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The paper, “A machine-learning fatigue life prediction approach of additively manufactured metals” by Hongyixi Bao, Shengchuan Wu, Zhengkai Wu, Guozheng Kang, Xin Peng, Philip J. Withers in Engineering Fracture Mechanics 242 (2021) 107508, p. 1-10., adopts a very interesting view of the correlation between fault geometry and fatigue properties. A simplified statistical description of irregular faults in large numbers is used. The variety of fault shapes that appear during the production of 3D objects from powder metal is described in terms of the distribution of size, volume, and position.

The studied test specimens are produced by selective melting during the build-up of a powder bed of a granulated titanium alloy. Each new layer is fused together with the underlying solidified material. The heat is introduced by a focused high-energy ultraviolet light beam. An almost inevitable problem is small defects, typically of grain size. Naturally, the strength of the structure and especially the fatigue properties take a beating. The authors examine the defects using synchrotron X-ray tomography. After fatigue experiments, the results are used for a machine learning method based on extended linear regression.

The statistical description based on a few geometric and morphology parameters if of course better than the size of a hypothetical crack that we often use for fracture mechanical analyses. The correlation of the more realistic geometrical description with the fatigue limits swallows the entire series of events from fault, fatigue crack initiation, and growth to final rupture.

I guess it could be interesting to benchmark test could be to use available analytical solutions of interacting, cracks, holes, spheres, edges, etc. If necessary numerical ditos could be used. Stress intensity factors for cracks and stress criteria for other faults with smooth boundaries.  

The paper is nicely written and offers very interesting reading. To me, the paper also calls for a reflection. Very few scientific studies combine basic science from different disciplines and create something directly industrially useful. This present paper is a good example of that. 

For industrial applications perhaps the Kalman filter could lead to a speedier optimization since it recursively adds adjustment of the previous result after each new mechanical test. In terms of calculation, it is advantageous because it does not require recalculation after each new test. The process provides a good overview and the series of tests can be interrupted as soon as an appropriate convergence rate-based criterion is met.

It would be interesting to hear from the authors or anyone else who would like to discuss or provide a comment or a thought, regarding the paper, the method, or anything related.

Per Ståhle

https://imechanica.org/node/25664

Weight functions are practical tools in linear elastic systems where several discrete or continuously distributed sources cause something, deformation, stress, or related stuff. In linear fracture mechanics, as also in the object of this blog, weight functions are used to calculate stress intensity factors. If the load is divided into discrete or continuous separate or overlapping parts which each gives a known contribution to the stress intensity factor, i.e. has a known weight, calculation for new loads may be reduced to simple algebra instead of extensive numerical calculations. This is of course something that is frequently used by all of us. It is just the expected result of linearity. However, in the paper:

“Asymptotic behaviour of the Oore-Burns integral for cracks with a corner and correction formulae for embedded convex defects” by Paolo Livieri and Fausto Segala in Engineering Fracture Mechanics 252 (2021) 107663, https://doi.org/10.1016/j.engfracmech.2021.107663

an important step is taken. The authors show how weight functions can be used for 3D cracks with irregular shaped cracks including sharp corners. As the reader probably knows, there are exact solutions for simple straight cracks, penny shaped cracks and its inverse, a ligament connecting two half-spaces. The geometries of all these are 2D but with the application of arbitrary point forces acting on the crack surfaces the problem becomes 3D. There may be more such solutions unknown to me, but for virtually all realistic cases we are referred to numerical methods. Closed form solutions are indeed rare, but weight functions offer direct access to exact formulations that may bring about analytical simplifications, such as a variety of series expansions, direct integration of extracted singularities (cf. J.R. Rice 1989) and much more. It opens a world of clarity that never comes about when dealing with numerical models.

The school book part with known weight functions and arbitrary load is readily understood, but that will be blown away while real cracks or material flaws usually are neither plane nor perfect circles. The paper gives a nice introduction to the subject and provides a manual for how to deal with the problematic crack edge irregularities including sharp outward corners. It involves approximations which of coarse may lead to possible inaccuracies.

