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# Sih's Strain Energy Density Approach in Fracture - why is it not very popular?

Most fracture classes and texts focus on the following different approaches: Griffith's energy approach, Irwin's stress intensity factor approach, the Barenblatt-Dugdale strip yield model (and subsequently, cohesive zone modeling) and Rice's J-Integral approach. As a graduate student studying fracture mechanics, I have often wondered why there seems to be very little discussion in the community with regard to Sih's strain energy density approach. Are there any fundamental limitations to the approach or are there "other" reasons behind this? Your thoughts are appreciated.

Thanks,

Dhruv

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## Comments

## Sih's SED theory

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.

a good question.

## Why does K work?

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

## K, S, J

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).

## Sih's SED Criterion

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.

## SED

Ajit,

Thanks for raising these points - they agree very well with my general reading (Mechanics of Fracture Initiation and Propagation by G.C. Sih). The combination of three-dimensional crack initation and propagation along with incremental plasticity for "large-scale" yielding, seems to be the nature of problems where SED theory would have an advantage, if my understanding is correct. I don't believe any of the G/K/J approaches can adequately be used for this class of problems (3D + incremental plasticity) which is why damage mechanics approaches are often resorted to. Again, I would appreciate any comments on the truth of these statements. I was drawn to SED theory because of the limitations of the G/K/J approach.

Also, thank you for exposing me to DED, I must confess I was unaware of its use as a fracture criterion and will look into this as well. I look forward to reading your paper - I think it would be a very worthy topic to address.

Thanks a lot.

## von Mises in Fracture

About K, G and J-I for incremental plasticity. There are a lot of pros and cons discussed in the literature. Many of the pro side of arguments come from tall personalities of mechanics. As such, these arguments are likely to be repeated over a long time to come.

Yet, personally, I think it is not posible to fully consistently advocate K (or G or J-I) as a *material property* if plasticity effects are going to be significant. Practically speaking, one very direct way in which the plasticity (or damage) effects make their presence felt in testing is via the specimen size effect. ... If plasticity in your situation is great enough, i.e., if you are going to get two different values of K just because you had two different sizes of specimens (or, if you are required to have a KIC specimen of 10 meters or 100 meters size) you might as well not at all use K (or G or J-I).

At the same time, unlike Sih's position, I also believe that the simple uniaxial tension test cannot accurately probe how the material is going to respond to the applied loading in the presence of a crack.

That is why I guess I would end up advocating the use of some parameter other than K (or G), *but* only as evaluated with a cracked specimen--not the uncracked one.

SED seems to be one such parameter. I thought DED would be another.

Actually, Dhruv, I am 101% sure you know DED already. In the context of the criteria for yielding in simple uniaxial tension, DED is known as the von Mises criterion. The only reason to use another name was to help us remind that the experimental evaluation here is assumed to be done on a specimen carrying a crack, that the purpose is study of fracture, not of yielding. A separate name helps remember that things like work hardening, micro-damage, stress-induced martensitic-transformations, irreversibility in unloading, hysteresis, etc. are all the implicit considerations. Simple yielding does not involve these matters just the way it does not probe the response to an already cracked configuration.

I do not know if anyone has even advocated DED as criterion in the context of fracture. When I wrote the above, I though it was a new suggestion. But, of course, I would be happy to know of precedence if any. One feels more secure that way! (BTW, this is advantage iMechanica. Since time-stamp is there, one can discuss ideas even in advance of writing the paper let alone journal publication.)

BTW, to make the term somewhat more "lively," let's agree to henceforth call it DSED (Distortional *Strain* Energy Density).

Thanks for your feedback and I look forward to seeing if you in your research find anything special with DSED--how its prediction compares with SED.

## The reason is simple

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.## More on fracture criteria and (D)SED

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.)

## SED for crack initiation angle

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

very good topic

## please let me know

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

## SED

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.

