Theory

Life prediction for fatigue cracks was made very much easier and far more quantitative, in the 1960's when Paris [1] postulated that the range of stress intensity factor might characterise sub-critical crack growth under fatigue loading in the same way that K characterised critical, or fast fracture.  He examined a number of alloys and realised that plots of crack growth rate against range of stress intensity factor gave straight lines on log-log scales. This implies that:

For the first time, it became possible to make a quantitative prediction of residual life for a crack of a certain size. This simply required finding limits on the integration in terms of crack size, which could be done by finding the final size which caused fast fracture from the relationship between fracture toughness and crack size:

Separation of the variables a and N and substitution for the range of stress intensity by the equivalent equation in terms of stress and crack size gives:

It was later realised that this so-called 'law' applied to growth rates in the range of perhaps 10-3 mm/cycle to 10-6 mm/cycle, and that the fatigue crack growth rate curve was sigmoidal in shape when growth lower and higher than this range were included. Typical data for austempered ductile iron in air, as a function of stress ratio (minimum stress in cycle divided by maximum stress in cycle - a measure of mean stress in the fatigue cycle) is shown in the figure below.
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The lower growth rate region is termed the threshold regime, because growth rates drop off steeply and the crack becomes essentially non-propagating. This represents a change in mechanism from double shear continuum growth to single shear non-continuum growth.  The higher growth rate regime is where values of maximum stress intensity in the fatigue cycle are tending towards the fracture toughness and static modes of fracture (cleavage, intergranular) are adding to the fatigue induced growth rates.

The Paris law remains a very useful relationship, however, because it covers the range of growth rates most useful to engineering structures, and because an extrapolation into the threshold regime gives a conservative estimate for the remaining life. This development was crucial to the adoption of defect-tolerance concepts and the implementation of a retirement-for-cause philosophy.

Further information from some early papers dealing with characterisation of fatigue crack growth rates in fracture mechanics terms can be found in references 2 to 6. There are a number of modern texts which deal with fatigue, and a useful starting point is reference 7.

References:
  1. P Paris and F Erdogan (1963), A critical analysis of crack propagation laws, Journal of Basic Engineering, Transactions of the American Society of Mechanical Engineers, December 1963, pp.528-534.
  2. T C Lindley, C E Richards and R O Ritchie (1976), Mechanics and mechanisms of fatigue crack growth in metals: a review, Metallurgia and Metal Forming, September 1976, pp.268-280.
  3. J Schijve (1978), Four lectures on fatigue crack growth, Engineering Fracture Mechanics, Vol. 11 No. 1 pp.169-206.
  4. R P Wei (1978), Fracture mechanics approach to fatigue analysis in design, Journal of Engineering Materials and Technology, April 1978, Vol. 100, pp.113-120.
  5. R O Ritchie (1980), Application of fracture mechanics to fatigue, corrosion fatigue and hydrogen embrittlement, Analytical and Experimental Fracture Mechanics, Proceedings of the International Conference held in Rome, June 1980, G C Shih (editor), Sijthoff and Nordhoff.
  6. R J Allen, G S Booth and T Jutla (1988), A review of fatigue crack growth characterisation by linear elastic fracture mechanics (LEFM) Parts 1 and 2, Fatigue and Fracture of Engineering Materials and Structures, Vol. 11 No. 1 pp.45-69 and No. 2 p.71-108.
  7. S Suresh (1998), Fatigue of Materials 2nd edition, Cambridge University Press, Cambridge, England.

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