The Strain Life ApproachThe Strain Life approach is used to predict the how long a product will survive due to cyclic loading.This article is intended to be an introduction of the Strain Life approach. The following is covered:1. Introduction to Strain Life
2. Elastic-Plastic Strain Time History and Ramberg-Osgood Relation
3. Stress-Strain Cycle Counting
4. Manson-Coffin-Morrow Curve and Damage Calculation
5. Mean Stress Influence and Damage Parameter
5.1 Smith-Watson-Topper Damage Parameter
5.2 Morrow Damage Parameter
6. Strain Life Results: Life Curves and Statistics1. Introduction to Strain LifeStrain Life differs from other approaches to fatigue because it takes into account both the elastic and plastic deformation of a material. Some approaches, like Stress Life, are only suitable when the stresses and strains remain in the elastic region of a material.To understand how strain life fatigue calculations work, it is helpful to understand the relationship between stress and strain. When a load is applied over an area, it creates stress. This stress in turn creates a strain, or elongation, as the part deforms (Figure 1).
Figure 1: Stress (force over area) and strain (change in length over original length) are used to predict how long an object can survive without a failure.
Subjected to cyclic stresses over time during usage, permanent deformation (when the part does not return to its original length when the load is removed) can occur as shown in Figure 2. In fact, the stress and strain may not maintain a linear relationship.
Figure 2: Time histories of stress (green) and strain (blue). Notice that the relationship is not always linear or proportional between stress and strain (left side). Permanent strain deformation also plays a role, where the test object gets a permanent offset (right side) over time.
Eventually enough cycles can be applied that the part fails (i.e., a crack forms, or object cannot carry a load). To understand when this failure could occur, it is necessary to understand the non-linear relationship between stress and strain as shown in Figure 3.
Figure 3: Right - When applying a stress to a material, initially the relationship between stress and strain is linear (elastic) and related by Young’s Modulus (blue area). As stress increases, the stress-strain relationship becomes non-linear (plastic) including necking and hardening (red area) until failure occurs.
At low stress levels, the relationship of stress to strain is linear (i.e., elastic) and related by the Young’s Modulus of the material. At higher stress levels, the stress to strain relationship is not linear (i.e., plastic). The Strain Life approach utilizes both elastic and plastic behavior.The Strain Life approach process is outlined in Figure 4.
Figure 4: There are three main steps to the Strain Life approach: 1) Determine the local stress-strain time history (blue – left), 2) Count the stress-strain cycles (middle – purple), 3) Calculate the damage from a strain life (EN curve) while including mean stress influences (right – yellow).
Starting with a measured strain, the strain life approach takes the following steps:
The local stress-strain time history can only predict the life at the location of the measurement. It cannot predict the life based on any other location.To determine if failure might occur at other locations on the structure, two possibilities exist:
To perform strain life analysis, check out the Knowledge article "Performing Strain Life Analysis in Simcenter Testlab Neo".2. Elastic-Plastic Strain Time History and Ramberg-Osgood RelationA typical starting point to determine the fatigue life of an object is with a measured strain time history (Figure 5) acquired while the part is being used. Experimentally, it is easier to measure the strain directly with a strain gauge, than to measure the stress.
- A corresponding stress at the same location as the measured strain is calculated (blue steps).
- This creates a local stress-strain time history which is cycle counted by plotting the stress against the strain (purple steps).
- The strain cycles are converted to damage to calculate the fatigue life using the Manson-Coffin-Morrow Strain Life curve (also called an EN Curve). This should also include effects of mean stress (yellow steps).
Figure 5: The Strain Life approach to predict damage or fatigue life requires local stress-strain time histories from the same location on the test object.
In order to calculate fatigue damage, the corresponding stress at the same location of the measured strain is needed. The Ramberg-Osgood relationship can be used to calculate the corresponding stress.The Ramberg-Osgood method’s place in the strain life process is outlined in blue in Figure 6.
Figure 6: With an elastic-plastic strain time history, the Ramberg Osgood relation is used to calculate local stress-strain time histories for cycle counting.
The Ramberg-Osgood relation (Figure 7) is an equation that describes the non-linear relationship between stress and strain, including materials that harden with plastic deformation.
Figure 7: The Ramberg-Osgood equation relates stress and strain of a material for both the elastic region (blue area, left) and plastic region (red area, right) behavior. Material properties needed include Young’s Modulus, cyclic hardening exponent, and cyclic hardening coefficient.
For every point in the measured strain time history, the corresponding stress (Figure 8) is calculated using the Ramberg-Osgood relation.
Figure 8: Ramberg-Osgood used to calculate corresponding stress for a measured strain at a given location on test object.
