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The J integral
For a crack in an elastic body subject to a load, the elastic energy stored in the body is a function of two independent variables: the displacement of the load, and the area of the crack. The energy release rate is defined by the partial derivative of the elastic energy of the body with respect to the area of the crack.
This definition of the energy release rate assumes that the body is elastic, but invokes no field theory. Indeed, the energy release rate can be determined experimentally by measuring the load-displacement curves of identically loaded bodies with different areas of the cracks. No field need be measured.
Many materials, however, can be modeled with a field theory of elasticity. When a material is modeled by such a field theory, the energy release rate can be represented in terms of field variables by an integral, the J integral.
This lecture describes the J integral, along with examples of calculation. Uses of the J integral are often better appreciated in the context of individual applications, which we will describe in later lectures.
The J integral can be developed for both linear and nonlinear elastic theories. The nonlinear elastic theory will be used in class, and the linear elastic theory will be used in a homework problem.
These notes belong to a course on fracture mechanics
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Comments
Your comments on 3-D J-integral ?
Hi Zhigang,
Thanks a lot for this handy and useful lecture notes on J-integral.
In engineering applications, we often need to calculate stress intensity factors
for a corner crack. A convenient approach is to deduce them from J- or similar
integrals. Numerically, such integral does not appear to be path-independent.
And I read somewhere (unfortunately cannot recall where) that theoretically the
path-independence can only be guaranteed for a 2-D crack.
Do you have any comments on the 3-D J-integral?
Xiaobo
J integral in 3D
Dear Xiaobo: The following paper should answer your question.
F.Z. Li, C.F. Shih, and A. Needleman, A comparision of methods for calculating energy release rates. Engineering Fracture Mechanics 21, 405-421 (1985).
J integral for a nonhomogenous material
Thanks Prof. Suo. It is a very good lecture. For a nonhomogenous material, the J integral is path-dependent. In this case, how to use J integral to calculate the energy release rate?
Re: J integral for a nonhomogenous material
Several points come to mind:
J-Integral application in interaction of two cracks
Dear Zhigang
You know, These days I'm working on the interaction of cracks but I've got a problem in applying J-Integral in the point where two cracks intersect.
Would you please let me know how can I possibly use J-Integral in intersection of two cracks.
Best,
Reza
Re: J-Integral application in interaction of two cracks
Dear Reza: You will get any useful result by using the J-integral around the point where two cracks intersect. For example, here is a paper on a crack impinging upon an interface. The J-integral is not useful here. There are many examples like this.
R. Huang, J.H. Prévost, Z.Y. Huang, and Z. Suo, "Channel-cracking of thin films with the extended finite element method". Engineering Fracture Mechanics,70, 2513-2526 (2003).
Prove that G=J for elastic materials
Dear zhigang
I have read the article (J.R. Rice, Mathematical analysis in the mechanics of fracture, Chapter 3) that proves G=J for elastic materails. and I faced a problem, why in two-dimensional crack we take A* any finite region in which the crack tip is imbedded?
can you please help me.
Sh.Eskandari
Re. Prove that G=J for elastic materials
Hi Sh. Eskandari,
Becaue J integral is path-indepedent for 2D crack in homogeneous materials, we can take any finite region to do the integral, as long as it contains the crack tip.
Kejie
Dear Kejie Yes it is
Dear Kejie
Yes it is obvious, but when we are going to prove that energy releae rate is equivalent to J integral , there is an area integral that it should be taken over all the body but in Prof. Rice article for some reason that i didn't get it we take the area as any finite region containing the crack tip.
if you can just read this part of the article and help with this.
thank you
Shahin Eskandari
Re. J integral
Hi Shahin Eskandari,
Sorry for the late reply, somehow I missed this thread. I have not read Prof. Rice's paper but I did see some proof in the textbooks. My understanding so far is, since the integral is path-indepedent (i.e., it's a constant for any finite region containing the singular crack tip), we can take any area to prove G=J.
This might be deviated from your question?
Kejie
Dear Kejie Thank you for
Dear Kejie
Thank you for your reply.
I think we can not use path-independency, becouse in the proovement we start from the rate of potential energy and then we show that it is equivalence to J-integra.
