Most of the errors that could arise would show up as problems with nonlinear elastic caluclations, but we get perfect path-indenpedence for all of our nonlinear elastic calculations and we have checked this for every case that we ran. So the only other possible location for an error wold be in our J2 flow theory integration (we use the procedure described in the ABAQUS theory manual). In the past, I have checked a crack calculation to make sure that the plastic strain increments are proportional to the deviatoric stresses at every integration point. Last night I used the routine to calculate the tension torsion test, and it agreed with the analytic result perfectly.

I think in the end, what is telling is that the result that there is an elastic sector in mode I has been found by several people, but the elastic sector does not exist in deformation theory calculations. This at least shows that the flow and deformation theories do not perfectly mimic each other in mode I.

What do you think?

Chad

## RE:RE: Path-dependent J integral

Dear Chad: Let's discuss your Fig. 3 and Bob's Fig. 9 when three of us meet in Penn State in June at the National Congress. To recap, the situation is as follows:

It is known that the size of the plastic zone is much larger than the crack opening displacement.

Before we explore the practical consequence of your Fig. 3, let us first reconcile the two calculations.

## I'm not sure I can wait that long

Zhigang,

Bob makes the statement, "The points showing J/Japp = 1 for R/b < 12 actually arise in the elastic solution to the problem and in the first few plastic increments, during which J has to be path-independent for contours separated by elastically-deforming material." So perhaps, at least for some of the points, the size of the plastic zone was not much larger than the COD.

An independent verification may be another route. If you have a student who is proficient with ABAQUS the calculation on a finite sized "pac man" geometry will show the effect. The only issue is that ABAQUS seems to do successive rings of domain integrals in a strange way so we might have to deconvolute the results if path-dependence is in fact found.

Chad

## Re: I'm not sure I can wait that long

Dear Chad: I'm forwarding your suggestion to the students in the class, and to my group. Thank you, Zhigang

## 2 more old papers on path-independent J integral

Dear Chad: I've just found two other papers that try to confirm the path independence of the J-integral.

C.F. Shih, Relationships between the J integral and the crack opening displacement for stationary and extending cracks. J. Mech. Phys. Solids 29, 305-326 (1991).

A.G. Varias, Z. Suo and C.F. Shih, Ductile failure of a constrained metal foil. J. Mech. Phys. Solids, 39, 963-986 (1991).

In both papers, the path independence of J along paths inside the plastic zone was confirmed. But this aspect was not the focus of the papers.

In the second paper, Varias did the calculation.

Hope that your work will clarify this situation.

## Path-independent J

My student and I tried to reproduce Shih's result (the year is 1981, not 1991) but were unable to do so. I don't have an explanation, but we tried the geometry that he describes and still found path-dependence. He also found that J on the farthest path was different from the prescribed value by 1%, but when we do deformation theory plasticity with a plastic zone size equal to 1/5th of the "pac man" we get a discrepancy of 4% (and our deformation theory results give perfect J path-independence and stress fields near the tip).

As for your paper, it looks like a situation of large scale yielding, is this correct? If so, we found similar behavior for J in large scale yielding, i.e. the path-dependence of J decreases as the plastic deformation progresses.

## Linking up the J discrepancy

Zhigang,

I was looking again at your figure 8 and more recently at Bob's figure 7. If you look at the N=0 results in Figure 7, blunting affects the stresses out to about 0.1 Rp (depending of course on s0/E). At this distance the reduction in J from the small deformation computations is only at about 10%. My guess is that the finite deformations affect J out to significanlty larger distances from the crack tip than just a few times the COD. It is just the precipitous drop in J that occurs very close to the tip. Obviously this is speculation at this point and additional calculations are required to confirm.

Chad

P.S. My student ran the pac-man on ABAQUS and we found the drop in J as in Figure 3 of our paper.

## Re: Linking up the J discrepancy

Dear Chad: I'm sorry for not coming back to discuss sooner. I've been preparing for Lecture 2 on elastic-plastic fracture mechanics, in between other things we all have to do. I gave the lecture this afternoon, and now move on to prepare for the next set of lectures.

As you will see in the notes posted online, toward the end of the lecture I discussed the significance of the HRR field, and included your paper and this ongoing discussion among a few papers for the interested students to read further.

I got a lot of questions in class. Maybe I just gave an unclear lecture. Maybe the students were really interetsted in the issues. I couldn't really tell.

## another paper which discussed path dependency

Dear Dr. Chad Landis, Dr. Zhigang Suo and Dr. Arash Yavari,

I have been following this interesting discussion about path-dependency of J-integral for contours very close to cracktip. I was basically learning from your disccussion rather than contributing to it, as I dont have much expertise in this filed.

Recently when I was reading some papers, I came across the following paper in which the authors made similar observations as Dr. Chad Landis. I thought it might be helpful to this discussion.