This stress-strain time history pair is called a “local” stress-strain time history.3. Stress-Strain Cycle CountingThe number and amplitude of potentially damaging cycles in the local stress-strain history need to be determined from the operating data. This is done by plotting the local stress-strain time history blocks against each other.The cycle counting step of the strain life process is highlighted in purple in Figure 9.
Figure 9: The elastic-plastic stress and strain time histories are plotted against each other (purple area) to determine the number of cycles and their respective amplitudes contained in the time history.
An example stress-strain plot is illustrated in Figure 10. From this plot, the number of cycles a part sees during usage can be determined.
Figure 10: In cycle counting for Strain-Life, the time history of stress (upper left, green) is plotted against the time history of strain (lower right, blue) at the same location on the structure. The resulting stress-strain plot (upper right, red) will contain “closed loops” (two in this case).
Note that the stress-strain plot (upper right, red line) contains curved lines, due to the non-linear relationship between stress and strain in the plastic region of the material.When a stress is applied and removed, the strain does not return to its original level because there is permanent plastic deformation. This behavior creates what appears to be loops in the stress-strain plot.There are two loops in Figure 10. These correspond to two complete strain cycles:
Some other observations:
- Cycle 1: Points 2 and 3 have a smaller range than points 1 and 4. Because 2 and 3 are encapsulated by 1 and 4, they form a “loop” or cycle.
- Cycle 2: Points 5 and 6 also form a “loop” or cycle with points 4 and 7.
For more information on residuals, turning points, and cycle counting, see the Knowledge base article “Rainflow Counting”.One of the strain cycles is shown in more detail in Figure 11.
- The formation of these loops only depends on the “turning points” (where the slope changes as indicated by numbers in the graph) in the stress and strain time histories.
- The two cycles are contained within a larger unclosed cycle (between turning points 1, 4, and 7). When calculating cycles from a time history, it is possible that some do not close fully. This is referred to as a residual cycle.
Figure 11: Closed loop cycle showing stress and strain ranges, strain cycle amplitude, and mean stress.
The result of the cycle counting is a histogram, or data distribution, which contains the strain cycle amplitudes, mean stresses of each cycle, and the number of cycles.A common method of viewing the results of cycle counting is a strain cycle histogram (Figure 12).
Figure 12: The strain cycle histogram shows the distribution of the amplitudes of the cycles (Y-axis) and their cumulative total (X-axis) contained in the measured strain signal.
The strain cycle histogram shows the amplitude of the strain cycles on the Y-axis. They are shown in descending order of amplitude from left to right. On the X-axis, the cumulative total of these cycles are shown. For example, there might be one cycle with 102 microstrain amplitude and three cycles with 100 microstrain amplitude. The graph would show two data points: one cycle at 102 microstrain, and four cycles (the accumulated total) at 100 microstrain. The stress for each cycle is still available, but not shown in this particular display.The cycles can now be converted into damage to calculate the fatigue life as described in the next section.4. Manson-Coffin-Morrow Curve and Damage CalculationThe Manson-Coffin-Morrow Strain Life curve (also called a EN curve) characterizes the number and amplitude of strain cycles a material can take before failing. Its place in the strain life process is shown in Figure 13.
Figure 13: The Manson-Coffin-Morrow (EN) curve is used to convert strain cycle amplitudes and occurrences into equivalent damage, with adjustments for mean stress influence. This part of the strain life process is highlighted in yellow.
The Manson-Coffin-Morrow relation describes the strain-life curve for a material (Figure 14) based on strength properties, ductility properties, and Young’s Modulus. A sample of a material, or coupon, is tested in a laboratory to determine these properties.
Figure 14: The Manson-Coffin-Morrow is a mathematical model relating total strain and cycles to failure. The total strain amplitude (green) is equal to the sum of elastic strain (blue) and the plastic strain (red) terms. Plastic strain behavior dominates at low cycle counts with high amplitude strain cycles. Elastic strain dominates at high cycle counts with low amplitude strain cycles.
The strain-life curve (also referred to as the EN curve) is property of the material itself. It contains both an elastic portion (blue dashed line above) and a plastic portion (red dashed line above). It relates the total cyclic amplitude (green line above) of the strain cycle to the number of cycles until the part fails. It is similar in concept to the SN Curve used in stress life calculations.Damage can be calculated using the cycles from the time history data and comparing it to Manson-Coffin-Morrow curve. The number (n) of strain cycles at a specific amplitude in the field data are compared to the number of cycles to failure (N) the material can withstand at that same amplitude from the EN Curve. The damage (D) is calculated as shown in Figure 15.
Figure 15: With two strain cycles (n) at a specific strain amplitude from field data, the corresponding number of cycles to failure (N) from the EN curve are used to calculate the fatigue damage (D). In this case, the two cycles have the same strain amplitude, but a different mean stress. Here the mean stress difference is ignored.
