Category: ESIS (Page 3 of 3)

A T-stress is generally not expected to contribute to the stress intensity factor because its contribution to the free energy is the same before and after crack growth. Nothing lost, nothing gained. Some time ago I came across a situation when a T-stress, violates this statement. The scene is the atomic level. As the crack is producing new crack surfaces the elastic stiffness in the few atomic layers closest to the crack plane are modified. This changes the elastic energy which could provide, contribute to or at least modify the energy release rate. If the energy is sufficient depends on the magnitude of the T-stress, the change of the elastic modulus and how many atomic layers that are involved. 

If I should make an estimate it would be that the energy release rate is the change of the T-stress times the fraction of change of the elastic modulus times the square root of the thickness on the affected layer. Assuming that the T-stress is a couple of GPa, the change of the fraction of change of the elastic modulus is 10% and the affected layer is around ten atomic layers one ends up with 100kPa m^(1/2). Fairly small and the stress and its change are taken at its upper limits but still it is there. The only crystalline material I could find is ice with a toughness of the same level. Other materials are affected but require some additional remote load.

Interestingly enough I came across a paper describing a different mechanism leading to a T-stress contribution to the energy release rate. The paper is:

Zi-Cheng Jiang, Guo-Jin Tang, Xian-Fang Li, Effect of initial T-stress on stress intensity factor for a crack in a thin pre-stressed layer, Engineering Fracture Mechanics, pp. 19-27.

This is a really read worthy paper. The reasons for the coupling between the T-stress and the stress intensity factor is made clear by their analysis. The authors have an admirable taste for simple but accurate solutions. The paper describes a crack with a layer of residual stress, that gives a T-stress in the crack tip vicinity. As the crack advances increasing more material end up behind the crack tip rather than in front of it. The elastic energy density caused by the T-stress is larger in front of the tip than it is behind it. The energy released on the way and can only disappear at the singular crack tip, not anywhere else in the elastic material. The reason for the energy release is the assumed buckling in the direction perpendicular to the crack plane. An Euler-Bernoulli beam theory is used to calculate the contribution to the energy release rate.

Having read the paper I realise that in a thin sheet buckling out of its own plane in the presence of a crack and a compressive T-stress there will be energy released that should contribute to crack growth. The buckling will give a more seriously distorted stress state around the crack tip, but never the less. In this case the buckling area would be proportional to the squared crack length in stead of crack length times the height of the layer as in the Jiang et al. paper. The consequence is that the contribution to the stress intensity factor should scale with the T-stress times square root of the crack length.

Suddenly I feel that it would be very interesting to hear if anyone, maybe the authors themselves, know of other mechanisms that could lead to this kind of surprising addition to the energy release rate caused by T-stresses. It would be great if we could add more to the picture. Anyone with information is cordially invited to contribute.

Per Ståhle

https://imechanica.org/node/21796

No doubt the energy release rate comes first. What comes next is proposed in a recently published study that describes a method based on a new constraint parameter Ap. The paper is:

Fracture assessment based on unified constraint parameter for pressurized pipes with circumferential surface cracks, M.Y. Mu, G.Z. Wang, F.Z. Xuan, S.T. Tu, Engineering Fracture Mechanics 175 (2017), 201–218 

The parameter Ap is compared with established parameters like T, Q etc. The application is to pipes with edge cracks. I would guess that it should also apply to other large structures with low crack tip constraint.

As everyone knows, linear fracture mechanics works safely only at small scales of yielding. Despite this, the approach to predict fracture by studying the energy loss at crack growth, using the stress intensity factor KI and its critical limit, the fracture toughness, has been an engineering success story. KI captures the energy release rate at crack growth. This is a well-founded concept that works for technical applications that meet the necessary requirements. The problem is that many or possibly most technical applications hardly do that. The autonomy concept in combination with J-integral calculations, which gives a measure of the potential energy release rate of a stationary crack, widens the range of applications. However, it is an ironin that the J-integral predicts the initiation of crack growth which is an event that is very difficult to observe, while global instability, which is the major concern and surely easy to detect, lacks a basic single parameter theory.

