Cracks only propagate during the tensile part of a stress cycle; a compressive stress forces the crack faces together, clamping it shut. Fatigue cracks frequently start from the surface, so if a thin surface layer can be given an internal stress that is compressive, any crack starting there will remain closed even when the average stress across the entire section is tensile. This is achieved by treatments that plastically compress the surface by shot peening (Figure 9.11): leaf-springs for cars and trucks are ‘stress peened’, meaning that they are bent to a large deflection and then treated in the way shown in the figure; this increases their fatigue life by a factor of 5. A similar outcome is achieved by sandblasting or burnishing (local deformation with a smooth, polished tool), or diffusing atoms into the surface, expanding a thin layer, which, because it is bonded to the more massive interior, becomes compressed.
FIGURE 9.11 Shot peening, one of several ways of creating compressive surface stresses.
9.7 Summary and conclusions
Static structures like the Eiffel Tower or the Golden Gate Bridge stand for centuries; those that move have a much shorter life span. One reason for this is that materials are better at supporting static loads than loads that fluctuate. The long-term cyclic load a material can tolerate, its endurance limit σe, is barely one-third of its tensile strength, σts. This sensitivity to cyclic loading—fatigue—was unknown until the mid-19th century, when a series of major industrial disasters made it the subject of intense study.
Empirical rules describing fatigue failure and studies of the underlying mechanisms have made fatigue failures less common, but they still happen, notably in rail track, rolling stock, engines and airframes. It is now known that cycling causes damage that slowly accumulates until a crack nucleates, grows slowly and suddenly runs unstably. This is pernicious behavior: the cracks are almost invisible until final failure occurs without warning.
The rules and data for their empirical constants, described in this chapter, give a basis for design to avoid fatigue failure.
The scientific studies provide insight that guides the development of materials with greater resistance to fatigue failure.
Surface treatments to inhibit crack formation are now standard practice. We return to these in the next chapter, in which the focus is on design issues.
9.8 Further reading
1. Hertzberg RW. Deformation and Fracture of Engineering Materials. 3rd ed. New York, USA: Wiley; 1989; ISBN 0- 471-63589-8.
(A readable and detailed coverage of deformation, fracture and fatigue.)
2. Suresh S. Fatigue of Materials. 2nd ed. Cambridge, UK: Cambridge University Press; 1998; ISBN 0-521-57847-7.
(The place to start for an authoritative introduction to the materials science of fatigue in both ductile and brittle materials.)
3. Tada H, Paris G, Irwin GR. The Stress Analysis of Cracks Handbook. 3rd ed. 2000; ISBN 1-86058-304-0.
(Here we have another ‘Yellow Pages’, like Roark for stress analysis of uncracked bodies—this time of stress intensity factors for a great range of geometries and modes of loading.)
9.9 Exercises
Exercise
E9.1 What is meant by the mechanical loss coefficient, , of a material? Give examples of designs in which it would play a role as a design-limiting property.
Exercise
E9.2 Distinguish between ‘low-cycle’ and ‘high-cycle’ fatigue. Find examples of engineering components that may fail by high-cycle fatigue. What is meant by the endurance limit, , of a material?
Exercise
E9.3 What is the fatigue ratio? If the tensile strength of an alloy is 900 MPa, what, roughly, would you expect its endurance limit to be?
Exercise
E9.4 The figure shows an curve for AISI 4340 steel, hardened to a tensile strength of 1800 MPa.
(a) What is the endurance limit?
(b) If cycled for 100 cycles at an amplitude of 1200 MPa and a zero mean stress, will it fail?
(c) If cycled for 100,000 cycles at an amplitude of 900 MPa and zero mean stress, will it fail?
(d) If cycled for 100,000 cycles at an amplitude of 800 MPa and a mean stress of 300 MPa, will it fail?
Exercise
E9.5 The high-cycle fatigue life, , of an aluminium alloy is described by Basquin’s law:
(stress in MPa). How many cycles will the material tolerate at a stress amplitude of 70 MPa and zero mean stress? How will this change if the mean stress is 10 MPa? What if the mean stress is –10 MPa? The tensile strength of the alloy is 200 MPa.