The damage (D) is calculated by dividing two (n) by ten (N). Damage (D) is equal to 0.20. Failure occurs when damage (D) is equal to one. The part could withstand the load history being applied five times in total (the inverse of D) before failing.Other Considerations when using the Manson-Coffin-Morrow curve (EN Curve):
In real life, there are typically thousands of strain cycles with many different amplitudes, so the damage needs to be calculated and summed for all the different cyclic amplitudes present in the input time history block. See the knowledge base article “Calculating Damage with Miner’s Rule” for more information.Trivia: The equation is named after L.F. Coffin and S.S. Manson. Coffin (Knolls Atomic Power in Niskayuna, New York) and Manson (NASA Lewis in Cleveland, OH) published their findings on strain life separately in 1953 and in 1954.5. Mean Stress Influence and Damage ParameterWhen there is a mean stress offset in the stress cycles, the damage calculation needs to be adjusted accordingly. Mean stress can make the part accumulate more or less damage depending if it is in compression or tension (Figure 16).
- Limits: The Manson-Coffin-Morrow equation itself has no limits. When doing a fatigue life calculation, a static stress (failure due to one cycle) and endurance limit (infinite cycles without failure) are needed to define an upper and lower limit for the equation.
- Mean Stress Influence: The mean stress of each cycle creates additional compression or tension that affects the fatigue life. In Figure 14, the two strain cycles were given equal weighting and the difference in mean stress was not considered. Methods for addressing the mean stress are discussed in the next section.
The Manson-Coffin-Morrow curve was created assuming cycles of zero mean stress:
Figure 16: Left – A material coupon undergoing cyclic stress with positive mean stress (tension). Right - A material coupon undergoing cyclic stress with negative mean stress (compression).
To account for the influence of mean stress in accumulating damage, a damage parameter (P) is used (Figure 17).
- A part under a positive mean stress, or tension, will last shorter versus zero mean stress. As it is cycling, the positive mean stress pulls the coupon apart.
- A part under a negative mean stress, or compression, will last longer versus zero mean stress. As it is cycling, the negative mean stress keeps the coupon together.
Figure 17: Damage calculation (D) adjusted for damage parameter (P).
The damage parameter (P) is a quantity that represents both stress and strain behavior in a single number. The fatigue life can be calculated using the damage parameter P as the basis, rather than only strain:
If it helps, think of the P values as a “parallel universe” to the strain-based damage calculation. The PN Curve is like the strain EN curve, and the P value of a cycle is like the strain cycle amplitude. The difference between the two “universes” is that mean stress influence is accounted for when using P based damage calculations, while mean stress is ignored when using strictly strain.An example PN Curve is shown in Figure 18.
- Cycles from Input Time History: Instead of a histogram of strain cycle amplitudes, a histogram of P values is calculated for each cycle. The P value of a cycle is calculated from both the amplitude of the strain and mean stress of the cycle. Different formulas (Smith-Watson-Topper, Morrow, etc.) for calculating the damage parameter of a cycle are described in the next section.
- Material Curve of Test Object: Instead of the Manson-Coffin-Morrow EN Curve, a PN Curve will be used. The PN curve relates P values to the Number of Cycles (N) to failure. The PN curve is calculated from the EN curve using the formulas (Smith-Watson-Topper, Morrow, etc.) in the next section, but the mean stress value is set to zero.
Using the PN Curve in conjunction with the P values of the cycles, fatigue damage can be calculated taking mean stress influence into account (Figure 19).
Figure 18: The PN curve (blue) relates the damage parameter (P) to the number of cycles to failure (N) for the material.
Figure 19: Left – Using the strain cycle amplitudes and ignoring mean stress, the predicted life for the highest cycle is calculated using the EN Curve. Right – A different value for the predicted life of the first cycle is calculated when using corresponding P value of the cycle and PN Curve. The P value and PN Curve approach takes into account the influence of mean stress.
In Figure 19, the difference between the strain only results and P value results are shown for the single largest amplitude of the cycle histogram. Depending on the mean stresses in the cycles, the fatigue life is either reduced or increased when using the damage parameter (P) versus a zero mean stress assumption. If the cycle is in net tension, then the fatigue life is reduced. If the cycle is in compressive stress, then the life is increased. The calculations for damage are done for all the P values in the histogram and summed.Two different methods for calculating the damage parameter (P) for a cycle are explained next. The Smith-Watson-Topper method is discussed first.5.1 Smith-Watson-Topper Damage Parameter (P)One method for calculating the damage parameter (P) for a cycle is the Smith-Watson-Topper (SWT) method, which is shown in Figure 20.
Figure 20: The Smith-Watson-Topper equation (left) is used to adjust for mean stress effects (magenta dashed line) in a closed strain cycle (red curve).
