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The present EFM paper selected for discussion applies artificial intelligence (AI) to fatigue crack growth. The subject is on the outskirts of my competence. To say the least, I am on thin ice when it comes to AI, machine learning, neural networks and similar. Still, I get the feeling that the selected paper describes an interesting step forward. I am sure that it will, sooner or later, be a reliable tool for predicting closing and opening loads at fatigue crack growth. 

The paper is: “Combining artificial intelligence with different plasticity-induced crack closure criteria to determine opening and closing loads on a three-dimensional centre cracked specimen” by R. Baptista and V. Infante in Engineering Fracture Mechanics, vol. 312, 2024.

Several cases are calculated using 3D FE models with elastic-plastic modelling. The results are focusing on the plastic zone and the crack growth. The latter refers to a comparison with the Dugdale model for which the stress reaches a limiting stress level. When the level is reached, the connected elements are removed by node relaxation. The process is similar to a cohesive zone. 

Interestingly, the Dugdale model was already solved and published by Russians. The Russian scientists were the coworkers Leonov and Panasyuk and independent of the Barenblatt. The reason for it being called the Dugdale model is probably that the Russian results were not well known in the Western world. Barenblatt’s solution is a decohesive zone in which the cohesive stresses decrease with increasing distances between the cohesive zone boundaries. The solution by Leonov-Panasyuk and Dugdale is a special case of Barenblatt’s model. The special case is for an infinitesimally thin, perfectly plastic sheet in plane stress. Also, the cohesive zone is straight ahead of the crack tip. The comparison is mathematical, while Barenblatt’s solution is the more elaborate for a reason. It concerns the fracture processes that gradually decrease the load-carrying capacity, which brings us closer to the real world. A decohesive zone is a model of the fracture process region. It would be interesting to see results for a Barenblatt zone. Already in a Dugdale thin sheet that suffers from necking ahead of the crack tip, the limit load for separation decreases with the decreasing plate thickness in the neck. The ultimate displacement before the completed fracture should be around the same as the original sheet thickness.

https://imechanica.org/node/27498

Shifting from macroscopic to microscopic plasticity helps us understand mechanisms that can help us develop high-strength metallic materials. Things that prevent dislocation dynamics or generation, such as other dislocations and grain boundaries in polycrystalline materials, lead to higher strength.

The interesting and well-written paper

“Dislocation penetration in basal-to-prismatic slip transfer in Mg: A fracture mechanics criterion” by Ryosuke Matsumoto in Engineering Fracture Mechanics, vol. 306, 2024, https://doi.org/10.1016/j.engfracmech.2024.110250,

analyses the penetration behaviour of pileup dislocations using molecular dynamics. The author guides us through different stages of dislocations that attack grain boundaries. 

The reviewer especially likes the analogy connection to mode II cracks and Takeo Yokobori’s analysis from the 1950’s. A good reference for basic mathematics is found in Some basic problems in the theory of elasticity by N Muskhelishvili in Russian from 1933 and English from 1954. The solution for a dislocation becomes a tiny cut where one end has a positive square root singular stress tensile stress and at the other end a negative ditto and a displacement discontinuity, i.e. the image of a single atomic layer inserted in an edge crack. Square root singularities do not decay as fast as the 1/r singularity but the negative singularity at one end of the cut and a positive singularity at the other end cancels the square root part and leaves a dominating 1/r singularity, as we recognise as the dislocation far field. The arrangement was used by many of us because it provided a length scale that advised us of the distance related to the short-range interaction between dislocations.

It would be interesting to hear from anyone who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. Perhaps the author can cast some light on future actions regarding the subject. If anyone wishes to comment and does not have an iMechanica account, please register to be able to file a comment. Many applications are rejected for reasons that are not revealed and nor understood by me. If it happens to you, please email me at per.stahle@solid.lth.se and I will post your comments in your name. The paper is available with open access. 

