What is a SN-Curve?
A SN-Curve (sometimes written S-N Curve) is a plot of the magnitude of an alternating stress versus the number of cycles to failure for a given material. Typically both the stress and number of cycles are displayed on logarithmic scales.
Given a load time history and a SN-Curve, one can use Miner's Rule to determine the accumulated damage or fatigue life of a mechanical part.
SN-Curves were developed by the German scientist August Wöhler (Figure 1) during the resulting investigation of an 1842 train crash in Versailles, France. In this crash, the axle of the train locomotive failed under the repeated “low level” cyclic stress of everyday usage on the railroad.
While investigating, Wöhler discovered that cracks formed and slowly grew on an axle surface. The cracks, after reaching a critical size, would suddenly propagate and the axle would fail. The level of these loads was less than the ultimate strength and/or yield strength of the material used to manufacture the axle.
Wöhler developed an apparatus to apply repeated loads to railroad axles and chart the relationship between load level and number of repeated cycles to failure. “Wöhler Curves” plot the relationship of alternating/cyclic stress levels against the number of cycles to failure.
Designing an axle to withstand the initial static loads associated with holding up a locomotive was well understood. It was fairly obvious if an axle could carry the weight of a train if it did not collapse immediately. The concept of low level cyclic stresses, repeated over a long time, was relatively new and not well understood. For many observers at the time, it seemed unpredictable when an axle might suddenly fail. It was not until Wöhler developed his SN-Curves that cyclic stress became better understood, and fatigue life could be predicted in a more consistent manner.
How is a SN-Curve created for a given material?
Today, these curves are often developed by using a metal coupon testing machine ( 2). A small metal coupon is placed into the machine and subjected to a cyclic (or alternating) stress time history until a crack or failure occurs in the metal coupon.
Several coupons must be tested at different stress levels to develop a SN-Curve. 3 illustrates a typical SN-Curve derived from testing metal coupons.
A SN-Curve functions as a “lookup table” between alternating stress level and the number of cycles to failure ( 4). Most SN-Curves generally slope downward from the upper left to the lower right. This indicates that high level amplitude cycles have fewer number of cycles to failure compared to lower level amplitude cycles.
In a fatigue test like this, the frequency at which the cycles are applied is not considered to be a factor in the number of cycles to failure. It is strictly the number of cycles, and not the rate at which the cycles are applied, that affect the SN-Curve results.
In real life, the frequency of the cycles can be a factor, especially if the loading frequency coincides with a natural frequency or resonance of the object which amplifies the magnitude of the cycles.
Plastic, Elastic and Infinite Regions
A SN-Curve can contain several different areas: a plastic region, an elastic region and an infinite life region as shown in 5.
There are three key values that separate the plastic, elastic and infinite life regions (Figure 5):
Several of the values on the SN-Curve can be found by doing a static stress-strain test on a material coupon (see 6).
For example, the Ultimate Strength stress is the value that causes a failure for one cycle. The Yield Strength stress level divides the plastic and elastic region.
Some materials, like steel, exhibit an infinite life region ( 7). In this region, if the stress levels are below a certain level, an infinite number of cycles can be applied without causing a failure (of course, no test has been performed for an infinite number of cycles in real life, but a million+ cycles is typical).
Critical components (ie, engine crankshafts and rods) are usually designed for infinite life because cycles are speed dependent and accumulate quickly. All the cyclic stress levels that the part is subjected to must be below the endurance limit to have infinite life.
Infinite life is not in effect under certain conditions:
Different metals have different endurance limits. Some typical endurance limits are show in Table 1.
Many non-ferrous metals and alloys, such as aluminum, magnesium, and copper alloys, do not exhibit well-defined endurance limits ( 8).
Where steel has a definite change in slope at the endurance limit, aluminum and other metals do not always have a distinct change.
To determine if a part or object is operating in the infinite life region, the Goodman-Haigh diagram is often used. In addition to the cycle amplitude, mean stress is also accounted for in the Goodman-Haigh approach.
In the elastic region ( 9), the relationship between stress and strain remains linear. When a cycle is applied and removed, the material returns to its original shape and/or length. This region is also referred to as the “High Cycle Fatigue” region, because a high number of stress cycles, at a low amplitude, can cause the part to fail.
Typical factors that influence the performance of a material in the elastic region are residual stresses and geometric considerations. For example, a severe geometry change in the material may be more likely to have a crack initiate than a smooth geometry change.
In the plastic region (Figure 10), the material experiences high stress levels, causing the shape and/or geometry to change due to the repeated application of stress cycles. This region is also referred to as the “Low Cycle Fatigue” region of the SN-Curve, where a low number of stress cycles, with a high amplitude, result in failure.
Material plasticity and geometry are big influences on the number of cycles to failure in the plastic region.
Calculating fatigue life or damage in the plastic region of a material with a SN-Curve is probably best avoided. If cyclic stress levels are in the plastic region, a strain life approach would typically be recommended instead, which includes an E-N (Strain vs Number of cycles) as part of the analysis. Strain life also takes into account the order or sequence in which loads are applied.