Exercise
E9.6 The low-cycle fatigue of an aluminium alloy is described by Coffin’s law:
How many cycles will the material tolerate at a plastic strain amplitude of 2%?
Exercise
E9.7 A material with a tensile strength = 350 MPa is loaded cyclically about a mean stress of 70 MPa. If the stress range that will cause fatigue fracture in 105 cycles under zero mean stress is 60 MPa, what stress range about the mean of 70 MPa will give the same life?
Exercise
E9.8 A component made of the AISI 4340 steel with a tensile strength of 1800 MPa and the S–N curve shown in Exercise E9.4 is loaded cyclically between 0 and 1200 MPa. What is the value and the mean stress, ? Use Goodman’s rule to find the equivalent stress amplitude for an -value of –1, and read off the fatigue life from the S–N curve.
Exercise
E9.9 Some uncracked bicycle forks are subject to fatigue loading. Approximate S–N data for the material used are given in the figure, for zero mean stress. This curve shows a ‘fatigue limit’: a stress amplitude below which the life is infinite.
(a) The loading cycle due to road roughness is assumed to have a constant stress range Δσ of 1200 MPa and a mean stress of zero. How many loading cycles will the forks withstand before failing?
(b) Due to a constant rider load the mean stress is 100 MPa. Use Goodman’s rule to estimate the percentage reduction in lifetime associated with this mean stress. The tensile strength σts of the steel is 1100 MPa.
(c) What practical changes could be made to the forks to bring the stress range below the fatigue limit and so avoid fatigue failure?
Exercise E9.10
An aluminium alloy for an airframe component was tested in the laboratory under an applied stress that varied sinusoidally with time about a mean stress of zero. The alloy failed under a stress range Δσ of 280 MPa after 105 cycles. Under a stress range of 200 MPa, the alloy failed after 107 cycles. Assume that the fatigue behaviour of the alloy can be represented by:
where b and C are material constants.
(a) Find the number of cycles to failure for a component subject to a stress range of 150 MPa.
(b) An aircraft using the airframe components has encountered an estimated 4 × 108 cycles at a stress range of 150 MPa. It is desired to extend the life of the airframe by another 4 × 108 cycles by reducing the performance of the aircraft. Use Miner’s rule to find the decrease in the stress range needed to achieve this additional life.
Exercise
E9.11 A medium-carbon steel was tested to obtain high-cycle fatigue data for the number of cycles to failure Nf in terms of the applied stress range Δσ (peak-to-peak). As the test equipment was only available for a limited time, the tests had to be accelerated. This was achieved by testing the specimen using the loading history shown schematically in the figure. A first set of N1 cycles was applied with a stress range Δσ1, followed by a second set of N2 cycles with a stress range Δσ2. The table shows the test programme. In every test the mean stress σm was held constant at 150 MPa, and each test was continued until specimen failure. A separate tensile test gave a tensile strength for the steel of 600 MPa.
Test Δσ1 (MPa) N1 Δσ2 (MPa) N2
1 630 104 – –
2 630 5 × 103 460 5 × 105
3 630 5 × 103 510 1.2 × 105
4 630 2.5 × 103 560 4.4 × 104
(a) Using Goodman’s rule, show that the stress range that would give failure in 104 cycles with zero mean stress is 840 MPa.
(b) Use Miner’s rule to find the expected number of cycles to failure for each of the stress ranges Δσ2 in the table, with the mean stress of 150 MPa.
(c) Convert the Δσ2 – Nf data obtained to the equivalent data for zero mean stress.
(d) Plot a suitable graph to show that all the fatigue life data for zero mean stress are consistent with Basquin’s law for high- cycle fatigue , and find the constant b.