For a working concept, geometry and load case must be classified with a second parameter in addition to KI or J. The most important quantity is no doubt the energy release rate, but what is the second most important. Several successful parameters have been proposed. Most of them describe some type of crack tip constraint, such as the T-stress, Q, the stress triaxiality factor h, etc. A recent suggestion that, as it seems to me, have great potential is a measure of the volume exposed to high effective stress, Ap. It was earlier proposed by the present group GZ Wang and co-authors. Ap is defined as the relative size of the region in which the effective stress exceeds a certain level. As pointed out by the authors, defects in large engineering structures such as pressure pipes and vessels are often subjected to a significantly lower level of crack tip constraint than what is obtained in laboratory test specimens. The load and geometry belong to an autonomy class to speak the language of KB Broberg in his book “Fracture and Cracks”. The lack of a suitable classifying parameter is covered by Ap.

The supporting idea is that KI or J describe the same series of events that lead to fracture both in the lab and in the application if the situations meet the same class requirements, i.e. in this case have the same Ap. The geometry and external loads are of course not the same, while a simpler and usually smaller geometry is the very idea of the lab test. The study goes a step further and proposes a one-parameter criterion that combines the KI or J with Ap by correlation with data.

The method is reinforced by several experiments that show that the method remains conservative, while still avoiding too conservative predictions. The latter of course makes it possible to avoid unnecessary disposal and replacement or repair of components. The authors’ conclusions are based on experience of a particular type of application. I like the use of the parameter. I guess more needs to be done extensively map of the autonomy classes that is covered by the method. I am sure the story does not end here.

A few questions could be sent along: Like “Is it possible to describe or give name to the second most important quantity after the energy release rate?” The paper mentions that statistical size effects and loss of constraint could affect Ap. Would it be possible to do experiments that separates the statistical effect from the loss of constraint? Is it required or even interesting?  

It would be interesting to hear from the authors or anyone else who would like to discuss or comment the paper, the proposed method, the parameter or anything related. 

Per Ståhle

https://imechanica.org/node/21722

Everyone loves an elegant engineering solution. It is particularly true when the alternatives are terrifying. In the paper:

”Brittle crack propagation/arrest behaviour in steel plate – Part I: Model formulation” by Kazuki Shibanuma, Fuminori Yanagimoto, Tetsuya Namegawa, Katsuyuki Suzuki, Shuji Aihara in Engineering Fracture Mechanics, 162 (2016) 324-340.

a team from University of Tokyo proposes a model for prediction of the arrest of propagating brittle cracks in steel plates. The approach, in spite of its simplicity, captures the physics of the fracture process. The model formulates the energy release rate in simple and comprehensible terms and gives accurate predictions. The theory is validated on several experiments described in a subsequent paper, a ”Part II: Experiments and model validation” also in Engineering Fracture Mechanics. The characteristics are those of a pilot study with the goal to provide a design tool for predicting crack arrest in steel plates.

In the model, the energy to complete the fracture process is at most what is left of the released energy when the work of plastic deformation and the part of the kinetic energy that is reflected away from crack tip region have been covered. The energy dissipation at plastic deformation is reduced at increasing crack tip velocity while the opposite applies to the dissipated kinetic energy. The energy required for the fracture processes is supposed to be constant. If it at some velocity is more than required then the energy is in balance only at a single stable higher crack tip velocity. If crack growth is initiated then the crack accelerates until the energy balance is obtained. When the crack subsequently loose driving force or require additional work, caused, e.g., by elevated temperature which decreases the material viscosity or by whatever, the crack decelerates until zero velocity or until the minimum energy release rate is obtained and the crack arrest comes abruptly.

Surface-ligaments are assumed to consume a serious part of the available energy. The slower the crack grows the wider these ligaments become which rapidly increases the plastic dissipation. Finally the energy balance and the stability of the crack tip velocity cannot be maintained and the crack will come to a stop.

