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Linear elastic fracture mechanics

Zhigang Suo's picture

These notes were initially written when I taught fracture mechanics in spring 2010.  The title of the notes was then "toughness".  In revising the notes for the class in 2014, I have changed the title of the notes to "Linear elastic fracture mechanics".

You can access all notes for the course on fracture mechanics


arash_yavari's picture

Dear Zhigang:

Do you agree that assuming K_Ic being a material property is fine as long as the crack is not too short? For very short cracks, toughness would depend on the crack size I think.


Zhigang Suo's picture

Dear Arash:  Thank you for the question.  Indeed, when the size of the crack is not very large compared to the size of the fracture process zone, the K-field will not be valid anywhere in the specimen.

That is, the measurment of Kc is valid only when the specimen satisfies the small-scale yielding condition.  Some practical guidlines are given in the notes in the paragraph titled "The small-scale yielding condition in practice".

L. Roy Xu's picture

Dear Zhigang,

Following Arash's comments, I would discuss my understanding after measuring all kinds of fracture toughnesses for metals, polymers and composites subjected to static and dynamic loading. An engineer always wants to use the smallest fracture toughness value for damage tolerance designs. While the plane-strain condition of the fracture toughness specimens exactly yields the smallest fracture toughness. So whether the initial crack is short or long, or whether the initial crack tip is very sharp or rounded (indeed, a rounded crack tip yields a high "fracture toughness"), if the measured fracture toughness is the smallest value-we treat it as a material constant for engineering designs.

Zhigang Suo's picture

Dear Roy:  Thank you so much for recounting your experience in measuring fracture toughness.  Do you have a list (in your head or on paper) that you follow in carrying out a masurement?  The students in the class and I myself would be interetsed in hearing more about practical experience.

L. Roy Xu's picture

Dear Zhigang,

Both Bo and you proposed some key issues to measure the fracture toughness. I'd attach our papers on fracture toughness measurements for very different materials:

1.JRPC paper (web link) on using the DCB, ENF and EDT tests to measure the mode-I and mode-II fracture touhnesses of composite materials. Here we notice that the friction has a significant effect on the mode-II fracture toughness. Also, there is no K-dominant zone for these thin composite beam tests.

2.ECF paper on developing a new short beam shear test to measure the mode-II fracture touhnesses of bonded materials and composite materials. We successfully remove the friction effect and the measured mode-II fracture is lower than we expected.

3. JCM paper o n using the SNB test to measure the mode-I fracture touhnesses of polymers and nanocomposite materials. The nanocomposite materials showed very limited fracture toughness increase over the pure epoxy matrix.

4.EFM paper on using a full-field optical technique to measure the dynamic mode-I fracture toughness of polymers. K-dominant zone is the key to analyze the dynamic stress field, and obtain the meaningful dynamic fractures toughness.

skravche's picture

Dear Luoyu,

Do you mean you found no effect of K-dominance zone size on the mode I fracture toughness of composite materials?

Because we, actually, found the opposite.

Would you be so kind to send me the link to the paper you've published? The current link for some reason leads me to nowhere.


Thank you,


Roy, how do you do?

Plane-stress vs. plane-strain refers to constraint effect on effective toughness in the through-thickness direction (in the direction of crack front).

If you are to find the lowest effective toughness, in-plane constraint condition should be considered too, e.g., in a J-Q approach (or, equivalently, K-T approach in LEFM).



Zhigang is right. When discussing K-dominated zone, two circles should be used. The outer one is to cut off any significant boundary effect, and the inner one to exclude the fracture process zone. The K-approach breaks down when the characteristic length scale of the inner nonlinear zone/core is comparable with distance between the tip and any boundary, including its own (such as a crack significantly curved/kinked) and any other crack.

The same argument should apply to J-approach as well.

Zhigang Suo's picture

Dear Bo:  You said better than I did.  The condition that the square-root singular field (i.e., the K-field) prevails in an annulus is central to why the LEFM works.'s picture

Michele Zappalorto PhD

The precise answer to this question can be find in the basic (at least in my mind) work due to Hui and Ruina:

C. Y. Hui, A. Ruina. Why K? Higher order singularities and small scale yielding”, International Journal of  Fracture, 72, 97­-120, 1995.

The .pdf of this paper can be downloaded at the following link:



Dear Michele,


Thanks for the article. It is one of those interesting articles I have missed while wrestling with fracture mechanics all these years. I feel sorry I couldn't finish reading it today. Allow me to post with what I have had--apologize if I misunderstood the arthurs.

I use neither of the reasons listed at the end of page 98 to justify the k-approach. Rather my understanding is that the k-term is the only term in the Williams' series expansion to yield a FINITE energy release rate when perturbing an elastic crack. All the more singular terms yield infinite G, and all the non-singular terms yield zero G. The energy of a system can be infinite (as pointed out by Hui and Ruina). However, the rate at which the energy is to be released by crack extension has to be finite. Recall how K is defined--it is a pure mathematical term, and defined with one stress state. In contrast, G is stemmed from a pertubation analysis (Griffith, 1920), involving consecutive stress states. It is by the above realization of finite G due to K-field that the K-approach gains equivalency to the G-approach. Irwin's contribution was not to merely identify K, but to relate K to G...   Finally, nonlinearity may be added and K becomes J, or other terms may be added so that size/constraint effects can be taken into account.

This is sort of discussion with respect to characteristic length scales involved in the fracture process. There would be discussion regarding characteristic time scales involved in it as well. This would lead to discussion on G/K-critetion vs. S-criterion I find taking place 2-3 years ago here, which I missed. That would be more interesting a topic as I have doubt on the applicability of G to dynamic fracture.



Bo's picture

Michele Zappalorto PhD

Dear Bo,

thanks to you for your helpful comments. I've missed that article for years too. But during the review process of a paper of mine, dealing with an analytical study of elastic-plastic stress and strain distributions ahead of a mode III loaded blunt crack (you can find here my work, if  you are interested: M. Zappalorto, P. Lazzarin. Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading, International Journal of Fracture, Volume 148, Number 2 / November, 2007, 139-154, direct link:  it happened to me that one of the reviewers asked whether the 1/r (strain singularity for an elastic-perfectly plastic material) was the stronger one, and suggested me to quote the paper by Hui and Ruina. That was really a luck.

I found the paper really interesting. A similar idea was also present in another prevoius paper by J. Rice (1974): you can find here the paper, if you are interested:

 Best regards,



Zhigang Suo's picture

Dear Bo:  Thank you for your helpful comments.  I also like to place G at the center of fracture mechanics.  K is a middleman, and is at best one of several ways to calculate G.

I have included an argument to reach the order of the singular field in later notes.  The argument consists of the following ingredients:

  1. The field is linear in applied stress
  2. G is quadratic in applied stress
  3. We look for a field such that the boundary conditions enter through G
  4. G/E is the only length scale in this problem.

It appears that this argument makes the square-root singularity obvious, and also bypasses the middleman K, which is now relegated to its proper place:  one of several ways to calculate G.

I've also extended the argument to obtain the order of singularity of the HRR field.

I'd love to learn how you feel about the argument.

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