Category: Research

Discussion of fracture paper #13 – Cohesive properties at ductile tearing

5 December 2024 / admin / 0 Comments

In this review of particularly readworthy papers in EFM, I have selected a paper about the tearing of large ductile plates, namely:

”Cohesive zone modeling and calibration for mode I tearing of large ductile plates” by P.B. Woelke, M.D. Shields, J.W. Hutchinson, Engineering Fracture Mechanics, 147 (2015) 293-305.

The paper begins with a very nice review of the failure processes for plates with thicknesses from thick to thin, from plane strain fracture, via increasing amounts of strain localisation and failure along shear planes, to the thinnest foils that fail by pure strain localisation.

The plates in the title have in common that they contain a blunt notch and are subjected to monotonically increasing load. They are too thin to exclusively fracture and too thick to fail through pure plastic yielding. Instead the failure process is necking, followed by fracture along a worn-out slip plane in the necking region. Macroscopically it is mode I but on a microscale the final failure along a slip plane have the kinetics of mixed mode I and III and, I guess, also mode II.

A numerical solution of the problem resolving the details of the fracture process, should perhaps be conceivable but highly unpractical for engineering purposes. Instead, the necking region, which includes the strain localisation process and subsequent shear failure is a region of macroscopically unstable material and is modelled by a cohesive zone. The remaining plate is modelled as a power-law hardening continuum based on true stress and logarithmic strains.

The analysis is divided into two parts. First a cross-section perpendicular to the stretching of the cohesive zone is treated as a plane strain section. This is the cross section with a shape in which the parable with a neck becomes obvious. Here the relation between the contributions to the cohesive energy from strain localisation and from shear failure is obtained. A Gurson material model is used. Second, the structural scale model reveals the division of the tearing energy into the cohesive energy and the plastic dissipation outside the cohesive zone. The cohesive zone model accounts for a position dependent cohesive tearing energy and experimental results of B.C. Simonsen, R. Törnqvist, Marine Structures, vol. 17, pp. 1-27, 2004 are used to calibrate the cohesive energy.

It is found that the calibrated cohesive energy is low directly after initiation of crack growth, and later assumes a considerably higher steady state value. The latter is attained when the crack has propagated a distance of a few plate thicknesses away from the initial crack tip position. Calculations are continued until the crack has transversed around a third of the plate width.

I can understand that the situation during the initial crack growth is complex, as remarked by the investigators. I guess they would also agree that it would be better if the lower initial cohesive energy could be correlated to a property of the mechanical state instead of position. As the situation is, the position dependence seems to be the correct choise until it is figured out what happens in a real necking region

I wonder if the investigators continued computing the cohesive energy until the crack completely transversed the plate. That would provide an opportunity to test hypothesises both at initiation of crack growth and at the completed breaking of the plate. The situations that have some similarities but are still different would put the consistency of any hypothesis regarding dependencies of mechanical state to the test.

I am here taking the liberty to suggest other characteristics that may vary with the distance to the original crack tip position.

The strains across the cohesive zone are supposed to be large compared to the strains along it. This is the motivation for doing the plane strain calculations of the necking process. Could it be different in the region close to the original blunt crack tip where the situation is closer to plane stress than plane strain? The question is of course, if that influences the cohesive energy a distance of several plate thicknesses ahead of the initial crack position.

Another hypothesis could be that the compressive residual stress along the crack surface that develops as the crack propagate, influence the mechanical behaviour ahead of the crack tip. For very short necking regions the stress may even reach the yield limit in a thin region along the crack surface. Possibly that can have an effect on the stresses and strains in the necking region that affects the failure processes.

My final candidate for a hypothesis is the rotation that is very large at the crack tip before initiation of crack growth. In a linear elastic model and a small strain theory, rotation becomes unbounded before crack growth is initiated. A similar phenomenon has been reported by Lau, Kinloch, Williams and coworkers. The observation is that the severe rotation of the material adjacent to a bi-material adhesive lead to erroneous calibration of the cohesive energy. Could this be related to the lower cohesion energy? I guess that would mean that the resolution is insufficient in the area around the original crack tip position.

Are there any other ideas, or, even better, does anyone already have the answer to why the cohesive energy is very small immediately after initiation of crack growth?

