Failure Under Cyclic Loading  Fatigue and Fracture  Crack Growth Rates 
1 Fatigue  Involves crack initiation followed by crack growth Requires cyclic  repeated stressing  normally cracks only develop under tensile stresses.
Fatigue has been recognised and researched for 120 years  so  why is it still a problem?
Welding now used extensively  and is a potent source of defects Higher mechanical efficiency is being required, leading to more highly stressed components. Comparatively recent techniques enable calculations to be made predicting the life or remaining life of a structure containing defects. There are two stages in fatigue: crack initiation and crack growth. For some materials, ferrous metals being an important group, low cyclic stresses, below the 'threshold limit', do not lead to crack initiation. Fatigue damage normally starts where cyclic stresses are most severe, this will often be at somewhere on a component surface where there is some imperfection or notch. For smooth specimens with a gradually changing section impurities or inhomogeneities in the grain structure provide crack initiation points. For these reasons it is important to take care when designing components which contain changes in section and features such as  key ways, screw threads, 'O' ring grooves, etc. to ensure that their effects are properly assessed. For components subjected to very high cyclic stresses, high purity steels may be used to minimise potential crack initiation locations, an example of this is the steel used by some manufacturers of ball and roller bearings. 2 Design philosophies: 2.1 Safe life:
There were however two significant problems with this approach:

2.2 Fail Safe:
In addition to the multiple load paths, crack stoppers are often used. These may consist
of materials with a high fracture toughness used to supplement the residual strength of the
surrounding structure and to prevent cracks propagating to failure.
2.3 Defect (or Damage) Tolerant Approach:
For aircraft the objective is to detect cracks in 'Principal Structural Elements' (PSE)
before they propagate to failure. By establishing inspection intervals for the PSEs based
on the time it takes for a crack to grow from an initial detectable size to the critical
crack length, safe operation can be maintained. This computation is quite complex and will
involve working from the detailed usage programme of the plane.
This methodolgy means that undamaged components are not retired and factors of safety can be reduced as fracture mechanics provides a more precise characterisation of crack behaviour, the large scatter factors associated with fatigue results and methods are not required. 
3 Fracture Mechanics 
Changes in component geometry such as those caused by grooves, give rise to stress concentration effects. As long as the geometry is known, these effects can be computed and taken into account during design. However many components contain defects, often cracks, where the end of the crack is very sharp, its radius is not known and can not be measured. For situations like these, the stress concentration can not be determined and an alternative approach is needed. The approach is to use 'fracture mechanics'. The way in which a cracked component is loaded can be idealised in one of three ways, see diagram on right. For the vast majority of engineering applications it is 'mode I', the 'opening mode', which is of interest. The stress near a crack tip can be characterised by a single parameter, the stress intensity factor, where Q is a factor depending of specimen and crack geometry ('geometry correction factor'), a is the crack depth (or half the crack length). Fracture occurs when the value of K reaches K_{c} the fracture toughness of the material. K_{c} can be found from laboratory tests and applied to real structures of different geometries. K_{Ic} (the subscript 'I' meaning mode I) is a function of the material thickness and reaches a minimum when the material is thick enough to provide full restraint (plane strain). Extensive values can be found in the literature (see below). Although strictly the stress intensity applies only to linear elastic fracture mechanics (LEFM), some crack tip plasticity is common in structural materials. Provided this plastic zone is only small <a/15 or < B/15 (in configurations shown on right: b = B) then the stress intensity is not significantly affected away from the crack tip and LEFM and K can be used. Structural materials absorb significant energy in the plastic zone near the crack tip which makes them tough, however brittle materials have no, or a very small crack tip plastic zone and have low toughness. Crack growth rates in components subjected to cyclic loading have been investigated in a number of structural materials and all structural metals have the same 'sigmoidal' shaped crack growth curves when the crack growth is plotted against the stress intensity factor range (K_{max}  K_{min}) on log axes. 3.2 How Fracture Mechanics is used in Design and Maintenance Operations Fracture mechanics may be used to assess if a defect present in a structure will cause failure during the load application or for cyclic
loading, how many load cycles will be required to grow the crack to the critical length that causes failure.