Anyway, can you tell me about the textbooks that you have seen the proovement.
Thanks.
Shahin Eskandari
Re.
Hi Shahin Eskandari
Please take a look at Anderson's book, fracture mechanics fundamental and applications, the appendix of chapter 3.
Kejie
Please help
Please help, i am in rush.
Evaluation of J integral by Ansys
Dear Everyone
I am working on Fracture Mechanics, and facing problem to evaluate J-Integral in 2D and 3D crack model and as well as in composite model by ANSYS.
If anyone have idea about it please send me tutorials or steps for the same problem.
Your help will be heartly appreciated.
Thanks.........
email ID: himanshu@iitp.ac.in
hpiitp@gmail.com
Have you heard of J integrals also in notches?
1.
Relationships between J-integral and the strain energy evaluated in a
finite volume surrounding the tip of sharp and blunt V-notches
International
Journal of Solids and Structures, Volume 44, Issues 14-15, July
2007, Pages 4621-4645
F. Berto, P. Lazzarin
| Related
Articles
2.
Elastic stress distributions for hyperbolic
and parabolic notches in round shafts under torsion and uniform
antiplane shear loadings
International Journal of
Solids and Structures, Volume 45, Issues 18-19, September
2008, Pages 4879-4901
M. Zappalorto, P. Lazzarin, J.R. Yates
| Related
Articles
3.
Some advantages derived from the use of the
strain energy density over a control volume in fatigue strength
assessments of welded joints
International Journal of
Fatigue, Volume 30, Issue 8, August 2008, Pages
1345-1357
P. Lazzarin, F. Berto, F.J. Gomez, M. Zappalorto
| Related
Articles
4.
Fracture of U-notched specimens under mixed
mode: Experimental results and numerical predictions
Engineering
Fracture Mechanics, Volume 76, Issue 2, January 2009,
Pages 236-249
F.J. Gómez, M. Elices, F. Berto, P. Lazzarin
| Related
Articles
5.
A review of the volume-based strain energy
density approach applied to V-notches and welded structures
Theoretical
and Applied Fracture Mechanics, Volume 52, Issue 3, December
2009, Pages 183-194
F. Berto, P. Lazzarin
| Related
Articles
6.
Local fatigue strength parameters for welded
joints based on strain energy density with inclusion of small-size
notches
Engineering Fracture Mechanics, Volume
76, Issue 8, May 2009, Pages 1109-1130
D. Radaj, F.
Berto, P.
Lazzarin
| Related
Articles
7.
Effect of the thickness on elastic
deformation and quasi-brittle fracture of plate components
Engineering
Fracture Mechanics, In Press, Corrected
Proof, Available online 14 April 2010
A.
Kotousov, P.
Lazzarin, F. Berto, S. Harding
| Related
Articles
8.
Fatigue design of complex welded structures
International
Journal of Fatigue, Volume 31, Issue 1, January 2009,
Pages 59-69
B. Atzori, P. Lazzarin, G. Meneghetti, M. Ricotta
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Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878
And in notches I generalized the well known HRR solution
An approximate, analytical approach to theHRR'-solution for
sharp V-notches
soton.ac.uk
[PDF]S Filippi, M Ciavarella, P Lazzarin -
International Journal of Fracture, 2002 - Springer
S.
FILIPPI1, M. CIAVARELLA2 and P. LAZZARIN3,∗ 1Department of Mechanical
Engineering
- University of Padova, Via Venezia 1, 35100 Padova (Italy); 2CNR-IRIS,
Computational Mechanics
of Solids, Str. Crocefisso 2B, 70125 Bari (Italy); 3Department of
Management and ...
Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878
J integral seen more in general from duality and simmetry loss
8.
Duality and symmetry lost in solid mechanics
Preview 
Purchase PDF (193 K)
Comptes
Rendus Mécanique, Volume 336, Issues 1-2, January-February
2008, Pages 12-23
Huy Duong Bui
| Related
Articles
Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878
How to plot J-integral path
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.
J-integral
Dears
I found in XFEM J-integral is little bit path dependent for three dimensionalcracks. its value changes up to 8%for penny crack with path.