Per Ståhle

https://imechanica.org/node/19424

Discussion of fracture paper #9 – Crack tip modelling

5 December 2024 / admin / 0 Comments

Dear Reader,

I recently took over as the ESIS blog editor. Being the second in this baton relay, I will do my best to live up to the good reader expectations that has been established by my precursor, who is also one of the instigators of the blog, Wolfgang Brock.

I did not follow the blog in the past. That I regret now that I go through the previous blogs. Here I discover many sharp observations of new methods and concepts paired with a great ability to extract both the essential merits and to spot weaknesses. Much deserve additional studies to bring things to a common view. We are reminded that common views, often rightfully, but not always, are perishable items.

Paper 9 in this series of reviews concerns phenomena that occur when a crack penetrates an interface between two materials with dissimilar material properties. In the purely elastic case it is known that a variation of Young’s modulus along the intended path of a crack may improve the fracture resistance of inherently brittle materials. If the variation is discontinuous and the crack is about to enter a stiffer material the stress intensity factor becomes unlimited with the result that fracture will never happen. At least if the non-linear region at the crack tip is treated as a point. To resolve the problem the extent of the non-linear region has to be considered.

The selected paper is: Effect of a single soft interlayer on the crack driving force, M. Sistaninia and O. Kolednik, Engineering Fracture Mechanics Vol. 130, 2014, pp. 21–41.

The authors show that spatial variations also of the yield stress alone can improve the fracture resistance. They find that the crack tip driving force of a crack that crosses a soft interlayer experiences a strong dip. The study is justified and the motivation is that the crack should be trapped in the interlayer. The concept of configurational forces (a paper on configurational forces was the subject of ESIS review no. 7) is employed to derive design rules for an optimal interlayer configuration. For a given matrix material and load, the thickness and the yield stress of a softer interlayer are determined so that the crack tip driving force is minimised. Such an optimum configuration can be used for a sophisticated design of fracture resistant components.

The authors discuss the most important limitations of the analysis of which one is that a series of stationary cracks are considered instead of a growing crack. The discussion of growing versus stationary cracks is supported by an earlier publication from the group. Further the analysis is limited to elastic-ideally plastic materials. A warning is promulgated by them for directly using the results for hardening materials.

The paper is a well written and a technically detailed study that makes the reading a good investment.

The object of my discussion is the role of the fracture process region in analogy with the discussion above of the elastic case. The process region is the region where the stresses decay with increasing straining. When the process region is sufficiently small it may be treated as a point but this may not be the case when a crack penetrates an interface. The process region cannot be small compared to the distance to the interface during the entire process. In the elastic case the simplification leads to a paradoxical result. The main difference as compared with the elastic case is that the ideally plastic fields surrounding a crack tip at some short distance from the interface have the same characteristics as the crack that has the tip at the interface, i.e. in the vicinity of the crack tip the stress is constant and the strain is inversely proportional to the distance to the crack tip. This means that the distance between the crack tip and the interface do not play the same role as in the elastic case. A couple of questions arise that perhaps could be objects of future studies. One is: What happens when the extent of the process region is larger than or of the order of the distance to the interface? If the crack is growing, obviously that has to happen and at some point the fracture processes will probably be active simultaneously in both materials. The way to extend the model could be to introduce a cohesive zone of Barenblatt type, that covers the fracture process region. The surrounding continuum may still be an elastic plastic material as in the present paper.

A problem with growing cracks is that the weaker crack tip fields does not provide any energy release rate at a point shaped crack tip. Would that limitation also be removed if the finite extent of the process region is considered?

With these open questions I hope to trigger those who are interested in the subject to comment or contribute with personal reflections regarding the paper under consideration.

Per Ståhle
Professor of Solid Mechanics
Lund University, Lund
Sweden

https://imechanica.org/node/17471

Discussion of fracture paper #7 – Configurational force approach

5 December 2024 / admin / 0 Comments

New paradigms may help understanding unsolved scientific problems by looking on them from a different perspective. Or they may lead to a new unification theory of so far separate phenomena. The concept of “material” or “configurational” forces tracing back to a seminal publication of Eshelby in 1970 and significantly extended and promoted by Maugin twenty years later provides a generalised theory on the character of singularities of various kinds in continua, among which the “driving force” at a crack tip is a special case. Whereas Eshelby’s energy momentum tensor resulting in the J-integral is a firm constituent of fracture mechanics, the concept of configurational forces has only hesitantly been applied to fracture problems, e.g. by Kolednik, Predan, and Fischer in Engineering Fracture Mechanics, Vol. 77, 2010. Whether this new “look” upon J helped discovering anything new about it remains disputable.

