Mean stress and damage parameters

The strain-life equation has been modified to include mean stress effects.

The concept of damage parameters

The Manson-Coffin-Morrow equation connects the number of cycles to crack initiation only to the total strain amplitude. To include effects of mean stress or other sequence effects, several modifications to the Manson-Coffin-Morrow equation may be found in literature.

Subsequently, we will use the concept of damage parameters to describe the different approaches. The idea of a damage parameter is to separate the additional effect of, for example, mean stress from the Manson-Coffin-Morrow equation by introducing an additional quantity which defines events of equivalent damages in a similar manner that the Goodman line and so on in the stress-life approach defines the equivalent stress amplitude.

You often find approaches in the literature that use terms plugged into the Manson-Coffin-Morrow equation. This has the drawback that it is complicated to understand the mean stress influence. The parameters described below will be presented in a slightly different format.

Smith-Watson-Topper type parameters

A common damage parameter is the Smith-Watson-Topper (SWT) damage parameter, PSWT.

Once a closed hysteresis loop has been identified, the value of the damage parameter is calculated from

where σa and εa are the stress and strain amplitudes of the hysteresis loop, respectively, E is Young's modulus, and σmax and σm are the maximum and mean stress of the hysteresis loop. It is related to the reversals to failure by

which is equivalent to the description often found in literature

The Smith-Watson-Topper parameter includes the influence of stress and strain amplitude and the mean stress to the damage. It has proven to be in accordance with physical tests in a variety of cases.

Note:

If the hysteresis loop is completely under pressure (that is, σmax < 0), then the Smith-Watson-Topper parameter is set to 0. This means independent of the size of the loop there will be no damage. Using the Mean Stress plot in Specialist Durability, you can check whether the largest loops are under pressure. In this case we recommend to double check with a Morrow type damage parameter.

Modifications of the Smith-Watson- Topper damage parameter

For shear stresses a modified version of PSWT has been implemented since the damage does not depend on the direction of τm

Here G is the shear module and calculated as

with the Poisson ratio ʊ given in the material parameters.

The Bergmann parameter with parametric mean stress influence

The Bergmann parameter

adds a free parameter k that enables you to adapt the mean stress influence to an actual problem. The parameter k should be chosen larger or equal to 0, where 0 means no mean stress influence and a value of 1 leads to the mean stress influence of the Smith-Watson-Topper parameter. A value greater than 1 leads to an even larger mean stress influence.

No mean stress influence

This option just uses the Manson-Coffin-Morrow relationship for damage calculation.

Linearized SWT parameter life curve

Usually the parameters for the Manson-Coffin-Morrow relationship are determined from tests between a maximum load and endurance limit. The plastic dominated part of the curve is extrapolated to very small cycle numbers. Because of this extrapolation, the corresponding plastic strains usually get too high.

To stay conservative in the region where there is usually no test data, the linearized version of the PSWT life curve takes this into account. It also uses a reduced endurance limit for variable amplitude loads: If the overall load leads to cycles above the endurance limit, cycles having a damage parameter value bigger than the half of the damage value at endurance limit are assumed to have a damaging effect.

Linearized Against Original PSWT Damage Parameter Life Curve

Morrow damage parameter

The idea of the Morrow parameter is to shift the elastic part of the Manson-Coffin-Morrow relation according to a mean stress influence:

where εa,p gives the amplitude of the plastic strain.

Mean Stress Influence of Morrow Type

Note:

There exist various mean stress influences of the Morrow type. The one used here is a modification of which may be found in Nihei, M., Heuler, P., Boller, Ch., and Seeger, T. "Evaluation of mean stress effects on fatigue life by use of damage parameters," Int J. Fatigue, 8(3), pp 119–126, 1986.

If σm / σʹf exceeds 0.999, the elastic strain amplitude is multiplied by 1000.

The Morrow parameter is frequently used in the United States. In cases of mainly compressive stress states the Morrow damage parameter may be more convenient than the Smith-Watson-Topper parameter.

It shows a higher mean stress influence than the Smith-Watson-Topper parameter and in most cases it is more conservative than the other parameters.

Parameter	Solution Parameter Name	Definition
Smith-Watson- Topper	P_SWT Original****P_SWT Linear
Smith-Watson- Topper for shear stresses	P_SWT Torsion****P_SWT Torsion Linear
Bergmann	P_Bergmann
No mean stress influence	None	P = εa
Morrow type	Morrow

Single amplitude damage parameter life curves

The number of reversals to failure is not calculated for each loop identified individually. Instead, a single amplitude damage parameter life curve is calculated in the beginning, giving a functional relation between a damage parameter value and the number of sustainable cycles at the given damage parameter value.

For the single amplitude damage parameter life curves, we have to distinguish between two types:

• Functional type: PSWT-orig, PSWT- Torsion, P-Bergmann, No Mean Stress influence, and P- Morrow

• Linearized type: PSWT-linear, PSWT-Torsion-linear, and PJ (Vormwald)

The multilinear approximation of the functional type

The approximation is done in four steps.

1. Subdivide the interval [1, NE] into M parts1 = N0 < N1 < ... < NM = NEequidistant in the log-scale, that is

2. Get the matching stress and strain amplitudes (σa,i, εa,i) from the Manson-Coffin-Morrow and the Ramberg-Osgood–relationship.From the R-ratio of the Manson-Coffin-Morrow relationship, the maximums and minimums of stress and strain are calculated (σu,i, σ1,i, εu,i, ε1,i).

3. Calculate the damage parameter values:Pi = P(σu,i, σ1,i, εu,i, ε1,i)

4. Interpolate the tuple (Pi, Ni) linearly in the log-log-scale.For values of N larger than NE, the interpolation between (PM-1,N M-1) and (PE, NE) is prolonged. This reflects the assumption that in the point of the endurance limit the Manson-Coffin-Morrow relation can be approximated just by the strain part.

Linearization

For the linearization, the interval [NS, NE], where NS and NE are given in the solution parameters (Start linearization at, End linearization at), is subdivided like in the latter section. We achieve the tuple

(P0, N0),....,(PL,NL)

as in the section above.

The linearization is done on the tuples transformation into the log scale:

(log P0, log N0),....,(log PL, log NL)

by linear regression. Since the point of the endurance limit (log PE, log NE) does not need to be part of the regression line

log P = a log N + b

we have to redefine this point. This is done by shifting log NE onto the regression line.

Parameter	Symbol	Preset to
Start Linearization At	NS	1000
End linearization At	NE	2 · 105

Use of damage parameter life curves

In the use of the equations, the fatigue life in reversals to failure, can be computed once the stress and strain amplitudes of a cycle have been determined. Closed cycles, or hysteresis loops, can be determined by methods described in Concepts in common. Once the damage for that event has been determined then the damage can be accumulated by using Miner's rule.

To define these parameters in Specialist Durability, use the Simulation Objects command and create a strain life damage parameter simulation object.

Modifying factors for the damage parameter life curve

In the stress-life approach the influence of size and surface effects are taken from the relation

σE,mod = nSize nSurface σE

For further details, please refer to the topic Non-local and surface effects.

To account for this, the damage parameter life curves are shifted by nsize nsurface. (Since the Vormwald parameter depends on the square of the stress amplitude, the square of the correction factors is used for the shift in this case.)

Shift of the PSWT Damage Parameter Life Curve Due to Surface Effects

Shift of the PSWT Damage Parameter Life Curve Due to Size Effects

Learn more

Local stress-strain behavior

Constant amplitude life curves

Endurance limit and static failure

Determining material properties

Axial versus torsion tests

Notch analysis

The strain-life analysis in Specialist Durability

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