1 of 33

Chapter 6

​

Fatigue Failure Resulting from Variable Loading

Lecture 10

The McGraw-Hill Companies © 2012

2 of 33

Chapter Outline

Shigley’s Mechanical Engineering Design

3 of 33

Shigley’s Mechanical Engineering Design

Introduction to Fatigue in Metals

  • Loading produces stresses that are variable, repeated, alternating, or fluctuating
  • Maximum stresses well below yield strength
  • Failure occurs after many stress cycles
  • Failure is by sudden ultimate fracture
  • No visible warning in advance of failure

4 of 33

Stages of Fatigue Failure

Shigley’s Mechanical Engineering Design

  • Stage I – Initiation of micro- crack due to cyclic plastic deformation
  • Stage II – Progresses to macro-crack that repeatedly opens and closes, creating bands called beach marks
  • Stage III – Crack has propagated far enough that remaining material is insufficient to carry the load, and fails by simple ultimate failure

Fig. 6–1

5 of 33

6 of 33

7 of 33

Schematics of Fatigue Fracture Surfaces

Shigley’s Mechanical Engineering Design

Fig. 6–2

8 of 33

Fatigue Fracture Examples

Shigley’s Mechanical Engineering Design

  • AISI 4320 drive shaft
  • B– crack initiation at stress concentration in keyway
  • C– Final brittle failure

Fig. 6–3

9 of 33

Fatigue Fracture Examples

Shigley’s Mechanical Engineering Design

  • Fatigue failure initiating at mismatched grease holes
  • Sharp corners (at arrows) provided stress concentrations

Fig. 6–4

10 of 33

Fatigue Fracture Examples

Shigley’s Mechanical Engineering Design

  • Fatigue failure of forged connecting rod
  • Crack initiated at flash line of the forging at the left edge of picture
  • Beach marks show crack propagation halfway around the hole before ultimate fracture

Fig. 6–5

11 of 33

Fatigue Fracture Examples

Shigley’s Mechanical Engineering Design

  • Fatigue failure of a 200-mm diameter piston rod of an alloy steel steam hammer
  • Loaded axially
  • Crack initiated at a forging flake internal to the part
  • Internal crack grew outward symmetrically

Fig. 6–6

12 of 33

Shigley’s Mechanical Engineering Design

Fatigue-Life Methods

  • Three major fatigue life models
  • Methods predict life in number of cycles to failure, N, for a specific level of loading
  • Stress-life method
    • Least accurate, particularly for low cycle applications
    • Most traditional, easiest to implement
  • Strain-life method
    • Detailed analysis of plastic deformation at localized regions
    • Several idealizations are compounded, leading to uncertainties

in results

  • Linear-elastic fracture mechanics method
    • Assumes crack exists
    • Predicts crack growth with respect to stress intensity

13 of 33

Stress-Life Method

Shigley’s Mechanical Engineering Design

  • Test specimens are subjected to repeated stress while counting cycles to failure
  • Most common test machine is R. R. Moore high-speed rotating-beam

machine

  • Subjects specimen to pure bending with no transverse shear
  • As specimen rotates, stress fluctuates between equal magnitudes of tension and compression, known as completely reversed stress cycling
  • Specimen is carefully machined and polished

Fig. 6–9

14 of 33

S-N Diagram

Shigley’s Mechanical Engineering Design

  • Number of cycles to failure at varying stress levels is plotted on log-

log scale

  • For steels, a knee occurs near 106 cycles
  • Strength corresponding to the knee is called endurance limit Se

Fig. 6–10

15 of 33

S-N Diagram for Steel

Shigley’s Mechanical Engineering Design

  • Stress levels below Se predict infinite life
  • Between 103 and 106 cycles, finite life is predicted
  • Below 103 cycles is known as low cycle, and is often considered quasi-static. Yielding usually occurs before fatigue in this zone.