The lack of exact solutions is a two-edged sword. It is an enticement that motivates studies but with the consequence that there is nothing exact to verify the result against. The authors compare with their own and others’ FEM results. I have no problem with that. Only the differences are of the order of what one expects from FEM which leaves us in limbo, not knowing if the weight function method is much better or twice as inaccurate. It is a consolation though, that the differences are a few percent only.

The authors suggest the method also requires comparison with experimental data. I agree with that in general, but I think it falls outside the scope of the study. It seems to me that it is a model selection problem, that has another context. I think it is good enough that the numerical model provides reliable results that are consistent with the mathematical description, i.e. the theory of elasticity.

The results are limited to convex corners, why this is so I could not understand. Is not a slight change of approach sufficient to include also concave corners?

It seems likely that the stress intensity factor becomes unlimited if an inward corner is approached from each side of the corner. In a real world, the stress intensity factor increases until something that blurs the picture pops up, e.g., that the corner is not sharp but rounded off or that the material is not linear elastic after all.

The situation slightly resembles a large crack with a tip very close to a free surface. For that, two different complete series expansions can be matched together with analytic continuation in a region where they both converge. The remote description is a power series with 1/r as a leading term. Around the crack tip a Williams series converges in circular region around the crack tip. Because of the different descriptions the stress intensity factor is given by the coefficient of the 1/r term times a-1/2, where the distance between the crack tip and the free surface, a, is the only length scale available. It is the direct result of dimensional considerations.

Perhaps an ansatz based on something that gives a singularity at the corner of the crack. The comparison with the crack approaching the free surface did not give any ideas per se. Possibly could it help if there was an unsharp corner. I feel that I am on thin ice here. Perhaps someone can give a hint of what to do. The first question is why did the authors exclude the concave corners.

Finally we hope that those who are interested in the subject would comment or contribute with personal reflections regarding the paper under consideration.  

Per Ståhle

https://imechanica.org/node/25419

The outstanding and brilliantly written paper, “Modeling of Dynamic Mode I Crack Growth in Glass Fiber-reinforced Polymer Composites: Fracture Energy and Failure Mechanism” by Liu, Y, van der Meer, FP, Sluys, LJ and Ke, L, Engineering Fracture Mechanics, 243, 2021, applies a numerical model to study the dynamics of a crack propagating in a glass fiber reinforced polymer. The paper is a school example of how a paper should be written. Everything is well described and carefully arranged in logical order. Reading is recommended and especially to young scientists. 

During recent reviewing of several manuscripts submitted to reputable journals I think I see a trend of increased shallowing of the scientific style. Often the reader is referred to other articles for definitions of variables, assumptions made, background, etc. and not seldom with references to the authors’ own previous works. The reading becomes a true pain in the … whatever. The present paper is free of all such obstacles. With the excitement of the distinct technical writing, the reading became enjoyable and with it the interest of the subject grew. 

The adopted theory includes fracture of the polymer matrix, debonding of the interface between reinforcements and matrix, and the energy dissipation due to viscoelastic-plastic material behaviour. A process region is defined as the region that includes sites with tensile stresses that initiates decohesion. It is interesting that many of the initiated cohesive sites never contribute to the global crack meaning that the definition of the process region also includes shielding of the crack tip. Perhaps unorthodox, but absolutely okay. As I said, it is a well written paper.

A series of numerical simulations with different specimen sizes and various load speeds are analysed. Instead of an explicit dynamic analysis a smart implicit dynamic solution scheme is established. A dynamic version of the J-integral is used as a measure of the energy release rate. 

