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### Miner's Rule Linear Damage Rule

**Strength Mechanics of Materials **

**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. A damage fraction, D, is defined as the fraction of life used up by an event or a series of events. Failure in any of the cumulative damage theories is assumed to occur when the summation of the damage fractions for all the events experienced by the structure is equal to or larger than a damage criterion, X. For Miner's Rule, X is assumed to be equal to 1.0. The obvious asset of this method is its simplicity.

The linear damage rule states that the damage fraction at a given stress level is equal to the cycle ratio. The cycle ratio is defined as the ratio of the number of cycles at a given stress level, n, to the fatigue life in cycles at that stress level, N. For example, the damage fraction due to one cycle of loading is 1/N. Stated another way, the application of one cycle of loading consumes 1/N of the total fatigue life of the structure.

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 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.

There is a general trend that for high-low tests the damage criterion values are less than 1.0, and for low-high tests the values are greater than 1.0. Thus, Miner's Rule is nonconservative for high-low tests. 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. Tests using random histories with several stress levels show good correlation with Miner's rule. However, Madayag proposed an alternative to Miner's Rule where the damage criterion is selected on a knowledge of the load history or on a desired factor of safety. For conservative estimates of the life of a structure, an X value of less than 1.0 (used in Miner's Rule) is usually used.

The linear damage rule has two main shortcomings when it comes to describing observed material behavior.

1) 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.

2) 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 (very little propagation fatigue).

The Palmgren-Miner cycle-ratio summation rule, also called Miner’s rule, is written:

∑ [ n_{i} / N_{i} ] = c

where n_{i} is the number of cycles at stress level σi and N_{i} is the number of cycles to failure
at stress level σ_{i} . The parameter c has been determined by experiment; it is usually
found in the range 0.7 < c < 2.2 with an average value near unity.

Using the deterministic formulation as a linear damage rule we write:

D = ∑ [ n_{i} / N_{i} ]

where D is the accumulated damage. When D = c = 1, failure ensues.

**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 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.

Most service loading histories of actual engineering parts have a variable amplitude loading, which at times can be quite complex. Several methods have been developed to deal with variable amplitude loading using baseline data generated from constant amplitude tests. These damage summation methods during the initiation phase of fatigue can be used in conjunction with either the stress-life or strain-life methods of constant amplitude fatigue analysis.

There are distinctly different approaches used when dealing with cumulative fatigue damage during the initiation and propagation phases of fatigue. The differences in these approaches are related to how fatigue damage can be defined during these two stages. During the propagation portion of fatigue, damage can be directly related to crack length. Methods that relate loading sequence (retardation)to crack extension are dealt with in the crack extension module. This module deals with fatigue damage during the initiation phase, which is more difficult to define. Although damage during the initiation phase can be related to dislocations, slip bands, microcracks, etc., these phenomena can only be measured in a highly controlled laboratory environment. Therefore, most damage summing methods for the initiation phase are empirical in nature. These methods relate damage to the expended life for a small laboratory specimen (life is defined as the separation of this specimen). Under this method, the separation of a small specimen is equal to the formation of a small crack in a large component or structure.

References:

Bannantine, Julie; Comer, Jess; Handrock, James (1990) Fundamentals of Metal Fatigue Analysis, Prentice Hall (New Jersey).

Palmgre, A. (1924) "Durability of Ball Bearings," ZVDI, Vol. 68, No. 14, pp. 339-341.

Miner, M. A. (1945) "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Vol. 12, Trans. ASME Vol. 67, pp. A159-A164.

Sines, G., Waisman, (eds.) (1959) Metal Fatigue, McGraw-Hill (New York).

Madayag, A. F. (ed.) (1969) Metal Fatigue: Theory and Design, Wiley (New York).

Collins, J.A. (1981) "Failure of Materials in Mechanical Design," Wiley-Interscience (New York).

Richart, F.E., Newmark, N.M., (1948) "An Hypothesis for the Determination of Cumulative Damage in Fatigu," American Society for Testing and Materials Proceedings, Vol. 48, pp. 767-800.

Marco, S.M. and Starkey, W.L. (1954) "A Concept of Fatigue Damage," Trans. ASME, Vol. 76, No. 4, pp. 627-632.

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