"The values ofJ. H. Kuang and Y. C. Chen.J-integral within the plastic zone"Engineering Fracture MechanicsVolume 55, Issue 6, December 1996, Pages 869-881They noted that the the J-integral will be path dependent if the selected integration contour can not fully enclose the plastic zone; otherwise the J-integral will be path independent.

I hope it is helpful.

Siva Nadimpalli

PhD Candidate, University of Toronto

## Thank you Siva

Dear Siva,

Thank you for pointing this paper out to me, I was not aware of it. I have added it to the revision of our paper.

Chad

## Finite deformation J path-dependence

We have now reproduced computations of the path-dependence of J for finite deformation. The plot of J versus the radius of the integration contour is included in the link. The finite deformation theory is the same as that described in McMeeking's work, which uses the Prandtl-Ruess equations to relate the Jaumann rate of the Kirchoff stress to the deformation rate. A forward Euler updated-Lagrangian finite element method is used, as described by McMeeking and Rice (IJSS 1975). The plot shows that the finite deformation results follow the linear kinematics results up to the point where the blunting becomes important. So as the ratio of the yields strength to the Young's modulus decreases, the J results follow the linear kinematics curve farther towards the crack tip. So, if this is the end of the story, it seems that there is nothing strange about adding finite deformation. There are two length scales, the radius of the plastic zone Rp, and the radius of the blunted crack tip Rt. If Rt << Rp then the linear kinematics result is followed closely until the radius of the J contour approaches Rt.

http://www.ae.utexas.edu/~landis/J-Finite.jpg

## Large Deformation Analytical Solution

I would like to suggest that a recent analytical solution of mine involving a nonproportional perfectly plastic solution for an elliptical hole problem, published in the

Journal of Elasticity,99, 117-130 (2010), might shed some light on large deformation crack problems and the limits of J -controlled growth. This problem represents highly idealized material with a crack traveling at a constant velocity under plane stress loading conditions and the Tresca yield condition. Despite having a constant crack velocity the problem is not steady-state because the radius of curvature of the hole at the crack tip is continually changing with time. The strain grows as – log( r), when approaching the crack tip, instead of 1/r as in the stationary crack problem. This change is a consequence of using as a strain measure the natural or logarithmic strain as opposed to engineering or linear strain.## Dear all, I am a new

Dear all,

I am a new Abaqus user. My work involve with fracture modelling by using Abaqus.

For 2D Edge Crack problem, I have requested 5 values of J contour integral in abaqus, but i can not plot the J-integral path (J - integral versus Radius). I know abaqus will choose the path of each contour integral automatically. But How to show the path of each contour integral?

Thank you so much,

Sutham A.

## I'm new too

I am ANSYS user. In crack analysis, I have set 5 contour integral to determine the J-integral. I have got that value, but the problem is how to separate al these value to the three mode of loading which is JI, JII, and JIII? Is anybody can help me?

## ASME Applied Mechanics Journal Vol 79 Issue 4 article

I assume you have read this article:

A Note on the Path-Dependence of theJ-Integral Near a Stationary Crack in an Elastic-Plastic Material With Finite DeformationJ. Appl. Mech. -- July 2012 -- Volume 79, Issue 4, 044502 (2 pages)

http://dx.doi.org/10.1115/1.4006255&idtype=cvips&gifs=Yes&ref=no

Any comments?

Jing Xu P.Eng., M. Eng.

Sr. Specialist - Stress Analysis & Design

ArcelorMittal Dofasco

## Jing Xu - Was your question

Jing Xu - Was your question directed to me? If so, I was one of the authors of the article in question. Did you have a concern about the article? Best regards - Chad

## Calrification

Mr. Landis

Sorry for the late reply. I didn't realized you were one of the author in the ASME journal paper.

I'm interested in this topic. Hope you would continue publish any of your new development in this forum.

Thanks,

Jing Xu P.Eng., M. Eng.

Sr. Specialist - Stress Analysis & Design

ArcelorMittal Dofasco

## J-integral independent of initial crack length?

Hello all,

I am a graduate student at the University of Delaware looking at modeling the bimaterial interface crack propagation. In this paper http://am.ippt.gov.pl/index.php/am/article/viewFile/244/99 the authors derived J-integral (equation (1.14)) for a DCB and it is independent of the initial crack length.

1. Is J-integral, in general, independent of the initial crack length for a given mode of loading?

2. In a mode II loading during crack propagation along the bimaterial interface, there can be friction between the materials. In that case would the J-integral be path dependent?

I would appreciate any comments.

Thanks,

Subramani

## J-Integral in Elastic-Plastic materials

Hello engineers,

According to the article "J-integral

and crack driving force in elastic–plastic materials" I want to know

whether it is correct to simply add elastic far field J-integral and plasticity

coefficient, Cp, to calculate J-integral in elastic-plastic materials.

Thanks,

Pooya Saniei