On constructing the analytical solutions for localizations in a slender cylinder composed of an incompressible hyperelastic material

Hui-Hui Dai a,*, Yanhong Hao b, Zhen Chen c

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

Kowloon Tong, Hong Kong

b Department of Mathematics, City University of Hong Kong, 83 TatChee Avenue, Kowloon Tong, Hong Kong

c Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO 65211-2200, USA

Abstract

In this paper, we study the localization phenomena in a slender cylinder composed of an incompressible hyperelastic material subjected to axial tension. We aim to construct the analytical solutions based on a three-dimensional setting and use the analytical results to describe the key features observed in the experiments by others. Using a novel approach of coupled series-asymptotic expansions, we derive the normal form equation of the original governing nonlinear partial differential equations. By writing the normal form equation into a first-order dynamical system and with the help of the phase plane, we manage to solve two boundary-value problems analytically. The explicit solution expressions (in terms of integrals) are obtained. By analyzing the solutions, we find that the width of the localization zone depends on the material parameters but remains almost unchanged for the same material in the post-peak region. Also, it is found that when the radius–length ratio is relatively small there is a snap-back phenomenon. These results are well in agreement with the experimental observations. Through an energy analysis, we also deduce the preferred configuration and give a prediction when a snap-through can happen. Finally, based on the maximum-energy-distortion theory, an analytical criterion for the onset of material failure is provided.

I am mainly an applied mathematician working on "Understanding and Exploiting Nonlinearity" in solids (nonlinear waves, instabilities and phase transitions, etc.). I am not an expert on fracture or damage mechanics and this is my first piece of work related to this area. However, I would certainly like to do more work in this area, in particular, fracture/damage caused by post-bifurcation behaviour (I view the damage caused by localization in a cylinder subject to tension/extension as one of such problems).

## My comments on SED approach

Dear Dr. Dhruv Bhate:

I read your comments and those of Dr. Ajit R. Jadhav with regard to SED and G/K/J approaches and their pros and cons. You put forward a good question and the discussions are very useful and helpful. While the non-technical issues can differ from the individuals, the technical issues are still open to discussion. I studied fracture mechanics when I was a student in the college. Now I teach fracture mechanics as a teacher. According to what I have read and know about the use of SED in contrast to G/K/J, I would like to point out a few lines below.

(a) I consulted Prof. Sih for the understanding and physical ground of SED theory several years ago. I know that SED theory has been successfully applied to a variety of engineering problems for different materials and at different scales in the open literature. The pre-1991 works on SED can be found in

G. C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer, Boston, 1991. In my opinion, energy release rate G and stress intensity factor K apply essentially to linear elastic systems while path-independent integral J to non-linear but still elastic systems. Systems with dissipation are excluded because of the presence of non-equilibrium, particularly when plasticity is included. Sih has pointed this out in a number of his more recent works that the neglect of the rate change of volume with surface has restricted continuum mechanics to equilibrium problems. This includes plasticity that assumes equilibrium by using the average bulk material properties but yielding addresses microscopic effects. Macro-equilibrium does not guarantee micro-equilibrium. Plasticity based on the average bulk properties cannot reconcile the non-equilibrium microscopic effects, a fundamental drawback and inconsistency of plasticity. The fact is that both J and G become negative for non-homogeneous systems, a character of non-equilibrium. In essence, G/K/J do not apply to localized physical systems such as the crack tip where all the action takes place. Examples can be found in the published literature, such as:[1] C. P. Spyropoulos. Energy release rate and path independent integral study for piezoelectric material with crack. International Journal of Solids and Structures, 41(3-4) (2004) 907-921.

[2] Y.E. Pak. Crack extension force in a piezoelectric material. Journal of Applied Mechanics, 57(3) (1990) 647-653.

[3] Z. Suo, C.M. Kuo, D.M. Barnett, J.R. Willis. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids, 40(4) (1992) 739-765.

(b) The idea of DSED (Distortional *Strain* Energy Density) approach is a complete misunderstanding. It should be emphasized that there are two basic ways in which a volume element absorbs energy that eventually leads to failure: an element can store energy by “dilatation” and “distortion”. The former is associated with the change in volume and the latter is associated with the change in shape. When an element of the material exceeds a certain threshold fracture could occur. The tradition of mechanics is to treat the surface properties as a separate entity from those in the bulk. It becomes troublesome when dealing with the behavior of objects whose volume-to-surface ratios can change from the very small to the very large.