A more accurate value for the shape factor 'Q', than shown below is used for the calculation in the applet, below. Geometry Correction Factors  Q Computation:Two configurations which are often of interest to designers are a centre crack through a plate and a surface breaking semi
elliptical crack perpendicular to the axis of a finite rectangular
component, both subject to tension. The applet below shows these configurations and information about the computation.
Use this applet to calculate:
For any engineering material it is frequently the case that increasing the UTS or yield strength results in a lowering of the fracture toughness.
Useful Conversions:


3.3 Geometry Correction Factors  for configurations shown in the right hand panel, above. These are for mode I loading and normally are only valid when b>>a. For a short single edge through crack of depth a, Q also = 1.12 Data, often obtained from finite element analysis, have then been plotted as curves and equations based on these curves have been derived and are used in computer software to calculate residual lives of cracked components. Such equations are used in the applet for the crack growth calculation of the elliptical surface crack, see 'Computation'. 3.4 Residual Life Assessment: Stage II of the sigmoidal crack growth curve can be represented by a 'power law' relationship giving the fatigue crack growth rate da/dN of a material in terms of the range of the applied stress intensity factor as: where C and m are constants for a particular material, environment and loading condition and is the range of the stress intensity factor occurring at the crack tip. This sometimes called the Paris law after P C Paris who first used it in the early 1960s. The power law eqn. can be integrated: between the limits of the initial (present) crack length and the final (maximum safe) crack size. Substituting for K: as Q generally depends upon the crack depth and shape, numerical integration
is normally needed.
For very small cracks, which are described by the initial part of the sigmoidal curve, the above results, which are based on the main, middle part of the sigmoidal curve, are conservative. The effects of R: R = minimum stress/maximum stress
The effects of temperature: As the temperature rises the yield stress decreases and for a given load the size of the plastic zone will increase, so even though LEFM may be applicable at ambient temperatures, it may not be applicable at elevated temperatures. It is generally preferable for pressure vessels to leak before they fracture. Fracture mechanics is used to check that when an eliptical crack in the wall has grown so that its depth extends through the wall, allowing leakage, the crack length is still below the critical length that would cause sudden rupture. 

3.5 Conversion of Crack Growth Data
The crack growth equation shown at the top of the right hand panel, shows it with the stress intensity factor range (dK) replaced by the terms used to to calculate it. The dimensionless geometry correction function is unaltered by the conversion and is omitted from the equation. The crack growth per cycle, plotted on the vertical axis: da/dN may be in units of m/load cycle, mm/load cycle or inches/load cycle. In the first 2 cases the stress range will be in MPa and the stress intensity factor range will be in MPa(m)^{0.5} In the first case, a, the crack length will be in m. (da/dN is in m/load cycle). In the second case, a, the crack length will be in mm. (N.B: da/dN is in m/load cycle). In the final case the stress range will be in ksi (thousands of pouds per square inch) and the stress intensity factor range will have the units of: ksi(inch)^{0.5}. (da/dN is in inches/load cycle). The stress intensity factor range is plotted on the horizontal axis. When converting between these units, m remains unaltered but 'C' must be converted and this is shown in the right hand panel. The applet Click Here provides an easy way of doing conversions of 'C' for different units. m is typically within the range of 3 to 4 

Some typical C and m values: Ferriticpearlitic steels: da/dN (m/cycle) = 6.9(10)^{12}(dK MPa(m)^{0.5})^{3.0} Martensitic steels: da/dN (m/cycle) = 1.35(10)^{10}(dK MPa(m)^{0.5})^{2.25} Austenitic stainless steels: da/dN (m/cycle) = 5.6(10)^{12}(dK MPa(m)^{0.5})^{3.25}  
Further reading 

David J Grieve, Revised: 16th March 2013, 29th January 2010, Original: 13th August 2004.