Forewarning: this is going to be fairly maths/science/theory heavy and so may not be too easy to read. If you love/hate this stuff blame prendrefeu and my big mouth, inspired by
this thread.
YOU HAVE BEEN WARNED I will assume some knowledge of
deformation mechanics and
fracture mechanics but hopefully most of it should be common sense. This will also predominately relate to metals as other materials tend to fail in different manners not covered by fatigue mechanics.
There may be errors so feel free to point them out and I will edit this if necessary, likewise if there is something you feel needs answering I will try to respond and maybe adapt this if I feel it suitable.
I will define fatigue here as the increase in crack length due to an oscillating force (e.g F = F
0 + ΔF sin(ω t), where ΔF is non-zero ) applied to a sample/object
Broadly failure by fatigue can be broken into 3 stages:
Stage 1 - initiation of a crack
Stage 2 - propagation of the crack
Stage 3 - critical failure of the crack
I will generally talk about a single crack but failure will occur when the first crack reaches it's critical length and thus there maybe multiple cracks propagating simultaneously.
Stage 1 - The initiation of a crack can happen in many ways, commonly in real materials this often due to a void in the material left over from the manufacturing process or can be caused by a sufficient tearing apart of the atomistic structure for a series of adjacent atoms to debonded. In the latter case, areas which have stress risers/concentrators are more susceptible due to their higher experienced localised stresses. In either case, once a crack is stable (energetically) it progresses onto stage 2
Stage 2 - As the crack is loaded it is slowly worked open until it reaches a critical crack length at which it fails, I will go over the mathematics of this further down
Stage 3 - once a crack reaches it's critical crack length, the remainder of the material is no longer strong enough to support the load, i.e. σ > σ
ys , the remaining material will endure rapid crack propagation and the component will break.
Stage 1 is either a process generated by manufacturing or a random process, thus it is of little interest as it is impossible to prevent unless you have perfect manufacturing and do not work the material close to it's limits (~0.5σ
ys)
Stage 3 is a rapid and often dramatic process where the component is no longer suitable for intended use and so just fails.
Stage 2 is the one of primary interest since it is the longest of the 3 stages and so plays the major part in the lifetime of a component when it is being designed and thus this is the one i will describe below.
In real life situations the force applied is not a constant oscillating force such as F = F
0 + ΔF sin(ω t). It often comprises of several different forces of different amplitude and frequency, all of which can contribute to fatiguing a sample the combined effects are given by Miner's law which states failure occurs at:
.m
.∑
.(n
i/N
i) = 1
i=1
Where:
m includes all loading types
n
i is the number of cycles at load i
N
i is the number of cycles at load i to cause failure
i.e fatigue stacks with the fraction of life "used" for a certain loading.
N
i can be determined from the Paris equation:
da/dN = C (ΔK)^m
Where:
a is crack length
N is from above (dropped subscript since this only considering a single loading type)
C is a material dependent constant (~10^-11)
ΔK is the stress intensity factor which is dependent on the geometry of the crack but can be approximated to Δσ*(pi*a)^0.5
m is a material dependent constant, ~4 for most materials
Δσ is the variation in stress applied for the loading cycle (assuming sinusoidal loading and σ
max < σ
ys)
By integrating from a
initial to a
final across N = 0 to N = N and solving for N you can get the number of load cycles for the critical crack length to develop.
Combining Paris and Miners you can get a good estimate of the number of load cycles a product can undergo. Taking this approach is designing for failure which is commonly used in consumer "white goods" as the load cycle is very well known and the frequency of which it is used can be well estimated too from which the life expectancy can be calculated (and the warranty set to one month prior
)
It is also used in checking critical parts (e.g. aircraft wings), by knowing the service intervals and thus how many load cycles between each service interval, you can check for a minimum crack length, which you know will grow to exceed the critical crack length in the next interval between servicing, thus preventing failure whilst also not retiring usable parts.
If you made it this far and understood it, congratulations.
TL:DR We know when planes wings might fall off because of science.