Tests for materials can be expensive to run. Ideally, the tests should be repeated many times and at many different stress levels. With enough experiments, the SN-Curve would consist of a series of confidence intervals around the main curve as shown in Figure 11.
Some materials have well known curves because they are very commonly used. Some materials do not. When a new alloy is developed, the SN-Curve may be completely unknown and testing will be required to determine the curve. Conventionally, five different stress levels with three repeats at each level is considered the minimum to determine a SN-Curve.
The book “FKM Analytical Strength Assessment” contains many material SN-curves. It is published by the VDMA (Verband Deutscher Maschinen- und Anlagenbau e.V.), a German engineering association of engineering companies, which includes Siemens.
Logarithmic Nature of SN-Curve: Double amplitude vs Double cycles example
Consider the following hypothetical time histories to be evaluated for damage:
First consider time history #1 and time history #2 shown in Figure 12. These time histories have:
Using Miner's Rule, one sees that the cumulative damage of time history #2 is double compared to time history #1 (see Figure 13).
Next compare the damage potential in load time history #1 vs time history #3 shown in Figure 14. These time histories have:
Using Miner’s Rule to analyze the load time history with a SN-Curve, one sees that the cumulative damage of time history #3 is 20 times compared to time history #1 (Figure 15), even though the amplitude of the cycles are only double in time history #3 compared to time history #1.
Why does doubling the stress level result in twenty times the damage? This is because the SN-Curve is actually a log vs log graph, which is easy to forget when viewing what appears to be straight lines in the log-log respresentation of the SN-Curve (Figure 16).
The relationship between stress level and number of cycles to failure is not linear, which has very important implications for fatigue life.
SN-Curve Slope: K-factor
The slope of the log-log SN-Curve is defined by “k-factor”. This "k-factor" governs the relationship between the stress level and the number of cycles to failure.
The "k-factor" was developed by Wöhler to easily relate the load (ie, stress) to the life (number of cycles to failure). Figure 17 shows how the Wöhler curve relates load to life via the k-factor in the elastic region of a SN-Curve.
Because of this log vs log relationship, it means that a small change in load amplitude can have a very large change in the fatigue life or damage. In Table 2 below, with k-factor of 5, a 15% change in load results in a factor of 2 change in damage/fatigue life.
The logarithmic relationship between alternating stress level and the number of cycles to failure is an important consideration in accelerating fatigue testing. As the k-factor gets larger, small increases in load (ie, stress) create larger and larger changes in life. This can be used to accelerate a durability test. By increasing the load a small amount, so that the failure mode is not changed, one can still get large reductions in test time.
As a general rule of thumb, one can associate the following k-factors with the following:
SN-Curve adjustments due to Mean Stress
When using SN-Curves, there can be extenuating circumstances where the SN-Curve must be adjusted to reflect certain situations.
Take the stress time history in Figure 18. The average or mean stress of the cycles is zero.
Why track the mean stress? In real world loading situations, there could be a mean stress other than zero acting on the part. For example, the suspension system of a car has to carry the static weight (or load) of the car. As the car drives on the road, cyclic stresses/loads are applied by bumps in the road while the car weight is applying a mean stress (which is not zero).
There are two different types of mean stress that a part may encounter: tension and compression.
In the case of tension (Figure 19), there is a positive mean stress. In a metal coupon test, a static tensile mean stress creates a load that tries to pull the coupon apart.
This additional tension reduces the number of cycles to failure. The part would fail sooner than the SN-Curve with mean stress of zero would predict.
The static mean stress could also be pushing the part together, creating compression (Figure 20). This compression would extend the life of the part, making it last longer than the zero mean stress SN-Curve would predict.
Mean stress effectively shifts the SN-Curve up or down (Figure 21). A tensile mean stress in effect shifts the SN-Curve downward so it takes fewer number of cycles to fail. A compressive mean stress shifts the SN-Curve upward so the number of cycles to failure is higher.
Typically, SN-Curves are developed for a specific “Stress Ratio” as shown in Figure 22. The “stress ratio” called R is the lower value of the stress divided by upper value of the stress in cyclic stress time history. It is a convenient way to designate the conditions for a SN-Curve test. For example, in the aerospace industry, many components are tested with a stress ratio of 0.1, which ensures a net tension on the component.
For fully-reversed loading conditions with mean stress of 0, R is equal to -1. For static loading, R is equal to 1. For a case where the mean stress is tensile and equal to the stress amplitude, R is equal to 0.
For more information, read the article on Mean Stress and Stress Ratios.
SN-Curve Adjustments due to Loading
How the load is applied to the structure makes a difference in the number of cycles to failure. The load can be applied in several ways: torsional, bending, axially, etc as shown in Figure 23.
The load scaling correction factors (Cf) are different and depend on the material. The correction factors are used to scale the stress up or down based on the type of load being applied. In the case of Bending vs Torsion for the material in Figure 23, the adjustment is 40%.
Other SN-Curve Adjustments
There are many other reasons why a SN-Curve may need to be adjusted. Other SN-Curve adjustments include part size, surface finishes, notches in the geometry, etc.
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