For ESIS,

Per Ståhle

https://imechanica.org/node/27203

Around 20 years ago, I gave a fracture mechanics lecture and talked about crack initiation that happens in the plane with the largest tensile stress. True, at least if the material has isotropic properties. The students already knew where an isotropic material would give the largest stress at bending and torsion. I planned to make a desktop experiment with an icicle and a carrot. This was during the autumn with an abundance of icicles everywhere. The carrot, I found at home.

I asked the students which one would be the anisotropic one. The majority said the carrot but one student, whom I already, halfway into the course, considered to be the smartest in the class, voted for the icicle. I asked why and she said, “Because you asked. If it is the carrot it would be boring”. She was the smarter one and I was the fool. The bending and twisting gave the expected isotropic result for the carrot but not for the icicle. 

It has been known since Laudise and Barns study from 1979, that the major part of the icicle is a single crystal. For all icicles I tested, the bending gave zig-zag crack paths and the torsion almost always resulted in a cup-cone fracture. For the latter, I still do not have a good explanation.

The interesting paper that brought these memories back is

“Thickness-independent fracture in columnar freshwater ice: An experimental study” by I.E. Gharamti, W. Ahmad, O. Poulakka, and J. Tuhkuri  in Engineering Fracture Mechanics, vol. 298, 2024, https://doi.org/10.1016/j.engfracmech.2024.109906.

It is a readworthy paper. It is only related to my in-class experiment via the freshwater ice. The study is very interesting and the motivation is marine applications. According to the authors, the tested ice is polycrystalline which is no surprise since the test specimens are measured in cubic metres and huge as compared with my middle finger-sized icicles.

What we learn is that the ice in general is polycrystalline and may be assumed to be isotropic. The experiments show that there is no detected influence of the ice thickness on the fracture behaviour. As I understand it the implication is that the material does not change properties as it grows thicker and the thickness change leads to a proportional increase in load-carrying capacity.

The expected transition from plane strain to plane stress occurs for specimens with thicknesses that are around, or smaller than the characteristic length scale of the cohesive zone ahead of the crack tip. In the paper, it is called the fictive crack based on Hillerborg’s notation for non-linear regions in concrete or similar materials. A more widespread and earlier introduced denotation is Barenblatt process region. 

The experiments were performed on specimens with pretty large thicknesses. With estimated process zone sizes being small as compared with the specimen thicknesses, the plane strain to plane stress transition that occurs for thin specimens is avoided. 

I have friends from the north of Sweden who claim that one or two inches of ice is enough to carry a human. It should depend on weight and shoe size. Since I am a Swede and the authors from the neighbouring country Finland share the same Baltic Sea water, it would be interesting to know if one or two inches is enough also on the Finnish side.

Only a couple of inches sounds risky. Also, it would involve the plane stress transition and the present linear scaling would possibly fail. It would be interesting to hear if anyone knows if a switch to plane stress leads to reduced or increased safety.

It would be interesting to hear from anyone who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. Perhaps the authors can cast some light on the Swedish sufficient ice thickness guess. If anyone wishes to comment and does not have an iMechanica account and fails to register, please email me at per.stahle@solid.lth.se and I will post your comments in your name. If the paper is not open-access it will be that in a couple of days. 

Per Ståhle

https://imechanica.org/node/27203

A most readworthy paper, “Static and dynamic mode I fracture toughness of rigid PUR foams under room and cryogenic temperatures” by E. Linul, L. Marşavina, C. Vălean, R. Bănică, Engineering Fracture Mechanics, 225, 15 February 2020, 106274, 1-10, is selected for this ESIS blog. It has received a lot of attention and was for an extended period of time one of the most read papers in EFM. The attention is earned because of the clear and concise writing about an intricate material that did not yet get as much focus as it deserves. 

As the title says, the paper concerns fracture mechanical testing of a solid polyurethane foam. The material has a closed pore structure. It is frequently used in the transport sector for its low density. It also has desirable performance at compression, giving a continuous and almost constant mechanical resistance. The beneficial properties are taken advantage of in applications such as sandwich composites, shock absorbers, packaging materials etc.

