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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 4621-4645F. Berto, P. Lazzarin

Preview PDF (849 K)

| 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 4879-4901M. Zappalorto, P. Lazzarin, J.R. Yates

Preview Purchase PDF (581 K)

| 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,Pages1345-1357

P. Lazzarin, F. Berto, F.J. Gomez, M. Zappalorto

Preview Purchase PDF (669 K)

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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 236-249F.J. Gómez, M. Elices, F. Berto, P. Lazzarin

Preview Purchase PDF (1385 K)

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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 183-194F. Berto, P. Lazzarin

Preview Purchase PDF (1254 K)

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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 1109-1130D. Radaj, F.

Berto, P.

Lazzarin

Preview Purchase PDF (796 K)

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7.

Effect of the thickness on elastic

deformation and quasi-brittle fracture of plate components

Engineering,Fracture Mechanics

,In Press, CorrectedProof

Available online 14 April 2010A.

Kotousov, P.

Lazzarin, F. Berto, S. Harding

Preview Purchase PDF (764 K)

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8.

Fatigue design of complex welded structures

International,Journal of Fatigue

Volume 31, Issue 1,January 2009,Pages 59-69B. 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.ukS[PDF]

Filippi, MCiavarella, 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

Comptes,Rendus Mécanique

Volume 336, Issues 1-2,January-February,2008

Pages 12-23Huy Duong Bui

Preview Purchase PDF (193 K)

| 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