The polymer matrix with its visco-plastic material behaviour given by a Perzyna inspired model has and exponent mp that is slightly larger than 7 should leave a dominating plastic strain field surrounding a sharp crack tip. The singular solutions for materials with mp > 1 have an asymptotic behaviour that does not permit any energy flux to point shaped crack tips. This is in contrast to many metals that have an mp < 1 which forms an asymptotic elastic crack tip stress field and simplifies the analyses. 

The authors eliminate the inconvenient singularities by introducing a cohesive zone to model initiation and growth of cracks. This provides a length scale that allow a flow of energy to feed decohesive processes and crack growth.  Camacho and Ortiz’ (1996) method for implementing cohesive zones is used.  The finite energy release rate required to maintain a steady state crack growth is observed to increase monotonically with increasing crack tip speed. The absence of a local minimum implies that sudden crack arrest, such as obtained by Freund and Hutchinson (1985) for an mp < 1 and a point shaped crack tip, cannot happen. For the present polymer the crack tip speed is always stable and uniquely given by the energy flux. Sudden crack arrest requires a finite minimum energy below which the crack cannot grow. 

Having said this I cannot help thinking that a material with the same toughness but represented by a large cohesive stress and small crack tip opening making the cohesive zone short, might like the point shaped crack tip receive visco-plastic shielding that decreases with increasing crack growth rate. This makes the crack accelerate and jump to a speed that is high enough to be be balanced by inertia. This probably means crack growth rates that are a considerable fractions of the Rayleigh wave speed.

I am not aware of any such studies. It would be interesting to know if there is. It would also be interesting to know if this, or a trend in this direction, has been observed. Perhaps the authors in their studies. I think that crack arrest could be of interest also for design of fibre reinforced polymer structures that are used in light weight pressure vessels, e.g., for hydrogen fuel in cars.

A comment regarding the J-integral that gives the energy dissipation of an infinitesimal translation in a given direction of the objects in the geometry enclosed by the integration path. Technically it means that the micro-cracks are climbing in the x1 direction. Could it be that it evens out if the micro-cracking appears as repeated? With the many micro-cracks that are modelled, I would like to think so.

Does anyone know or have suggestions that could lead forward? Perhaps the authors of the paper or anyone wishes to comment. Please, don’t hesitate to ask a question or provide other thoughts regarding the paper, the method, the blog, or anything related,

Per Ståhle

https://imechanica.org/node/25094

The paper “Estimating static/dynamic strength of notched unreinforced concrete under mixed-mode I/II loading” by N. Alanazi and L. Susmel in Engineering Fracture Mechanics 240 (2020) 107329, pp. 1-18, is a readworthy and very interesting paper. Extensive fracture mechanical testing of concrete is throughly described in the paper. The tests are performed for different fracture mode mixities applied to test specimens with different notch root radii at various elevated loading rates. 

According to the experimental results the strength of concrete increases as the loading rate increases. The mixed-mode loading conditions refers to the stress distribution around the original notch. Fracture starts in all cases at a half circular notch bottom. Initiation of a mode I cracks are anticipated and were clearly observed in all cases. The position of maximum tensile stress along the notch root as predicted by assuming, isotropic and linear elastic material properties correlates very nicely with where the cracks initiate. The selected crack initiation criterion is based on stresses at, or alternatively geometrically weighted inside, a region ahead of the crack tip. The linear extent of the region is material dependent. The criterion, used with a loading rate motivated modification, is strongly supported by the result.

The result regarding the rate dependence is different from what is observed for ductile metals, where at high strain rate dislocation motion is limited. This reduces the plastic deformation and increases the near tip stress level. It therefore decreases the observed toughness as opposed to what happens in concrete. Should the stress level or energy release rate exceed a critical value, the crack accelerates until the overshooting energy is balanced by inertia as described by Freund and Hutchinson (1985). Usually this means a substantial part of the elastic wave speed. For concrete I guess this must mean a couple of km/s. This is outside the scope of the present paper but a related question arises: What could be the source of the strain rate effects that are observed? Plasticity/nonlinearities are mentioned. I would possibly suggest for damage as well. We know that reinforced ceramics are affected by crack bridging and micro-crack clusters appearing along the crack path or offside it. If such elements are present then both decreased and increased toughnesses may be anticipated according to studies by Budiansky, Amazigo and Evans (1988) and Gudmundson (1990). Could concrete be influenced by the presence of crack bridging elements or micro-cracks or anything related? If not, what could be a plausible guess?  