Now there is a revival of this concept

K. Özenç, M. Kaliske, G. Lin, and G. Bhashyam: Evaluation of energy contributions in elasto-plastic fracture: A review of the configurational force approach. Engineering Fracture Mechanics, Vol. 115, 2014, pp. 137–153.

It is admittedly difficult to contribute some novel aspect to more than forty years of research on J in elastoplastic fracture mechanics. Though a clear perception of the nature of “path dependence” of J is often enough still missing in some publications to the point of the user’s manual of a major commercial FE code, there is no lack of theoretical knowledge. Background, applicability and limitations of J are quite clear. Those looking for deeper insight will be disappointed: The present publication just answers questions and solves problems which arose with the chosen approach of material forces.

“The path dependency of the material force approach in elasto-plastic continua is found to be considerably depending on the so-called material body forces.” This is well-known and trivial as the derivation of path independence of J is, among others, based on the absence of body forces. It does not need “numerical examples … to clarify the concept of path dependence nature of the crack tip domain (?) and effect of the material body forces”. Correction terms re-establishing path independence have been introduced years ago, see e.g. Siegele, Comput. Struct., 1989.As many continuum mechanics people, the authors start with a display of fireworks introducing the general nonlinear kinematics of large deformations which can be found in every respective textbook. In the end, this impressing framework is simmered down again to “small strain elasto-plasticity and hyperelasto-plasticity”, whatever “hyperelasto-plasticity” is supposed to mean. This does not become much clearer by the statement “the Helmholtz free energy function of finite elasto-plasticity is introduced in order to obtain geometrically nonlinear von Mises plasticity”. Finite, i.e. Hencky-type plasticity and incremental plasticity, i.e. the von Mises, Prandtl, Reuss theory are alternative approaches, where the latter is more appropriate for describing irreversible, dissipative processes. What a “geometrically nonlinear” material behaviour is remains the secret of the authors. They presumably applied the so-called “deformation theory of plasticity” which actually describes hyperelastic behaviour based on the existence of a strain-energy density as stress potential. Thus “path dependence” should not be an issue at all as the requirements for deriving path-independence are met. The rest is numerics!

So where are the problem and its solution after all? Can “material forces” be calculated by the finite element method – who doubts? Is the implementation of this concept in a commercial FE code a major scientific achievement – who knows?

»W. Brocks’s blog

https://imechanica.org/node/16356

Applicable limit of the stress intensity factor for steep yield strength distribution

5 December 2024 / admin / 0 Comments

“The number of bad papers is multiplying. … a new, dramatic problem arises: how to select in the mud the papers conveying innovative ideas?” wrote Piero Villaggio in his Editorial “Crisis of mechanics literature?”, Meccanica, Vol. 48, pp. 765–767. He identified, among others, two factors, “the necessity of multiplying published papers in a large international competition” and “the abuse of self-quotations in order to remedy the perverse rule imposed by the impact-factor”. Disregarding “journals ready to publish everything“, the editors of top-ranking scientific journals have to face up to the question how to ensure a constantly high quality of the published manuscripts. A strict and carefully executed review process is a mandatory requirement. However, reading published articles, I sometimes wondered how a reviewer could let pass a manuscript like this?

One indispensable demand is a minimum standard of English expression. If the reader cannot discriminate what is inapt expression and what is lack of understanding of the problem, he or she will put the paper aside and stop reading. The author has scored on the publication list in any case, but the scientific benefit is null!