(c) Moreover, yield criteria based on distortional energy density are known to be sub-case of SED that contains both the von-Mises or distortional and dilatational effects. The proportion of both effects are automatically accounted for linear and nonlinear effects. Neglecting one or the others is equivalent to dropping out the surface effect against the volume effect or vice versa. This was mentioned in (1) where both surface and volume effects must be weighed simultaneously. Their relative effects are about four to one, not even one order of magnitude. For non-linear and non-homogeneous problems the relative influence can be determined from the stationary values of SED (maxima and minima of SED). Reference can be made to:

[4] G.C. Sih, E.P. Chen. Dilatational and distortional behavior of cracks in megnetoelectroelastic materials. Theoretical and Applied Fracture Mechanics, 40(1) (2003) 1-21.

(d) The recent works have further shown that SED can be used at any different scale levels such as nano. micro and macro while G/K/J are limited to the macro scale using the average bulk material properties. If one is interested, a group of papers can be found in the related technical journals.

(e) It is surprising that SED has not been used more often to solve general engineering problems, not only with cracks. I have applied it to classical thermoelastic crack problems and the results are straight forward:

[5]

X. S. Tang. Mechanical/thermal stress intensification for mode I crack tip: fracture initiation behavior of steel, aluminum and titanium alloys. Theoretical and Applied Fracture Mechanics, 2008, 50(2): 92-104.[6]

X. S. Tang. Mechanical/thermal stress intensification for mode II crack tip: fracture initiation behavior of steel, aluminum and titanium alloys. Theoretical and Applied Fracture Mechanics, 2008, 50(2): 105-123.(f) I leave the non-technical issues of SED versus to G/K/J to those who may have a better idea why the majority prefers using G/K/J while I will be glad to answer questions with reference to what I have said. I can also consult Sih on the future use of SED that I am not familiar to.

Prof. X.S. Tang

School of Civil Engineering and Architecture

Changsha University of Science and Technology, China.

Nov. 12, 2008

## My comments on SED

It is very interesting to learn about the topics discussed. According to what I have read and know about the use of SED as well as G/K/J, a few lines below are pointed out:(1) In my opinion, energy release rate G and stress intensity factor K andpath-independent integral J are global parameters. The strain energy density theory considers the complete energy field both local and global. G/ K can be applied to linear elastic system, the J-integral can be applied to nonlinear elastic deformation system. In Linear Elastic Fracture Mechanics (or under the Small Scale Yielding condition ), the J-integral is exactly equal to G. It is obvious that the SED criterion can predict precisely the same critical stress as what is determined by using either K or G in LEFM.(2) The non-equilibrium microscopic effects exist in plastic deformation systems. Average volume material parameters, which are applied to analyze plastic deformation problems, can not describe the non-equilibrium microscopic effects. K/G/J can not be applied to plastic deformation system. Negative J and G was found in non-homogeneous systems, so G/K/J can not be applied to localized deformation systems. SED theory takes the complete both local and global energy field into account, SED can be applied to both plastic deformation systems and non-homogeneous systems.(3) Since SED theory considers the complete energy field both local and global, which varies from point to point in materials, only the SED criteria can be applied to multi-scale modeling and simulation. It can find the location of crack initiation, the crack path and the point of final termination.(4) Distortional energy density are known to be sub-case of SED that contains both distortional and dilatational effects. For non-homogeneous material, the proportion of both effects had been determined from the stationary values of SED by Sih.(5) The critical value of the strain energy stored in a unit volume corresponds to the area under the true stress and strains curve. They are readily available in the handbooks.(6) Subroutines in the computer program can be determined for finding the growth direction of the pre-existing crack or a point in a material without crack. SED has been applied to solve general engineering problems.

