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Strain energy release rate of beam specimen using J-integral

Hi All,

Are there any good references showing the detailed derivations of elastic strain energy release rate using J-integral instead of differentiating compliance for end notch beam samples : DCB, 3/4 point bend ...? many thanks ...

Zhigang Suo's picture

You may find much what you want in Hutchinson and Suo, Mixed cracking in layered materials.

Henry Tan's picture

Hutchinson and Suo’s work reported in the above paper are idealized to a perfectly bonded interface. Real world applications of layered materials need to account for the interface debonding.

I am not familiar of the literatures contributed to this direction. Any suggestion?


I agree with you. Beam on elastic foundation might be a good correction for adding adhesive effect for DCB and some other beam specimens.

As the adhesive layer is very thin, the energy stored in adhesive layer is negligible. Then those indealized solutions for energy release rate become "real solutions".


Zhigang Suo's picture

Henry:  I'm not entirely sure what you are looking for, but if you are looking for beam-like specimens with bridging zones, here is a paper that you might find useful.  The experiments suggested in the paper have since been carried out for adhesive joints, I believe. 

Z. Suo, G. Bao and B. Fan, " Delamination R-curve phenomena due to damage", J. Mech. Phys. Solids. 40, 1-16 (1992).

Dr. Suo,

Thanks a lot for reminding me this famous summary paper. It was a good education every time I read it when I was a graduate student.

The reason I asked is because I came across a paper mentioning the derivation of G using J-integral without showing details. I tried on scratch pad but ran into "path-dependent" results for DCB. I definitely have missed some terms. I remember there was such analysis in several books for an edge crack under presciibed diaplacement. I wonder if you have such derivations for more complicated cases in your lecture notes that could share.


Rui Huang's picture


Glad to see you here. As far as I know, most of the solutions for energy release rate given in Hutchinson and Suo's review were derived based on the energy method rather than J-integral. My personal experience is that the energy method is much more powerful. In fact, I am not even convinced that the J-integral can give you the correct energy release rate for interfacial cracks between dissimilar materials. 




Thanks for your input. I came to know this good forum when I was bowsing your personal web.

Could you elaborate a little more on why the J-integral might not work for interfacial crack between dissmilar materials? My understanding is that J-integral still works for interfacial crack. The J-intergral is path-independent even for the contour passing dissimilar material. If so, J-integral should still be able to catch the crack on the interface. What do you think?


Amir Siddiq's picture

Hi Mingji

We worked on the interface fracture analysis in the recent past and also came across path dependence of J-integral due to plasticity around the crack tip.

One of  the best works (which I found quite interesting) regarding the path dependence of J-integral can be downloaded from

But for the estimation of the energy release rates for interface cracks between two dissimilar materials we always had to compare the values between the contour integral values with energy methods. We also used VCCT for stationary crack tips and results were quite promising.

Also, there are many references available discussing the limitations of different methods, such as, from Prof. Shih (more tempted towards energy methods).


De Xie's picture

VCCT is a very powerful numerical tool to compute strain energy release rate when it is used with FEA.  The math required is minimum, easy to implement, not sensitive to mesh size, one-step analysis.  Engineers like it.  Once you try it, you will like it.

 Xie D and Biggers, Jr. SB, Progressive crack growth analysis using interface element based on the virtual crack closure technique, Finite Elements in Analysis and Design, 42(2006): 977-984.

Xie D, Waas AM, Shahwan KW, Schroeder JA and Boeman RG, Computation of strain energy release rate for kinking cracks based on virtual crack closure technique, CMES: Computer Modeling in Engineering & Sciences, 6(2004): 515-524.

Hi mr Xie

I am a italian student and i am using Genoa for VCCT. I have some problem simulating the propagation of delamination in a plate with an initial central circular zone of delamination subjected to compression.

I select as VCCT path all the nodes of the plate expept the ones belonging to initial circular delamination.

Is it possible that in Genoa VCCT  the crack path must be a line and not a surface?
I hope to be clear




ravitejk4u's picture

hello all,

i am doing my Mtech in iit kanpur, i am working on FGMs. can i model the FGM in abaqus by using layered model with the property is varying with each and every layer. or are there any special element for which we can assign property variation along the thickness? 

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