So what is fracture mechanics? In short it’s a method of determining the time it takes a crack in a part to grow to failure under a specific loading condition. The crack growth stage of fatigue can make up a significant portion of a products life. This happens in products ranging from bicycles, to airplanes, to steam turbine blades.

At the heart of fracture mechanics is the stress intensity factor K defined as:

Where:

f(g) is a correction factor based on crack geometry. This value tends to be between 1 and 1.4.

a is the crack length

s is the remote stress

Fatigue crack growth is divided into 3 regions as shown in the figure below. In this figure, crack growth rate (da/dN) is plotted on the vertical axis in log scale and _{max}-K_{min}) is plotted on the horizontal axis in log scale. Region I is associated with crack threshold effects (the area where a crack first begins to grow), Region II is an area of linear growth (

For design purposes the focus is on Regions I and II. Crack growth is so fast in Region III that it does not have a significant effect on the total crack propagation life. Noted on the graph is DK_{th}, the threshold stress intensity which is determined through testing. This value marks the beginning of crack growth. K_{c} is the critical stress intensity and values higher than this predict fracture.

When performing a turbine blade analysis we often want to determine if a dynamic stress condition is severe enough to grow a crack (Refer to previous posts on blade analysis). To determine that, we run a fracture mechanics analysis for Region I of the above graph. If the stress condition and initial flaw size is not capable of growing a crack then we need not be concerned with removing the near resonant condition.

This analysis starts with calculating the stress intensity factor range:

In the case of an edge crack on a turbine blade airfoil the a correction factor f(g)=1.12 is typically used. Ds is the stress range, or dynamic stress for turbine blades (again refer to my last post on dynamic stress analysis). If DK is greater than DK_{th}, then a crack will propagate under the given loading condition.

One other thing to consider is the R ratio. Test results for DK_{th} values are very dependant on the conditions which they were tested at. A particular DK_{th} value will only apply to a loading condition that has the same R ratio as the test. The R ratio calculation is shown below:

Where s_{m} is the mean steady stress and s_{d} is the alternating or dynamic stress. If the R ratio for the DK_{th} test is different than calculated above, the DK_{th} will need to be adjusted to account for the difference. One common method for the compensation of DK_{th }is

Where is the value at R=0, g is a material constant and is typically between 0.3 and 1. Steels are typically around 0.5. Using this relationship, and assuming that is a constant, you can calculate the DK_{th} value for any R ratio.

Thanks for reading,

I found the following test project on github (https://github.com/vctrhg/crackgrowth), and could be interesting.

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