I have no professional experience of the material but I have come across it a few times and I recognise its character. The excellent description in the introductions confirms the feeling of something that I am familiar with, i.e., the crushing under compressive load and the brittle fracture in tension. Judging from the listed yield stresses given in the paper, I guess that one can manually make an indent e.g. with a finger.

Before fracture the material may be treated as linear elastic with the elastic limit reached only in a small region at the crack tip, which is controlled by the stress intensity factor KI. The linear extent of the non-linear region is supposed to be below at most a tenth, or so, of the crack length. The exact limit depends of course on the specific geometry. 

The ASTM convention described in STP 410 by Brown and Srawley in 1966, claims for structural steels that ligaments, thickness, and crack length should not be less than 2.5(KIc/yield stress)2. It is not mentioned in the paper but the results show that the specimen in all cases fulfill these requirements with an almost four-folded safety, i.e. ligament, thickness and crack length exceed 9.6(KIc/yield stress)2. 

The validity of the obtained toughnesses KIc becomes important when it is applied to real structures with cracks that could be too small. This is not within the scope of the present paper. When is a crack too short for linear fracture mechanics? It may not be the most urgent thing to study, but I guess that it has to be checked before the results are put into general use. I am particularly excited over how it compares with the STP 410 recommendations. 

When the scale of yielding or damage becomes excessive the fracture process region generally loses its KI autonomy. It happens when the shielding of the fracture process region increases which leads to an increased energy release rate required for crack growth. An analysis would require a more elaborate continuum mechanical model in combination with a box or line model of the fracture process region. The material model would be a challenge I guess. 

I did a minor literature search for both establishing the limits of linear fracture mechanics and the application of non-linear models beyond these limits for solid foam materials but didn’t find anything definite. I could have missed some. Who knows?

Per Ståhle 

https://imechanica.org/node/25514

It is common practice when solving boundary value problems to split the solution into a symmetric and an antisymmetric part to temporarily reduce the number of variables and the mathematical administration. As soon as the symmetric problem is solved, the antisymmetric problem, or vice versa, is almost solving itself. Any problem can be split into a symmetric and an antisymmetric part which is a relief for anyone who analyses mixed cases.

It gives a clearer view but it is an academic exercise while nature usually doesn’t have any comprehension of symmetry and antisymmetry. Fracture is no exception. The fracture processes will be activated when sufficient conditions are fulfilled. Even the smallest deviation from the pure mode I or II caused by geometry or load will not affect the conditions at the crack tip in any decisive way. Everything is almost pure mode I or II and it may be convenient ignore the small deviation and still treat the problem as a pure case. This seems simple enough but the paper reviewed tells that it has been a tripwire for many. The selected paper is the recently published:

“An improved definition for mode I and mode II crack problems” by M.R. Ayatollahi, M. Zakeri in Engineering Fracture Mechanics 175 (2017) 235–246.

The authors examine a power series expansion for an Airy stress function about the crack tip. The series give stress as a sum of powers r-1/2, 1, r1/2, r, etc. of the distance to the crack tip. Each term has an known angular dependence. The application is to a plane crack with any in-plane load. The series starts with a square root singular term while it is assumed that the crack tip is sharp and the material is linear elastic. The assumption requires that the geometrical features of the crack tip and the nonlinear region is not visible from where the expansion with some accuracy describes the stress field. The problem that the authors emphasise is that the splitting in symmetric and antisymmetric modes that leads to two similar expansions of the radial power functions with symmetric and antisymmetric angular functions. The representations so far has been called pure if the solution is strictly symmetric or strictly antisymmetric, i.e. the notation has been pure mode I and pure mode II. The problem is that not only seldom, has a vanishing mode I stress intensity factor misled investigators to drop all symmetric terms of the series expansion. Also mode II has been unfairly treated in the same way. The most striking problem is of course when the constant stress acting along the crack plane, the T-stress, by mistake is neglected. The authors are doing a nice work sorting this out. They describe a range of cases where one stress intensity factor vanishes but for sure the crack tip stress state is neither strictly symmetric nor strictly antisymmetric. They also provide quite many examples to demonstrate the necessity to consider the T-stress even if the mode I singular stress term is absent. I commend the authors for doing a conscientious work. 