Fig. 6–10

16 of 33

Shigley’s Mechanical Engineering Design

Strain-Life Method

  • Method uses detailed analysis of plastic deformation at localized regions
  • Compounding of several idealizations leads to significant

uncertainties in numerical results

  • Useful for explaining nature of fatigue

Reversal

2Nf = # of Reversals

17 of 33

Strain-Life Method

Shigley’s Mechanical Engineering Design

  • Fatigue failure almost always begins at a local discontinuity
  • When stress at discontinuity exceeds elastic limit, plastic strain occurs
  • Cyclic plastic strain can change elastic limit, leading to fatigue
  • Fig. 6–12 shows true stress-true strain hysteresis loops of the first five stress reversals

Fig. 6–12

18 of 33

Shigley’s Mechanical Engineering Design

Relation of Fatigue Life to Strain

  • Figure 6–13 plots relationship of fatigue life to true-strain amplitude
  • Fatigue ductility coefficient ε'F is true strain corresponding to

fracture in one reversal (point A in Fig. 6–12)

  • Fatigue strength coefficient σ'F is true stress corresponding to fracture in one reversal (point A in Fig. 6–12)

Fig. 6–13

19 of 33

Relation of Fatigue Life to Strain

Shigley’s Mechanical Engineering Design

  • Fatigue ductility exponent c is the slope of plastic-strain line, and is the power to which the life 2N must be raised to be proportional to the true plastic-strain amplitude. Note that 2N stress reversals corresponds to N cycles.
  • Fatigue strength exponent b is the slope of the elastic-strain line, and is the power to which the life 2N must be raised to be proportional to the true-stress amplitude.

Fig. 6–13

20 of 33

Slope of log-log Plot

Consider general equation:

​

​

We take the logarithm of each side:

​

​

​

​

The function log(y) is a linear function of log(x) and its graph is a straight line with slope of n which intercepts the log(y) axis at log(A).

21 of 33

Shigley’s Mechanical Engineering Design

Relation of Fatigue Life to Strain

  • Total strain is sum of elastic and plastic strain
  • Total strain amplitude is half the total strain range
  • The equation of the plastic-strain line in Fig. 6–13
  • The equation of the elastic strain line in Fig. 6–13
  • Applying Eq. (a), the total-strain amplitude is

22 of 33

Relation of Fatigue Life to Strain

  • Known as Manson-Coffin relationship between fatigue life and total strain
  • Some values of coefficients and exponents given in Table A–23
  • Equation has limited use for design since values for total strain at discontinuities are not readily available

Shigley’s Mechanical Engineering Design

 

23 of 33

Shigley’s Mechanical Engineering Design

Linear-Elastic Fracture Mechanics Method

  • Assumes Stage I fatigue (crack initiation) has occurred (point A)
  • Predicts crack growth in Stage II with respect to stress intensity (Starts at point B)
  • Stage III ultimate fracture occurs when the stress intensity

factor KI reaches some critical level KIc

24 of 33

Shigley’s Mechanical Engineering Design

Stress Intensity Modification Factor

  • Stress intensity factor KI is a function of geometry, size, and

shape of the crack, and type of loading

  • For various load and geometric configurations, a stress intensity modification factor β can be incorporated

​

​

  • Tables for β are available in previous lectures (CH5)
  • Figures 5−25 to 5−30 present some common configurations

Figure 5−25

Figures 5−30

25 of 33

Crack Growth

Shigley’s Mechanical Engineering Design

  • Stress intensity factor is given by
  • In case of cycling, a stress range Δσ, the stress intensity range per cycle is
  • Testing specimens at various levels of Δσ provide plots of crack length vs. stress cycles

Fig. 6–14

Stress intensity factor

Stress intensity modification factor

26 of 33

Crack Growth

Shigley’s Mechanical Engineering Design

  • Log-log plot of rate of crack growth, da/dN, shows all three stages of growth
  • Stage II data are linear on log-log scale
  • Similar curves can be generated by changing the stress ratio R = σmin/ σmax

Fig. 6–15

27 of 33

Shigley’s Mechanical Engineering Design

Crack Growth

  • Crack growth in Region II is approximated by the Paris equation
  • C and m are empirical material constants. Conservative

representative values are shown in Table 6–1.

28 of 33

Shigley’s Mechanical Engineering Design

Crack Growth

  • Substituting Eq. (6–4) into Eq. (6–5) and integrating,
  • ai is the initial crack length
  • af is the final crack length corresponding to failure
  • Nf is the estimated number of cycles to produce a failure after the

initial crack is formed

29 of 33

Crack Growth

Shigley’s Mechanical Engineering Design

  • If β is not constant, then the following numerical integration

algorithm can be used.

The following example will explain a simple procedure to evaluate β

30 of 33

Example 6-1

Shigley’s Mechanical Engineering Design

Fig. 6–16

31 of 33

Example 6-1

Shigley’s Mechanical Engineering Design

5-37

32 of 33

Example 6-1

Shigley’s Mechanical Engineering Design

Fig. 5–27

1.07

33 of 33

Example 6-1

Shigley’s Mechanical Engineering Design