Does anyone know or have suggestions that could lead forward? Perhaps the authors of the paper or anyone wishes to comment. Please, don’t hesitate to ask a question or provide other thoughts regarding the paper, the method, or anything related.

Per Ståhle

https://imechanica.org/node/24762

Landau and Ginzburg formulated a theory that includes the free energy of phases, with the purpose to derive coupled PDEs describing the dynamics of phase transformations. Their model with focus on the phase transition process itself also found many other applications, not the least because many exact solutions can be obtained. During the last few decades, with focus on the bulk material rather than the phase transition, the theory has been used as a convenient tool in numerical analyses to keep track of cracks and other moving boundaries. As a Swede I can’t help myself from noting that both of them received Nobel prizes, Landau in 1962 and Ginzburg in 2003. At least Ginzburg lived long enough to see their model used in connection with formation and growth of cracks. 

The Ginzburg-Landau equation assumes, as virtually all free energy based models do, that the state follows the direction of steepest descent towards a minimum free energy. Sooner or later a local minimum is reached. It doesn’t necessarily have to be the global minimum and may depend on the starting point. Often more than one form of energy, such as elastic, heat, electric, concentration and more energies are interacting along the path. Should there be only a single form of energy the result becomes Navier’s, Fourier’s, Ohm’s or Fick’s  law. If more than one form of energy is involved, all coupling terms between the different physical phenomena are readily obtained. By including chemical energy of phases Ginzburg and Landau were able to explain the physics leading to superfluid and superconducting materials. Later by mimicking vanished matter as a second phase with virtually no free energy we end up with a model suitable for studies of growing cracks, corrosion, dissolution of matter, electroplating or similar phenomena. The present paper 

“Phase-field modeling of crack branching and deflection in heterogeneous media” by Arne Claus Hansen-Dörr, Franz Dammaß, René de Borst and Markus Kästner in Engineering Fracture Mechanics, vol. 232, 2020, https://doi.org/10.1016/j.engfracmech.2020.107004, 

describes a usable benchmarked numerical model for computing crack growth based on a phase field model inspired by the Ginzburg-Landau’s pioneering work. The paper gives a nice background to the usage of the phase field model with many intriguing modelling details thoroughly described. Unlike in Paper #11 here the application is on cracks penetrating interfaces. Both mono- and bi-material interfaces at different angles are covered. It has been seen before e.g. in the works by He and Hutchinson 1989, but with the phase field model results are obtained without requiring any specific criterium for neither growth nor branching nor path. The cracking becomes the product of a continuous phase transformation. 

According to the work by Zak and Williams 1962, the stress singularity of a crack perpendicular to, and with its tip at, a bimaterial interface possesses a singularity r^-s that is weaker than r^-1/2 if the half space containing the crack is stiffer than the unbroken half space. In the absence of any other length scale than the distance, d, between the interface and the crack tip of an approaching crack, the stress intensity factor have to scale with d^(1/2-s). The consequence is that the energy release rate either becomes unlimited or vanishes. At least that latter scenario is surprisingly foolish whereas it means that it becomes impossible to make the crack reach the interface no matter how large the applied remote load is.

In the present paper the phase field provides an additional length parameter, the width of the crack surfaces. That changes the scene. Assume that the crack grows towards the interface and the distance to the interface is large compared with the width of the surface layer. The expected outcome I think would be that the crack growth energy release rate increases for a crack in a stiffer material and decreases it for a crack in a weaker material. As the surface layer width and the distance to the interface is of similar length the changes of the energy release rate does no more change as rapid as d^(1-2s). What happens then, I am not sure, but it seems reasonable that the tip penetrates the interface under neither infinite nor vanishing load. 