If I should bring up something where different positions may be assumed it would be the selection of the series. The powers of r-1, r-3/2, r-2 etc are never mentioned and I agree that it is not always necessary. It should be commonly known that a sharp crack, a linear elastic material and traction free crack surfaces says it. There cannot be any stronger singularities than r-1/2. However, isn’t one consequence that close enough to the crack tip any constant stress should be insignificant as compared to the singular stress terms. If so, it should not have any significant effect on the stresses closest to the crack tip and neither affect the fracture processes nor the selection of crack path. On the other hand, if the constant term has a real influence on the course of events, that would as far as I understand mean that the nonlinear region has to have a substantial extent so that its state is given by both singular terms and the T-stress. The contradiction is then that the stronger singular terms r-1, r-3/2, etc. cannot be neglected. These terms are there. Already the r-1 term seems obvious if the crack has grown because of the residual stress caused by plastic strain along the crack surface that in the wake region behind the crack tip.  

Also, the region of convergence, which is at most the length of the crack, is another pothole. Outside the convergence region a different series or an analytical continuation, may be used. For the series expansions the symmetric and antisymmetric solutions have to be treated as well, with the difference that there are constant stresses in both symmetric and antisymmetric modes that have to be included.  

It would be interesting to hear if there are any thoughts regarding this.

Per Ståhle

P.S. On the courtesy of Elsevier there is a 3 month promotional access to the latest article in the blog, meaning the articles are freely available to everyone. Now everyone who wishes to comment or discuss the paper here can do so.  (Dr. Kumar, I hope you are reading this).

https://imechanica.org/node/21428

This is the story of threefold failure, which doubtlessly is the subject of fracture mechanics, a story of failure in various regards, however. First, it comments on an article dealing with failure assessment, second it reports on the personal failure of the blogger to understand this article, and finally it bemoans the failure of a seminal idea.

Chasing for “prey”, I came upon a contribution on the assessment of “crack-like defects under combined primary and secondary loads”, namely

P.M. James: Re-derivation of plasticity interaction for combined loading under significant levels of elastic follow-up. Engineering Fracture Mechanics, Vol. 126, 2014, pp. 12–26,

and was intrigued by the expression “elastic follow-up”, of which I had never heard before. I started asking friends and colleagues who are engaged in fracture mechanics but they couldn’t help me. Collins Compact English Dictionary explains “follow-up” as “something done to reinforce an initial action” – which wasn’t really helpful, either. The author of the above contribution states that “elastic follow-up can be considered to occur in cases where the secondary load acting over a sufficiently large length scale such that localised relaxation (e.g. in the vicinity of a crack) does not diminish the influence of the remote stresses” – which left me stranded, still not knowing which “effect” is actually addressed, particularly because I do not have the slightest idea what “primary and secondary loads” are. Assuming (!) that the respective effect (which one?) “can be described by a single parameter” the author presents a quantitative measure, the “elastic follow-up factor”, Z, at least, which traces back to a preceding article of an internationally acknowledged expert of integrity assessment,

R.A. Ainsworth: Consideration of elastic follow-up in the treatment of combined primary and secondary stresses in fracture assessments. Engineering Fracture Mechanics, Vol. 96, 2012, pp. 558–569,

where I read: “when elastic follow-up is high this leads to secondary loads acting as primary”, which appeared as mystical as the explanation cited above, just inverting cause and effect. Obviously, nobody who has not internalised the concept of primary and secondary loads or stresses will ever be able to understand this “effect”.

