The *Palmgren-Miner rule*, also known as Miner's Rule, the *Linear Damage Rule (LDR)* or the *Cumulative Damage Hypothesis (CDH)*, is one of the most widely used methods for fatigue life prediction under variable amplitude (VA) loading. The rule was proposed by M.A. Miner in 1945, and it provides a simple approach to assess cumulative fatigue damage over multiple load cycles with different stress amplitudes. It is based on the assumption that fatigue damage accumulates in a linear fashion until failure occurs.

## How Miner’s Rule Works

Miner’s Rule calculates fatigue damage based on the number of cycles at each stress level. For a given stress level, the *damage fraction* can be computed as the ratio of the number of cycles at that stress level to the total number of cycles to failure under the same stress level (derived from S-N curves).

The basic form of Miner’s Rule is expressed as:

\[D = \sum_{i=1}^{n} \frac{n_i}{N_i}\]

Where:

- \(n_i\) = number of cycles experienced at stress level \(S_i\)
- \(N_i\) = number of cycles to failure at stress level \(S_i\) (from the S-N curve)
- \(D\) = total accumulated damage

According to Miner’s Rule, failure occurs when the total accumulated damage \(D\) equals or exceeds 1:

\[D \geq 1 \quad \text{(Fatigue failure)}\]

## Key Assumptions of Miner’s Rule

The Miner's rule is based on the following assumptions:

- 1. Linear Accumulation of Damage
- The rule assumes that damage accumulates linearly, i.e., each load cycle contributes to damage independently of other cycles, and there is no interaction between cycles of different stress amplitudes.
- 2. No Load Sequence Effects
- Miner’s Rule assumes that the order of applied loads has no effect on fatigue life. For example, applying a high-stress cycle before a low-stress cycle is assumed to produce the same damage as applying the low-stress cycle first.
- 3. No Memory of Past Loading History
- The rule suggests that once a load cycle is completed, the material "forgets" the previous history, which means that the damage from a certain load cycle is independent of prior cycles.

## Pros of Miner’s Rule

- Miner’s Rule is easy to apply and understand. It requires only basic inputs such as the number of cycles at different stress levels and the S-N curve for the material.
- It provides a straightforward way to assess fatigue damage in structures and components subjected to VA loading without complex computational tools.

## Cons and Limitations of Miner’s Rule

*load sequence effects*. In reality, the order in which different stress levels are applied can have a significant impact on fatigue life. For example, high-amplitude stress cycles followed by low-amplitude cycles often cause more damage than the reverse sequence due to

*plasticity-induced crack closure*and other microstructural effects.

- Fatigue damage is not always linear. For certain materials and loading conditions, damage tends to accumulate more rapidly or more slowly than predicted by Miner’s Rule. As a result, Miner’s Rule can
*overestimate or underestimate damage*depending on the material and the loading history. *High-cycle fatigue*tends to be better modeled by Miner’s Rule, whereas*low-cycle fatigue*often involves significant plastic deformation, which the rule does not account for.

*complex loadings*. Real-world loads are often irregular and multi-axial, involving combinations of torsion, tension, and bending, which Miner’s Rule cannot adequately account for.

*fatigue limit*, meaning that stress cycles below a certain threshold do not cause fatigue damage. The rule does not account for this, leading to overly conservative estimates of damage when low-stress cycles are present.

*creep*or

*corrosion*plays a role in fatigue, as these factors accelerate crack initiation and growth, interacting with the cyclic loading in complex ways that Miner’s linear accumulation of damage does not capture.

## Alternatives and Modifications to Miner’s Rule

Because of its limitations, several modifications and alternatives to Miner’s Rule have been proposed to improve accuracy in fatigue damage assessment under variable loading:

*non-linear damage accumulation*, where the damage rate increases or decreases depending on stress amplitude or loading history. These models are more complex but tend to be more accurate for materials that do not follow a linear damage rule.

*load sequence effects*, which can modify the damage caused by subsequent load cycles based on the preceding load history. These models often introduce corrective factors that consider whether high loads occur before or after low loads.

*multi-axial loading*conditions, Miner’s Rule is extended or replaced by models that account for the interaction between different types of loading (e.g., torsion combined with tension). These models are better suited to handle real-world complex loading patterns.

*two stages*: crack initiation and crack growth. This approach models crack initiation using S-N curves and then predicts crack growth using fracture mechanics, which takes into account

*stress intensity factors*and

*crack propagation*laws like Paris’ Law.

## Practical Application of Miner’s Rule

Despite its limitations, Miner’s Rule is still widely used as a practical tool in many engineering disciplines, especially when an *approximation* of fatigue life is sufficient, and the loading conditions are not overly complex. For safety-critical applications where more precision is needed, the rule may be supplemented by more advanced methods or laboratory testing.

## Conclusion

Miner’s Rule offers a *simple and practical* method for estimating fatigue damage under variable amplitude loading, making it valuable in many engineering applications. However, it has significant *limitations*, particularly in its inability to account for load sequence effects and non-linear damage accumulation. In cases where a more accurate prediction of fatigue life is required, *non-linear models* and *load sequence-dependent methods* are often more appropriate alternatives.