Considering that one has to keep track of the complicated sequence of processes that keep the crack growing, it seemed obvious to me that this would end up in a horrible and time consuming analysis. Then, to my surprise, the investigators present an ingenious solution that simplifies the analysis a lot. It is based on three assumptions: 1) that the crack front is assumed to be straight through the plate, 2) that the unbroken side-ligaments are regarded as integrated parts of the crack front, and 3) that the evaluation of the state of the crack front in done at the plate mid-plane.

In the subsequent part II the functionality of the model is verified. The validation is performed on different grades of steel that are exposed to different load levels. The authors believe that this model can be used to establish a design strategy for steel plates. I too believe that, even if more possibly needs to be done to qualify the method as a design standard.

I understand that the authors are familiar with the series of wide plate experiments on crack arrest in very large specimens  (around 11x1x0.1 m3) reported by Naus et al., NUREG/CR-4930 ORNL-6388, Oakridge Laboratories, USA, 1987. 

In the aftermath of the experiments a variety of models where proposed. An interesting observation made by D. Alexander and I.B. Johansson at Oakridge Labs when they examined the crack surfaces was that remains of plastic deformation framed the cloven grains. The guess was that this was remaining parts of broken ligaments between the crack surfaces and that these ligaments were ripped apart during the fracture process. The area covered by these remains was clearly increasing with decreasing crack tip velocity. Just before crack arrest they could cover as much as 10 to 20 % of the ”brittle” part of the crack surface. I have a feeling that this may mean something. The plastic ligaments per se consume large amounts of energy and with increasing fractions they might influence the crack tip velocity at arrest. Only 10% may seem as small or even insignificant, but considering that the plastic ligaments that bridge a crack may consume many times more energy than the pure cleavage of the remaining 90%, even 10% must be important. It would be interesting to know if the authors observed any remains of plastic ligaments. If so, did the fraction of them change in any systematic way during crack growth? 

Any contribution to this blog is gratefully acknowledged.

Per Ståhle

https://imechanica.org/node/20605

A nice demonstration of toughening by introducing multiple secondary cracking of planes parallel with the primary crack is found in the paper:

”Fracture resistance enhancement of layered structures by multiple cracks”  by Stergios Goutianos and Bent F. Sørensen in Engineering Fracture Mechanics, 151 (2016) 92-108.

The 14th paper belong to the category innovative ideas leading to improved composites. We already know of combinations of hard/soft, stiff/weak or brittle/ductile materials that are used to obtain some desired properties. The results are not at all limited to what is set by the pure materials themselves. It has been shown that cracks intersecting soft material layers are exposed to elevated fracture resistances (see eg. the paper 9 blog). Differences in stiffness can be used to improve fatigue and fracture mechanical properties as found in studies by Surresh, Sou, Cominou, He, Hutchinson, and others. Weak interfaces can be used to diverge or split a crack on an intersecting path. A retardation is caused by the additional energy consumed for the extended crack surface area or caused by smaller crack tip driving forces of diverging crack branches. 

A primary crack is confined to grow in a weak layer. The crack tip that is modelled with a cohesive zone remains stationary until the full load carrying capacity of the cohesive forces is reached. Meanwhile the increasing stress across an even weaker adjacent layer also develops a cohesive zone that takes its share of the energy released from the surrounding elastic material. At some point the cohesive capacity is exhausted also here and a secondary crack is initiated. Both cracks are confined to different crack planes and will never coalesce. The continuation may follow different scenarios depending on the distance between the two planes, the relative cohesive properties like cohesive stress, critical crack tip opening, the behaviour at closure etc. of the second layer. All these aspects are studied and discussed in the paper.

The investigators have successfully found a model for how to design the cohesive properties to obtain structures with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them. A theoretical model is formulated. With it they are able to predict whether or not the toughness of a layered structure can be increased by introducing weak layers as described. 