In engineering mechanics, students are taught Cauchy’s stress principle of 1823, which was a breakthrough in the science of strength of materials enabling engineers to reduce various loading configurations to simple entities, viz. stresses, and to measure strength limits on simple test specimens. Actually, we measure deformations and relate them to stresses by constitutive laws. There is no room or need for primary and secondary stresses within in this framework, least of all for primary and secondary loads.

I scanned further literature on the problem finding numerous contributions. The whole world seemed to know what “elastic follow-up” is, except me. A contribution on “creep-fatigue tests  including elastic follow-up“ in the International Journal of Pressure Vessels and Piping of 2000 presents some uninspiring “illustration of follow-up behavior”. The essential hint resulted from the title of an article, “generalization of elastic follow-up model”, in Nuclear Engineering and Design of 1995: what, if this “effect” was not a physical phenomenon, a “behaviour”, but a model used in assessment codes? Finally, ITER Structural Design Criteria for In-Vessel Components (Appendix C) gave the enlightening explanation: “Neuber’s rule is applicable if the remote stress field away from the notch is elastic. If the remote stress-strain field itself undergoes plastic deformation, then a further correction is necessary, because the remote strain is greater than the elastically calculated strain.” It simply says that “elastic follow-up” is a correction term in an elastic analysis incorporating plasticity.

The code also gives a comprehensible definition of “primary and secondary stresses”, which appear to be model artefacts rather than having physical significance: “Consider a cylindrical bar of length L, cross-sectional area A, which is subjected to an axial load such that the extension would be uel if it behaved elastically. … There are a number of ways of applying the specified load, the two simplest being a displacement u = εel.L and a force F = E A εel.. imposed. As long as the behaviour is linear elastic, a strain εel.. is effectively obtained for both loads. When the behaviour ceases to be linear elastic, the two loadings no longer cause the same strain. For the imposed displacement loading u, the real strain remains the same as the elastically calculated strain εel., which means that no correction is necessary and the elastically calculated stress = E u/L is a pure secondary stress. For the imposed force load, the real strain corresponds to the real stress = F/A on the stress-strain curve. This stress is a pure primary stress that can be seen to cause real strain  which is much higher than the elastically calculated εel.”

This I can comprehend as it fits in my terminology and my view of the world of mechanics.

Now what is the conclusion resulting from this story?

·         If terminology creates insurmountable barriers of understanding even among people having similar scientific interests and background, namely fracture mechanics and structural integrity, we have to be concerned about the language we use in our publications.

·         If no distinction is made between models and physical phenomena, misunderstanding and misconception are programmed.

Finally, the present contribution marks the failure of the constitutive idea for the present blog. Its aim was to create a forum ofscientific exchange, realising that scientific achievements require time and chance for free, impartial and uncensored discussions among people. The European Structural Society (ESIS) and an international publisher of scientific journals appeared as an ideal combination for launching such a project. However, encouraging young scientists to frank discussions about their findings will work in a large-minded and democratic atmosphere, only, where they must not fear sanctions. Representatives of a society who themselves do not stand divergent opinions give a poor example. This is my last blog entry I shall be able to write.

»

• W. Brocks’s blog

https://imechanica.org/node/16898

Some material scientists and experimentalists are generally sceptical of simulations and reproach the theoreticians with lacking knowledge of real materials. Sometimes they may just be ignorant of the mathematics behind the models but sometimes they appear to be right. An example: Introducing “damage” into a constitutive equation simply as an internal variable which obeys some evolution law, not having the foggiest notion about the specific nature of damage – whether brittle, ductile, creep, fatigue – and its micromechanical mechanisms, promotes scepticism about the benefits of modelling in general, and deservedly so.

This is not my particular point today, however, but it is related. My problem today is the handling of length scales in

H. Krull and H. Yuan: Suggestions to the cohesive traction–separation law from atomistic simulations. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 525-533.

