# Blog

With the ESIS blog we want to promote and intensify relevant scientific discussions on recent publications in Engineering Fracture Mechanics. The blog is hosted by IMechanica on:

http://imechanica.org/node/9794

The editors, Professors Karl-Heinz Schwalbe and Tony Ingraffea, support this initiative. ESIS hopes that this blog will achieve the following objectives:

• To initiate scientific discussions on relevant topics by highlighting, leaving comments, suggestions, questions etc. related to recent publications;

• To suggest re-reading, re-examination, comparison of results from the past, that may be overlooked by the authors or have fallen into general oblivion;

• To promote and give reference to groups with similar or related scientific goals and to promote collaboration, as well as bridging gaps between different disciplines;

• To focus attention on new ideas that may face a risk of drowning in the noise of today's extensive scientific production;

Per Ståhle

## Latest posts from ESIS Blog on IMechanica

#### Discussion of fracture paper #30 - Weight functions, cracks and corners

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

13.Sep.2021

#### Discussion of fracture paper #29 - Fast Crack Growth in Fibre Reinforced Composites

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 m p 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 x 1 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

06.Apr.2021

#### Discussion of fracture paper #28 - Rate effects and dynamic toughness of concrete

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 rates 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

30.Nov.2020

#### Discussion of fracture paper #27 - Phase-field modelling of cracks and interfaces

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 ^(1-2 s ). 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

17.Oct.2020

#### Discussion of fracture paper #26 - Cracks and anisotropic materials

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 2 a . Outside the circle r =2 a the present series for r <2 a should be possible to extend by analytic continuation. My question is: Could it be useful to have the alternative series for the region r >2 a 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

13.Aug.2020