I could not find any observation of this mentioned in the paper so this becomes just pure speculation. It could be of more general interest though, since it could provide a hint of the possibilities to determine the critical load that might lead to crack arrest.

Comments, opinions or thoughts regarding the paper, the method, or anything related are encouraged.

Per Ståhle

https://imechanica.org/node/24661

All materials are anisotropic, that’s a fact. Like the fact that all materials have a nonlinear response. This we can’t deny. Still enormous progress has been made by assuming both isotropy and linear elasticity. The success, as we all know, is due to the fact that many construction materials are very close to being both isotropic and linear. By definition materials may be claimed to be isotropic and linear, provided that the deviations are held within specified limits. Very often or almost always in structural design nearly perfect linearity is expected. In contrast to that quite a few construction materials show considerable anisotropy. It may be natural or artificial, created by humans or evolved by biological selection, to obtain preferred mechanical properties or for other reasons. To be able to choose between an isotropic analysis or a more cumbersome anisotropic dito, we at least once have to make calculations of both models and define a measure of the grade of anisotropy. This is realised in the excellent paper

“The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes” by Majid R. Ayatollahi, Morteza Nejati, Saeid Ghouli in Engineering Fracture Mechanics, vol. 231, 15 May 2020, https://doi.org/10.1016/j.engfracmech.2020.106982.

The study provides a thorough review of materials that might require consideration of the anisotropic material properties. As a great fan of sorted data, I very much appreciate the references the authors give listed in a table with specified goals and utilised analysis methods. There are around 30 different methods listed. Methods are mostly numerical but also a few using Lekhnitskiy and Stroh formalisms. If I should add something the only I could think of would be Thomas C.T. Ting’s book “Anisotropic Elasticity”. In the book Ting derives a solution for a large plate containing an elliptic hole, which provides cracks as a special case.

The present paper gives an excellent quick start for those who need exact solutions. Exact solutions are of course needed to legitimise numerical solutions and to understand geometric constraints and numerical circumstances that affect the result. The Lekhnitskiy and Stroh formalisms boil down to the “method of characteristics” for solving partial differential equations. The authors focus on the solution for the vicinity of a crack tip that is given as a truncated series in polar coordinates attached to a crack tip. 

As far as I can see it is never mentioned in the paper, but I guess the series diverges at distances equal to or larger than the crack length 2a. Outside the circle r=2a the present series for r<2a should be possible to extend by analytic continuation. My question is: Could it be useful to have the alternative series for the region r>2a to relate the solution to the remote load?

Does anyone have any thoughts regarding this. Possibly the authors of the paper or anyone wishes to comment, ask a question or provide other thoughts regarding the paper, the method, or anything related.

Per Ståhle

https://imechanica.org/node/24513

The subject of this blog is a fracture mechanical study of soft polymers. It is well written and technically detailed which makes the reading a good investment. The paper is:

“Experimental and numerical assessment of the work of fracture in injection-moulded low-density polyethylene” by Martin Kroon, Eskil Andreasson, Viktor Petersson, Pär A.T. Olsson in Engineering Fracture Mechanics 192 (2018) 1–11.

As the title says, it is about the fracture mechanical properties of a group of polymers. The basic idea is to identify the energy release rate that is required to initiate crack growth. To distinguish between the energy required for creating crack and the energy dissipated in the surrounding continuum, the former is defined as the unstable material which has passed its largest load carrying capacity, and the remaining is the stable elastic plastic continuum. The energy required for creating crack surface is supposed to be independent of the scale of yielding.

