Javad, I am not sure which FE code you use. If Abaqus, which version? In the somewhat older versions the treatment of residual stresses in the calculation of J was incorrect. I believe the problem is fixed in the newer versions. That said, if the residual stresses are properly accounted for, then the J-integral should remain contour-independent (of course it may drop a bit towards the tip in the large strain region).

Hi Mohammad Javad Razavi,

This may depends on the software which you have used in the calculation and the method which you have used to apply residual stresses. Different software may use different J definition. If the Rice J definition is used, it is path-independent for mechanical load but not for cases with residual stresses. For ABAQUS old versions (V6.10 and lower), the definition was the Rice J plus thermal load correction, which does not work for cases with residual stresses. For ABAQUS versions v6.11-14 and Version 2016, a J definition followed Lei et al. (International Journal of Fracture 106, p.159-206) is used, which allows J to be evaluated when residual stresses exist. However, errors and code bugs have been found (see  "Validation of the contour integral J function in ABAQUS v6.11-v6.14 for cases involving residual stresses" by Y Lei in ResearchGate) and the correctness of J calculation may depend on the way you apply residual stress field and the choice of residual stress step in the calculation.

To get help, you should tell people the software which you are using for the calculation and the method you have used to apply residual stresses.

Negative J's are very common both in monotonic and in cyclic loading. This is sometimes due to errors in coding, sometimes due to the fact that J is only applicable to deformation plasticity, or due to the existance of multiple load paths. It can also happen when you try to compute J for a 3D crack where the crack intersects a free surface. At this point the r to the power -0.5 singularity does not hold and the integral is meaningless. Basically  there are only a few classes of problems where J has a physical meaning.

J-integral represents the energy release, which in independent of the stress sign but depends on the crack development sign. If it is positive under a positive variation of the crack length it is negative for the crack closing.

 Looking closer at your query it would  ap0ear that your compressive stresses have generated an additional load path so that the path independence is lost.

Also reading others comments it should be noted that accrediting the J integral to Rice is misleading as the integral was developed a good many years earlier by Eshelby in the UK.

According to the existence of compressive residual stresses around the crack tip, the value of J-integral is not constant for different contours. Is this result reasonable? 

Yes, it is. 

Theoretically, J-integral should be always equal or greater than zero regardless of crack closure. 

Dear Dr. Jivkov,

Dear Dr. Lei,

Thank you so much for your kind response;

I'm using ABAQUS 6.11 and I am applying the stress field of indentation process. 

I tried both modelling indentation and applying residual stress field, but unfortunately the result was same and the J was negative!

Dear Professor Jones,

Dear Professor Slepyan,

Dear Dr. Zuo,

Thank you for your kind responses.

Actually, I am modelling the indentation process by considering the elastoplastic behaviour of material. The indentation process is undertaken in the first simulation step and the axial load is applied to the CT specimen in second step. Even under tensile far field load the J values are negative for the first nine contours and it would be positive for contours equal or greater than 10. 

As I mentioned before, I considered the surface contact. Shall I consider this negative J as closure?

In this case, according to the path dependent behaviour of J, shall I use an average value of positive J?

Thank you in advance.

Consider a variation as a small displacement of the crack tip together with the contour. J-integral is valid (as the path-independent) if the stress-strain field inside the displaced contour is the same, that is if it moves with the contour. This is true for a uniform elastic material. 

In the first part of your message you mention a tensile far field load. In that conditions the crack have to be open and the J_integrals positive and the contact surface is not necessary. In the second part of your message you mention a  contact surface between crack flanks. In a contact problem a very small crack closure can occur and therefore the J_integrals can became negative. Check the displacement discontinuity in the contact zone. The sign of it play a central role in your problem.

Can happen due to defective modeling. Direction of path of the integral needs to be checked.

Check CTOD. A +ve (opening) value should indicate +ve J. If problem persists, then the magnitude of residual compressive stress applied is quite high that outweighs the effect of tensile load.

To obtain a constant J-integral when residual stresses exist, try this way. I assumed that you applied residual stresses to all elements in your FE model. So you obtained the wanted residual stress distribution along the crack path prior to crack opening. Now, do not apply residual stresses to elements which are in crack tip contours. If your FE mesh around the crack tip is fine enough, the loss of residual stresses in those elements will not have significant impact on residual stress distribution along the crack path. This can be checked if you compare two residual stress distributions using these two residual stress applying methods. In this way, the problem for J-integral calculation becomes J-integral when residual stresses do not exist.