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# Fatigue

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These notes were prepared when I taught fracture mechanics in 2010, and were updated when I taught the course again in 2014.

Notes on other parts of the course are also online.

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## Comments

## 2 useful reviews

Zhigang you may find useful2 papers which reconstruct also this story, and add also short cracks, misunderstandings,

One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach

as well as this one, which merges fatigue with fracture mechanics (Paris).

The very bizzare story of the paper: FFEMS-4283 - A simplified "damage ...

Notice that the two approaches have been so distant (and this explains the reluctance to accept my latter paper) that you can hardly find material constants for a given material, and for both approaches.

Noone has so far developed a satisfactory unified theory. Which was an attempt of both of my papers.

## to know about "how to apply direct loading"

Qaleem:

MY QUESTION IS - HOW TO APPLY CYCLIC LOADINGS IN ABAQUS,SAY FOR INSTANCE I'VE RUN 60000 CYCLES AT 10 HZ AT A MAXIMUM-MINIMUM LOAD OF 18 KN - 2KN

## Crack Initiation

Dear Zhigang

crack initiation time or cycles is something debatable. I would better accept a title in terms number of cycle to detect a specific crack size rather crack initiation. It is perhaps the trend of the British School to accept that very small cracks in terms of microstructural features pro-exist in any material. As such what we call crack initiation is the time to detect with our means a observable size.

Chris

## Crack Initiation

Chris,

If we cannot detect a crack with any available equipment (e.g. x-ray diffraction), then as a safety factor we just assume that the initial flaw size is whatever the minimum flaw size that that piece of equipment can detect. This is the general approach of practicing mechanical engineers. Also, we usually think that there are some pre-existing flaws in the material, but whether these are "cracks" or not may depend on your definition of a crack (in my opinion at least). Also, during this initiation period to which Zhigang is referring, other flaws can be nucleated. This idea was initially proposed by Wood (1958). He suggested that cyclic straining in ductile materials leads to different amounts of net slip on different glide planes. This irreversible process results in a roughening of the surface of the material, forming valleys and hills knowns as "intrusions" and "extrusions," respecitvely. This can create "micronotches" and can promote further slip and fatigue crack nucleation. Similar processes can occur at persistant slip bands, grain boundaries, etc. Thus, it appears that there may be some mechanism by which fatigue cracks can be nucleated. I believe this is the initation period to which Zhigang is referring.

-Matt

## Dear Matt If we

Dear Matt

If we cannot detect a crack with any available equipment (e.g. x-ray

diffraction), then as a safety factor we just assume that the initial

flaw size is whatever the minimum flaw size that that piece of equipment

can detect. Agree

Also, we usually think that there are some pre-existing flaws in the

material, but whether these are "cracks" or not may depend on your

definition of a crack (in my opinion at least). Agree

Also, during this initiation period to which Zhigang is referring,

other flaws can be nucleated. This idea was initially proposed by Wood

(1958). He suggested that cyclic straining in ductile materials leads

to different amounts of net slip on different glide planes. This

irreversible process results in a roughening of the surface of the

material, forming valleys and hills knowns as "intrusions" and

"extrusions," respecitvely. Not always. Single crystals will promote such phenomenon. Alloys will not. The number of triple points in an alloy systems is huge.

This can create "micronotches" and can promote further slip and fatigue

crack nucleation. This is a very basic definition and falls within single crystals again.

Thus, it appears that there may be some mechanism by which fatigue

cracks can be nucleated. Indeed for small laboratory specimens. Yet crack initiation in large manufactured components is somehow different.

## Flaw Nucleation

I agree that the ideas I have listed are very qualitative and not very well characterized. However, I am not sure why this would only be true in single crystal materials. I am talking about intrusions and extrusions forming micronotches on the surface of the crystal. So I think this idea should be fairly general, and should apply to single crystals, polycrystals, alloys, etc. Also, I am unclear why you claim that fatigue cracks can be nucleated only for small laboratory specimens and not for large manufactured components. Could you please clarify this?

Thanks,

Matt

## I tend to agree with Matt -Chris you cannot always distinguish!

Chris, you need to draw a line where you can make general conclusions. I agree with Matt. We cannot make one theory for every material --- specifically you seem an expert of Al 2024. But then? Matt is interested in single crystal. How much more do we need to study? Wouldn't be better to apply Ashby kind of reasoning, or my ones? Matt, please also comment

## to know about "how to apply direct loading"

Qaleem

## Chris, Zhigang, and Matt ... some comments

Dear Chris, Zhigang, and Matt ... some comments on the long debate on the crack initiation vs crack propagation.

In reality, this has caused a lot more literature, than real advancement.

Fracture mechanics in the Paris sense has produced, in his own words, almost as many problems as it has solved. See one review paper he wrote in 1999, I think I have a reference in my paper One, no one, and one hundred

thousand crack propagation laws: A generalized Barenblatt and Botvina

dimensional analysis approach

In that paper, you will see that there is no good prediction using Paris' law possible, simply because Paris' law is so empirical and the constants so much dependent on many factors, that one should do better NOT to use it!

The curious story about Paris is perhaps similar to the story of my paper The very bizzare story of the

paper: FFEMS-4283 - A simplified "damage ... although I do not expect the success Paul Paris made with his suggestion that if a paper is rejected by 3 journals, it is revolutionary!

In fact, Paul Paris was quite lucky to have his data fit so well the power law, ONLY BECAUSE he did not have ENOUGH data! See the further discussion in my paper. He had made a very remarkable suggestion, because as you say, people were not prepared to use an elastic factor for the crack propagation which is certainly NOT an elastic phenomenon.

However, how unfortunate Crack propagation has been since then! Air Force returned recently to what Ted Nicholas in his well written book calls damage tolerance for high cycle fatigue, which is NOT too different from the early classical approach using diagrams.

The idea itself of the Kitagawa diagram has been very successful to study interaction between classical fatigue and fracture mechanics, but limited to the non-propagation of cracks.

When cracks do propagate, we DO NOT have a clear definition of what is the initial size.

This is why I suggested to use the same "idea" of the El Haddad - Topper initial crack into the "extended" Kitagawa diagram. A first version appeared to Int J Fatigue which I don't like yet was published without any problem!

M. Ciavarella, F.Monno, (2006) On the possible generalizations of the Kitagawa—Takahashi

diagram and of the El Haddad equation to finite life, International Journal of Fatigue 28 1826—

1837

The new version, which I like much more, has the courage to remove some assumptions everybody assumes as "granted", and infact Susmel and Taylor arrive at somewhere near my idea, but not removing the "tabu" of a threshold limit "dependent on number of cycles", which in fact they define without noticing.

See discussion in the "public review" of my paper The very bizzare

story of the

paper: FFEMS-4283 - A simplified "damage ...

Regards

mike

## On applying fracture mechanics to fatigue

Dear Mike: Thank you for the comments, and for the references. For the students who happen to read this thread of discussion, I would like to add the following points of discussion.

## to know about "how to apply direct loading"

Qaleem

## Dear Zhigang, I fully understand this is advanced material.. but

But I disagree Paris' law is as general as a stress-strain curve for a material.

It is MUCH LESS than that.

It is just curve fit under SPECIAL conditions. As soon as you apply the crack extension problem to ANY OTHER condition, say new dimension of the specimen, new thickness, even new dimension or shape of the crack, new loading conditions (R-ratio, random loading, etc. etc.) then the constants C, m in the Paris equation invariably change. This is especially true when the crack is small (and generally the cracks start small), for which the LARGEST amount of funding was spent in aeronautical industry (partly to the advantage of Paul Paris who, I'm told, at one point was flying with Boeing private jets), yet nothing relevant was found, which was not known in classical fatigue!

This, in Barenblatt's notation, is because Paris' power law is NOT fundamentally given as a law of physics.

Of course, you are rigth to say that there is an analog there when we use power-law in stress-strain curves. HOWEVER, stress-strain curves are extremely more stable with respect to any change of parameters as above.

