I am also wondered the same. In our analysis we are trying to use SED theory to determine the process zone size at the fretting fatigue contact edge where the crack initiation occurs. The results are in good agreement with those conventional approaches.

a good question.

I read Sih's work a long time ago, but am fuzzy about the details now.  In a review article published in 90s, I tried to explain why Irwin's K works when the small scale yilding conditions apply.  The explanation was a standard one, and I'm not really sure who invented it.  I think I learned it from Rice's class.  To me, this explanation makes all other criteria either wrong or equivalent to K, so long as the small-scale yielding condition applies.  Do I miss anything?

A related thread of discussion:  Fracture Criteria 

1.the stress intensity facor,K, is equivalent to the strain energy density factor(functions),S, when the crack problem is linear/elastic

2.but when the crack problem is nonlineal, it is hard(or unfit) to use K to evaluation the crack problems, at this time J intergral becomes . of course S can also evaluate the crack problems.

3. S is a vector quality but K is a scale , this means that in general K is the function of (r,u,stress), while S is function of (r,u,stress, angle1,angle2). angle1 and angle 2 means that the angle around point of crack tip.

4. in practice, S is hard to calculated than K(elastic) or J(viscoplastic).   

Dhruv,

Thanks for raising a very interesting topic. It has been more than a decade since I thought about this issue. I can't give specific references to what I write. Hope this is OK.

Sih was not the first to think of strain energy density (SED) as a parameter in characterization of mechanical behavior. SED was already known as a yield criterion, comparable to the Tresca and von Mises yield criteria.

But, apparently, Sih was the first to assert the applicability, even desirability, of SED in predicting *fracture*.

Sih apparently based his reasons on the desirability of having the uniaxial tension test as the fundamental means of determining mechanical properties of materials. Essentially, his arguments relied on the simplicity of the test.

In particular, I believe, Sih found the idea of having a very special geometry such as what KIC testing requires, to be a contrived one. KIC testing has the size aspect built into it. In any case, Sih thought it to be not fundamental enough. In thinking so, he squarely ran into the major research program that had earlier been established by Irwin in the US.

It seems that Sih meant SED to be used regardless of the extent of plastic deformation or damage near the crack tip--i.e. in both LEFM and outside of it: e.g., in EPFM, fatigue, etc.

In LEFM (or under the SSY assumption), it is obvious that the SED criterion would predit precisely the same critical stress as what is obtained using either K or G. Incidentally, even the J-integral exactly equals G under these conditions. So, the differences between all these approaches only begin to matter when one goes outside the province of the simplest LEFM case.

For plastically deforming materials like metals, SED might be expected to give results that are *somewhat* at variance from the experimental data. The reason is, many interesting physical processes at the microstructural level respond to only the distortional component. The hydrostatic stress state suppresses dislocation movement and plastic deformation. So, where plasticity in general matters, whether at the crack-tip or away from it, one can expect the distortional energy density (DED) criterion to do somewhat better. (Though, here, both SED and DED would do better than the Tresca criterion.)

K, G are by definition global in nature. Even the J-Integral is in a sense global in that it refers to an encompassed volume--not a point. In contrast, at least while applying the SED (or DED) criterion, I suppose, it is the local stress- and strain-fields which matter more acutely. Even if this is not the case (i.e. even if I am wrong here--which, could easily be the case here) historically, the former three approaches have gone better with the analytical solution approaches.

For the same reason, one would suppose, SED could be better used in the computational approaches, esp. those involving design *optimization* for savings in materials. Such a use of SED increasingly seems to be the case regardless of what underlying computational technique has been used--whether FEM, BEM, or any other.

Of course, even K, G, and J-integral approaches have their own limitations. For instance, the famous "nonlinearity" of the J-Integral refers to the constitutional nonlinearity (as in the rubber stress-strain curve), not differential (as in the Navier-Stokes equation). K is more of a convention. It is a conceptually appealing way to isolate crack size effect away from geometry, but has no fundamental basis. G does have a basis but is global in nature and hence, doesn't really add a lot to our understanding.

Overall, Sih's argument of using only the tension test for toughness determination does have that merit of *simplicity*. Yet, IMHO, unfortunately the proposal misses a crucial fact:  stress fields are *tensor* fields. Fundamentally, when it comes to tensor fields, you cannot extract information by considering only one section alone. Just one section--as the simple uniaxial tension test involves--would have been sufficient had the stress fields been vectors fields. But they are not. As an implication of this fundamental fact, distortion has to be separated from the mere volume dilation. SED carries this fundamental limitation. (The other limitation is much more superficial. Where analytical treatment involving SED is tractable, we already have K, G, J-I.)

Now, DED can overcome that fundamental limitation of SED. But why DED is not a widely accepted criterion, I have no idea...

... May be, I will write a more detailed answer on this matter as a technical paper. Any individual here may feel free to suggest me a suitable journal, and of course, to criticise the thoughts here. Thanks in advance.

