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Dynamic fracture is a never-ending story. In 1951, EH Yoffe obtained an analytical solution for a crack of constant length travelling at constant speed along a plane. She used a Galilean transformation to get a solution for arbitrary speeds. The situation seems strange with a crack tip where the material breaks and a lagging tip where the material heals. However, there are applications. One that I encountered was several mode II cracks that travel in the contact plane between a brake pad and a brake disc. The moving cracks were blamed for the causing squeaking noise when braking.

A step towards a more versatile solution was given by KG Broberg in 1960 for crack growth with both tips moving apart from each other at the same speed. The self-similar state leads to simplifications while all coordinates are scaled with speed and time. The solution suffers from the crack growth rate instability that is expected as long as the crack growth rate is less than the speed of the Rayleigh wave, i.e. always as long as you don’t push the crack tip with something. I can recall an epoxy experiment and a high-power laser that gave the crack tip a push.

Finally in 1972 LB Freund solved the problem of a semi-infinite crack and an arbitrary crack growth rate which more or less culminated the entire subject. Another milestone is the incorporation of plastic strain rate dependencies, dealt with LB Freund and JW Hutchinson in 1989, that explained the observed sudden arrest from a finite crack growth speed.

These are a few milestones as regards the mathematical physics of dynamic fracture. Maybe a new milestone is reached with the article,

“From macro fracture energy to micro bond breaking mechanisms – Shorter is tougher” by Merena Shaaeen-Mualim, Guy Kovel, Fouad Atrash, Liron Ben-Bashat-Bergman, Anna Gleizer, Lingyue Ma and Dov Sherman in Engineering Fracture Mechanics, vol. 289, 2023, https://doi.org/10.1016/j.engfracmech.2023.109447,

that begins with a brilliant and detailed review of three successful models for the initiation of crack growth in brittle materials. The first is AA Griffith’s theory from 1920, for the initiation of crack growth, which has served us well for decades. The limitation is that it only applies to stable quasistatic crack growth. The second model is due to LB Freund and his work from 1972. This certainly put the subject and us on a different level. The third model is based on molecular dynamics. Here the present paper gives much insight and could be a springboard for initiates.

Observations and results are based on data from fracture mechanical testing of brittle single-crystal silicon samples. The focus is on the relationship between the energy balance and the crack tip speed. Also, the details of the crack front contribute. Analysis of fracture processes on a microscopic scale enabled the development of an interesting model for the required binding energy. The involved mechanisms are the low-energy migration and high-energy kink nucleation along the crack front.

In particular, the energy release rate at crack initiation, and its derivative with respect to the crack length, play an important role. The macroscale experiments, the microscale atomistic model and the energy release rate gradient lead to several conclusions. A primary conclusion is that the energy release rate required for breaking the bonding in the crack plane is not constant. Instead, it is limited by the classical Griffith energy and an upper limit, related to the lattice trapping at as much as 3 times the Griffith energy. Also an interesting transition phase between breaking up from the virgin crack and growth. During this phase, the sequence of bond-breaking mechanisms varies, leading to an increase in the cleavage energy.

Also, the variation of the cleavage energy shows that shorter cracks require higher energy to grow, and are stronger than what is predicted by the classic Griffith’s surface energy.

I consider this to be an interesting and important paper. This is why I only have two questions. First, were there ever any dislocations that were nucleated and thrown out from the crack tip? Second, there is of course no such thing as a pure mode for a kink but is the kink propagating under mixed mode closer to mode I or possibly closer to mode II?

It would be interesting to hear from the author or anyone else who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. If you do not have an iMechanica account and fail to register, please email me at per.stahle@solid.lth.se and I will post your comments in your name. The paper will be open-access in a couple of days. 

Per Ståhle

https://imechanica.org/node/27054

The Nobel laureate Andre Geim made graphene by playing with pencil leads and Scotch tape and coauthored a paper on how to get the Nobel prize the fun way. Before that, he co-authored with his hamster, Ter Tisha, a paper on diamagnetic levitation and demonstrated it on a frog. He was honoured with the Ig Nobel prize for the paper and later became the only person so far who got both the Harvard Ig version and the real Alfred version of the Nobel prize. Geim is one of my favourite scientists, which led me to read the paper 

“The applicability and the low limit of the classical fracture theory at nanoscale: The fracture of graphene”, by Jie Wang, Xuan Ye, Xiaoyu Yang, Mengxiong Liu and Xide Li, Engineering Fracture Mechanics 284 (2023) 109282, p. 1-23,

The paper turned out to be well-written and really interesting. The subject itself is indeed exciting. Several obstacles arise. Already the dimensions of the object that normally never would be disputed in conventional continuum mechanical analyses pop up as problematic. Length and width are easy but what is the thickness?

