4.2 Fatigue
4.2.4 Effect of Mean Stress on Fatigue
The integrity of a component surface or subsurface plays a key role in determining the component fatigue life. Today’s ever-improved material process control has minimized material anomalies for some high-quality alloys made by conventional casting or forging processes. For example, in a high-quality aerospace-grade titanium alloy, the occurrence of the detrimen- tal hard alpha phase is less than twice in every 1 million kilograms of mate- rial; and in a nickel-base superalloy, an oxide white spot can only be found a few times in every 1 million kilograms. However, statistically the occurrence of material anomalies in powder metallurgy alloys is usually much higher. It is not unusual to find thousands of small micro anomalies in a single turbine disk made of powder metallurgy alloy. This has led the aerospace industry to adopt different methodologies to predict the fatigue life for components made of powder metallurgy alloys.
Many parts reproduced by reverse engineering are subject to cyclical stresses of fluctuating magnitudes in service. Their fatigue lives can be esti- mated based on the linear cumulative fatigue damage described by Equation 4.12, the Palmgren-Miner’s rule:
n N
n N
n N
n N
i i 1
1 2
2 3
3 1
+ + + =…
∑
= (4.12)where N1 is the number of cycles to failure under stress σ1, n1 is the number of cycles the component is exposed to while under stress σ1, Ni is the number of cycles to failure under stress σi , and ni is the number of cycles the com- ponent is exposed to while under stress σi. The total fatigue life is ∑ni. The Palmgren-Miner’s rule states that the total fatigue life can be estimated by adding up the percentage of the life that is consumed at each stress level to which the component has been exposed. There are many exceptions to this simple linear damage summation rule; however, it does provide a first order of engineering approximation to estimate the fatigue life when the compo- nent is subject to irregular alternating stresses.
It is worth noting that in service, a component is often subject to multiaxial loads, such as axial and radial stresses, at the same time, and multiple fatigue modes, such as tension, torsion, and bending cyclic stresses, simultaneously.
The most reliable life perdition of a component is based on a direct compo- nent test with a real-life simulated loading condition.
increases, where σmax and σmin are the maximum and minimum stresses, respectively. The curves that show the dependence of the alternating stress range on mean stress are generally referred to as Goodman diagrams. The Goodman diagrams are presented in various formats. One is schematically illustrated in Figure 4.5, showing the basic principle of a Goodman diagram.
The mean stress is plotted along the x-axis as the abscissa, and the total stress is plotted along the y-axis as the ordinate, where σu and σy are the ultimate tensile and yield strengths, respectively; σe is the fatigue endurance limit; σm
and σa are the mean and alternating stresses, respectively; and σr is the alter- nating stress range. Also plotted in the diagram is a supplementary line with a 45° inclination showing the middle mean stress between the maximum and minimum alternating stresses. It shows the allowed stress boundary for a fatigue life, with the maximum stress on top and the minimum stress at the bottom. If the yield strength is the design criterion for failure, the maximum and minimum stress boundaries converge to yield strength with decreasing stress amplitude when the mean stress increases.
Several modified Goodman diagrams are schematically illustrated in Figure 4.6, where the ordinate y-axis is the alternating stress and the abscissa x-axis is the mean stress. The diagram shows two additional stress boundaries, the Gerber parabolic curve and the Soderberg line, where σyt, σut, and σyc are yield strength in tension, ultimate tensile strength, and yield strength in com- pression, respectively. It is assumed that both tensile and compressive yield
0 σm σy σu
σu σy σmax
σa
σa σr +
– σe
σe
Stress
Meanstress
Max. stress
Min. stress σmin
σm
Mean stress
FIgurE 4.5 Goodman diagram.
strengths are the same and simplified as σy in the ordinate axis. Following the classic Goodman theory, the straight line connecting the fatigue endurance limit, σe, and the ultimate tensile strength in tension, σut, presents a bound- ary of fatigue limit. Any combination of mean stress and alternating stress that falls below this boundary meets the fatigue safety criterion. Gerber took a more liberal approach to reflect a better match with the experiment data. The Gerber parabolic curve connects σe and σut with a parabolic curve instead of a straight line. Soderberg suggested a more conservative approach. He replaced the ultimate tensile strength with the yield strength in tension, σyt; therefore, the Soderberg line bends the σe–σut connection to σyt. The alternating stress is reduced as the mean stress in tension increases, and it eventually reduces to zero when the mean stress reaches the ultimate tensile strength in the Gerber parabolic curve, or yield strength in the Soderberg line. However, the mean stress in compression shows little effect on fatigue strength. The alternat- ing stress essentially remains the same when the compressive mean stress increases within a boundary. Therefore, a straight line usually applies in the compressive mean stress region to reflect the marginal effect of compressive mean stress on fatigue strength until it reaches the σy–σyc boundary line, when the alternating stress is subject to the limitation of yielding. Mathematically these diagrams can be expressed as Equation 4.13, where σa, σe, σm, and σu
are alternating stress, fatigue endurance limit, mean stress, ad ultimate tensile strength, respectively, and x is an exponent constant. When x = 1, Equation 4.13 represents the Goodman linear diagram, and when x = 2, the Gerber parabolic diagram. Equation 4.13 represents the Soderberg diagram when σu is replaced by σy. Currently, there is no established methodology to decode which theory was used by an OEM in fatigue life analysis. This is still a dilemma that reverse engineering faces today. The best solution is to make an educated judgment call based on industrial standards, corporate knowledge, and tests if necessary.