The authors call it the essential work of fracture, as I believe was coined by Mai and Cotterell. If not the same, then this is very close to the energy dissipation in the fracture process region, as suggested by Barenblatt, and used by many others. Material instability could, of course also be the result of void or crack nucleation at irregularities of one kind or another outside the process region. How much should be included as essential work or not, could be discussed. I guess it depends on if it is a necessary requirement for fracture. The fact that it may both support and be an impediment to fracture does not make it less complicated. In the paper an FE model is successfully used to calculate the global energy release rate vis à vis the local unstable energy release in the fracture process region, modelled as a cohesive zone.

What captured my interest was the proposed two parameter cohesive zone model and its expected autonomy. With one parameter, whatever happens in the process region is determined by, e.g., K, J, G. The single parameter autonomy has its limits but more parameters can add more details and extend the autonomy and applicability. For the proposed cohesive zone, the most important parameter is the work of fracture. A second parameter is a critical stress that marks the onset of the fracture processes. In the model the critical stress is found at the tip of the cohesive zone. By using the model of the process region, the effect of different extents of plastic deformations is accounted for through the numerical calculation of the surrounding elastic plastic continuum.

The work of fracture is proportional to the product of the critical stress and the critical separation of the cohesive zone surfaces. The importance of the cohesive zone is that it provides a length scale. Without it, the process region would be represented by a point, the crack tip, with the consequence that the elastic plastic material during crack growth consumes all released energy. Nothing is let through to the crack tip.

Stationary cracks are surrounded by a crack tip field that releases energy to fracture process regions that may be small or even a singular point. If the crack is growing at steady-state very little is let through to a small fracture process region and to a singular point, nothing. In conventional thinking a large cohesive stress leads to a short cohesive zone, and by that, the available energy would be less. A variation of the critical stress is discussed in the paper. Presently, however, the two parameter model is more of a one parameter ditto, where the cohesive stress is selected just as sufficiently plausible. 

What could be done to nail the most suitable critical cohesive stress? With the present range of crack length and initiation of crack growth nothing is needed. The obtained constant energy release rate fits the experimental result perfectly. Further, it is difficult to find any good reason for why the excellent result would not hold also for larger cracks. As opposed to that, small, very small or no crack at all should give crack initiation and growth at a remote stress that is close to the critical cohesive stress. As the limit result of a vanishing crack, the two stresses should be identical. I am not sure about the present polymer but in many metals the growing plastic wake requires significant increase of the remote load. Often several times rather than percentages. So letting the crack grow at least a few times the linear extent of the plastic zone, would add on requirements that may be used to optimise both cohesive parameters. 

I really enjoyed reading this interesting paper. I understand that the paper is about initiation of crack growth which is excellent, but in view of the free critical cohesive stress, I wonder if the model can be extended to include very small cracks or the behaviour from initiation of crack growth to an approximate steady-state. It would be interesting if anyone would like to discuss or provide a comment or a thought, regarding the paper, the method, the autonomy, or anything related. The authors themselves perhaps.

Per Ståhle

https://imechanica.org/node/23886

Carbon fibre reinforced polymers combines desired features from different worlds. The fibres are stiff and hard, while the polymers are the opposite, weak, soft and with irrelevant fracture toughness. Irrelevant considering the small in-plane deformation that the fibres can handle before they break. It is not totally surprising that one can make composites that display the best properties from each material. Perhaps less obvious or even surprising is that materials and composition can be designed to make the composite properties go beyond what the constituent materials are even near. A well-known example is the ordinary aluminium foil for household use that is laminated with a polymer film with similar thickness. The laminate gets a toughness that is several times that of the aluminium foil even though the over all strains are so small that the polymer hardly can carry any significant load. 

In search of something recent on laminate composites, I came across a very interesting paper on material and fracture mechanical testing of carbon fibre laminates::

“Innovative mechanical characterization of CFRP using acoustic emission technology” by Claudia Barile published in Engineering Fracture Mechanics Vol. 210 (2019) pp. 414–421

What caught my eye first was that the paper got citations already during the in press period. It was not less interesting when I found that the paper describes how acoustic emissions can detect damage and initiation of crack growth. The author, Barile, cleverly uses the wavelet transform to analyse the response to acoustic emission. In a couple of likewise recent publications she has examined the ability of the method. There Barile et al. simulate the testing for varying material parameters and analyse the simulated acoustic response using wavelet transformation. This allow them to explore the dependencies of the properties of the involved materials. 