## My comment on SED theory

The discussion on the Sih’s SED theory is very interesting, so I would like to have some words on this topic. The SED is a fracture criterion which is proposed based on a different physical standpoint. It predicts the fracture behaviors according to the strain energy density rather than the stress intensity factor. The strain energy density associates the energy with space. The SED provides a different view on this problem.It is well known that the stress and strain at the vicinity of fracture tip are of r-(1/2) order singularity. It is impossible to directly use the stress or strain to predict the fracture behavior. The stress intensity factor K is a finite variable which governs the stress and strain field. So the K/G theory uses K as an index to predict the fracture behavior. Moreover there is a relationship between K and the energy release rate. Different from the K/G theory, the SED theory reveal the fracture behavior from the energy density concept. Since both the stress and strain have the r-(1/2) order singularity, the strain energy density has the r-1 order singularity, just shown as the specific formula of SED, dW/dV = S/r with W being the energy, V the volume, S the energy density factor, r the distance from the fracture tip. The energy density factor S is a finite variable. It governs the energy density field as the stress intensity factor governs the stress strain field. So the beauty of SED lies in that it predict the fracture based on the energy density concept which unites the three stress intensity factor, KI, KII and KIII, into the general expression of strain energy density factor for the linear case. But the idea applies to any non-linear material. Hence, the SED has great potential in predicting fracture behaviors, especially the complicated fracture problems.

To reinforce what was said above the relation dW/dV = S/r holds not just near the crack tip but even for S being a function of r where the K factors are no longer valid. That is the factor S(r) applies to distance further away from the crack front. This is because dW/dV can be computed in general for any problems by factoring out R as the distance referenced from the location of crack initiation. This is a necessary condition for any non-trivial solution in mathematics as well as in physics that a finite local distance must be observed that is r¹0 otherwise you will end up with a trivial solution.

Zhennan Zhang

## Comments on SED

In a previous lifetime I was heavily involved with a couple of engineers at Douglas, now Boeing who applied the SED model to fracture of center cracked panels. Center cracked panel (CCP) is the model for cracks in skin of a structure such as an aircraft or a cylindrical pressure vessel and the fracture behavior of CCP do not follow the K or G or J theory at all -- results from CCP tests cannot be used to determine Kic because the results depend heavily on the size of the test panel -- both thickness and width.

The two engineers, both dead now, were George Bockrath and Jim Glasco -- and they tried to apply the SED model to predict failure in a manner analogous to the manner used by the fracture mechanics community at the time (1980's). They were suprisingly successful. They tried to get a closed form solution of the strain energy density at each point in the stress-strain field at the crack tip. They were not completely successful -- it turns out continuity theory only worked in the field where the stresses were below "ultimate" -- that is the true stress/strain where necking begins in the uniaxial tensile stress test. In the stress/strain field at the crack tip where the strains or stresses exceed this value they couldn't get data to agree with their calculations.

All was not lost -- it turns out if you take the remaining area from were continuity theory breaks down to where r=0 as a set volume (they considered it to be a coarse slip band) you can use the strain energy density from the true stress/strain unixial tensile test as a constant. For high strength low toughness materials (aluminums, high strength steel and titanium) this volume of material times the average strain energy density from the true stress/strain tensile test from UTS to fracture correlates well to Kic. Their analysis actually matches well with the analysis for J integral, which accounts for all the plastic energy and what's left over is Kic (George and Jim performed an area integral of the SED around the crack tip, J integral performs a line integral of the tensor -- equivelent analysis really).

Using this model, they were able to predict critical failure stress vs crack length for CCP tests for a wide variety of metals (aluminum, steel, beryllium and titanium), thicknesses and stress conditions and also for more practical applications (pressure vessels, ship hull plates and other structural plates). Their theory worked well for CCP not only under LEFM but also for conditions where the applied stress was well above general yield, predicting fracture with remarkable accuracy.

I went on to other things in the 1990's and lost track of them. They did not publish extensively and I haven't seen any use of their theory recently. If this thread continues I would like to know if their work is still being used. Prior to writing this I Googled Bockrath and Glasco and got a couple of hits -- the one that is most pertinate is "Theory of Ductile Fracture". I also found a couple of papers from China that looked like they were based on George and Jim's work.

Bill Crumly