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Paris' exponent m<2 and behaviour of short cracks

ESIS's picture

I came across a very interesting paper in Engineering Fracture Mechanics about a year ago. It gives some new results of stochastic aspects of fatigue. The paper is:

”On the distribution and scatter of fatigue lives obtained by integration of crack growth curves: Does initial crack size distribution matter?” by M. Ciavarella, A. Papangelo, Engineering Fracture Mechanics, Vol 191 (2018) pp. 111–124.

The authors remind us of the turning point the a Paris' exponent m=2 is. Initial crack length always matters but if the initial crack is small, the initial crack is seemingly very important for the if m>2.  For exponents less than 2, small initial cracks matters less or nothing at all. If all initial cracks are sufficiently small their size play no role  and may be ignored at  the calculation of the remaining life of the structure. Not so surprising this also applies to the stochastic approach by the authors. 

 What surprised me is the in the paper is the fuzz around small cracks. I am sure there is an obstacle that I have overlooked. I am thinking that by using a cohesive zone model and why not a Dugdale or a Barenblatt model for which the analytical solutions are just an inverse trigonometric resp. hyperbolic function. What is needed to adopt the model to small crack mechanics is the stress intensity factor and a length parameter such as the crack tip opening displacement or an estimate of the linear extent of the nonlinear crack tip region.

I really enjoyed reading this interesting paper and get introduced to extreme value distribution. I also liked that the Weibull distribution was used. The guy himself, Waloddi Weibull was born a few km's from my house in Scania, Sweden. Having said that I will take the opportunity to share a story that I got from one of Waloddi's students Bertram Broberg. The story tells that the US army was skeptic and didn't want to use a theory (Waloddi's) that couldn't even predict zero probability that object should brake. Not even at vanishing load. A year later they called him and told that they received  a cannon barrel that was broken already when they pulled it out of its casing and now they fully embraced his theory. 

Per Ståhle


Mike Ciavarella's picture

dear Per

 thanks for your interest in our paper.  We simply play some simple statistics using a pure form of Paris' "law", and remark that the case m=2 is "singular" --- with many "damage tolerant" materials (metals, mostly) being in close to that range.  For very high m, like in ceramics, there is wide scatter of static strength, but it makes little sense to use damage tolerant design.  

Short cracks have been studied for long time but they do not follow Paris law, and indeed are not sensitive even to Irwin stress intensity factors.  You can classify them depending on their size with respect to other characteristic sizes in the material, the simplest one being the "Topper-El Haddad" constant a0.   In the end, despite large efforts in the 1980's, there is no engineering treatment for short cracks, and they propagate possibly much faster than Paris' law, making it difficult to design for fatigue using a purely propagation approach, as it is well known.   We point out that "damage tolerant" approach circumvents this problem, by assuming there are large cracks in critical spots of structures, so that Paris' law type of approach may be used -- some people suggest this may be too conservative, but here we are.

Anyway, your story about people being reluctant to accept a theory which predicts non-zero probability of failure even under no load is interesting, and reminds me that somebody told me "damage tolerant" design at one point became not very well accepted by insurance companies which are reluctant to accept there can be "cracks" in a structure.  I don't know the end of that story, but for sure the situation is even worse. Even if we design with damage tolerant, the material constants have statistical distributions (like Paris C and m, as we discuss in the paper), so you can never rule out that there is a small chance even the most conservative of the damage tolerant design may lead to 1 failure every million cases or so.   People say that design proceeds today exactly leading to very high safety margin, but this requires knowing the tails of the distributions of loads, material properties, etc. with fine details.   So although it is possible to enunciate this as a goal, I don't beleive this can be done in reality.  What happens is that there are also several independent tests and one hopes they are sufficient to indicate a problem.   But new lessons always emerge with new solutions, like in the case of 737-Max regarding electronic systems in this case.

Thanks again for your interest.


ESIS's picture

Dear Mike, In some cases engineers lack the necessary qualifications.

I realise that it can be really difficult to quantify the tail of the distribution. Unintended human error could perhaps be estimated, but what is the risk that someone deliberatly do harm or inventive craftsmen "solve" problems that may jeopardise the structural safety. A PhD in social science or behavioural science is needed. 

I saw that you recently formulated a unified crack propagation law for both short and long cracks. 

On unified crack propagation laws, A.Papangelo, R.Guarinoc, N.Pugnocde, M.Ciavarella,

I will take a look att what you did. From what I could understand you did not abandon the stress intensity factor and point shaped crack tip.

I could only read the abstract because of the cancelled agreement with Elsevier. You may know that we are locked out as well as (I think all) other universities in Sweden, Norway, Germany, UCLA and perhaps more. If you want to promote your paper please share personal copies with your friends. Please count me in.

Oh, by the way, thanks for the comment. 

Best regards, Per

Mike Ciavarella's picture

dear Per

   in the paper you refer to we simply tried to make some attempt at unifying formulations, based on "short cracks" as viewed empirically from some Japanese authors (mainly Nisitani) who claimed and measured crack growth which seems to correspond to classical SN curve behaviour with UNcracked specimen, and the classical Paris' law when crack becomes long.   But the very fact that Nisitani's formulation hasn't found so much success in the literature, worries me, and I would not raise my hopes too much that this is the end of the story.  I will send you the paper, but I would not be surprised if you find it does not raise your hopes either :)

I notice my friends have put the paper openly on their web site :)

regards, mike

p.s. you know that there are illegal website where you can download all elsevier papers, operating from kazastan?  This is why perhaps so many universities are not so willing to pay anymore.

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