Graphene, being a single carbon atom thick sheet of graphite is as thin as it can be. Still, measuring the thickness itself poses a challenge. A surface is a geometric object, that is between what is inside and outside. How do we define the position of the surface of a single atom or a sheet with a single atom thickness? Perhaps, a known mass density of graphite, the total mass of the graphene sheet and the atomic ditto of carbon could define the thickness. We could go for the distance between the crystal planes in a graphite crystal. However, I am sure that another Nobel laureate, Lev Landau, would have said that the chemical potential that makes the graphite form a solid, also narrows the crystal planes in the graphite. I guess it means, at least theoretically, that the space between the atoms increases with a decreasing number of atomic layers. 

The thickness is a paramount quantity. As long as tension is applied across a crack in a strip there will be compression along the crack surface. The stress initiating crack growth is independent of the thickness while the stress causing buckling is proportional to the square of (sheet thickness/crack length). Once the sheet buckles and bends the region around the crack will get the shape of a finch beak. The stress expansion will not include the square root singularity the stress intensity factor, as I see it, loses its meaning. 

This is not diminishing the importance of the present paper but it would be an interesting continuation to see what happens if the present model includes out-of-plane motion. Either the buckling comes before the initiation of crack growth. If not the crack may grow until the remote stress, which decreases proportionally to the inversed square root of the crack length meets the buckling stress, which decreases much faster and proportionally to the inversed squared crack length. 

It would be interesting to hear from the author or anyone else who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. If you do not have an iMechanica account and fail 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/26911

The advantage of simplicity is that mechanics and physics can be understood and predicted just by using pen and paper. In the end, numerics may have to be used but then you should already have a pretty good idea of what happens. The other way around, starting with numerics and a limited toolbox of models will seldom lead to anything new. 

The paper, “Experimental determination of coupled cohesive laws with an unsymmetrical stiffness matrix for structural adhesive joints loaded in mixed-mode I+III, by Stephan Marzi in Engineering Fracture Mechanics, 283 (2023)“, adopts a very interesting view of the cohesive zone models that are based on the energy release rate and the crack tip opening displacement. Instead of using a classical interpolation between single-mode test results that often fails because of changes in the adhesive’s physical behaviour, a new model is suggested. The new model is based on direct measurements of various mode I and III mixes, which inevitably ensure that the adhesive behaviour alteration is captured. The method comes with a few requirements that lead to possible limitations that are cleverly discussed.

The paper is well-written and offers very interesting reading. To me, the paper also calls for a reflection. An adhesive is usually stuck between two materials that do not fail easily. This is different from isotropic materials for which the mode I and III mix leads to larger freedom of choice of crack propagation and crack plane morphology. 

Plates of glass or any other brittle material that are exposed to a mixed mode or for that matter pure mode III results in a decrease of the tougher in favour of the less tough mode by tilting the crack plane. A well-known situation is when a window glass is exposed to a remote tearing, appearing as mode III similar results in a propagating mode I crack that is tilted towards the glass surfaces even if it initially was perpendicular.

The selected tilt angle ought to be the one that minimises the required energy release rate. While the crack depth initially may be the plate thickness, i.e., straight through the plate, the tilt to obtain more of the least tough fracture mode may be hampered by an increasing the crack depth. The total dissipated energy in creating a new crack surface is what counts.

Having written this I realise that the tilting crack plane model may be oversimplified. The crack surfaces surely must have a waviness that correlates with the variation of the stress-strain conditions as we move from the front side surface to the back side of the plate. 

I guess that a brittle and thicker adhesive layer could go the same way, but a thinner layer might, in lack of space, develop a favourable zigzag pattern. Ductile adhesives may also shift modus operandi depending on which is the preferred failure mechanism. However, this is on a micro-scale with details that are not required to make use of Marzi’s ingenious method for obtaining cohesive properties.

