Stress-Life Diagram (S-N Diagram)
The basis of the Stress-Life method is the Wohler S-N diagram, shown schematically for two materials in Figure 1. The S-N diagram plots nominal stress amplitude S versus cycles to failure N. There are numerous testing procedures to generate the required data for a proper
S-N diagram. S-N test data are usually displayed on a log-log plot, with the actual S-N line reprenting the mean of the data from veral tests.
梦见陌生人死了
Figure 1: Typical S-N Curves
Endurance Limit
Certain materials have a fatigue limit or endurance limit which reprents a stress level below which the material does not fail and can be cycled infinitely. If the applied stress level is below the endurance limit of the material, the structure is said to have an infinite life. This is characteristic of steel and titanium in benign environmental conditions. A typical S-N curve corresponding to this type of material is shown Curve A in Figure 1.
蝴蝶的眼睛Many non-ferrous metals and alloys, such as aluminum, magnesium, and copper alloys, do not exhibit well-defined endurance limits. The materials instead display a continuously decreasing S-N respon, similar to Cuve B in Figure 1. In such cas a fatigue strength S f for a given number of cycles must be specified. An effective endurance limit for the materials is sometimes defined as the stress that caus failure at 1x108 or 5x108 loading cycl es.
The concept of an endurance limit is ud in infinite-life or safe stress designs. It is due to interstitial elements (such as carbon or nitrogen in iron) that pin dislocations, thus preventing the slip mechanism that leads to the formation of microcracks. Care must be taken when using an enduranc
e limit in design applications becau it can disappear due to:
• Periodic overloads (unpin dislocations)
感统器材• Corrosive environments (due to fatigue corrosion interaction)
• High temperatures (mobilize dislocations)
The endurance limit is not a true property of a material, since other significant influences such as surface finish cannot be entirely eliminated. However, a test values (S e') obtained from polished specimens provide a baline to which other factors can be applied. Influences that can affect the endurance limit include:
• Surface Finish
• Temperature
• Stress Concentration
• Notch Sensitivity
• Size
• Environment
• Reliability
Such influences are reprented by reduction factors, k, which are ud to establish a working endurance strength S e for the material:
Power Relationship
李氏起源
When plotted on a log-log scale, an S-N curve can be approximated by a straight line as shown in Figure 3. A power law equation can then be ud to define the S-N relationship:
where b is the slope of the line, sometimes referred to as the Basquin slope, which is given by: Given the Basquin slope and any coordinate pair (N,S) on the S-N curve, the power law equation calculates the c ycles to failure for a known stress amplitude.
育苗技术
Figure 3: Idealized S-N Curve
The power relationship is only valid for fatigue lives that are on the design line. For ferrous metals this range is from 1x103 to 1x106 cycles. For non-ferrous metals, this range is from
1x103 to 5x108 cycles. Note the empirical relationships and equations described above are only estimates. Depending on the level of certainty required in the fatigue analysis, actual test data may be necessary.
Fatigue Ratio (Relating Fatigue to Tensile Properties)
Through many years of experience, empirical relations between fatigue and tensile properties have been developed. Although the relationships are very general, they remain uful for engineers in asssing preliminary fatigue performance.
The ratio of the endurance limit S e to the ultimate strength S u of a material is called the fatigue ratio. It has values that range from 0.25 to 0.60, depending on the material.
For steel, the endurance strength can be approximated by:
and:
In addition to this relationship, for wrought steels the stress level corresponding to 1000 cycles, S1000, can be approximated by:
Utilizing the approximations, a generalized S-N curve for wrought steels can be created by connecting the S1000 point with the endurance limit, as shown in Figure 4.
Figure 4: Generalized S-N Curve for Wrought Steels
Mean Stress Effects
Most basic S-N fatigue data collected in the laboratory is generated using a fully-reverd stress cycle. However, actual loading applications usually involve a mean stress on which the oscillatory stress is superimpod, as shown in Figure 5. The following definitions are ud to define a stress cycle with both alternating and mean stress.
The stress range is the algebraic difference between the maximum and minimum stress in a cycle:
The stress amplitude is one-half the stress range:
The mean stress is the algebraic mean of the the maximum and minimum stress in the cycle: Two ratios that are often defined for the reprentation of mean stress are the stress ratio R and the amplitude ratio A:
For fully-reverd loading conditions, R is equal to -1. For static loading, R is equal to 1. For a ca where the m ean stress is tensile and equal to the stress amplitude, R is equal to 0. A stress cycle of R = 0.1 is often ud in aircraft component testing, and corresponds to a tension-tension cycle in which the minimum stress is equal to 0.1 times the maximum stress.
Figure 5: Typical Cyclic Loading Parameters
The results of a fatigue test using a nonzero mean stress are often prented in a Haigh diagram, shown in Figure 6. A Haigh diagram plots the mean stress, usually tensile, along the x-axis and the
oscillatory stress amplitude along the y-axis. Lines of constant life are drawn through the data points. The infinite life region is the region under the curve and the finite life region is the region above the curve. For finite life calculations the endurance limit in any of the models can be replaced with a fully reverd (R = -1) alternating stress level corresponding to the finite life value.
企划文案Figure 6: Example of a Haigh Diagram
A very substantial amount of testing is required to generate a Haigh diagram, and it is usually impractical to develop curves for all combinations of mean and alternating stress. Several empirical relationships that relate alternating stress to mean stress have been developed to address this difficulty. The methods define various curves to connect the endurance limit on the alternating stress axis to either the yield strength, S y, ultimate strength S u, or true fracture stress σf on the mean stress axis. The following relations are available in the Stress-Life module:
Goodman (England, 1899):
Gerber (Germany, 1874):
招财鼠Soderberg (USA, 1930):
Morrow (USA, 1960s):
A graphical comparison of the equations is shown in Figure 7. The two most widely accepted methods are tho of Goodman and Gerber. Experience has shown that test data tends to fall between the Goodman and Gerber curves. Goodman is often ud due to mathematical simplicity and slightly conrvative values. Other obrvations related to the mean stress equations include:
• All methods should only be ud for tensile mean stress values.
• For cas where the mean stress is small relative to the alternating stress (R << 1), there is little difference in the methods.
• The Soderberg method is very conrvative. It is ud in applications where neither fatigue failure nor yielding should occur.
• For hard steels (brittle), where the ultimate strength approaches the true fracture stress, the Morrow and Goodman curves are esntially equivalent. For ductile steels
(σf > S u), the Morrow model predicts less nsitivity to mean stress.
• As the R approaches 1, the models show large differences. There is a lack of experimental data available for this condition, and the yield criterion may t design
limits.
Figure 7: Comparison of Mean Stress Equations
los
