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Obituary for prof. Paul Paris the inventor of the fatigue crack propagation "Paris' law"

Mike Ciavarella's picture

I do not think in Imechanica there has been an obituary for Paul Paris, whose work in fatigue crack propagation is extraordinary and one of the few important and lasting contributions ----- today our aircrafts, both civil and military, all rely on "damage tolerance", which is based on Paris' law.   Paris' law, as every great innovations, was published in 1961 only after rejection by three of the leading journals in the fracture mechanics field.  It bears the name of "law" although of course it is not like Newton's law.  It is a power law, based perhaps on what Barenblatt calls "incomplete" some self-similarity considerations.

The problem was that people could not beleive that Irwin elastic stress intensity factor (the range of it), could predict fatigue crack propagation rate. In fact, previous laws had attempted to predict da/dN from plastic mechanisms, and in the end resulted, like Frost-Dugdale, in a law which is essentially Paris law, but with m=2.  The australian Air Force and even USAF is returning to this assumption today (leading to exponential crack growth), but this is another story (see Jones, 2014).   

An obituary can be found here

Today Paris' law is so esthablished that it is difficult to get research funding on this fundamental problem.  But each time I return to fatigue in my classes, I immediately tell students about a problem I encountered and identified in 2006 (Ciavarella and Monno, 2006), which still puzzles me.  When we integrate Paris, we get also a finite life prediction.  But the size effect predicted by Paris in the Kitagawa generalized diagram is very hard and funny to beleive.  While the fatigue threshold DK_th gives a -1/2 classical size effect, and similarly does static failure KIc, Paris intergrated gives obviously 1/m-1/2, which for fixed life N smoothly converges to either limits ONLY for infinite m!    And for m=2 the case is not much worse than, say, 3, but still unrealistic, and Paris integrated becomes (almost) horizontal, with the most striking difference.

I know that C in Paris tends to show size effects, and I have even written about it in JMPS (Ciavarella et al, 2008), which often are neglected, and normally are correlated with m (when you increase C, you tend to see decrease of m), but this is also found in a single size of initial cracks, by simply varying the specimen, like Virkler did in his famous work using 68 nominally identical specimen.  

Has Paris' law been measured in a wide enough range of sizes?  Nobody has explained this.   The Paris law regime is simply inconsistent with the other size effects, and m=2 makes things even worse than any other m.  Maybe if we do there is something new to say.

By the way, I also wrote a paper in the very first article of a new journal in 2011 (Ciavarella, 2011), where I show that if I had a generalized El Haddad law the way makes much more sense in the Kitagawa diagram, then the Paris law would show size effects, although it becomes less easy to write.   Since most people find Paris with a single specimen, and do not start with different crack sizes (typically a CT specimen is used), probably this effect has been overlooked.

But this doesn't answer my question, and now that Paul Paris died, I feel so sad that I cannot ask him directly this challenging question.

Prof. Michele Ciavarella



Paris, P. and Erdogan, F. (1963), A critical analysis of crack propagation laws, Journal of Basic Engineering, Transactions of the American Society of Mechanical Engineers, December 1963, pp. 528–534.

Ciavarella, M., & Monno, F. (2006). On the possible generalizations of the Kitagawa–Takahashi diagram and of the El Haddad equation to finite life. International journal of fatigue, 28(12), 1826-1837.

Ciavarella, M., Paggi, M., & Carpinteri, A. (2008). One, no one, and one hundred thousand crack propagation laws: a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth. Journal of the Mechanics and Physics of Solids, 56(12), 3416-3432.

Ciavarella, M. (2011). Crack propagation laws corresponding to a generalized El Haddad equation. International Journal of Aerospace and Lightweight Structures (IJALS), 1(1).

D. A. Virkler, B. M. Hillberry and P. K. Gael, The statistical nature of fatigue crack propagation. J. Engng Mater.Technol. 101, 148-153 (1979). 

Jones, R. (2014). Fatigue crack growth and damage tolerance. Fatigue & Fracture of Engineering Materials & Structures, 37(5), 463-483 



Zhigang Suo's picture

Here is a copy of the 1961 paper by Paris et al. signed by the man himself.  Paris studied fatigue crack in metals, and showed that the extension per cycle is a function of stress intensity factor. 

In a 1958 paper, Thomas studied fatigue fracture in rubbers, and showed that the extension per cycle is a function of energy release rate.  I also posted the Thomas paper.

It is interesting to see diffeent individuals independently come up with the same idea. 

Mike Ciavarella's picture

Dear Zhigang

  I did not know of the Thomas paper, probably you're looking at elastomer and hydrogel fatigue recently and this is most interesting.

I would like to point to you that there is a not so subtle difference in Paris law as da/dN=DK^m, and a possible da/dN=DG^n as some people have written.

It is true that Irwin proved the equivalence KI^2/E = G, but in terms of range, DK=Kmax-Kmin, and hence DG=Gmax-Gmin.

Hence, it is not the same to write (Kmax-Kmin)^m and (Gmax-Gmin)^n = (Kmax^2/E - Kmin^2/E) ^n ---- for example.  There is no way the two are equivalent in general: this has to do mostly with R ratio effects.

Notice this is already true under small scale yielding, but laws based on DG^n or indeed DJ^n where J is J integrals have been proposed in an attempt to extend Paris to the elasto-plastic fracture mechanics, or for composite materials.  Some people have made some interesting remarks recently that the Paris law works much better than the DG or DJ laws.

If you are interested I will find the reference.

Mike Ciavarella's picture

Here are the two papers I had in mind.  I suspect they are also useful for rubber and hydrogel if that is your interest.  Hope this helps.




Fatigue crack growth in fibre-composites and adhesive joints is discussed.