It would be interesting to hear from the author or anyone else who would like to discuss or provide a comment or a thought, regarding the paper, the method, or anything related. If you belong to do not have an iMechanica account and fail to register, please email me at per.stahle@solid.lth.se and I will post your comment in your name. 

Per Ståhle

https://imechanica.org/node/26674

The fracture of concrete and other semi-brittle materials offers some simplifications that simplify the analytical analysis. The simple check that reveals if something broken requires an elastic or an elastic-plastic fracture mechanical analysis by just trying to fit the pieces together sometimes fails. The suggestion is that if they do not fit together, we have an elastic-plastic fracture and if they do we have an elastic fracture. We may jump to the false conclusion that linear elastic fracture mechanics can be applied. The fracture processes are confined to a narrow zone stretching ahead of the crack tip for concrete and similar materials. A Barenblatt process zone seems ideal but it requires knowledge of how the cohesive capacity decays with increasing stretch across the crack plane. The version proposed by Dugdale* is intended for plastic necking in thin sheets and requires only yield stress and sheet thickness. Out of a variety of other proposals, the double-K model seems to have achieved widespread attention and appreciation because of its engineering approach providing practical simplicity. The review paper,

“The double-K fracture model: A state-of-the-art review”, by Xing Yin, Qinghua Li, Qingmin Wang, Hans-Wolf Reinhardt, Shilang Xu, Engineering Fracture Mechanics 277 (2023) 108988, p. 1-42,

gives a thorough overview including the theoretical background of the method. It is approved by the Chinese organisation of standards and the international organisation for construction materials experts RILEM for fracture mechanical testing of a restricted group of materials. 

The method is based on two critical stress intensities, one for initiation of crack growth and a second for the switch to fast uncontrollable crack growth. A large number of experimental techniques and numerical methods to improve measurements and their evaluation accuracies are nicely organised into a large number of subsections. The review is a rewarding reading that gave me great pleasure and introduced me to the difficulties and advances in numerical techniques to approach the fracture mechanics of one of the most important groups of materials. The nearly three decades of history put much into perspective. 

One thing that puzzled me regarding the unstable crack growth considering observations during the 1980s when it was discovered that small cracks are prone to jump the stable crack growth part. Instead, unstable crack growth was initiated earlier than what was expected from linear fracture mechanics analyses. In modelling the event using cohesive zones replacing the plastic deformation and the fracture processes, the tip of the already growing cohesive zone tip becomes unstable while the crack length is unchanged. The increasing load resulted in unstable crack growth shortly thereafter. The larger the crack the shorter the time gap between the initiation of unstable growth of the tip of the cohesive zone and that of the crack tip.  

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 per.stahle@solid.lth.se.

The link that leads to the paper is presently not fully open access paper but it will be within a couple of days.

Per Ståhle

*D.S. Dugdale’s paper from 1960 was published the year after G.I. Barenblatt’s original Russian paper from 1959, which was published in English in 1963.

https://imechanica.org/node/26445

The subject of this blog is a well-written and technically detailed study of thermal crack initiation where an adhesive joint between two dissimilar materials meets a free surface. The method that is used goes under the group designation finite fracture mechanics. The paper is:

“Predicting thermally induced edge-crack initiation using finite fracture mechanics” by S. Dölling, S. Bremm, A. Kohlstetter, J. Felger, and W. Becker. in Engineering Fracture Mechanics 252 (2021) 107808.

Reading it is a good investment for anyone interested in the analysis of real fracture mechanics, when one does not know all the details of the original crack, or if it even existed at the time before the load was applied.

A body with a bi-material adhesive joint that meets a free surface generally experiences high stresses. Under idealised conditions, a stress singularity occurs. Only pathological loads are exceptions. Crack initiation is assumed to be caused by thermally induced stresses. A criterion based on a coupled stress and energy criterion within the framework of the so-called finite fracture mechanics has been chosen. The essential part of the energy is that required for the initiation of the crack which then commences crack growth. The energy required for initiation is the work needed to stretch the adhesive layer thickness to the point, where the adhesive fails.

The principal numerical method is a boundary integral method. The guidance through the basics of boundary integral methods is educational and enjoyable and greatly appreciated. The method in itself is direct and very intuitive compared to finite difference and finite element methods.