Ductile tearing is governed by the initiation, growth and coalescence of voids in an elasto-plastic or viscoplastic material. Koplik and Needleman (1981) have been the first to perform unit-cell calculations of void growth to analyse this mechanism, and numerous studies by other authors followed varying the void shape, accounting for inclusions etc.. They helped improving and generalising constitutive equations of ductile damage like the Gurson model of porous metal plasticity. Unit-cell calculations have also been used to derive traction-separation laws for cohesive zones. The physical processes take place at length scales of micrometers to millimetres, accordant to the dimensions of the microstructure. Continuum models still apply at this length scale.

Atomistic simulations and molecular dynamics are based on models that relate binding energies or forces to spatial configurations in order to calculate accelerations of particles via Newton’s law. They describe interactions at a length scale of nanometers, which is at least three orders of magnitude below the relevant length scale of ductile tearing. What occurs at this length scale has absolutely nothing in common with plastic deformations due to dislocation motion and ductile tearing of metals, and hence the authors’ conclusions are apocryphal and unsubstantiated:

>     “The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale.” 

 What do the authors actually mean by vague formulations like “in the form of void nucleation …” and “similar to ductile fracture”? Voids nucleate at particles, for instance. Which particles are of atomic or sub-atomic size? What is their criterion for a process being “similar to” another?   

>   “Understanding the failure process based on atomistic simulation can provide detailed information for the cohesive law.”     

This is correct, of course, provided the correct failure process is modelled.  

>   “The relationship between atomic traction and atomic separation including elastic deformations confirms the exponential function form suggested by Needleman.”   

This is neither a miracle nor striking news, as the cohesive law proposed by Needleman in Int. J. Fracture 42 (1990) is based on the universal atomistic binding energy function proposed by Rose et al. in Phys. Rev. Letters 47 (1981), and he actually analysed “tensile decohesion along an interface” in an elastic medium.     

>  “The computations show that void nucleation and growth are controlled by tensile stress and hydrostatic stress. The Mises stress is not involved in material failure.”     

This is the final death sentence to any simulation: that it yields results which are contrary to everything that is known about the real process. Apart from this, the hydrostatic stress and the maximum tensile stress depend on each other under fully plastic conditions of plane strain. But plasticity is out of the scope of the MD simulations, anyway.

After all, the simulations could indeed provide useful information on a cohesive law for a process which is not void growth and coalescence in ductile metals, however. It is the authors’ business to present an actual decohesion mechanism following their model. 

https://imechanica.org/node/10945

Williams derived it in 1939, Irwin addressed it in 1957 as one of two parameters characterising “the influence of the test configuration, loads and crack length upon the stresses”, and Rice used it in 1974 to calculate the effects of the specimen geometry on the plastic zone in small scale yielding: the non-singular term in the series expansion of the stress field at crack tips called T-stress. It gained importance in the early 1990s in numerous investigations on constraint effects and two-parameter approaches. With the upcoming of damage mechanics, the number of publications on T-stress went down and everything seemed to be said about their significance. The few papers on T-stresses appearing in the first decade of the 21st century mostly concerned their calculation for various configurations – even for “cracks in anisotropic bimaterials” (EFM 75, 2008), which is outside of their theoretical foundation. Surprisingly, two new (partly quite similar) papers on this subject appeared recently:

J.C. Sobotka, R.H. Dodds: Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 343–363.J.C. Sobotka, R.H. Dodds: T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 1182–1200.

Is this a renaissance of early fracture mechanics concepts or just a latecomer? Let us have a look on the details.

T-stress effects on stress fields at stationary cracks for small-scale yielding have been extensively investigated in the 1990s using a so-called boundary-layer model, i.e. a disk-shaped volume centred at the crack front, which is subjected to a K-field and a constant stress parallel to the ligament. It needs a particular Eulerian analysis to represent steady-state growth on a fixed mesh in a boundary-layer framework. The application of the respective “streamline integration” introduced by Dean and Hutchinson in 1980 to 3D panels is the basic achievement of the two papers, allowing to study thickness effects and variations of plastic zones, stress fields and crack opening displacement over the thickness, the first one for T = 0, the second for T ¹ 0.