I shall outline some examples in the paper by

Tetsuo Yasuoka, Yoshihiro Mizutani, Akira Todoroki: Applicable limit of the stress intensity factor for steep yield strength distribution, Engineering Fracture Mechanics, Vol.110, 2013, pp. 1–11,

not to blame the authors but to ask the reviewer(s) of this manuscript whether they have actually understood cryptic sentences like “The crack was divided into discrete bar elements in this model. Each bar element involved the stress, yield strength and displacement. The remote tensile stress and the yield strength distribution were discretized using the principle of superposition”. What is “the SIF of the jth bar element subjected to the loading stress σj“? What shall I imagine by “this rectangle means (!?) CTOD” in Fig. 4, if it is an area under a stress distribution curve in the ligament?

That the substance of a submitted manuscript is correct to the best knowledge of the reviewer should be a matter of course. This actually may be a time-consuming task to check including literature research. There are some simple sanity checks, however. One would be: How can there be normal stresses acting on the free surface of a crack in Fig. 2? Newman’s respective Fig. 2 (ASTM STP 748 [1981]) which the authors quote shows compressive stresses due to crack closure, but this is not examined in the present manuscript.

Did the reviewers of the above-mentioned manuscript ask the authors any of these questions – or did they just wave the paper through?

A final delicate question a reviewer might ask is how substantial and significant the presented results are: what did he or she really learn from this contribution? A considerable number of submitted papers could be rejected with this argument. Taking a look on the references in the present manuscript raises doubts. Despite an own publication of 2012, the rest is mostly from the 60s and 70s of the last century. What did the scientific community miss over the last 40 to 50 years not having read this manuscript?

W. Brocks’s blog

https://imechanica.org/node/15504

Discussion of fracture paper #5 – Yield ciriterion or failure criterion

5 December 2024 / admin / 0 Comments

What is the difference betwee a failure criterion and a yield condition?

You may meet natural and engineering scientists who blame their colleagues from social sciences or humanities for working unscholarly, not adhering to an explicit and unique terminology but substituting scientific cognition by adopting novel terms. Those sitting in a glasshouse should not throw stones, however. Imprecise terminology and hazy definitions are not at all a “privilege” of social scientists. When I started learning fracture mechanics, I discovered that nearly every anomaly in the real failure behaviour of components which did not fit into the common concept was attributed to “constraint” – but few people had a precise idea what constraint actually is and how to quantify it. The multifarious usage of “damage” in the current literature is an actual example, and “plasticity” is another.

Though von Mises, Drucker, Hill and many others established a precise foundation of phenomenological plasticity, it has become a bad habit to call any inelastic, nonlinear mechanical behaviour “plastic”. One will find applications of the Mises-Prandtl-Reuss equations to polymers, and the authors do not even query, much less justify this approach. In my previous blog, #4, I criticised Mäkelä and Östlund (Engineering Fracture Mechanics, Vol. 79, 2012) for modelling the deformation of paper by means of plasticity. One year later I find an “application” to wood.

Henrik Danielsson and Per Johan Gustafsson: A three dimensional plasticity model for perpendicular to grain cohesive fracture in wood, Engineering Fracture Mechanics Vol. 98 2013, pp.137–152.

The authors’ misconception is a different one. The deformation behaviour of wood is considered as linear elastic and, of course, orthotropic. But they add a new facet to the term “plasticity”, namely the irreversible and unstable material softening in some process zone: “Initiation of softening, i.e. the formation of a fracture process zone, is determined by an initial yield function F according to the Tsai–Wu failure criterion”. This is a failure criterion, correct, and the respective limit surface in the stress space may be assumed as convex as the yield surface in the theory of plasticity. For the sake of a thermodynamically consistent theory, one may also define a corresponding damage potential, but this is not a plastic potential! Once again: The theory of plasticity deals with the stress-strain relationship of ductile materials, having metals in mind, where plastic flow occurs by sliding along crystallographic planes or by twinning. “A physical theory of plasticity starts with these microscopic details and attempts to explain why and how plastic flow occurs” (Khan & Huang: Continuum Theory of Plasticity, Wiley, 1995, p. 310). Following Drucker, classical phenomenological plasticity describes stable, i.e. strain-hardening, material behaviour.