J-integral validity
Hi All,
I have a question in mind for a while and unfortunately
have not yet found a clear answer to it. I would highly appreciate if you
provide me with your idea about this question:
Is J-integral valid inside the process zone ahead of a
sharp crack? and if is valid, Is it equal to the value obtained from a contour
outside the process zone?
Thanks
Ahmad Khayer D.
To be precise the
To be precise the J-integral is only path independent for isothermal, non-linear elastic problems where there are no body forces. For monotonic loading where there is plasticity then it is path independent only under deformation plasticity. For in-elastic problems where there you hasve cyclic loading the J-integral is not path dependent. In such situations the crack tip can be described by a related path integral called T*, that was first formulated by Satya Atluri. When J is path independent then J=T*. The importance of the J integral is related to its ability to characterise the near tip stresses and strains for a non-linear elastic material. The so called HRR solution. The experimental work of Professor Albert Kobayashi has shown that this is not true for cyclic loading.
As an aside many people mistakenly think that the J-Integral was developed by Jim Rice. This is not true. It was first proposed in the early 50's by Professor Eshelby at the University of Sheffield. Prof Rice popularised it many years later. The 3D variant was first developed in a joint paper by Professors George Sih and George Irwin, both at Lehigh, shortly thereafter. You can chase down the relevant references with a simple Google search. The reference to the original Sih-Irwin paper is in one of Professor Sih's papers in his Journal, viz: Theoretical and Applied Fracture Mechanics.
Hope this helps
Rhys Jones
if you can just read this
if you can just read this part of the article and help with this?
J-integral for the ENF specimen
Dear Colleagues,
My problem is that if I define a zero-area path around the crack tip of the mode-II ENF specimen and
calculate the traction vectors, there is a negative sign (2nd page)
that I do not understand and this is what leads to the wrong result. Can
You take a look at my note and tell me what is wrong?
page 1
page 2
Thanks
Andras
M-integral domain size
Hi
i'm modeling crack with EFG method and using M-integral to calculate stress intensity and i have some quastions:
is domain integral form of j-integral still path-independence?
what is the best 'q' function?
what domain size is good for the M-integral?
i found diffrence between answers with several domain size is it mean that m-integral is path-dependence?
and my bigest problem is sometimes i got worse answer with finer mesh!!! (also with uniform distribution)
and in some domain size my enriched basis got worse answer than the ordinary basis!!!
is my model unstable or something like that???
what should i do???
i checked everything!!! my code works well what do you suggest????
I´m very interested in this
I´m very interested in this
files
Dear Prof. Suo,
I have read your posted outlines for fracture meachanis and they are really helpful. Thank you for sharing them! But one problem is that I couldn't find all the outlines for all the listed topics. I could not find any pdf file for this one, for example. Are you still posting those outline?
Best,
Notes on J-integral are re
Notes on J-integral are re-posted. Not sure why they had disappeared.
Hi Zhigang,
Hi Zhigang,
Thank you for the useful explanation. I would like to ask you if you can model a crack on a 3D curved surface like a pipe. How could this be modeled ? can i still use the contour integral method?
Thank you,
Diana.
a curved crack front in 3D
One can model 3D curved crack, and talk about energy release rate point-by-point along the front. Here is a paper on the subject by Li, Shih and Needleman (1985). I have not tried to use the J-integral to calculate energy release rate for a curved crack front. Hopefull other people can shed light.
Re: Curved crack front in 3D
Interaction domain integrals for curved crack fronts in 3D are derived by Gosz and Moran (Engg. Fracture Mechanics, Vol 69, 2002, pp. 299-319). The link to the paper is here.
applicability for J-integral at interfaces (two dissimilar mat.)
Dear Prof. Suo,
Rice introduced the J integral concept which is equal to the energy release rate for a uniform, linear or nonlinear elastic material free of body forces and subjected to a 2d deformation field (plane strain, plane stress etc.).
In ANSYS and Abaqus, the implementation is always referenced to the work of Shih (1986). There has been numerous publications on the use of J-integral to estimate Gc (critical energy release rate) for cracks which are located at the interface of dissimilar materials.
I could not find the relevant information if this approach is applicable, if yes, under what circumstances? Could you please give me some information or references about the implementation of Jintegral at the interface of two materials?
I am looking forward to hearing from you,
Yalcin