The Sih's SED is one of the criteria regarding crack kinkings (change of crack direction from original orienation).  Now, study of crack kinking in engineering background is almost none.   Most frequently use of fracture mechanics  is to determine only the crack initiation for self similar cracks (assume crack will extand along its original orientaion). In this late case, Sih's SED is not necessary.

Since the topic is not HOT, it is not surpprise at all to see it is less used.  Another reason could be a personal factor that Sih does not have many of his students as facultes who keep producing papers and cite his works.

There are also some other criteria for crck kinking (pure mode I, zero mode II, ect.).  I have a paper on this.

 Xie D, Waas AM, Shahwan KW Schroeder JA, and Boeman RG, Fracture criteria for kinking cracks in triple material bonded joints, Engineering Fracture Mechanics, 72(2005): 2487-2504.

All the three "big" criteria--i.e., the K, G and J--implicitly take a catastrophe-theoretical view of fracture. They assume an uncontrolled and indefinite growth of crack once the system *is* at the critical point. (As an aside, none of the big three criteria basically addresses the issue of how the system *reaches* criticality. They only answer how the system behaves once it happens to be *at* criticality.)

Owing to this nature, each of the big three criteria is a global parameters. Each is applicable to the specimen as a whole, in a "one-off" manner.

In contrast, SED (or DSED, or other criteria like the maximum shear stress criterion) is a field variable. It varies from point to point in the specimen. (Think if K, G, or J would vary from point to point.)

The big three criteria avoid the meaninglessness of the singular point at the crack tip by actually referring, in their definitional process, to how the stress is distributed over a *finite* volume *elsewhere*--viz., ahead of the crack tip.

At first glance, the SED-class of criteria seem unable to avoid the difficulty arising due to the singularity. But this is only an illusion. The singular stress field itself is just a model--an analytical model--that in turn carries its own assumptions--the assumptions that permit the model to carry a singularity in itself.

In contrast to the "big three" criteria, only the SED-class of criteria would be suitable for multi-scale modeling and simulation. The reason is that only these criteria are field variables--not global parameters.

My idea of testing for SED in the presence of a crack, mentioned earlier in this thread, is somewhat *wrong*.

For experimental determinations, such an idea would fundamentally involve a conceptual mis-match, and hence it would necessarily lead to a form of circularity. The issue here is quite similar to asking the following question: What precisely is the local state of strain at point in the neck in an uniaxial tension specimen? Here, any answer you give necessarily depends on some prior assumptions about how stresses and strains are supposed to be distributed in the region of the neck. Now, the funny thing is that any such assumptions, in turn, require a knowledge of the constitutional law for that material! Thus, there is a basic circularity here--you have to a priori assume a constitutional law in the process of experimentally measuring precisely that very constitutional law!

Now, of course, ASTM (and all the rest of us) still "practically" do believe that it is possible to measure the local strains in necking using just the simple uniaxial tension test. It then should be similarly possible to advocate a practically acceptable manner of pulling the wool over the eye when it comes to measurement of SED, DSED etc. in fracture testing too!

One necessary ingradient of SED/DSED testing, then, would be to specify the volume of the material ahead of the crack tip over which the variable has been experimentally measured (or computationally evaluated.) The specification of the volume could be in relative terms, say, relative to crack size. (Come to think of it, this is at least as decent a suggestion as plain assuming how the traction separation law decays in the cohesive modeling!)

Another point. It may be possible to derive SED/DSED property data more simply from the simpler bi-axial or tri-axial tension tests (rather than through the conventional fracture toughness tests). This way, experimental evaluation of the critical SED/DSED would still refer to material response under multi-axial stress state, but the experiment would not involve the *globally applicable* size considerations that arise in ensuring conformance to singular field requirements, as in the conventional fracture toughness testing.

Thanks to De Xie for his post. The thought-line pursued by him helped me in makng clear the nature of my mistake (that I mention in this post.) 

It is really interesting to read and know new related topics. You can refer this paper also for the application of SED in determining the crack initiation angle.

"Fretting fatigue crack initiation: An experimental and theoretical study
International Journal of Fatigue, Volume 29, Issue 7, July 2007, Pages 1328-1338
M.S.D. Jacob, Prithvi Raj Arora, M. Saleem, Elsadig Mahdi Ahmed and S.M. Sapuan"

very good topic

how we can simulate the bursting of turbine disk using fracture mechanics ( using ABAQUS)

In a recent paper (International Journal of Solids and Structures 45 (2008) 2613–2628), we have used SED (or DSED) as a fracture criterion. The reason is that we manage to construct the analytical solution for the localization in a three-dimensional slender cylinder which gives the point-wise strain energy distribution and then it is easy to use the DSED. I thought that the work is related to the discussions on this topic so I provide the title and abstract below.

a Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 TatChee Avenue,