Just open a Bet365 account today and make a deposit http://abonuscode.co.uk Make a deposit of £10-£200 and then enter the 10-digit bonus code


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:


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 #33 The Interaction Integral

This blog concerns an interesting review of the interaction integral methodology. It deserves to be read by everyone dealing with analyses of cracks. If one's focus is on mathematical analysis or numerics is irrelevant. The review is for all of us. The review paper is, ”Interaction integral method for computation of crack parameters K–T – A review", by Hongjun Yu and Meinhard Kuna, Engineering Fracture Mechanics 249 (2021) 107722, p. 1-34. Already the introduction gives a thorough historic background of the incremental improvements and additions made to tackle an increasing sphere of problems. The starting point is J. Rice´s J -integral, which has served us well for more than half a century. It gives the energy release rate in the near tip region at crack growth in a homogeneous material. Inhomogeneities, bi-material interfaces, and more requires amendments. A drawback with the J -integral is that it provides the energy release rate independent of a mode mixity. When it was shown by Stern, Becker and Dunham that an auxiliary field in equilibrium, also providing path independence, added to the original field allowed decoupling of the mixed modes and their respective stress intensity factors, the interaction integral was established. Out of the large variety of other solutions to the problematic mode separation, the interaction integral seems to be the most effective and suitable for FEM implementation. The review in its introduction takes the reader on an odyssey through the five decades of inventive selections of auxiliary fields giving solutions to a large variety of static and dynamic problems and introducing domain integrals that improve the accuracy. In their paper, Yu and Kuna include the theoretical background and explain the basic amendments introduced to allow the treatment of many problems, including anisotropy, dynamics, mechanics coupled with other physical processes, etc. The inspiring reading gives a great starting point for anyone who wishes to explore new possibilities the method provides. There is also a useful section showing the implementation and example cases with data. I very much liked the declaration of advantages and especially the limitations that give a direct sense of reliability. The 351 references are also much appreciated. The usage of the M-integral for calculations of intensity factors for thermo-elastic and piezoelectric materials (cf. L. Banks-Sills et al. 2004 and 2008) is interesting. I assume that stress-driven diffusion and other transport phenomena that are governed by Laplace's equation coupled with elastic deformation could have direct use of the M-integral. By the way, the interaction M-integral should not be confused with the M -integral that gives the energy release rate for expanding geometries (cf. L.B. Freund 1978). Regarding the auxiliary field applied to K -dominance problems, there is an annular ring around the crack tip in which stress is represented by a full series of r ^( n /2) terms, where the n includes both negative and positive integers. Essentially only the term with n =-1 connects the remote boundary with the near tip region. Outside the annular ring, terms with n ≥-1 dominate and inside the ones with n ≤-1 dominate. The selection for the auxiliary field seems so far to be one of n ≥ -1, -2, or -3. What I ask myself is cannot the e.g. the Dugdale model with its exact series expansion solution including an arbitrary number of terms n ≤ -1 be of use to cover elastic-plastic problems. I know that the direct superposition fails but could perhaps be given an analytically matched zone length. Perhaps I am on the wrong path. Please, enlighten me authors, readers, anyone. All comments, opinions or thoughts regarding the paper, the method, or anything related are encouraged. If you belong to the unfortunate that do not have an iMechanica account, please email me at pers@solid.lth.se and I will see what I can do. Per Ståhle



Discussion of fracture paper #32 Fatigue and machine-learning

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



Discussion of fracture paper #31 - Toughness of a rigid foam

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



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



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



Cookies make it easier for us to provide you with our services. With the usage of our services you permit us to use cookies.
More information Ok Decline