Their results coincide well with the experimental results by Rask and Sørensen (2012) and they have found a model for how to design the cohesive properties to obtain a structure with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them.  

The part that I would like to discuss concerns an estimation of an upper bound of the enhancement of the fracture toughness. The derived theoretical model is based on the J integral taken along a path that ensures path independence. Two different paths are evaluated and compared. Along a remote path the J-value is given as a function of external load and deformation. The structural stiffness is reduced as the crack advances in the direction of the primary crack. In the linear elastic case the J-value is half of the work done by the external load during a unit of crack growth. In an evaluation taken along a local path, J receive contributions from the primary crack tip and the two crack tips of the secondary crack. All three tips are supposed to move a unit of length in the direction of the extending primary crack. 

As observed by the authors the secondary crack does not contribute to the energy release rate while what is dissipated at the propagating foremost crack tip is to the same amount produced at the healing trailing crack tip. Both crack tips propagate in the same direction so that the crack length does not change. 

An observation from the experimental study was that all crack tips have different growth rates and especially the trailing tip of the secondary crack was found to be stationary. Therefore the contribution from that crack tip to the local energy release rate is annulated which leaves less available to the primary crack. To me this seems right. However, when the two remaining advancing crack tips grow does not the respective contributions to J have to be reassessed to reflect their different growth rates? If we assume that the secondary crack grow faster than the primary crack then the enhancing effect is underestimated by the J-integral. Upper bound or lower bound – I can’t decide. I would say that it is a fair estimate of where the fracture resistance will end up. 

In conjunction with the evaluation of the work done by the external load during a ”unit of crack growth” it seems to be an intricate problem to correlate the unit of crack growth with the different crack tip speeds. Some kind of average perhaps.

Any contribution to the blog is gratefully acknowledged.

Per Ståhle

https://imechanica.org/node/20004

Cracks typically follow paths that locally give a mode I crack tip load. At mixed modes crack are extended via a kink in a direction that locally restores mode I. In isotropic materials this is known to more or less, true for static and dynamic loads. Exceptions are cracks that are subjected to high compressive load, e.g., at contact between train wheels and rails or at cracks caused by seismic movements. Other exceptions are cracks growing in anisotropic materials, at grain boundaries or other weak, or by deformation weakened, interfaces. 

The recently published 

“Method for calculating G, GI, and GII to simulate crack growth in 2D, multiple-material structures” by E.K. Oneida, M.C.H. van der Meulen, A.R. Ingraffea, Engineering Fracture Mechanics, Vol 140 (2015) pp. 106–126, 

is an interesting paper about the calculation of mixed mode loads and crack paths by use of a combination of several concepts for crack tip modelling. The developed method has general applicability in that it includes cracks that pass, join or deflect from interfaces, cracks at bifurcation points where three materials meet, and of course the crack paths embedded in homogeneous materials. A scheme is presented that uses two slightly altered local meshes to obtain the variation of the energy release rate due to a small variation of the crack path. The  M-integral by Yau, Wang and Corten, J. Appl. Mech., 1980 for separation of mode I and mode II is used. The paper is nicely completed with a demonstration of a crack propagation framework, which combines the developed methods. The result is a convincing simulation of crack growth through a composite material. The path that maximises the strain energy release rate relative to the toughness, is followed.

When the fracture processes are confined to a small region it may be safe to use a sharp crack tip. However, occasionally it leads to an unreasonable behaviour, such as when the energy release rate disappears as the crack tip passes through a bimaterial interface from a weaker to a stiffer material (cf. discussion of paper 9 in this series). Similarly, let’s say that a crack meets a conceivable branching point with two branches or paths to chose between and both paths are having equal loading and equal toughness. This seems to be a dead heat. However, say that initiation of the fracture processes need sufficient hydrostatic stress and sufficient subsequent deformation to complete the fracture and that the relation between these quantities are different along the two paths. Then even though the toughnesses are equal, the growing crack is likely to follow the path that first allow initiation of the fracture processes and the other path will never be activated. Perhaps there are exceptions but in general it seems to me that a crack tip model with more details is needed for these cases. 