The Smith-Watson-Topper (SWT) equation calculates a damage parameter (P). This damage parameter calculates a stress based value based on the following inputs:
Stress Amplitude (sigma a): Amplitude of the closed cycle.
Mean Stress (sigma m): Stress offset on cycle from zero.
Strain Amplitude (epsilon a): Amplitude of the closed cycle.
Young’s modulus (E): Ratio of the stress over strain from the elastic region of a material.
Possible results for the damage parameter P from SWT are:
More Damage in Tension: A positive mean stress is added to the alternating stress, creating a new P-value greater than the original stress cycle amplitude. This higher amplitude creates more damage.
No Adjustment: If the mean stress is zero, then the result of the SWT calculation is an unadjusted alternating stress. The equation under the square root is simplified to the alternating stress squared, which when square rooted is the original alternating stress amplitude.
No Damage in Compression: If the mean stress (sigma m) is greater than alternating stress (sigma a) and negative (i.e, compressive mean stress), then the P-value of the Smith-Watson-Topper is the square root of a negative number. This results in no damage being created by the cycle.
The Smith-Watson-Topper method only considers tensile mean stress and ignores compression. Other methods, like Morrow, accumulate damage under compressive mean stress.
5.2 Morrow Damage Parameter (P)
The Morrow equation is shown in Figure 21. Just like Smith-Watson-Topper, a damage parameter (P) is calculated which takes into account the mean stress influence.
Figure 21: Calculation of the Morrow damage parameter.
Unlike Smith-Watson-Topper, which adjusts the stress, the Morrow method creates a strain based P-value. And unlike Smith-Watson-Topper, damage can accumulate for compressive mean stress.
8. Strain Life Results: Life Curves and Statistics
After calculating the fatigue damage using the strain life method, the results are typically shown as Life Curves and Life Statistics (Figure 22).
Figure 22: The “Life Curve” output of the strain life method is used to determine the “Life Statistics”. The number of times that the analyzed time input can be applied until the part fails is calculated.
The “Life Curves” comprise three different functions. These functions are the based on the results of the damage calculations.
Histogram: A cycle histogram (red curve in Figure 22) contains the cumulative number of strain cycles and their respective amplitudes present in the input time signal.
Design Point: A single point (green dot in Figure 22) defined by the intersection of the predicted number of cycles to failure (the X value position, dotted blue) and maximum cycle amplitude (the Y value position, dotted orange). The number of cycles to failure is based on the fatigue life calculations, using both the input time history and the material properties to calculate how many cycles of the current time history that the part can survive.
Life Curve: By scaling the input time history both higher and lower, the sensitivity of the design point to changes in cycle amplitude can be determined. By calculating design points for various amplitude scaling, both lower and higher than the original cycle amplitudes, a full life curve (blue curve in Figure 22) is created. This life curve will have a very similar slope to the material property curve.
From these life curves, a single number output called the “Design Point Block” is calculated. The “Design Point Block” is the number of times, or repeats, that the input time history can be applied before failure occurs. As shown in Figure 22, this is the ratio of the number of cycles to failure (blue arrow) versus the cycles in the input signal (red arrow).
The Life Statistics output of the Simcenter Testlab Neo Strain-Life method (Figure 23), shows the number of “Design Point Blocks” in numerical form.
Figure 23: The Design Point Blocks (highlighted in orange) is the number of times the input signal can be repeated before the part fails. In this case, it is 125.38 repeats of the input signal.
For example, if the originally measured strain time history was 60 seconds, and the design point block repeat was 125.38, then the part could survive 7522 seconds of the original time history being applied repeatedly to the part. Keep in mind, this is only applicable to location where the measurement was performed. It could fail sooner at other locations that were not measured or analyzed.
It is also possible that the part would not survive a single application of the input time history block as shown in Figure 24.
Figure 24: The Design Point Block is less than one (0.93 in this case), because failure occurs during the application of the original, non-repeated, input time history.
In this case, the part fails during the application of the original input time history. Usually this would already be known from the original test!
If the amplitudes are low enough, then the part could live forever as shown in Figure 25.
Figure 25: If the amplitude of the histogram cycles of the measurement do not exceed a specific level, there is no Design Point. The input time history could be applied an infinite number of times without causing failure.
When the projection of the amplitudes from the histogram does not intersect the life curve, the design point is undefined. In this case, without a Design Point, an infinite number of repeats (Figure 25) of the original time history can be applied without the part failing.
The Strain Life approach predicts how long a product will survive due to for both elastic and plastic loads.
The Strain Life approach requires a local stress-strain time history (observed at same location) to determine the cycles and corresponding damage to the product.
To account for both the strain and mean stress, the fatigue life calculations performed using a damage parameter (P).