This for your students, not to be fascinated too much with Paris' law. As I said, it has

If I were you, I would rather teach fatigue than crack propagation fracture But of course your students in Harvard will be intrigued by our discussion, and perhaps get fascinated. In this sense, I hope this helps your course! Next time, I will invite you in Italy, to teach to my students... :)

## stress-strain curve is just as "unstable"

Well, stress-strain curve is just as "unstable". It varies with temperature, humidity, loading rate, specimen size, shape of a body (if strain-gradient is important, for example)...

I do not mean to advocate that we abandon stress-strain curve. Stress is an important idea. Strain is another important idea. Their relation characterizes an aspect of a material, under certain conditions.

The above statements read just fine if stress is replaced by "stress intensity factor", and strain is replaced by "crack extension per cycle".

## fatigue

could you please tell me how can apply cyclic loading to 2D model steel joint.And how can i connect them by welded type element.That should be in ansys commands.

regards

## Re: fatigue

I'm sorry. I have no expertise with ansys

## to know about "how to apply direct loading"

Qaleem:

can you please help me. MY QUESTION IS - HOW TO APPLY CYCLIC LOADINGS IN ABAQUS,SAY FOR INSTANCE I'VE RUN 60000 CYCLES AT 10 HZ AT A MAXIMUM-MINIMUM LOAD OF 18 KN - 2KN

## That is a good challenge, Zhigang !

I like your competitive approach.

We need to be quantitative here. Let's forget about why stress-strain is more "physically stable", and be practical. How much do you expect the Elastic modulus E to vary with humidity? Negligible change!

Try to find out how much C, m change. A lot, and try to see the implication for that: for E, perhaps the error in the final result remains proportional to the error in input, since the sigma=E eps is a linear function.

Try now to consider the NATURE of Paris law, it is given by a derivative equation

da/dN = C (dK)^m,

where m is never lower than about 3 (in fact, when crack closure is removed, it appears to be near 3 for many materials, but this is another complication). For some ceramics, m is as high as 10-50.

In that case, even if you make a mistake in your computation, you can easily make an error in the prediction of MANY ORDERS of magnitude.

This is the ADDITIONAL risk of Paris law. This is incidentally why it is REASONABLE to use it for ligth alloys, and certainly NOT for ceramic materials.

Maybe some figures would help here, but they are simply in my One, no one, and one hundred

thousand crack propagation laws: A generalized Barenblatt and Botvina

dimensional analysis approach

So the real question is, with respect to your students, were they prepared to use only Paris in crack propagation, but

1) do they know they cannot use is for short or small cracks?

2) If so, are they prepared to understand when crack is small or short?

3) were they prepared to use it for ligth alloys, or were misunderstanding it also for ceramic materials?

4) would they be prepared, in case they do research, to rather look at other approaches, than stretching this too much

Perhaps a good single reference to all this is the Fleck and Abshy review paper, which is Fleck and Ashby.

But if you have time, have a quick look at my abstract:

Barenblatt and Botvina with elegant dimensional analysis arguments have elucidated

that Paris’ power-law is a weak form of scaling, so that the Paris’

parameters C and m should not be taken as material constants. On the contrary, they are

expected to depend on all the dimensionless parameters of the problem,

and are really “constants” only within some specific ranges of all

these. In the present paper, the dimensional analysis approach by

Barenblatt and Botvina is generalized to explore the functional

dependencies of m and C on more dimensionless parameters than the original Barenblatt and

Botvina, and experimental results are interpreted for a wider range of

materials including both metals and concrete. In particular, we find

that the size-scale dependencies of m and C

and the resulting correlation between C and m

are quite different for metals and for quasi-brittle materials, as it

is already suggested from the fact the fatigue crack propagation

processes lead to m=2–5 in metals and m=10–50

in quasi-brittle materials. Therefore, according to the concepts of

complete and incomplete self-similarities, the experimentally observed

breakdowns of the classical Paris’ law are discussed and interpreted

within a unified theoretical framework. Finally, we show that most

attempts to address the deviations from the Paris’ law or the empirical

correlations between the constants can be explained with this approach.

We also suggest that “incomplete similarity” corresponds to the

difficulties encountered so far by the “damage tolerant” approach

which, after nearly 50 years since the introduction of Paris’ law, is still not a reliable calculation of

damage, as Paris himself admits in a recent review.

Article Outlineapproach

generalized

of the functional dependencies of the Paris’ law parameters

between the Paris’ law parameters

complete and incomplete similarity laws

based on Young's modulus

based on the stress ratio or on the maximum stress-intensity factor

and conclusions

Figures

Fig. 2. Dependence of the crack growth rate on .

## An analogy helps only if you let it help

An analogy helps only if you let it help. Indeed, Young's modulus of a metal is insensitive to humidity. But we have a plenty of familiar examples that show drastic changes:

Instead of abandoning Young's modulus and yield strength in these cases, we may choose to study them, or use them with care.

I believe that you and I agree that fatigue is a complex phenomenon, and fracture mechanics does not solve all probelms in fatigue.

## what is "material constant" and a "fundamental law of physics"?

Zhigang

I wish I were one of your students, indeed can I apply to be one? You make your lessons very interesting.

The question now seems to be the two related ones:

1) what is a "material constant", and what is not

2) what is a "fundamental law of physics", and what is not

Fracture thoughness is a material constant, which depends on many factors, and conditions. But Paris' constants are not material constants, and not only because they depend on many conditions, but simply because the "law" which they come from is a purely empirical law, whereas toughness is related to resilience, and this in turn is related to laws of physics!

Turning back to Young's modulus, this can vary, but the variation can be REASONABLY obtained from basic laws of physics, I am sure you can explain all the phenomena above, and model them.

Viceversa, YOU CANNOT explain the variation of C,m Paris' constants from the laws of physics, no way! This suggests many people wasted time trying to do so, and I am suggesting your students to do better than that, in case they are tempted to.

If you could derive Paris law from basic material constants (and there are various formulations see my paper, none of which works in general), then Paris' law would be a law of physics, and Paris' constant would be derived from other basic constants, at least in principle.

In fact, Paris himself in recent years has attempted to do that, with the Hertzberg law (Hertzberg was I beleive a student of his), which surprisingly, removing crack closure, does obtain a generality which so far is unexplained, i.e. m=3.

I have tried to discuss Hertberg's law in my paper, but there is room for improvement.

Maybe somebody has any idea over why Hertzberg law is so general? Room for a phd thesis, perhaps in Harvard.

Otherwise, we risk to be so confused about science as to use laws of thermodynamics to disprove Darwin, like these people:

http://www.darwinismrefuted.com/thermodynamics.html

:)

## Zhigang so you give up? Lack of time, of interest or replies?

Zhigang, please do not give up on this discussion. It was quite interesting, although I understand not easy now.

Perhaps you can see more ideas and an ever earlier background on Paris' law, of which you are even too in love with, here...

The

discussion in my paper with Alberto Carpinteri on Paris' law

2

sec ago

## Paris hypothesis vs. Paris law

Dear Mike: I have not found time to read your papers, but I would like to. Just swamped. From the above exchanges, I believe that you and I agree on the following two points:

The Paris hypothesis. Under the small-scale yielding condition, the extension per cycle, da/dN, depends on mechanical boundary conditions of the specimen through the history of the stress intensity factor, K. This hypothesis, was stated in Paris's original paper. The hypothesis is not data fitting. Rather, it is an application of a fundamental idea of fracture mechanics. One can plot the curve between da/dN and Kmax. (Let's set Kmin = 0 to shorten the discussion). This curve is a material property, independent of the shape of the specimen or the distribution of the load. Like many material properties, this curve will vary with temperature, etc.The Paris law. People some time choose to fit the experimentally determined da/dN - Kmax curve to a power law. So far as we know, this curve fitting has no theoretical basis.I wrote in my notes the following:

"The above equation (the power law in Point 2) is now known as the Paris law. The equation is a rather restrictive expression of Paris’s original hypothesis (Point 1)."

