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.
While Griffith, Irwin, Kies, and others were working on using the energy balance to predict when cracks would grow;, some researchers continued to work on calculating the stress around a crack tip. In particular, people wanted to know how the stress near a crack depends on the crack length and on the load being placed on a specimen as a whole.
The groundwork for this research had been laid by Professor Sir Charles Inglis, a British engineer. In his 1913 paper ‘Stresses in a plate due to the presence of cracks and sharp corners’, Ingliss published a series of equations that allow you to determine the stress in the vicinity of a crack. There was one small problem however. According to Ingliss’ equations, the sharper the tip of a crack is, the higher the stress. In fact, if the crack is perfectly sharp (i.e. the radius of the crack tip is zero), Ingliss’ equations predict that the stresses will be infinite! In mathematical terms, this is referred to as the ‘crack-tip singularity’.
In real life of course the stress doesn’t become infinite, because the material plastically deforms or breaks before that can happen. Nevertheless, being able to describe mathematically how quickly the theoretical stresses approach infinity in the vicinity of the crack, can still give you useful information about the actual stresses.
Various people set about the task of coming up with such a description. The most successful was the Danish-American Harold M. Westergaard. His ‘Westergaard functions’, which he published in 1939 still form the basis of the modern stress analysis in the vicinity of cracks. The Westergaard functions are complex functions, which in the jargon of mathematics doesn’t mean that they are (necessarily) complicated, but rather that they make use of imaginary numbers. This makes the Westergaard functions somewhat unwieldy in daily practice.
Fortunately a more readily useable set of equations was developed by the Caltech professor (later University of Utah Dean of Engineering) Max L. Williams (for the mathematically inclined readers: Williams found a series approximation to the Westergaard functions). From Williams’ functions (first published in 1957) follows one of the most important parameters of present-day fracture mechanics: the stress intensity factor (SIF).
The next part is going to dive into some mathematics and equations. If you don’t like that kind of thing, feel free to skip ahead.
Before looking at Williams’ functions and what the SIF is in more detail, we first need to talk about the difference between Cartesian and polar coordinates.
Cartesian coordinates are the ones you are probably most familiar with if you’re not really into math. Cartesian coordinates work just like giving someone directions in an American city. E.g. you might say, to get to the supermarket, go 3 blocks East and then 4 blocks North. In other words, Cartesian coordinates specify a by point in space by specifying how far you have to travel along two perpendicular directions to get there.
Polar coordinates on the other hand, work just like directions at sea. E.g.: to get to the harbour sail North-East for 5 nautical miles. In other words, polar coordinates work by specifying an angle θ with respect to some reference, and a distance r you need to go in that direction.
As you probably guessed, the reason I brought up polar coordinates is because the Wiliams’ solutions make use of polar coordinates to describe the location you want to determine the stress. The origin of this coordinate system is the tip of the crack. The angle θ gives the angle counter-clockwise with respect to the horizontal, and r gives the distance to the crack tip. The stress applied to the specimen as a whole is represented by the symbol S, and the crack length is represented by a.
In this coordinate-system, the stress in vertical direction σy at the location (r,θ) is given by the Williams’ solution as:Now that might look scary at first, but fortunately the rules of mathematics allow us to chop this equation into three pieces.
Notice that the blue box, which contains all the scary sines and cosines, depends only on θ. That means the blue box will always give the same result if we are looking at the same spot relative to the crack tip. This is true no matter how long the crack is, or how much stress is being applied to the specimen! Similarly, the red box depends only on r. So once again it only depends on which location we have chosen to look at.
Together then the red and blue boxes describe how the stress varies as you move around with respect to the crack tip. This behaviour is always the same, regardless of how we are loading our specimen, and regardless of how long the crack is. The only part of the equation that is affected by those two parameters is the purple box. If you compare two different specimens, and look in the same location (relative to the crack tip) on each specimen, the formula in the purple box will account for all differences in the stress state.
This portion of the Williams’ series: stress applied to the specimen, times the square root of pi, times the square root of crack length, became known as the stress intensity factor. It is usually denoted by the symbol K. K is a powerful parameter, because by itself it is sufficient to describe how the stress at the crack tip is affected by crack length and by the load on the specimen as a whole.
By comparing K-values between two specimen geometries or loading conditions we can easily compare what the stress state near the crack tip will be, without having to calculate the stress in every location. If K for a certain combination of crack length and loading is higher than for a different combination, we know the stress at the crack tip will be higher. Even more than that, the stress at any point near the crack tip is proportional to K, so if K is two times as large, the stress will also be twice as large. So just by calculating K we can say how the crack length and the external loading will affect the crack-tip stress.
Next time we’ll see how K allowed researchers to predict the growth of cracks under fatigue load.