Exercise E9.12
The leaf spring of a heavy vehicle suspension is 1.2 m long, 75 mm wide and 30 mm thick. A central thickened section is 150 mm long and 34 mm thick with a fillet of radius 5 mm, as shown in the figure. It is supported by a pin joint at one end and a
‘shackle’ at the other. This can be assumed to be equivalent to ‘simple’ supports. The spring is deflected sideways in a testing machine using a hydraulic actuator attached to the central section. The actuator causes the spring to oscillate cyclically with an amplitude of ± 50 mm. The material is AISI 4340 steel, for which the Young’s Modulus is 210 GPa and the S–N curve is provided in Example E9.4. You will need to use elastic solutions for the deflection and stresses in beams under three-point bending, from Chapter 5, and for the stress concentration factor, from Chapter 7.
(a) Find the load amplitude applied by the actuator to give the specified deflection (ignore the stiffening effect of the central thickened section).
(b) Find the bending moment and maximum surface stress on the thinner part of the spring, at the location of the fillet (initially ignoring the change in thickness). What stress concentration factor should then be applied to the stress amplitude for this location?
(c) If the spring is initially cycled about its unstressed (equilibrium) position, estimate the number of cycles of loading that it will last before it fails by fatigue at the fillet.
(d) If the test is performed with a mean load of 30 kN, find the mean stress (including the stress concentration), and hence use Goodman’s rule to estimate the number of cycles of loading that the spring will last before failure.
Exercise
E9.13 A material has a threshold cyclic stress intensity of 2.5 MPa.m1/2. If it contains an internal crack of length 1 mm, will it be safe (meaning no failure) if subjected to continuous cyclic range of tensile stress of 50 MPa?
Exercise
E9.14 A plate of width a = 50 mm contains a sharp, transverse edge-crack of length c = 5 mm. The plate is subjected to a cyclic axial stress σ that varies from 0 to 50 MPa. The Paris law constants (equation (9.10)) are m = 4 and A = 5 × 10−9, where σ is in MPa.
The fracture toughness of the material is K1c = 10 MPa.m1/2.
(a) At what ‘critical’ crack length will fast fracture occur? (Assume the geometry factor in the expression for K is Y = 1.1) (b) How many load cycles will it take for the crack to grow to the critical crack length?
Exercise E9.15
Components that are susceptible to fatigue are sometimes surface treated by ‘shot peening’. Explain how the process works, and why it is beneficial to fatigue life.
9.10 Exploring design with CES
Exercise
E9.16 Make a bar chart of Mechanical loss coefficient, . Low loss materials are used for vibrating systems where damping is to be minimised—bells, high-frequency relays and resonant systems. High loss materials are used when damping is desired—sound- deadening cladding for buildings, cars and machinery, for instance. Use the chart to find:
(a) The metal with the lowest loss coefficient.
(b) The metal with the highest loss coefficient.
Do their applications include one or more of those listed above?
Exercise
E9.17 Use the ‘Search’ facility to search materials that are used for:
(a) Bells.
(b) Cladding.
Exercise
E9.18 Use a ‘Limit’ stage, applied to the Surface treatment data table, to find surface treatment processes that enhance fatigue resistance. To do this:
(a) Change the selection table to Process universe Level 2 Surface treatment, open a ‘Limit’ stage, locate Function of treatment and click on Fatigue resistance > Apply. Copy and report the results.
(b) Repeat, using the Level 3 Surface treatment data table.
Exercise E9.19
Explore the relationship between fatigue ratio and strength for a heat-treatable low alloy steel AISI 4340. The endurance limit σe is stored in the database under the heading ‘Fatigue strength at 107 cycles’.
(a) Plot the fatigue ratio σe/σy against the yield strength σy. (b) Plot the fatigue ratio σe/σts against the tensile strength σts.
Use Level 3 of the database, apply a ‘Tree’ stage to isolate the folder for the low alloy steel, AISI 4340, then make the two charts, hiding all the other materials. How do you explain the trends?
1August Wöhler (1819–1914), German engineer, and from 1854 to 1889 Director of the Prussian Imperial Railways. It was Wöhler’s systematic studies of metal fatigue that first gave insight into design methods to prevent it.