Fitting the da/dN vs. delta K curve to a power law is like fitting stress-strain curve to a power law. The fitting is done for convenience, and should be used with care.

Do we agree, or are you also challenging Point 1?

## Dear Zhigang This curve

Dear Zhigang

This curve is a material property, independent of the shape of the

specimen or the distribution of the load. I honestly dissagre. Try the near threhold area and you will see that depends very much on specimen. Also please do not neglect the importance of T stres. In addition Stage II (paris) delivers scatter. This scatter is larger in FCC rather than BCC systems. Why?

Chris Rodopoulos

## You think you are critical of Paris' law now, but not enough!

I remember seeing a plot made using various codes using Paris law in the best forms, including Nasa etc.

It was there in a draft of my One, no one .... paper, but eventually I omitted it.

However, it is so nice now to explain it.

Yes, I found it! In the original version of the paper, it was later deleted by my co-authors, I wanted to have this interesting result which was never properly published by collegues in Pisa.

However, the size-scale effects do not seem directly adressed by these models,and it remains partly unknown if their consideration of out-of-plane constraint

takes this into account. In general, even Paris himself (Paris et al 1999) doesn’t

seem to be satisfied by the degree of accuracy expected by predictions made

by these models, and particularly on crack closure, he says: “..On the other

hand, with respect to formulating an accumulation of damage model or method,

it has created as many problems as it has solved.” For example, in a recent

survery over existing crack closure models including spectrum loading (Lazzeri,

Pieracci and Salvetti , 1995; Lazzeri and Salvetti , 1996) have compared the

life predictions of various empirical models for an aircraft spectrum under a

flight-by-flight load history at a mean flight load of 75 MPa, with results shown

in Fig.10. It would seem all the models underestimate the life to failure, and

in this sense are conservative, but it does not really appear a good prediction

in any of the ranges, with crack lengths expected to be various mm length

different from the test values even for the apparently most accurate model

(FASTRAN-II, here), despite all the methods are used consistently to their

fitting parameters.

Vasudevan et al. (2001) recollect that there are over 70 different crack initia-

tion models, and more than 40 empirical models proposed for the long crack

growth predictions. They recognize the difficulty using the damage tolerance

approach: “Despite all these developments, current fatigue life prediction meth-

ods stem from several sources: (1) the assumption of plasticity induced crack

closure, (2) the lack of terms in the model that relates to the environmental

effects and slip deformation behavior, and (3) several adjustable parameters

needed to fit the observed data. inadequacies in the prediction methods are

compensated by the use of several adjustable parameters which are correlated

using the component test data. .... Commonly, the inadequacies in the pre-

diction methods are compensated by the use of several adjustable parameters

which are correlated using the component test data. In practice, vehicle safety

is guarded by the use of safety factors in design, the selected use of the com-

ponent data, periodic NDE inspections, the use of statistics to assign data

scatter, material quality control, etc.”

HOW CAN YOU BELEIVE PARIS LAW IF THE BEST OF THE BEST SOFWARES AROUND, INCLUDING ALL POSSSIBLE DEVIATIONS FROM PARIS LAW; MAKE THIS KIND OF INCREDIBLY DIFFERENT PREDICTIONS? ISN'T THIS THE SINGLE BIGGEST MISTAKE IN RESEARCH INTHE LASTE 40 YEARS IN MECHANICS?## Is this spectrum loading?

Is this spectrum loading? Are the prediction based on FE? If yes then I cannot see the point. I can show you many similar.

## Of course, as you can see if you just read the figure!!!

Not only they are spectrum loadings, but it is even written which! Please read before embarking into many comments faster than ligths' speed.

## I am afraid Zhigang, but I am challenging indeed Point 1 !

Zhigang,

as I suspected, you are very acute, and indeed what you call Point 1 is what I call Equation 1 in my paper with Alberto Carpinteri, who today incidentally is the President of the International Congress of Fracture, if I am not wrong. Indeed, Paris firstly proposed da/dN = f (DK), without promising a power law. But you are falling into the big mistake many people made.

In short, you expect that, like "small scale yielding", LEFM implies the Point 1. Unfortunately, this is

notthe case. ;( In fact,you need to add many more assumptions.Your point 1 seems more general than point 2 only superficially. There is no way to prove Point 1 without proving point 2 -- unless you suggest me otherwise! If you plot today data points for f(DK), you can easily see points all over the places, at least for points away from Region I and Region II, and also depending on many other factors.

You are falling in the same error that hundreds of people made when attempted to derive a more "general" equation than Power law. There are probably 5000 or more papers making this naive attempt, so you are ingood company.

For Point 1 to hopefully hold, you need to add that you are in the so-called Region II of propagation, i.e. NOT in Region I, nor in Region II.

Region I is called short crack, Region II is fast propagation.

But this is not enough. You need to have many other dimensionless parameters in appropriate ranges. If they are all in the appropriate range, then you will ALSO see Paris' law, i.e. Point 2.

So, I would not make this distinction. Paris law, if it holds, it is power law. Please do not distinguish the two points. If you cannot read all the Barenblatt books, perhaps try to read my introduction at least... By now I think you understand...

1.

Introduction

More than 40 years ago, Paris

and Erdogan (1963)

suggested using the stress-intensity factor range, , to obtain the rate of crack advance per cycle, , proposing a very general and simple correlation:

(1)

that was considered so revolutionary that received a

strong opposition from the scientific Community (see Paris

et al., 1999).

Actually, two years before, Paris

et al. (1961)

proposed a fatigue crack growth criterion where

was considered to be proportional to . Only in his doctoral thesis, Paris

(1962)

analyzed the experimental data by A.J. McEvily and found an

impressively good power-law fit for some Al-alloy with an exponent which

could not correspond to any of the previous laws. In Paris

and Erdogan (1963)

it was therefore suggested that Eq. (1)

should have a power-law form, with

mas a free parameter (see the Paris’ own recollection in a recent

tribute to Professor A.J. McEvily's contributions, Lados

and Paris, 2007):

"One of the present authors made use of McEvily's data in his doctoral

thesis plotted on a double logarithmic basis to develop an empirical

law of crack growth

These graphs showed that the data correlated reasonably well with other

data

This led to the familiar power law for the crack propagation (, ):

(2)

where

depends on load ratio and 2

is at least in the range of 3–4 for these data."

Actually, the first original “competitor” of his law was the Head's law

(

Eqs. (2) and (4)[(2),

(4)]

in Paris

and Erdogan, 1963), which could in retrospective be put in a

power-law form (2)

with either

or , depending on whether plastic zone in the denominator is

constant as originally suggested by Head, or is considered to be

dependent on Irwin's plastic-zone size, as Paris argued. McEvily's data

on 7075-T6 and on 2024-T3 compared well with a power-law fit with

in the Paris’ original Thesis plots in 1962. But Paris did not propose a

fixed value for

m, because other data analyzed in the secondpaper by Paris

and Erdogan (1963)

fitted power-laws with

or

like in Head's law!

Hence, it was actually Paris’ strong belief on the

use of the stress-intensity factor, which Paris himself recollects

being his idea already in 1957 in a summer internship at NASA ( Paris

et al., 1999), and the McEvily data suggesting

that really led to propose to free up the exponent in the equation.

