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Discussion of fracture paper #36 - The Double-K Fracture Model

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.

31.Dec.2022

Discussion of fracture paper #35 - What is Finite Fracture Mechanics?

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

24.Sep.2022

Discussion of fracture paper #34 - The Physics of Hydrogen Embrittlement

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

25.Jun.2022

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

07.Apr.2022

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

31.Dec.2021