These are thoroughly performed analyses yielding substantial information on the local fields. What they do not answer is the question on their relevance for actual fracture problems. In the extensive discussions on ductile tearing resistance, the T-stress has been proposed as a parameter characterising the “constraint”. This definitely works in small-scale yielding, but can steady state crack extension occur under small-scale yielding conditions? The authors argue with crack growth in thin panels of high-strength aluminium alloys as they are used in aerospace structures. They claim in the introduction that a T-L orientation of the cracked panel, “tends to favor a local ‘flat‘ mode I fracture process rather than a local ‘slanted‘ mixed-mode process. … Essentially steady conditions evolve as the crack front advances further over distances of several thicknesses, characterized by a flat-to-nearly-flat tearing resistance curve.“ Both statements are indeed essential in the context of their investigations but unfortunately, they are not substantiated by experimental evidence. And a final question: how significant is the T-stress in a cracked thin panel under tension? So what about a continuation including test data and their analysis?

This is a premiere: my first contribution to the new ESIS’ blog announced in January. Why comment on papers in a scientific journal after they have passed the review process already? Not to question their quality, of course, but animating a vital virtue of science again, namely discussion. The pressure to publish has increased so much that one may doubt whether there is enough time left to read scientific papers. This impression is substantiated by my experience as a referee. Some submitted manuscripts have to be rejected just because they treat a subject, which conclusively has been dealt years before – and the authors just don’t realise. So much to my and Stefano’s intention and motivation to start this project.

Here is my first “object of preference”:

Ehsan Barati, Younes Alizadeh, Jamshid Aghazadeh Mohandesi, “J-integral evaluation of austenitic-martensitic functionally graded steel in plates weakened by U-notches”, Engineering Fracture Mechanics, Vol. 77, Issue 16, 2010, pp. 3341-3358.

The comment

It is the concept of a finite “control” or “elementary volume” which puzzles me. It is introduced to establish “a link between the elastic strain energy E(e) and the J-integral” as the authors state. Rice’s integral introduced for homogeneous hyperelastic materials is path-independent and hence does not need anything like a characteristic volume. This is basically its favourable feature qualifying it as a fracture mechanics parameter relating the work done by external forces to the intensity of the near-tip stress and strain fields.

Fig. 2 (a) schematically presents this control volume in a homogeneous material, and the authors find that “the control volume boundary in homogeneous steel is semi-circular”. But how is it determined and what is the gain of it?

Introducing a characteristic volume for homogeneous materials undermines 40 years of fracture mechanics in my eyes..

One might argue that the introduction of this volume is necessary or beneficial for functionally graded materials (FGM). The authors state however that “comparison of the J-integral evaluated by two integration paths has shown that the path-independent property of the J-integral is valid also for FGMs”. Whether or not this is true (there are numerous publications on “correction terms” to be introduced for multi-phase materials), it questions the necessity of introducing a “control volume”. There is another point confusing me. The J-integral is a quantity of continuum mechanics knowing nothing about the microstructure of a material. The austenite and martensite phases of the FGM differ by their ultimate tensile strength and their fracture toughness. Neither of the two material parameters affects the (applied) J, only Young’s modulus does in elasticity. Hence it does not surprise that J emerged as path-independent! The authors compare J-integral values of homogeneous and FG materials for some defined stress level at the notch root in Fig. 10. The differences appear as minor. Should we seriously expect, that a comparison of the critical fracture load predicted by Jcr and the experimental results (Fig. 16) will provide more than a validation of the classical J concept for homogeneous brittle materials?

Not to forget: The authors deserve thanks that they actually present experimental data for a validation of their concept, which positively distinguishes their paper from many others!

W. Brocks

https://imechanica.org/node/9793