Curiously, while Paris’ law was perceived clear and strong enough to

lead to the so-called damage tolerance approach (see, e.g. Suresh,

1998), the progress in subsequent years has seen a proliferation of

“generalized laws”, mainly to model the various observed deviations

from the power-law regime. However, while ambitions gradually decreased

to have a single simple law, the enthusiasm had nevertheless been

already pervaded industries and research centers, so Paris’ law

continued to be perceived as a “law” almost with the status of a physics

law, and only few authors, including Barenblatt and Botvina (BB), have

really returned to warn that this is not the case. Today, it is well

known that the power-law holds in an intermediate range of

(region II), where there should be a limited dependence on the material

microstructure, loading ratio and environmental conditions, so that

Cand

mappear in this regime essentially as “material constants”. On the other

hand, it is admitted that a region I exists, where there is a decrease

in the crack growth rate until below a threshold stress-intensity

factor, , long cracks do not propagate anymore. This threshold

significantly depends on the material microstructure, environmental

aspects, as well as on the loading ratio. Similarly, when , another deviation from the Paris’ law regime is observed,

since the Griffith–Irwin crack growth instability is approached.

Actually, the condition

is not completely sufficient to guarantee crack arrest since short

cracks propagate also below this threshold, whereas for very high

nominal stresses, the dependence of

is no longer on

but perhaps on , where

is the EPFM (elasto-plastic fracture mechanics) parameter. More

importantly, the dependence of the phenomenon of fatigue crack growth on

the material microstructure in region II can be relevant, so that some

results established for metals may not be extrapolated to other

materials. From this preliminary introduction, it clearly emerges that

the Paris’ law in its original form holds only within a very limited

range of conditions.

Moreover, while a pure “crack propagation”

approach to damage tolerance is possible to control large enough cracks

under not too severe loading conditions, for long lives or high

frequency loading, it is still not possible to propose inspection

intervals safe enough when cracks are essentially in the initiation

phase for most of their life. Hence, it is difficult to account for all

the deviations from the Paris’ simple regime, and no computational model

is entirely satisfactory today, even in the opinion of Paris himself ( Paris

et al., 1999). In parallel to the damage tolerance approach,

research is still active on “damage tolerance in HCF” (high cycle

fatigue) where most of the design approach is based on threshold and

fatigue limits, returning in part to the original SN curves “empirical”

approach, and not using Paris’ type of laws (see for example a recent US

Air Force initiative in the excellent reviews by

Nicholas, 1999,[2006

Nicholas, 1999,

Nicholas, 2006]).

It is clear that the observation of the strong

power-law nature of crack propagation, originally recognized by pure

observation and great intuition, comes from the underlying

self-

similarityof crack propagation connected to the self-similarity of the crack

geometry, when the crack length

is larger than the microstructural dimensions, yet smaller than any

other dimensions. However, this process is not a

singleprocess, but a series of potentially different processes depending on

several potential dimensionless parameters, length scales, and the crack

length itself. This explains partial success of the earliest attempt to

generalize Paris’ law considered different material or fitting

constants in addition to the nominal load and the crack length. To cite a

few, we mention the models based on perfect plasticity mechanism, those

considering damage ahead of the crack tip in terms of low cycle fatigue

using the Coffin–Manson relationships, as well as the models taking

into account the crack tip cyclic stresses and strains, or using Miner's

law to analyze the effect of the increasing amplitude of loading while a

material point approaches the propagating crack tip (see e.g.

Glinka,1982; Kaisand and Mowbray, 1979; Majumdar and Morrow, 1969; Weiss, 1968

[

Glinka, 1982,

Kaisand and Mowbray, 1979,

Majumdar and Morrow, 1969,

Weiss, 1968]).

Barenblatt

and Botvina (1980)

(BB in the following, see also

Barenblatt, 1996, 2006[Barenblatt, 1996,

Barenblatt, 2006]), considered the problem of scaling of fatigue

crack growth in the general context of scaling processes and other

authors have more recently re-examined the idea (see e.g.

Ritchie,[2005; Spagnoli, 2005; Carpinteri and Paggi, 2007

Carpinteri and Paggi, 2007,

Ritchie, 2005,

Spagnoli, 2005]). BB in particular noticed that

completesimilarity

would imply

in Eq. (2),

which is not observed if not as nearly a limit case. They introduced

the concept of

incomplete similarity, analyzing the dependence ofthe Paris’ law parameter

mon the dimensionless number , where

is the tensile strength,

is the fracture toughness and

his the specimen thickness. They were very careful to “provisionally”

suggest a certain special form of the dependence for metals: namely,

that

mshould be constant for

Zless than about unity and then linearly increase with

Z.Following Ritchie

and Knott (1973), BB proposed a possible interpretation by

observing that large specimens imply more “static” modes of failure, as

it is well known that constraint at the crack tip is only highest for

large enough width and thickness of the specimen (see also the

prescription in ASTM E399-90, (2002) 3

for toughness measurement, as further remarked by Ritchie,

2005).

Ritchie

(2005)

also made interesting further comparisons of data points using this

approach. However, Ritchie's plot seems to imply a much wider scatter

than the original plots in the BB's paper, which of course were for

different materials, and this suggested us a generalization of the BB's

approach to look for more general dependencies on dimensionless

quantities, and also analyzing the constant

Crather than just

m. We also consider not only the original datapoints of the BB's and Ritchie's papers, but also other fatigue data for

concrete obtained by Bažant

and Shell (1993)

and Bažant

and Xu (1991)

and recently reexamined by Spagnoli

(2005).

In the present paper, therefore, by revisiting and

generalizing the BB's dimensional analysis approach, we will show an

anomalous relationship between

mand

Zin concrete as compared to metals. A novel interpretation is made by

noting that the slope of the linear

mvs.

Zrelationship is strongly correlated to another dimensionless parameters

(apart from

R, as well known), namely the ratio between theelastic modulus and the material tensile strength, . Moreover, by looking at the corresponding relationship

between

Cand

Z, we find an inverse relationship between these twoparameters, which can be mathematically treated according to either

complete or incomplete self-similarity, depending on the material being

considered. Eliminating

Zbetween the two relationships, one between

mand

Zand the other between

Cand

Z, a correlation betweenCand

mis found. As far as metals are concerned, we will show that such a

relationship is very close to the correlations proposed in the past by

several authors on empirical basis. On the contrary, we will demonstrate

that concrete behaves quite differently from metals, emphasizing the

important role played by the material microstructure even in the Paris’

law regime (region II), leading to incomplete self-similarity in

Z.As suggested by BB, Paris’ equation can be applied

only within certain ranges of variations of the dimensionless

parameters governing the problem, and we give some further hints in

particular on more parameters than those considered by BB. The drawback

in not recognizing these aspects is therefore the dangerous risk of

extrapolation.This is even more evident when considering that also the initiation

process (or processes) leads to fatigue power-laws, such as those by

Basquin and by Coffin–Manson, which evidently are perceived as more

empirical, but actually depend on other dimensionless numbers, as

recently shown in the context of scaling phenomena (

Brechet et al.,[1992; Chan, 1993

Brechet et al., 1992,

Chan, 1993]). Certainly, a challenging task will be to interpret

all these power-laws within a unified framework, comprising also the

well-known Hall–Petch relationship, which relates the material yield

strength to microstructural quantities.

## We disagree on the Paris hypothesis then.

Now we have made good progress. We agree on Point 2.

But we disagree on Point 1. What do you mean by a short crack? Short compared to what?

## A few basic questions

Hi Mike,

I am following this post. There are a few confusion in reading, which I thought were basic to me. It would be wonderful if you can clarify them in a simple way.

I understand fatigue is complex in general, and a number of uncertainties remain today. Complications exist especially in regime I (K_th regime or crack growth initiation), and regime III (K_c or fast crack propogation), I belive many models and considerations have been devoted to these two regimes. Also other factors, like loading history including overloading and varied amplitude loading, large-plastic zone (compared with crack size), crack closure, etc., introduce additional complications. However, it seems (to me) the fatigue in regime II (steady crack growth) under small-scale yielding conditions is well-establised. So now let's focus our eye on regime II.

