Fatigue

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.

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 

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

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...  :)

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 Outline

1. Introduction

2. BB's
approach

3. BB's
generalized

4. Analysis
of the functional dependencies of the Paris’ law parameters

5. Correlations
between the Paris’ law parameters

6. Other
complete and incomplete similarity laws

6.1. Representation
based on Young's modulus

6.2. Representation
based on the stress ratio or on the maximum stress-intensity factor

7. Discussion
and conclusions

Acknowledgements

References

Figures

 

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

 

 

 

 

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, 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

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 not the 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 m
as 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 second
paper 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 C
and m
appear 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
-similarity
of 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 single
process, 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 complete
similarity
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 of
the Paris’ law parameter m
on the dimensionless number , where
is the tensile strength,
is the fracture toughness and h
is the specimen thickness. They were very careful to “provisionally”
suggest a certain special form of the dependence for metals: namely,
that m
should be constant for Z
less 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 C
rather than just m. We also consider not only the original data
points 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 m
and Z
in concrete as compared to metals. A novel interpretation is made by
noting that the slope of the linear m
vs. Z
relationship is strongly correlated to another dimensionless parameters
(apart from R, as well known), namely the ratio between the
elastic modulus and the material tensile strength, . Moreover, by looking at the corresponding relationship
between C
and Z, we find an inverse relationship between these two
parameters, which can be mathematically treated according to either
complete or incomplete self-similarity, depending on the material being
considered. Eliminating Z
between the two relationships, one between m
and Z
and the other between C
and Z, a correlation between C
and m
is 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.

 

 

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 all of my paper?

Mike

 

3.
BB's generalized

According to dimensional analysis, the physical
phenomenon under observation can be regarded as a black box
connecting 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 b
and c) as well as any microstructural length scale, such as
those 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
E

Characteristic structural size
h
L

Initial crack length
a
L

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 category
comprises the variables governing the testing conditions, such as the
stress-intensity factor range, , the loading ratio, R, and the frequency of the
loading 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 Z
defined by Barenblatt
and Botvina (1980), and to the inverse of the square of the brittleness
number 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 Z
and 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, or self
-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 kind
in 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 C
is dependent on two material parameters, such as the fracture
toughness, , and the yield stress, , as well as on the loading ratio, R, and on the
length scales h
and a.

Hence, the phenomenon of fatigue crack growth
presents different length scales, i.e. the specimen size, h, the
plastic-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 mechanics
is 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 C
in concrete, it is interesting to note that Carpinteri
and Spagnoli (2004)
considered incomplete self-similarity in
to prove that C
is 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 C
has 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 C
is 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 the
crack-tip opening displacement (CTOD) and the dislocation barrier
spacing d, and assuming the propagation is for the dislocation
cell element of size s, also giving the striation spacing and a
more 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 C
should 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).

 

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 m
and C
were 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 m
is dependent on . More specifically, the parameter m
tends 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 m
is approximately constant.

 

Fig. 1:
-dependence of the Paris’ law parameter m. (a) A typical
curve for steel (, , ). (b) Effective Paris’ slope m
vs.
computed from (a).

 

The slope m
of 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 m
is 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 m
may 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 m
is a linear function of Z, being the slope of such a
relationship different from a material to another. For very low values
of Z, they found that m
turns out to be almost constant and independent of Z. To explain
such a trend, Barenblatt
and Botvina (1980)
supposed that the relationship between m
and Z
has three regimes:
for small Z, m
linear with Z
for
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 m
and Z
progressively decreases from aluminium alloys to steels. For low-carbon
steels, m
becomes nearly independent of Z
and 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 Z
is 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).

 
 

 

 

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.

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 

Start from Part I and II.

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

Hello

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

thank you.