They convincingly show that it is possible to both detect damage and damage mechanisms. In addition, a feature of the wavelet transform as opposed to its Fourier counterpart is the advantages at analyses of transients. By using the transform they were able to single out the initiation of crack growth. Very useful indeed. I get the feeling that their method may show even more benefits.

A detail that is unclear to me, if I should be fussy, is that there are more unstable phenomena than just crack growth that can appear as the load increases. Also regions of damage and in particular, fracture process regions may grow. When the stress intensity factor K alone is sufficient there is no need to consider neither size nor growth of the fracture process region. The need arises when K, J, or any other one-parameter description is insufficient, e.g. in situations when the physical size of the process region becomes important. Typical examples are when cracks cross bi-material interfaces or when they are small relative to the size of the process region. When the size seems to be the second most important feature, then the primary parameter may be complemented with a finite size model of the process region to get things right. There is a special twist of this in connection with process region size and rapid growth. In the mid 1980’s cohesive zones came in use to model fracture process regions in FEM analyses of elastic and elastic-plastic materials. Generally, during increasing load, cohesive zones appear at crack tips and develop until the crack begins to grow. One thing that at first glance was surprising, at least to some of us, was that for small cracks the process region first grows stably and shifts to be fast and uncontrollable, while the crack tip remains stationary. Later, of course the criterion for crack growth becomes fulfilled and crack growth follows.

Is it possible to differentiate between the signals from a suddenly fast growing damage region or fracture process region vis à vis a fast growing crack?

It would be interesting to hear from the authors or anyone else who would like to discuss or provide a comment or a thought, regarding the paper, the method, or anything related.

Per Ståhle

https://imechanica.org/node/23731

I came across a very interesting paper in Engineering Fracture Mechanics about a year ago. It gives some new results of stochastic aspects of fatigue. The paper is:

”On the distribution and scatter of fatigue lives obtained by integration of crack growth curves: Does initial crack size distribution matter?” by M. Ciavarella, A. Papangelo, Engineering Fracture Mechanics, Vol 191 (2018) pp. 111–124.

The authors remind us of the turning point the a Paris’ exponent m=2 is. Initial crack length always matters but if the initial crack is small, the initial crack is seemingly very important for the if m>2.  For exponents less than 2, small initial cracks matters less or nothing at all. If all initial cracks are sufficiently small their size play no role  and may be ignored at  the calculation of the remaining life of the structure. Not so surprising this also applies to the stochastic approach by the authors. 

What surprised me is the fuzz around small cracks. I am sure there is an obstacle that I have overlooked. I am thinking that by using a cohesive zone model and why not a Dugdale or a Barenblatt model for which the analytical solutions are just an inverse trigonometric resp. hyperbolic function. What is needed to adopt the model to small crack mechanics is the stress intensity factor and a length parameter such as the crack tip opening displacement or an estimate of the linear extent of the nonlinear crack tip region.

I really enjoyed reading this interesting paper and get introduced to extreme value distribution. I also liked that the Weibull distribution was used. The guy himself, Waloddi Weibull was born a few km’s from my house in Scania, Sweden. Having said that I will take the opportunity to share a story that I got from one of Waloddi’s students Bertram Broberg. The story tells that the US army was skeptic and didn’t want to use a theory (Waloddi’s) that couldn’t even predict zero probability that object should brake. Not even at vanishing load. A year later they called him and told that they received  a cannon barrel that was broken already when they pulled it out of its casing and now they fully embraced his theory. 

Per Ståhle

https://imechanica.org/node/23169