The term ΔG is shown not to be a valid crack-driving force (CDF).

However, the term Δ√G is shown to provide a valid CDF.

For a given Δ√Gda/dN now correctly increases with an increasing R-ratio.

Use of the Hartman–Schijve equation collapses the data onto a ‘master’ curve.


Jones, R., Kinloch, A. J., & Hu, W. (2016). Cyclic-fatigue crack growth in composite and adhesively-bonded structures: The FAA slow crack growth approach to certification and the problem of similitude. International Journal of Fatigue88, 10-18.

Jones, R., Hu, W., & Kinloch, A. J. (2015). A convenient way to represent fatigue crack growth in structural adhesives. Fatigue & Fracture of Engineering Materials & Structures38(4), 379-391.


Mike Ciavarella's picture

The "inconsistency" that I find in the size effect of Paris' law may have been obscured in past experiments by the fact that, for example looking at fig.8 of Ciavarella, M., & Monno, F. (2006), it may well start to occur for crack sizes of several millimeters.  In this range, perhaps the real value of C in Paris that I expect is higher than what I expect at smaller crack sizes.   

This would result in the usual problem of "size effects" --- what is measured in the lab may be unconservative of what you obtain in the field.   But given this occurs for relatively large crack sizes, perhaps people have not noticed, or have confused the apparent "increase" of C with the expected increase of crack growth near failure that is measured anyway in Paris law.

Mike Ciavarella's picture

You can find many famous professors giving memories of Paul Paris here.  I have also added a pdf in my main post.

[HTML] Dr. Paul Croce Paris August 7, 1930-January 15, 2017 A Eulogy

AR Ingraffea - 2017 - Elsevier

Mike Ciavarella's picture

A very interesting recent paper by Jones et al. Crack growth: Does microstructure play a role? challenges the community to explain some apparent paradoxical dependences (rather, independence) of crack growth curves on microstructure, contrary to common belief.

The experimental data presented in this paper reveals that even if the growth of long cracks in two materials, with different microstructures, have different da/dN versus ΔK curves the corresponding small crack curves can be similar. We also see that long cracks in a large range of steels with different microstructures, chemical compositions, and yield stresses can have similar crack growth rates. The materials science community is challenged to explain these observations. The experimental data also suggests that the threshold term ΔKthr

 in the Hartman-Schijve variant of the NASGRO crack growth equation appears to have the potential to quantify the way in which small cracks interact with the local microstructure. In this context it is also noted that the variability in the life of operational aircraft is controlled by the probability distribution associated with the size and nature of the material discontinuities in the airframe rather than the probability distribution associated with the scatter in the growth of small cracks with a fixed initial size. 




I read the "challenge" paper and it certainly contains an interesting perspective. From a theoretical point of view, there is nothing surprising really.  Paris' law is an "incomplete similarity" law, and as such, C and m are fitting parameters of a power law could depend on many possible dimensionless parameters, but nothing prescribes that they should depend on all of them! 

On the contrary, in some cases they do not depend on microstructure and chemical composition!   Very nice, this means that most people are correct to say Paris coefficients are "material properties", and indeed they are less variable than yield stress.   This much to say against the Frost-Dugdale law which was instead a plasticity-based law.

Perhaps m=2, as usual, is an interesting limit case:  in this case the "incomplete similarity" in the Barenblatt-Botvina sense, could indeed become complete and C and m should be independent on any other dimensionless parameters except one  ---- the most obvious choice could be yield strentgh, but also elastic modulus, or failure strength are candidates.   So perhaps I would reformulate the point.  If really m=2, then we should indeed observe such a very strong form of Paris' law.

However this rigorous way forward would lead us back to the fact that m is unlikely to be 2, as you are not able to find this single quantity with dimension of stress which governs the crack growth.   Whereas Jones is one of the firm believers of the Hartman Schjive equation with m=2.

REMARK: the paper shows that dK_thr does depend crucial on microstructure, and hence the fatigue life also.  Again, nothing theoretically fundamental here, but this means that they should not say  In this context it is also noted that the variability in the life of operational aircraft is controlled by the probability distribution associated with the size and nature of the material discontinuities in the airframe rather than the probability distribution associated with the scatter in the growth of small cracks with a fixed initial size.

Finally, the paper contains many data with m>>2, ... In general, the paper is a nice remix of interesting facts.

Mike Ciavarella's picture

p.s. when I say that for m=2 C and m should be independent on any other dimensionless parameters except one  ----  I mean DK/sigmaC, not just sigmaC obviously.

This comes from the fact that DK^2=Dsigma^2 a and hence an obvious complete similarity da/dN = C (Y Dsigma/DsigmaC)^2 a, where C is a fundamental material property now.

In general, the discussion in Jones et al's recent paper is related to my "problem with Paris' law":  for while Paris' law is found to be essentially independent on microstructure, static properties (and to some extent fatigue limit) are.  Hence the difficulty to obtain a clear picture of a Kitagawa generalized diagram, or of size-effects in fatigue.  Within a "damage tolerance" framework, we are ok in that the clever people who created "damage tolerance" in USAF I think (Gallagher?) have eliminated a lot of uncertainty by deciding that aircrafts need to survive quite large crack possibly present in a structure, for which Paris law gives reasonable prediction of the growth.

The fact that fatigue life depends on most cases heavily on the short crack growth, where scatter in dKthr is the main factor affecting a crack growth curve  ---- it is unclear if due to stastistical variations in a given nominal material, or to large dependence on microstructure as well, gives some hopes that "damage tolerance" could be extended some day to less conservative design process, but the issue is still not entirely convincing.  Indeed, since anyway in service we could detect cracks only when they are relatively large, what is the point of worrying about understanding short crack growth?

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