The authors also include a simple dimensional analysis with an elegant demonstration of how the completed study can be extended without requiring additional numerical calculations. The simplifications are based on scaling with the respect to the few length parameters present. The method is evaluated using both boundary integral methods and a finite element method. The result of the coupled stress and energy criterion is also compared with a cohesive zone model that introduces a critical stress and a critical displacement.

I really enjoyed reading this interesting paper. A long time ago when I myself studied small cracks using cohesive zone models in the late 70s under the guidance of Profs. K.B. Broberg and G.I. Barenblatt, our result displayed an unexpected instability. Instead of the expected unstable crack growth, the tip of the cohesive zone kick-started in an uncontrolled rapid crack growth while the crack tip remained stationary. 

Possibly the coupled stress and energy criterion could give a different result. Our result is of course affected by our selected cohesion which continuously decreases with increasing separation of the cohesive zone boundaries. After a short reflection, it seems natural that the cohesive zone tip should start first and the rapid growth comes with the rapidly increasing energy release rate of the short crack. A long time has passed since then, and over the years I have heard about similar observations made by others. 

All comments, thoughts or opinions, regarding the finite fracture model, cohesive zones, the paper, or anything related are encouraged to submit a comment. If you belong to the unfortunate that do not have an iMechanica account and fail to get one, please email the text to me at per.stahle@solid.lth.se and I will post your comments under your name.

Per Ståhle

https://imechanica.org/node/26236

Hydrogen embrittlement causes problems that probably will become apparent to an increasing extent as hydrogen is taken into general use for energy storage and as a fuel for heating and electricity production. According to Wikipedia, the phenomenon has been known since at least 1875. The subject of this blog 

“The synergistic action and interplay of hydrogen embrittlement mechanisms in steels and iron: Localized plasticity and decohesion”, by Milos B. Djukic, Gordana M. Bakic, Vera Sijacki Zeravcic, Aleksandar Sedmak, and Bratislav Rajicic Engineering Fracture Mechanics 216 (2019) 106528, pp. 1-33

is an in-depth and comprehensive review article that deservedly is frequently cited. It deals with the progress made over the past 50 years. For those who want to get into the subject, the paper is an excellent starting point with 243 references and nice descriptions of known mechanisms and methods used for risk assessments. The paper is not open access yet but will be that within a couple of days with courtesy from EFM.

The presumable outdated observations by William Johnson from 1875 are not mentioned in the review article. I assume that not much happened before the second half of the 20th century. Johnson’s findings were published in “Proceedings of the Royal Society of London” on New Year’s Eve 1875. He conducted measurements of the strength of conventional tensile test specimens. The strength, after bathing the sample in an acid, dropped by up to 20%. As the classically trained experimental physicist Johnson was, he did not stop at strength but also measured the effect of hydrogen on electrical conductivity and on the diffusion rate of the hydrogen. In the latter case, the distribution of hydrogen in the test rod revealed itself as bubbles forming on the fracture surfaces of the test rod. During the test, the rod was dipped to different depths in the acid bath. When the fracture occurred in a part below the surface of the acid bath, the entire cross-section was covered with bubbles from leaking hydrogen and when the fracture occurred at a distance equal to the specimen radius above the bath, only the two thirds closest to the centre of the fracture surface were covered with hydrogen bubbles. The observation gives a wonderful picture of how the diffusion of hydrogen deviates towards the free outer surfaces. Brilliant results with the simple scarce experimental resources of the time.

I traditionally have an inquiry for the authors or any reader regarding something that puzzles me. This time it strikes me that in the review article nothing is mentioned about other affected material properties. I know that the review article focuses on the embrittlement of steel. However, since it is rightly regretted that too little is known to facilitate a formulation of a theory that can provide reliable models for prediction, perhaps observations of other things such as diffusion rates and electrical conductivity may provide more light to the underlying physics. Any suggestions?

All comments, opinions, thoughts regarding the paper, or anything related are encouraged. If you belong to the unfortunate that do not have an iMechanica account, please email me at per.stahle@solid.lth.se and I will see what can be done.

Per Ståhle 

https://imechanica.org/node/26076

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.

https://imechanica.org/node/25891