In reading your comments, here are my questions:

1. Do you agree that under small-scale yielding condition, the crack tip is uniquely defined by Kmax (Kmin=0), and the crack growth rate da/dN=function(Kmax)? Let's assume the loading amplitude is constant for simplicity. What do you mean "In fact,

you need to add many more assumptions" explicitly?2. As pointed out by Zhigang, the fundamental idea of da/dN=f(Kmax) (maybe not?) and Pair's data fitting are separate. I agree there is no clear physical understanding of Paris's law, it works for many materials however. So could you explain why "There is no way to prove Point 1 without proving point 2"?

Thanks a lot

Kejie

## thanks for your questions. Here I reply

Kejie

I agree my paper with Paggi and Carpinteri is too long to be digested at once.

It is the first paper which really contains so many different aspects of Paris' law, and I am glad that this discussion came along, since otherwise it would have been under the dust for perhaps 20 years.

It was my major effort during a sabbatical leave at Ecole Polytechnique, and Marco Paggi did a lot too, together with Prof. Carpinteri, whom you certainly know is one of the leading experts of Fracture (not by change, the President of ICF), especially in static one.

Now, regarding your questions, they are answered in one part of my paper which I have already attached. But here I separate even further the parts.

1. Do you agree that under small-scale yielding condition, the crack tip

is uniquely defined by Kmax (Kmin=0), and the crack growth rate

da/dN=function(Kmax)? Let's assume the loading amplitude is constant for

simplicity. What do you mean "In fact,

you need to add many" explicitly?more assumptions

Small scale yielding is not obvious to define. I did not include that in my list, which was just that we were in regime II. So now, you are adding that. OK. This corresponds to my part of the paper here:

Notice that our choice of dimensionless ratios is not unique, and more distinctions and microstructural length scales

would be needed to include all the possible categories of crack in well-known classifications (

Suresh and Ritchie, 1984; Ritchie[Miller, 1999, Ritchie and Lankford, 1986, Suresh and Ritchie, 1984]). For example, if we choose instead of the ratio (or, equivalently, we consider the ratio between two of our dimensionless ratios), we would have another form of transition, where the Irwin parameter is no longer very useful, and EPFM should be used, introducing the -integral in the crack growth equation. Notice that this can occur both for long and for short cracks (in the sense of ), and indeed it is better to talk of physically orand

Lankford, 1986; Miller, 1999

mechanically small cracks ( Miller, 1999). Obviously, very large nominal loads are required to have a very large plastic-zone size at the tip (close to the full-yielding condition) when the crack size is in its turn also of the order of . The final case is when cracks are

microscopically(microstructurally small) for which continuum mechanics breaks down andshort

microstructural fracture mechanicsis needed (

Hobson et al., 1986; Navarro and de los Rios, 1988[Hobson et al., 1986, Navarro and de los Rios, 1988]); this is perhaps the most complex category, since crack deceleration or self-arrest is dependent on the grains size and orientations, and possible decelerations or “minima” in and multiple small-crack curves can be found ( Ritchie and Lankford, 1986).

However, you are probably familiar to Metals only,as was Paul Paris. If you look at concrete, you find unexpected results. (and not only in concrete, just wait...)

As regards the scaling laws for

Cin concrete, it is interesting to note that Carpinteri and Spagnoli (2004)considered incomplete self-similarity in to prove that

Cis structural-size dependent. Afterwards, Spagnoli(2005) reinterpreted the same fatigue data in terms of incomplete self-similarity in , simply noting that the initial crack length was

proportional to the structural size in the tested specimens (geometric similarity). However, this is rigorously true if and only if the

situation (3) occurs. In such a case, we have either or, equivalently, .

In other words, the Paris constant C is NOT independent on size of the crack. That is VERY UNFORTUNATE INDEED FOR A PARIS LAW!

Another example of anomalous scaling is represented by the fatigue crack growth representation by Frost (1966)

(16)

where is the initial crack length and is a material constant. Differentiating this law with respect to

N, we see that this parametric representation of fatigue has and , i.e. . Yet, this is still used apparently by theAustralian Air

Force ( Molent et al., 2005) with some success.

So here you see that some people use something different from Paris, and yet satisfactorily --- the Australian Air force is not the last of the air forces...

The anomalous crack-size dependence for the parameter

Chas also important consequences for the scaling of the fatiguethreshold ( Paggi and Carpinteri, 2008). In fact, if we determine the value of by inverting the Paris’ law in correspondence of a conventional value of the crack growth rate, , then we have

(17)

This implies that, since , we find . For and , we have , which corresponds to the scaling law for proposed by Frost (1966) and Murakami and Endo (1986).

Again, anomalous scaling. Or maybe we should not say anomalous?

Last, but not least....

Notice that another form of anomalous scaling of

Cis due to microstructure, which we have not included in the presenttreatment. Indeed, Chan (1995) derived a crack growth equation which depends on the dimensionless

number

(18)

which is derived from using the Coffin–Manson equation (the plastic term only, hence the appearing of the parameter

and the exponent

b) with a strain range derived from the crack-tip opening displacement (CTOD) and the dislocation barrierspacing

d, and assuming the propagation is for the dislocation cell element of sizes, also giving the striation spacing and amore precise basis to the similar but more empirical approach in Glinka (1982), Kaisand and Mowbray (1979), Majumdar

and Morrow (1969) and Weiss (1968). The resulting crack growth equation is

(19)

which, for

as often approximately observed in many metals, leads to and the equations derived by Rice-Weerman and Mura (see also Chan,

1995). Chan (1995) argues that the reason why this law is not observed in most cases—from which the common belief that

Cshould not depend on microstructure in region II of propagation—is due to the fact that, for decreasing dislocation barrier spacing, yieldstress and fatigue ductility usually increase, so that only spans a limited range and its dependence is not seen. This is,

however, not always the case, as proved by Chan (1995) for special types of steels (HSLA, high-strength low-alloy steels).

So here there is an explanation on why perhaps Paris' is seen to work in most metals!Simply, we do not see the correct dependence... How lucky, or unlucky ??? But the day we work with special steels, Paris doesn't work.Are we prepared to this kind of problems? Another day we will work with hydrogels, and we will write a big paper on JMPS

Paris' law does not hold for hydrogel!Well, that will be perhaps accepted, since everyone beleive Paris is a law of physics. For me, I would ask, as reviewer, to accept the paper, but change the title into

Paris' law, being not a law, does not hold for hydrogel, and we are not that surprised!How do you think I explained all this? does it answer also your question 2?

2. As pointed out by Zhigang, the fundamental idea of da/dN=f(Kmax)

(maybe not?) and Pair's data fitting are separate. I agree there is no

clear physical understanding of Paris's law, it works for many materials

however. So could you explain why "There is no way to prove Point 1

without proving point 2"?

If there is a dependence of C (anomalous, in the spirit of assuming Paris as a law), then we cannot write da/dN =f(Kmax). Do you agree? Unless we write C=g(Kmax) which is not the case, as for example in high strength steels, we saw

C == g( ) and this is not what you like. Concrete, even another dependence. Sorry!

## Reply

Hi Mike,

Thanks for the reply. Here are several points I am thinking

1. It's very risky after you abandon the small scale yielding condition, it's easy to convolute every single complication for different materials together. At this point I think no fatigue model can capture different behaviors for every material, Pari's law can not, others can not either. Some models are better comparably, because they work well in some range and easy to use.

2. Aside from many complications in crack growth initiation regime and fast propogation regime, I agree in regime II there are still many uncertainties. A few examples as you already pointed out, i.e., the grain size dependence, yes, if you introduce grain size, you essentially introduce another length scale, this is not considered in Pari's law. For "microstructurally small" fracture where continuum mechanics does not even work, yes, the intrinsic length scale must be introduced. For large plasticity zone, we might need to consider the J integral or dislocation emission, etc.... Beyond these details, I think Paris's classical work is suitable for education purpose, especially for the course which is not devoted to the fatigue subject. This won't affect our learning and further use with care, just as we learned stress-strain relation for metals, it won't limit us in studying the stress-strain for hydrogel which behaves totally differently.

