This module calculates the fatigue life of a part under variable amplitude loading using constant amplitude fatigue life data.

The fatigue life of the structure can be represented by the following equation:

where:

	- Total fatigue cycles to failure.
	- Number of cycles to initiate a crack.
	- Number of cycles to propagate a crack to failure.

This module deals with fatigue damage during the crack initiation phase.  Damage during the initiation phase can be related to dislocations, slip bands, microcracks, etc.  Since these phenomena can only be measured in a highly controlled laboratory environment, most damage summation approaches for the initiation phase are empirical in nature. These methods relate damage to the expended life for a small laboratory specimen.  For this purpose, life is defined as the separation of a specimen, which is equivalent to the formation of a small crack in a large component or structure.

The service loading time history of an actual engineering part can be quite complex, as shown in Figure 1. Several methods have been developed to deal with variable amplitude loading using baseline data generated from constant amplitude tests. These damage summation methods can be used in conjunction with either the Stress-Life or Strain-Life methods of constant amplitude fatigue analysis to calculate the fatigue life of a component subject to variable amplitude loading.

Figure 1:  Variable amplitude time history

There are a number of methods such as rainflow cycle counting for extracting constant amplitude cycles from a non-uniform time history.  These algorithms decompose an irregular sequence of peaks and valleys into an equivalent set of loading blocks, as shown in Figure 2. These constant amplitude blocks are used by the damage summation techniques described below.

Figure 2:  Stress spectrum

Linear Damage Rule (Miner's Rule)

The Linear Damage Rule was first proposed by Palmgren in 1924 and was further developed by Miner in 1945. Today the method is commonly known as Miner's Rule.  

The Linear Damage Rule is based on the concept of fatigue damage.  A damage fraction, D, is defined as the fraction of life used up by an event or a series of events.  Failure is predicted to occur when:

where:

	- Index of each set of applied load cycles at constant stress level Si
	- Damage fraction accumulated during the load cycles of interval i at constant stress level Si
	- Damage criterion (a constant).

The Linear Damage Rule states that the damage fraction at a given constant stress level S_i is equal to the number of applied cycles n at stress level S_i divided by the fatigue life N at stress level S_i.  That is, 

where:

	- Damage fraction accumulated during the load cycles of interval i at constant stress level Si
	- Number of applied load cycles at constant stress level Si
	- Fatigue life at constant stress level Si, obtained from the S-N curve.

For Miner's Rule, the damage criterion X is assumed to be equal to 1.0, and failure is predicted to occur when:

Considerable test data has been generated in an attempt to verify Miner's Rule. Most test cases use a two step load history. This involves testing at an initial stress level S₁ for a certain number of cycles, then the stress level is changed to a second level S₂ until failure occurs. If S₁ > S₂, it is called a high-low test, and if S₁ < S₂, a low-high test. The results of Miner's original tests showed that the damage criterion X corresponding to failure ranged from 0.61 to 1.45. Other researchers have shown variations as large as 0.18 to 23.0, with most results tending to fall between 0.5 and 2.0. In most cases, the average value is close to Miner's proposed value of 1.0.

One problem with two-level step tests is that they do not accurately represent many service load histories. Most load histories do not follow any step arrangement and instead are made up of a random distribution of loads of various magnitudes. However, tests using random histories with several stress levels show good correlation with Miner's rule. Even so, for conservative estimates of the life of a structure an X value of less than 1.0 is usually used.

The Linear Damage Rule has two main shortcomings when it comes to describing observed material behavior:

• Load sequence effects are ignored. The theory predicts that the damage caused by a stress cycle is independent of where it occurs in the load history.  An example of this discrepancy was discussed earlier regarding high-low and low-high tests.
• The rate of damage accumulation is independent of the stress level. This trend does not correspond to observed behavior. At high strain amplitudes cracks will initiate in a few cycles, whereas at low strain amplitudes almost all the life is spent initiating a crack (i.e., very little propagation fatigue).

Despite these limitations, the Linear Damage Rule is still widely used.  This is due both to its simplicity and the fact that more sophisticated methods do not always result in better predictions.  

Nonlinear Damage Rule - Marco and Starkey Method

Many nonlinear damage theories have been proposed which attempt to overcome the shortcomings of Miner's Rule. The following is a general description of a nonlinear damage approach which was proposed by Richard and Newmark and was developed further by Marco and Starkey. 

The theory uses a nonlinear damage exponent P to augment Miner's Rule. The damage fraction is thus given by:

The value of P is considered to be greater than 0.0 and less than or equal to 1.0, with the value increasing with stress level. Note that with P = 1.0, this method is equivalent to Miner's Rule.

The nonlinear method described has good correlation to observed material behavior and can be used to sum damage in high temperature applications where there is interaction between creep and fatigue. However, like all nonlinear theories, it requires a material constant that requires a considerable amount of testing to determine and may not be available for a given material or application.