The authors continue “The change in size of the yield surface f is described by the softening parameter K which is a function of an internal variable that memorizes the plastic loading and determines the softening behavior”, and they introduce a “dimensionless deformation δeff“, as internal variable, which is „related to the plastic straining of the material” (wood?), whatever this is supposed to mean. It is not just the “size” of the failure surface that changes, by the way, as Fig 2 shows. In the context of cohesive models, δ is commonly called “separation”, i.e. a jump in the discontinuous displacement field, and Fig 3 is a typical traction-separation law. So why introduce a terminology divergent from the established one?

Roberto Balarini stated in a blog node/7622 : “Cohesive models are linear elasticity”. In contrast, the present authors apparently assert that cohesive models “are” plasticity. What is so difficult in understanding the model of a cohesive zone? Cohesive models “are” neither elasticity nor plasticity. They describe the nonlinear decohesion process in a continuum that obeys any kind of constitutive equations, for instance plasticity, visco-plasticity or, as in the present case, orthotropic elasticity.

More generally: What is so complicated in applying a unique terminology which is established in the scientific community, and how about the reviewers of manuscripts like this: Are they not aware of the correct terminology themselves or do they just don’t care about it? Remember: The corruption of reasoning starts with a false handling of language!

W. Brocks’s blog

https://imechanica.org/node/14387

Discussion of fracture paper #4 – Is paper ductile?

5 December 2024 / admin / 0 Comments

In my previous blog, I complained about colleagues developing constitutive models without having any notion about the specific nature of deformation and damage and their micromechanical mechanisms. Unfortunately, this happens more often than one might (or would like to) believe, as a recent example testifies.

P. Mäkelä and S. Östlund: Cohesive crack modelling of thin sheet material exhibiting anisotropy, plasticity and large-scale damage evolution. Engineering Fracture Mechanics, Vol. 79, 2012 pp. 50-60.

Looking at the title and the Abstract:

“In this work, a cohesive crack model suitable for static fracture mechanics analysis of thin sheet materials exhibiting anisotropy, plasticity, and large-scale damage evolution was developed”,

the reader might think of “metal sheets”. But far wrong: the authors talk about paper! And the unbelieving reader starts realising this at the end of the Introduction:“The elastic–plastic model was calibrated by tensile testing and the cohesive zone model was calibrated by stable tensile testing for one grade of paper material”,and in section 2.4, where the tests are described.

The theory of plasticity deals with the stress-strain and load deflection relationships for ductile materials. Is paper “ductile”, and what means “ductile”, actually? Talking about ductility, people commonly think of the behaviour of metals, and it were metals for which phenomenological plasticity has been developed. Metals have a crystalline structure, so the plastic flow occurs by sliding along crystallographic planes or by twinning. “A physical theory of plasticity starts with these microscopic details and attempts to explain why and how plastic flow occurs” (A.S. Khan & S. Huang: Continuum Theory of Plasticity, Wiley, 1995, p. 310). What are the mechanisms of deformation and damage in paper? The authors do not tell us!

They just present a pretty conventional phenomenological model of orthotropic plasticity based on a transformation of the stress tensor and the von Mises yield criterion with an associated flow rule, and apply this to paper specimens. The “excellent prediction” of the test results by the model, which the authors claim with respect to Fig. 6, is not at all impressive, as it just shows that the uniaxial stress-strain curves can be described by an exponential function as in Eq. (6). Finally, they combine this with some exponential cohesive softening law to describe the tearing of the paper and as the respective cohesive parameters were fitted to the test results, there is no reason why this should not “predict” failure of centre-cracked sheets for varying crack lengths, a/W, satisfactorily.

What a “large-scale damage evolution” announced in the title is supposed to be, remains obscure, as a cohesive zone describes localised and no “large scale” damage.

That the authors try to surprise us with repeated statements like

“The accuracy of the cohesive crack model is largely dependent on accurate constitutive modelling.”
“The performance of the cohesive crack model is generally most dependent on the accurate formulation and calibration of the cohesive zone model.”
“The key to accurate cohesive crack modelling is constituted by the ability to accurately determine the cohesive material behaviour.”

underlines the lack of information about paper in the present manuscript.

Final remark: The present blogs intend to encourage discussion on fracture mechanics and related subjects. No reaction by the authors has yet come to any of them. Does this support the suspicion that publishing does not intend to contribute to science but just to increase the individual scoring of scientists?

W. Brocks’s blog

https://imechanica.org/node/11741