# Applying the stress intensity factor to fatigue

Note for new readers, this is part of a series of posts on the history of fracture mechanics, you can find the first post in this series here.

Fatigue is the phenomenon that a structure gradually degrades due to repeated load cycles, eventually leading to structural failure at much lower loading than for an undamaged structure. The structural degradation usually takes the form of cracks, which initiate at some point during the lifetime of the structure and then grow by a small increment every time a load cycle is applied.

That fatigue is an issue that needs careful consideration has been known to structural engineers ever since the investigation into the 1842 Versailles rail disaster. In the wake of this disaster, which killed between 50 and 200 people, various engineers investigated the phenomenon of fatigue. Eventually this led to the discovery by the German railroad engineer August Wöhler that the lifetime of a certain part subjected to repeated loads is proportional to the amplitude of the load.

This discovery allowed engineers to test materials and generate so-called SN-curves (a.k.a. Wöhler curves) showing how long a part can safely be used. It also showed how fatigue problems could be prevented: by using more material to make a part the forces on the part can be spread over a larger area, which lowers the stress (force per area) and results in a longer life.

This approach works well for railway carriages and bridges, but does not work so well for aeroplanes. To fly efficiently, aeroplanes need to be as light as possible, so simply adding more material doesn’t work, it would make the plane too heavy. Furthermore Wöhler curves only tell you how long it will take for an undamaged specimen to fail. If your part is damaged (e.g. because the process used to drill rivet holes in your structure is not perfect) you can’t use Wöhler curves to predict what will happen.

As the world entered the jet age, and aeroplanes needed to go faster and higher, and thus be lighter, this created a pressing need to be able to predict growth of cracks. Especially after two Comet jetliners crashed in 1954 due to fatigue cracking, which I will discuss in detail some other time.

Various researchers worked on trying to predict the rate of fatigue crack growth for a given load cycle, but the breakthrough was realised by Paul C Paris, at the time Assistant Professor of Civil Engineering at the University of Washington, and faculty associate at the Boeing Co. Paris reasoned that if the stress amplitude on a specimen as a whole governed the life time of a specimen, then perhaps the stress amplitude at the crack tip could be used to predict the crack growth rate (how much the crack grows in one cycle). As we saw last time the stress at the crack tip depends on the stress intensity factor, K. Rather than taking the amplitude of K, Paris instead took the range, ΔK = Kmax – Kmin. I.e. the difference between K for maximum applied stress in the cycle and K for minimum applied stress in the cycle.

Paris took a bunch of experimental data from other investigations and calculated ΔK for these tests. He then plotted ΔK against the crack growth rate and found that there was indeed a very strong calculation. In fact he found that the data could be very well described by the equation (now often called the ‘Paris law’):

$\frac{da}{dN}=C\Delta K^n$

In other words, the crack growth rate (da/dN) is equal to some number C times ΔK to the power n. What are C and n? To this day no one has come up with a satisfying explanation. However if you do a bunch of experiments you can work out which values of C and n give you the correct results for a certain material. Even though we’re not exactly sure of the physical meaning of C and n, we know they will remain constant if we keep the material and certain other conditions the same, and thus we can use them to make predictions of the crack growth rate.

Paris’ ideas were quite revolutionary at the time and he had quite some trouble getting them published. This was at least partially because he could not give a satisfactory explanation of the physics underlying his ‘law’. Why should da/dN depend on ΔK to some power? No one has yet given a satisfactory explanation, it just seems to work. Because of these issues and probably due to conservatism in the community of fatigue researchers as well, Paris’ first paper on this topic (written with two fellow engineers from Boeing: Mario Gomez and William Anderson) : ‘A rational analytic theory of fatigue’  appeared not in an academic journal, but instead in ‘The Trend in Engineering’, the magazine for alumni of the University of Washington College of Engineering.

Eventually however, it became clear that physical underpinning or not, Paris’ law worked. His later papers (written both individually and together with F. Erdogan) were accepted in the regular academic literature and his proposed theory of fatigue became the standard method of predicting fatigue crack growth. In fact, if you’ve ever flown on an aeroplane, you’ve done so safely due in no small part due to the work of Paris and his colleagues.