3. Honestly saying, I dont think your guidance in reading the paper is helpful. Several reasons, (1), the formatting is not readable because some formulas and symbols are lost during copy. (2). More important, while the text should be long to be careful in the paper, the essential idea must be simple and short. I think communication of the idea would be much more helpful. (3). It's hard to ask a person to read a full paper by spending an afternoon, who is not immersing in the subject. But many people (including me) are more interested in the basic ideas in the paper, after appealing their interest they will read the paper.

Hope it will process this discussion a little bit.

Kejie

## I don't see the point. Make some questions and I reply please.

As you have hard time reading my paper, now I have hard time reading your "thinking". Please go to the point with some precise question. MC

## Small scale yielding condition

Mike,

I do not understand your comment that the small scale yielding condition is hard to define. What do you mean by this? What are the difficulties?

Thanks,

Matt

## Matt, sorry I missed this question earlier

Matt

thanks for your question. What I meant by "small scale yielding" being difficult to define is that you are probably thinking of applying the classical ideas of static fracture mechanics, where the material properties of interest are much less than in fatigue.

Small scale yielding by definition is used to define the case when the size of plastic region is small with respect to other dimensions of the specimen including the crack. When the conditions are not met, one should use Elasto-Plastic Fracture Mechanics.

But here, in the process of fatigue crack propagation, one can define certainly a small scale yielding condition and a large scale yielding condition, but that is hardly useful in improving Paris' law. In fact, there are many more deviations from Paris' regime than just deviations from small scale yield. This is because of the many more "constants", and the fact that Paris is purely empirical fitting equation.

Not surprisingly, possible

categories of cracks are many and disputed, see the well-known classifications (

Suresh and[Miller,Ritchie, 1984; Ritchie

and Lankford, 1986; Miller, 1999

1999, Ritchie

and Lankford, 1986,

Suresh and Ritchie, 1984]). Small scale is replaced by "physically" or

"mechanically small" cracks ( Miller,

1999). "

microscopically short(microstructurally small) for which continuum mechanicsbreaks down and

microstructural fracture mechanicsis needed (Hobson et al., 1986; Navarro and de los Rios, 1988[Hobsonet al., 1986, Navarro

and de los Rios, 1988]); this is perhaps the most complex category,

since crack deceleration or self-arrest is dependent on the grains size

and orientations, and possible decelerations or “minima” in and multiple small-crack curves can be found ( Ritchie

and Lankford, 1986).

So as you see the matter is confused!

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## Ok, now you can jump to paragraph 3 of my paper ;(

Zhigang, now since you have no time, you can skip perhaps par.2 of my paper (although it would be useful), and go to par.3, where short crack is defined rigorously as well as empirically. If you have even less time, try to interpret this sentence ----- In general, depending on the values assumed by

and , the following situations may occur. And see situation 2. But really, at this point wouldn't be easier to read

allof my paper?Mike

3.

BB's generalized

According to dimensional analysis, the physical

phenomenon under observation can be regarded as a

black boxconnecting the external variables (called input or governing

parameters) with the mechanical response (output parameters). In case of

fatigue crack growth in region II, we assume that the mechanical

response of the system is fully represented by the crack growth rate, , which is the parameter to be determined. This output

parameter is a function of a number of variables:

(7)

where

are quantities with independent physical dimensions, i.e. none of these

quantities has a dimension that can be represented in terms of a

product of powers of the dimensions of the remaining quantities.

Parameters

are such that their dimensions can be expressed as products of powers

of the dimensions of the parameters . Finally, parameters

are dimensionless quantities.

As regards the phenomenon of fatigue crack growth,

it is possible to consider the following functional dependence, by

extending a little the BB choice to include other fatigue material

constants but still omitting for simplicity other possible choices, such

as the Coffin–Manson constants ,

(fatigue strength and ductility factors, and the corresponding

coefficients

band

c) as well as any microstructural length scale, such asthose related to dislocations or to the grain size:

(8)

where the governing variables are summarized in Table

1,

along with their physical dimensions expressed in the length-force-time

(LFT) class.

Table 1: Governing variables of the fatigue crack growth phenomenon

VariableDefinitionSymbolDimensions

Tensile yield stress of the material

Material fracture toughness

Frequency of the loading cycle

T

Stress-intensity range

Threshold stress-intensity factor

Fatigue limit

Elastic modulus

ECharacteristic structural size

hL

Initial crack length

aL

Loading ratio

–

From this list it is possible to distinguish

between three main categories of parameters. The first category regards

the static and cyclic material properties, such as the yield stress, , the fracture toughness, , the threshold stress-intensity factor range, , the fatigue limit, , and Young's modulus,

E. The second categorycomprises the variables governing the testing conditions, such as the

stress-intensity factor range, , the loading ratio,

R, and the frequency of theloading cycle, . Finally, the last category includes geometric parameters

related to the tested geometry, such as the characteristic structural

size,

h, and the initial crack length,a.Considering a state with no explicit time

dependence and assuming

and

as independent variables, then Buckingham's

Theorem gives

(9)

where the dimensionless parameters are

It has to be noticed that

takes into account the effect of the specimen size and it corresponds

to the square of the dimensionless number

Zdefined by Barenblatt

and Botvina (1980), and to the inverse of the square of the

brittlenessnumber s

introduced in

Carpinteri (1981a, b, 1982, 1983, 1994)[Carpinteri, 1981a,

Carpinteri, 1981b,

Carpinteri, 1982,

Carpinteri, 1983,

Carpinteri, 1994]. Since the plastic-zone size, , scales with

according to Irwin, it follows that . Therefore, this dimensionless parameter rules the

transition from small-scale yielding, when , to large-scale yielding, when

(see also Ritchie,

2005).

The parameter

is responsible for the dependence of the fatigue phenomenon on the

initial crack length, as recently pointed out by Spagnoli

(2005). In fact, if we introduce the El

Haddad et al. (1979)

length scale:

(10)

then it follows that

and we can define a dimensionless number

which is analogous to

Zand governs the transition from short-cracks, when , to long-cracks, when . Here it has to be remarked that in general the El Haddad

length scale

is also a function of the loading ratio.

At this point, we want to see if the number of the

quantities involved in the relationship (9)

can be reduced further from five. For example, starting from , this parameter can be considered as non-essential when,

for very large or very small values of the corresponding dimensionless

parameter , a finite non-zero limit of the function

exists:

(11)

In this case we speak about

complete self-

similarity, orself-

similarity of the first kind( Barenblatt,

1996), in the parameter . On the other hand, if the limit of the function

tends to zero or infinity, the quantity

remains essential no matter how small or large it becomes. However, in

some cases, the limit of the function

tends to zero or infinity, but the function

has a power-type asymptotic representation:

(12)

where the exponent

and, consequently, the dimensionless parameter , cannot be determined from considerations of dimensional

analysis alone. Moreover, the exponent

may depend on the dimensionless parameters . In such cases, we speak about

incomplete similarity,or

self-

similarity of the second kindin the parameter

( Barenblatt,

1996). It is remarkable to notice that the parameter

can only be obtained either from a best-fitting procedure on

experimental results or according to numerical simulations.

As regards the parameter , the corresponding dimensionless parameter

is usually small in region II of fatigue crack growth. However, since

it is well known that the fatigue crack growth phenomenon is strongly

dependent on this variable, a complete self-similarity in

cannot be accepted. Hence, assuming an incomplete self-similarity in , we have

(13)

where

may depend on .

Repeating this reasoning for the parameters ,

and , we find the following generalized representation:

(14)

where, again, the exponents

may depend on . Comparing Eq. (14)

with the expression of the Paris’ law, we find that our proposed

formulation encompasses the classical Paris’ equation as a limit case

when the Paris’ law parameters

and

are given by

(15a)

(15b)

As a consequence, from Eq. (15b)

it is possible to note that the parameter

Cis dependent on two material parameters, such as the fracture

toughness, , and the yield stress, , as well as on the loading ratio,

R, and on thelength scales

hand

a.Hence, the phenomenon of fatigue crack growth

presents different length scales, i.e. the specimen size,

h, theplastic-zone size, , and the transition crack length corresponding to the

breakdown of LEFM concepts and to the activation of short crack effects,

. The crack length interacts with such length scales and the

fatigue response is influenced by them in the different stages of crack

growth.

