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 is only a problem when the failure is unexpected
Fatigue contributes to 80 - 90 % of all failures
Offshore fatigue contributes to 20 - 25% of failures

Fatigue has been recognised and researched for 120 years - so - why is it still a problem?

A major reason is that it is complex.
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:
Developed in the late 1950s and 1960s for the aircraft industry.
Still widely used, based on S-N curves, but although mean values are available for many materials, experimental curves contain a lot of scatter.
Some effects are important and fairly well understood, effects of surface roughness, components size, notches.
The effects of mean stress may need to be considered as most data has been generated for R = - 1 (zero mean stress) a little data is available for R = 0 ( zero - tension loading). (R = minimum stress/maximum stress). For different loadings, it is necessary to carry out a transformation using:
Goodman Line, Goodman criterion "Calculator" Gerber Parabola, Soderberg Line or Smith Curve.

There were however two significant problems with this approach:

  • The structure was not protected if it contained a manufacturing or maintenance induced defect.
  • Owing to the spread of results, a conservative safety factor was required and many components were prematurely retired. Even testing to 4 times the required life did not prevent some aircraft losses.

2.2 Fail Safe:
Developed in the 1960s for aircraft design to overcome limitations of the 'Safe Life' methodology.
The idea is to multiple load path structures, such that if an individual element should fail, the remaining elements would have sufficient structural integrity to carry the additional loads from the failed element until until the damage is detected through scheduled maintenance.
Designers and operators live safely with cracks. This was not a feasible approach until the 1960's when fracture mechanics started to be able to provide a quantative description of the residual life of a cracked component.

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.
An example of a crack stopper is a stringer in a pressurised aircraft fuselage.

2.3 Defect (or Damage) Tolerant Approach:
Developed in the 1970s for aircraft design and based upon fracture mechanics techniques.
This is useful for complex structures with inherent defects, it is assumed that all structures contain growing cracks and failure can occur when actual conditions are different to those modelled.
For this approach to be used facilities must be available for measuring crack lengths. Generally defects need to be bigger than the grain size of the metal for the fatigue strength to be lowered.

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.
Having determined the number of flight hours to failure, this is normally divided by two to give an inspection interval, this means that should a PSE develop a crack it should be inspected at least once before the crack propagates to failure.

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 -
3.1 Introduction

In the 1960s when higher strength steels started to be introduced, it was noted that components often failed suddenly at loads that were well below those that would have caused yielding. Investigation these failures led to the development of '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 Kc the fracture toughness of the material.

Kc can be found from laboratory tests and applied to real structures of different geometries. KIc (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 (Kmax - Kmin) 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.
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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:
i) The stress intensity factor for a specific applied stress with a crack of a particular length present.
ii) The number of load cycles to grow a crack when subjected to a specified stress range.


For any engineering material it is frequently the case that increasing the UTS or yield strength results in a lowering of the fracture toughness.

Material UTS Yield strength, MPa KIC, MPa(m)0.5
A533B Alloy steel -- 500 175
4340 steel 2332 1450 51
4340 1827 1503 59
4340 1764 1530 201
52100 -- 2070 14
AISI 403 stainless steel 821 690 77
Maraging steel 18% Ni 1783 -- 174
Ti-6Al-4V -- 910 115
Ti-6Al-4Zr-2Sn-0.5Mo-0.5V 890 836 139
2219-T851 Al 454 340 32
2024 Al -- 455 26
6061-T651 Al 352 299 29
7079-T651 Al 569 502 26
7178 Al -- 490 33

Useful Conversions:
1 psi (1 lb/inch2) = 6895 Pa
1 ksi (1000 lb/in2) = 6.895 MPa
1 ksi(inch)0.5 = 1.099 MPa(m)0.5

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.
An applet is provided to do this, see 'Computation', above right. This re-computes the geometry factor and crack growth rate at whatever intervals the user specifies.
This page shows a hand calculation of crack growth and the effects of different geometry functions and of different intervals of re-calculating the geometry function and stress intensity factor.

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
An increase in the R ratio of the cyclic loading causes the growth rate for a given stress intensity factor range to be larger. The effect with brittle materials is greater, whereas with ductile, low strength ductile structural metals the effect is slight.

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:

Ferritic-pearlitic 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 -

  • 'Mechanical Behaviour of Materials', by N E Dowling.

  • 'A Compendium of Stress Intensity Factors', by D P Rooke and D J Cartwright, HMSO, London, 1976.

  • 'Fatigue Design: Life Expectancy of Machine Parts', by E Zahavi, published by CRC Press, 1996.

  • References - 'Stress Concentration Design Factors', by R E Peterson, J Wiley & Sons, New York, 1953.

  • 'A Compendium of Stress Intensity Factors', by D P Rooke and D J Cartwright, HMSO, London, 1976.

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