σ σ σ
a e σm
u x
= −
1 (4.13)
σut σyt σm
0 Mean stress σyc
σy
σe σa Stress amplitude
Gerber parabolic curve
Soderberg line
FIgurE 4.6
Revised Goodman diagrams.
In engineering analysis the Goodman diagram is often plotted with con- stant fatigue life curves and referred to as a constant fatigue life diagram.
Figure 4.7 is a simulated Goodman diagram depicted based on the data extracted from Military Handbook K-5 of an engineering alloy. It is for gen- eral discussion purposes only. It is a master Goodman diagram showing various R and A ratios for smooth unnotched and notched specimens. This master diagram summarizes the relationship between fatigue life and the following parameters: maximum stress, minimum stress, alternating stress, mean stress, stress and strain ratios, symbolized as σmax, σmin, σa, σm, R and A, respectively. There are two sets of coordinate systems in this diagram.
The inside one is established by turning the referenced Goodman diagram 45° counterclockwise. The y-axis of the internal coordinate system repre- sents the alternating stress. It is coincident with the R = –1 (A = ∞) line where σmin = –σmax, σa = σmax, and σm = 0. The x-axis of the internal coordinate sys- tem reflects the mean stress. It is coincident with the R = 1 (A = 0) line, which represents a simple tensile test condition, σmin = σmax = σm, and σa = 0. The y-axis of the external coordinate system is marked with the maximum stress.
The x-axis of the external coordinate system is marked with the minimum stress; tension is on the right as a positive value, and compression is to the left as a negative value. The stress condition R = 0 and A = 1 is represented by the vertical line perpendicular to the minimum stress x-axis at σmin = 0, and to the right the dashed line represents another stress condition, R = 0.2, A = 0.67. There are also two sets of constant fatigue life curves, one in solid and another in dashed lines. The solid curves are boundaries confining the safe combined stress conditions for smooth unnotched specimens. The dashed curves apply to the notched specimens. The lower fatigue lives for the notched specimens are due to the effects of stress concentration at the notches.
R = –1, A = ∞
40 80 120 –120
–80 –40
120
40 80
160
–120 –80 –40 40 80 120 160
R = 1, A = 0 R = 0, A = 1
Maximum stress
Alternating stress
Minimum stress (ksi) 0
R = 0.2, A = 0.67
Mean stress 105
104 104
105
FIgurE 4.7
Master Goodman diagram of AISI 4340 steel. Data from Military Handbook 5, U.S. Department of Defense (Dieter, G. F., Mechanical Metallurgy, McGraw-Hill, New York, 1986, p. 386).
This master Goodman diagram is useful for predicting fatigue life with any two independent parameters. If the mean stress is 80 ksi and R = 0 (or A = 1), the estimated fatigue life for a smooth unnotched specimen can be predicted at the intersection of the two lines of constant σm = 80 ksi and R = 0 (or A = 1). In this case, the estimated fatigue life will be less than 104 cycles.
The superimposed curves and scales make Figure 4.7 a very busy diagram.
However, the complexity also allows predictions of fatigue life to be made using alternate data sets. The maximum and minimum stresses, σmax and σmin, are the two most commonly cited stress data in fatigue analysis. Their inclusion as the ordinate and abscissa coordinates in Figure 4.7 makes the fatigue life analysis easier, particularly for those who are not familiar with the terminologies used in fatigue analysis, such as R or A ratio. If a notched specimen is subject to a minimum stress, σmin = 20 ksi in a fatigue loading condition with R = 0.2, the corresponding maximum, mean, and alternating stresses will be σmax = 100 ksi, σm = 60 ksi, and σa = 40 ksi, respectively. The fatigue life can be estimated as approximately 3 × 104 cycles by locating the intersection of the two lines of constant σmin = 20 ksi and R = 0.2 in Figure 4.7.
Alternatively, the same fatigue life can also be found by locating the inter- section of the two lines σmin = 20 ksi and σmax = 100 ksi, or σm = 60 ksi and σa = 40 ksi, etc., because all these lines intersect at the same point.