In general, depending on the values assumed by

and , the following situations may occur.

(1)

Incomplete self-similarity only in

: this may happen when

is neither too small nor too large, i.e. the size of the process zone

is comparable with the structural size and therefore we have a

transition from small-scale to large-scale yielding. On the other hand,

the crack length is long enough such that a long-crack regime can be

considered. In this case, the scaling law in Eq. (15b)

gives , i.e. a Paris’ law parameter dependent on the structural

size-scale.

(2)

Incomplete self-similarity only in

: this may occur when the crack length is comparable with the El Haddad

transition crack length, . This usually occurs in the short-crack regime. In this

case, the scaling law in Eq. (15b)

gives , i.e. a Paris’ law parameter dependent on the initial crack

length.

(3)

Incomplete self-similarity both in

and in

: this is an intermediate situation where both the transitional sizes

are comparable. Therefore, in this situation, both microstructural and

structural aspects could affect the fatigue response.

In all of these situations, the parameter

can depend on

and .

Notice that our choice of dimensionless ratios is

not unique, and more distinctions and microstructural length scales

would be needed to include all the possible categories of crack in

well-known classifications (

Suresh and Ritchie, 1984; Ritchie and[Lankford, 1986; Miller, 1999

Miller, 1999,

Ritchie and Lankford, 1986,

Suresh and Ritchie, 1984]). For example, if we choose instead of

the ratio

(or, equivalently, we consider the ratio between two of our

dimensionless ratios), we would have another form of transition, where

the Irwin parameter is no longer very useful, and EPFM should be used,

introducing the

-integral in the crack growth equation. Notice that this can occur both

for long and for short cracks (in the sense of ), and indeed it is better to talk of physically or

mechanically small cracks ( Miller,

1999). Obviously, very large nominal loads are required to have a

very large plastic-zone size at the tip (close to the full-yielding

condition) when the crack size is in its turn also of the order of . The final case is when cracks are

microscopically short(microstructurally small) for which continuum mechanics breaks down and

microstructural fracture mechanicsis needed (

Hobson et al., 1986; Navarro and de los Rios, 1988[Hobson et al., 1986,

Navarro and de los Rios, 1988]); this is perhaps the most complex

category, since crack deceleration or self-arrest is dependent on the

grains size and orientations, and possible decelerations or “minima” in

and multiple small-crack curves can be found ( Ritchie

and Lankford, 1986).

As regards the scaling laws for

Cin concrete, it is interesting to note that Carpinteri

and Spagnoli (2004)

considered incomplete self-similarity in

to prove that

Cis structural-size dependent. Afterwards, Spagnoli

(2005)

reinterpreted the same fatigue data in terms of incomplete

self-similarity in , simply noting that the initial crack length was

proportional to the structural size in the tested specimens (geometric

similarity). However, this is rigorously true if and only if the

situation (3) occurs. In such a case, we have either

or, equivalently, .

Another example of anomalous scaling is

represented by the fatigue crack growth representation by Frost

(1966)

:

(16)

where

is the initial crack length and

is a material constant. Differentiating this law with respect to

N,we see that this parametric representation of fatigue has

and , i.e. . Yet, this is still used apparently by the Australian Air

Force ( Molent

et al., 2005) with some success.

The anomalous crack-size dependence for the

parameter

Chas also important consequences for the scaling of the fatigue

threshold ( Paggi

and Carpinteri, 2008). In fact, if we determine the value of

by inverting the Paris’ law in correspondence of a conventional value

of the crack growth rate, , then we have

(17)

This implies that, since , we find . For

and , we have , which corresponds to the scaling law for

proposed by Frost

(1966)

and Murakami

and Endo (1986).

Notice that another form of anomalous scaling of

Cis due to microstructure, which we have not included in the present

treatment. Indeed, Chan

(1995)

derived a crack growth equation which depends on the dimensionless

number

(18)

which is derived from using the Coffin–Manson

equation (the plastic term only, hence the appearing of the parameter

and the exponent

b) with a strain range derived from thecrack-tip opening displacement (CTOD) and the dislocation barrier

spacing

d, and assuming the propagation is for the dislocationcell element of size

s, also giving the striation spacing and amore precise basis to the similar but more empirical approach in Glinka

(1982), Kaisand

and Mowbray (1979), Majumdar

and Morrow (1969)

and Weiss

(1968). The resulting crack growth equation is

(19)

which, for

as often approximately observed in many metals, leads to

and the equations derived by Rice-Weerman and Mura (see also Chan,

1995). Chan

(1995)

argues that the reason why this law is not observed in most cases—from

which the common belief that

Cshould not depend on microstructure in region II of propagation—is due

to the fact that, for decreasing dislocation barrier spacing, yield

stress and fatigue ductility usually increase, so that

only spans a limited range and its dependence is not seen. This is,

however, not always the case, as proved by Chan

(1995)

for special types of steels (HSLA, high-strength low-alloy steels).

## Once again, What do you mean by a short crack?

Dear Mike: Please give an explicit answer to the question I raised above:

"What do you mean by a short crack? Short compared to what?"

## OK. you need fatigue threshold and fatigue limit

You need to compare to a factor of the order

a0 = (DK_th / D_sigma_lim) ^2

which some people call the El Haddad Topper "intrinsic crack" because it doesn't affect the effective fatigue limit of an "uncracked material".

In other words, if you consider that a material has to fulfill both the classical fatigue limit condition (based on stress) and the fatigue threshold limit (a limit to the range of stress intensity factor), you can fullfill both by considering the "asymptotic matching equation"

DK < DSigma Sqrt (a + a0)

This is the basis of the "interaction diagram" also called Kitagawa- Takahashi.

See e.g. for human dentin a nice paper by Rob Ritchie (the former supervisor of Subra Suresh)

http://www.lbl.gov/ritchie/Library/PDF/kitagawa_takahashi_human_dentin.pdf

Any clearer now?

## Is a0 in your previous comment the plastic zone size?

I'm not sure of all the terms used in your first equation above. a0 looks like a plastic zone zise. If a0 is indeed the plastic zone size, and if a short crack means a crack shorter than a0, then the small-scale yielding condition is violated.

Is a0 in your previous comment the plastic zone size associated with K_th?

## Well, most likely not !

Zhigang, a0 as you can see from the definition, is NOT associated to any measure of yield strenght, and so at such it cannot be even loosely associated to a plastic zone.

It would be a (cyclic) plastic zone size

at the stress range level of the fatigue limit--- if the fatigue limit of the material were coincident,by chance,with the cyclic yield strenght. But this is not generally the case, since fatigue limit is generally not related to cyclic yield.For all this, the best introduction in a single paper is the review paper:

Overview no. 112: The cyclic properties of

engineering materials

Acta Metallurgica et Materialia,Volume 42, Issue 2,February 1994,Pages 365-381N.A. Fleck, K.J.

Kang, M.F. Ashby

Abstract

The basic fatigue

properties of materials (endurance limit, fatigue

threshold and Paris law constants) are surveyed, inter-related and

compared with static properties such as yield strength and modulus. The

properties are presented in the form of

Material Property Charts.The charts identify fundamental relationships between properties and,

when combined with

performance indices(which capture theperformance-limiting grouping of material properties) provide a

systematic basis for the optimal selection of materials in fatigue-limited

design.

from which I have even extracted a figure in my paper on Paris' constant m. In the original Fleck and Ashby paper, you can find the Ashby plot of the a0 constant, so that you can estimate it for many materials.