Cases when cracks deflect from a weak interface are, I believe, similarly problematic. Whether a crack will follow a weak plane under a mixed mode load or kink out of that plane should to a large extent depend on the affinity to initiate a fracture process outside the interface.

I understand that the paper is concerned with indivisible fracture toughness which is excellent, but in view of the sketched scenarios above, I wonder if the model can be extended to include modelling of the process region with a finite physical extent, e.g. by using a cohesive zone model, that provides a two parameter model for the process region. One difficulty that I immediately come to think of is that the strain energy singularity is annulated by the cohesive stresses so that the M-integral possibly will fail. Still, if the foremost part of the process region, i.e. the tip of the cohesive zone rather than the crack tip, is the path finder then maybe a stress criterion could be a suitable candidate. Are there other possibilities? Could the point shaped crack tip be kept while using a stress criterium at some fixed distance ahead of the crack tip? Or would “fixed distance” per se require process region autonomy?

Per Ståhle

https://imechanica.org/node/18931

According to the Swedish Plant Inspectorate the major part of all reported fracture related failures in Sweden are due to stress corrosion. I guess it is more or less a reality everywhere. The association with accidents is probably because it comes without warning and usually at surprisingly low loads. Just a mm sized spot of decomposing grease is enough to create a locally extremely acid environment. In an otherwise friendly environment this often not even considered as a possibility by the designer.

The paper for this discussion is:

”Further study on crack growth model of buried pipelines exposed to concentrated carbonate-bicarbonate solution”, B.T. Lu, Engineering Fracture Mechanics vol. 131 (2014) pp. 296-314. 

A stress corrosion cracking model is developed. The main character of the fracture processes is a repeated breaking and healing of a passivating oxide film. When it is intact it prevents the metal from being dissolved by an aggressive environment, and when it is broken, metal ions escape from the surface and the crack thereby advances. The bare metal surface quickly becomes covered by a new thin oxide film when it is exposed to air and moist. To keep up with the oxidisation rate a sufficient strain rate has to be maintained in the crack tip region.

The authors study the combined effect of cyclic loading leading to stress corrosion cracking and mechanical fatigue with good results. The model is used successfully in describing the behaviour of several experimental results reported by different groups. 

In ESIS review no. 3 the importance of knowing the length scales of fracture processes was emphasized. In the present paper this is fully understood. The crack tip is confined to a point that is under KI control. To deal with the problem of assigning a strain rate to the singular stress field, the strain rate a short distance (a few microns) ahead of the crack tip is selected. It seems to be an accepted practice by more than the present author and the precise distance is regarded to be a material parameter. However, I feel a bit uncertain about the physical reasons for the actual choice. 

Is it possible that there is no length scale that is simultaneously relevant to both the mechanical and the chemical processes. Assume that the width of the blunted tip is a few microns as it is given by KI. We also have an oxide film of a few nm that covers the blunted surface. A distance of a few nm is not likely to be exposed to any gradients of the strain field where the meaningful distances are of the order of microns. In this case the film thickness seems irrelevant. The dissolution of the metal takes place around the crack tip and keeps the growing crack blunt. With the only length scale relevant to the mechanical state being provided by the stress intensity factor the result would be a self-similar shape and a constant stress and strain field in the crack tip region.

A consequence would be that the crack growth rate would be independent of the remote load. Something like that can be seen in the paper “Q.J. Peng et al. Journal of Nuclear Materials 324 (2004) 52–61” that is cited in the present paper. Fig. 2 test 3 shows almost constant growth rate in spite of an almost doubled remote load. 

A length scale of a few microns is introduced in the discussed paper. What could be the relevance of the choice? Is a length scale always necessary?

Per Ståhle

https://imechanica.org/node/17865