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Energy release rate. Fracture energy

Zhigang Suo's picture

These notes were prepared when I taught fracture mechanics in 2010, and were updated when I taught the course again in 2014. I hope to start a conversation at a new post entitled Division of Labor.

Notes on other parts of the course are also online.

PDF icon Energy release rate 2014 02 14.pdf820.51 KB


Konstantin Volokh's picture

Dear Zhigang, sorry for boring youSealed You touch my favorite topics Laughing.

Unfortunately, the identitiy σc√a=constant , predicted by Griffith, is not correct generally. It is only correct for cracks with the same sharpness (curvature or radius) at the tip of the crack.Cracks with the same length and different tips will not obey the identity. The latter explains the scattering of the toughness magnitude in tests.

Kejie Zhao's picture

Hi Konstantin,

You mentioned that "σc√a=constant is correct for cracks with the same sharpness". Here are two panels (one homework problem confusing meCool), panel 1 has a hole, and a sharp crack tip on the hole surface, the crack size is much smaller than the hole size. The second panel has a notch-like crack. Are the strengths of two panels the same? Thanks




Konstantin Volokh's picture

Hi Kejie,

I am talking about the tip of the crack not its shape. Feel the difference!


P.S. Please look at Paper #2 on for more details

P.P.S. Fortunately, I do not need to do homeworks anymore Cool

Kejie Zhao's picture

Hi Kosta,

Sorry for the unclearness. I mean the two panels have the same sharp tips. Thanks


Konstantin Volokh's picture

Kejie, I know the answer but I do not want to intervene your learning process...

Kejie Zhao's picture

Hi Kosta,

Thanks. The deadline is over so I think it's ok to talk about this homework question nowLaughing. My thought (may be way too naive) is if we have a big hole and small cracks, the panel strength will be factor of 3 off, compared to the panel only with small cracks (the crack sizes are the same). However, one confusion is that if the hole becomes flatter and flatter, following the same way, the strength will be found to be 0, which is obviously wrong. One possible interpretation is, there are two length scales, one is for the hole, and one is the crack, if these two length scales are close, the "concentration factor off" strength will not be correct. Please let me know how you think.Thanks


Zhigang Suo's picture

Dear Kosta:  The shape of the tip may cause variation in the measured strength.  But before you assign any significance of this effect in practice, consider the following factors.

  1. For a brittle material like silica, the flaw needs to be well annealed to be rounded at the tip.  Otherwise, there is a tendency to have an atomistically sharpe tip.  You may not wish to emphasize a high value of strength just because you happen to have a large and smooth tip.  The practice would be risky.  In determining fracture energy, one tries to make a sharp crack.
  2. For a ductile material, the fracture resistance is a function of the extension of the crack (the R-curve).  While the initial fracture energy may be affected by how the initial tip is prepared, the steady-state value will be independent of the initial shape of the tip.
Konstantin Volokh's picture

Dear Zhigang,

1. All real cracks have a tip not a mathematical singularity. I believe that the radius/curvature is a reasonable approximation for the tip. Of course, one (mathematician) can invent all kinds of pathological tips... It does not matter. The point is that the tip sharpness is crucial. Practically that means that the measurement of the initial fracture toughness is problematic.

2. This is interesting. Do you have references considering experiments confirming that the steady-state value is independent of the initial shape of the tip?

L. Roy Xu's picture

Energy release rate vs. fracture toughness: mixed-mode and mode-II cases

I prefer "the fracture toughness" rather than "the fracture energy"because it is commonly used in fracture measurements such as ASTM test standards. Konstantin's view is very important because only a mathematical sharp crack tip will lead to a constant stress singularity order (-0.5) and it is shown as square root a (half crack length) in fracture toughness calculations. Unfortunately, it is very hard to make these sharp crack tips. If the crack tip is rounded or just a notch tip, the stress singularity order is not a constant then we cannot get a material constant: fracture toughness.

Griffith's model addresses a mode-I crack only. Here I'd extend our discussion to mixed mode and pure mode-II cracks under static loading. My PhD student Arun Krishnan and I are developing an efficient way to measure the pure mode-II fracture toughness of interfaces as shown in Fig. 1. We observed two fracture modes:

1) if the interfacial bonding is strong. the initial crack, although loaded in shear, kinks from the original crack path and forms a mode-I crack (its symmetrical stress field was verified by our optical techniques);and

2) a pure mode-II crack propagates along the interfacial bonding ( a self-similar crack) if the interfacial bonding is weak.

Therefore, we can obtain the correct pure mode-II fracture toughness (material resistance) is case 2). Simply because the material resistance (fracture toughness or fracture energy) for a mode-I crack case in case 1) and a mode-II crack in case 2) is very different for the same material. 

Efficient short-beam shear specimen to measure pure mode-II fracture toughness


Now if we want to compare the energy release rates and fracture toughnesses in order to predict whether case 1) or case 2) will occur at first, based on Zhigang and Prof. Hutchison’s famous review paper in 1992, the energy release rate of the initial crack in Figure 1 could be easily calculated and it is a function of the potential crack path or function of θ. For the kinked mode-I crack, the energy release rate,  G(θ=θc) should exceed the mode-I fracture toughness of the polymer material ΓPMIC, i.e.

Equation 1

Where the crack kinking angle θc is around 70 degrees based on the Erdogan and Sih’s classical paper in 1963 (the T-stress effect was not included).  For the pure mode-II crack initiation and propagation, the energy release rate, G(θ=0) should exceed the mode-II fracture toughness of the interface ΓITIIC  , i. e.,

 Equation 2

So the competition of the energy release rate and the fracture toughness leads to different failure modes for a mixed-mode crack, which is more complicated than the Griffith’s original problem. Here, if equation (1) is satisfied at first, a crack loaded in shear will kink as a mode-I crack. If equation (2) is satisfied at first, a crack loaded in shear will propagate as a mode-II crack along the interface. The major purpose of my note is that I found some mistakes in recent journal papers to measure the pure mode-II fracture toughness: their initial cracks kinked during experiments but they still use the crack initiation loading to characterize the “mode-II fracture toughness”.

Zhigang Suo's picture

Dear Roy:  Many thanks for the comments.  The notes posted here are for a course I'm teaching now.  We are second week into the semester.  As listed in the page for the course, stress intensity factor and mixed-mode fracture will be discussed later in the course. 

I have followed more or less the historical development.  The development of fracture mechanics offers wonderful lessons for students in mechanics, possibly more valuable than its immediate purpose for engineering design.  I'd like to reenact some of the early excitement and disappointment, and even confusion.  Most of all, I'd like to contrast Irwin's pragmatic approach to some of the other approaches to fracture.  Hopefully the spirit of Griffith and Irwin will inspire the students to create useful things in the broad field of mechanics.

Indeed, a real crack will never be mathematically sharp, and the stress will never be singular.  Even a crack in silica cannot be sharper than atomic dimension.  The tip of a crack in a metal may be blunted, and voids may open, etc.  However, this reality should not prevent us from using the singular stress field.  Irwin's use of the singular field was discussed in the following paper:

G. Bao and Z. Suo, "Remarks on crack-bridging concepts," Applied Mechanics Review 45, 355-366 (1992). 

Kejie Zhao's picture

Here are some nice movies on cracks illustrated by bubble raft.

Zhigang Suo's picture

Dear Kejie:  I finally got around to watch some of the movies.  Very instructive indeed!  Thank you very much for the link.

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