## Short Crack

Dear all

please do not give dimensional nature to short cracking. For me short cracking represents the unsteady or non uniform exchange of energies between the creation of a new surface and that for the development of plasticity.To make that clear think that BCC metals have a low tendency to short cracking compared to FCC. Also large component will rarely show short cracking. Small components will do. Short crcking can be found close to notches, holes, geometrical complexities, etc. A large flat panel loaded at low stress level will not do. Also short cracking dissapears with stress ratio. I reckon this last phenomenon should provide an answer to the Paris hypothesis.

Chris Rodopoulos

## Well, the word itself short seems to suggest a length scale !

Please explain better. Perhaps you are thinking of an energy term, which however you need to include or divide by another term, resulting in a length scale. If a crack is short or not, it must be a length !! ;)

## But perhaps easier to understand is par.4 with FIGURES!!

Since one figure is much better than 100 equations (and 1 video is much better than 100 figures, which means we should move papers and imechanica ONLY to videos...), you could also see fig.1 and fig.2 of par.4 for easy understanding.

4.

Analysis of the functional dependencies of the Paris’ law parameters

The original data of the Paris’ law in Paris

et al. (1961)

and Paris

and Erdogan (1963)

only showed the intermediate range of

vs.

curve in a bi-logarithmic diagram, where

mand

Cwere sufficient to characterize the whole curve. However, immediately

afterwards, it was recognized that the slope is changing when

is in the near-threshold region (region I) or in the rapid-crack

propagation region (region III), as experimentally evidenced by

Radhakrishnan[(1979, 1980)

Radhakrishnan, 1979,

Radhakrishnan, 1980]. This well-known result can be reinterpreted

in the framework of BB's dimensional analysis stating that

mis dependent on . More specifically, the parameter

mtends to infinity when either

or when . This trend is shown in Fig.

1, where the effective slope of a typical fatigue crack growth

curve is computed as a function of . Clearly, the Paris’ law applies only in region II, where

mis approximately constant.

Fig. 1:

-dependence of the Paris’ law parameter

m. (a) A typicalcurve for steel (, , ). (b) Effective Paris’ slope

mvs.

computed from (a).

The slope

mof the

vs.

relationship in a bi-logarithmic plane is also dependent on the

dimensionless parameter . The crack growth rate depends on the crack length regime

in region I, as schematically shown in Fig. 2.

Short cracks are characterized by

and the

vs.

curve may have a negative slope in region I. On the contrary, the

classical positive slope

mis found for long cracks having .

Fig. 2: Dependence of the crack growth rate on .

The main point raised by BB was, however, that,

because of incomplete similarity, the Paris’ law parameter

mmay depend on , which corresponds to the square of the brittleness number

Z.Analyzing aluminium alloys, 4340 steel and low-carbon steels, Barenblatt

and Botvina (1980)

firstly found that

mis a linear function of

Z, being the slope of such arelationship different from a material to another. For very low values

of

Z, they found thatmturns out to be almost constant and independent of

Z. To explainsuch a trend, Barenblatt

and Botvina (1980)

supposed that the relationship between

mand

Zhas three regimes:

for small

Z,mlinear with

Zfor

and again

for large

Z.The BB's data concerning aluminium alloys, 4340

steel and low-carbon steels are herein reanalyzed in Fig.

3, along with the data for ASTM steels and for normal and high

strength concretes. All the data refer to a loading ratio . As can be seen, the slope of the linear relationship

between

mand

Zprogressively decreases from aluminium alloys to steels. For low-carbon

steels,

mbecomes nearly independent of

Zand the slope becomes negative valued for normal and high strength

concretes. Clearly, the BB's interpretation of the slope variability is

not consistent with the analyzed data. In fact, although the range of

variation of

Zis almost the same for high strength concrete, 4340 steel and ASTM

steels, their slopes are significantly different.

Fig. 3:

Z-dependence of the Paris’ law parameter

m. (a) Aluminium alloys (Yarema[and Ostash, 1975; Ostash et al., 1977

Ostash et al., 1977,

Yarema and Ostash, 1975]). (b) 4340 steel ( Heiser

and Mortimer, 1972). (c) ASTM steels ( Clark

and Wessel, 1970). (d) Low carbon steel (

Ritchie and Knott,[1974; Ritchie et al., 1975

Ritchie and Knott, 1974,

Ritchie et al., 1975]). (e) High strength concrete (data from Bažant

and Shell, 1993

reinterpreted by Spagnoli,

2005). (f) Normal strength concrete (data from Bažant

and Xu, 1991

reinterpreted by Spagnoli,

2005).

## Fatigue

Dear All

It is not in my nature to promote my papers but please read

C. A. Rodopoulos (2006) Predicting the Evolution of Fatigue Damage Using the Fatigue Damage Map Method, Theor.Appl.Fract.Mech., 45, 252-265.

C.A. Rodopoulos and G. Chliveros (2008) Fatigue damage in polycrystals – Part 1: The numbers two and three, Theor.Appl.Fract.Mech., 49(1) 61-67.

C.A. Rodopoulos and G. Chliveros (2008) Fatigue damage in polycrystals – Part 2: Intrinsic scatter of fatigue life, Theor.Appl.Fract.Mech., 49(1) 77-97.

Maybe I can help with the discussion. I have to tell you that they are not very simple.

## chris you can easily post the links with better formatting

image

Please provide us some direct links, perhaps the essential figure, and the abstract. Nobody here reads full papers unless forced to. So, like I did with my papers, you need to guide us.

For example, why should I read it? I am convinced details do not matter, and it is all about interaction and asymptotic matchings....

## I am terribly sorry but I

I am terribly sorry but I cannot do that. Otherwise we should published figures only. Details do matter. As a matter of fact every word we put in an academic text matters.

## ok, discuss your details, but carefully. I will read, provided..

... provided you guide me kindly as I guided Zhigang who didn't have time to read a single paper of mine.

So let's start.

Why should I read your paper? Which one first? Which part I can start from? Where there is contrast to what I am saying? If short crack become instantly long crack, this maybe because the length is CHANGING, not because the problem is not dominated by a length anyway!!!!

## Yes you need to help us, as none of us has much time !

I guess Chris you need to illustrate the basics here, as I do usually in my posts, with a figure or so. Otherwise nobody will these days make the effort to even try to retreive these papers. These days if you don't provide actual links, you cannot expect people to go to library or even to search for these papers --- and after that, you should not be surprised if these papers have so few citations... So it is a good chance to indeed make them more popular, as probably there is good stuff which we could merge with my own papers and those of others...

## Short Cracking

Mike

short cracking is a phenomenon not a size case. If it's size why dissapears at R>0.5 in 2024 alloy and many other materials. To prevent wrongful answers plasticity induced closure is also a phenomenon not always present.

Chris

## I suspect you have studied only Al2024. Is that general?

Since your approach is quite difficult, you have studied only one material. Now, my approach is rather more general, although probably more approximate. It seems we don't have a single material where we can compare. Do we?

## Mike I have checked

Mike

I have checked more than 400 materials for a Aircraft Manufacturer.

## Start from Part I and II.

Start from Part I and II.

## Chris you are quite optimist if you think people respond ...

To " orders " like that! Maybe a direct phd student of yours could.... but others... unlikely! So we are back to square one, your papers not cited by more than few people, my paper not reviewed, and Paris' "equation" still taken by mistake as a "law"...

I have a terrible suspect.... don't tell me you were the reviewer !!!!!! ;)

## Mike it was not an

Mike

it was not an order. of course you can do what ever you want. At the end of the day you are not listening to anybody and neither you care. I stop the discussion here. There is no point. You are using a scientific bog to create private debates.

I cannot see the point. Forgive me I am not very smart.

## I am not sure what you mean. But I understand you give up.

I am quite disappointed by your decision. First you raised questions, now nobody is telling you that you are not smart. I don't follow.

## fatigue, in abaqus

Hello

I need the steps to simulate fatigue under abaqus (2D)

thank you.