Fatigue is the weakening of a material under repeatedly applied loads and can lead to material failure even when the applied loads are below the material’s ultimate tensile strength. During fatigue, microscopic cracks form at stress concentration areas — for example, grain boundaries, areas of altered microstructure, or any sharp corners within the material. With every stress cycle, these cracks will grow and form striations, eventually reaching a critical size at which point the crack will grow rapidly and the material will ultimately fail.

## S-N Curves

High-cycle fatigue is defined as a situation in which the maximum stress in the cycle is low enough to cause mostly elastic deformation. In this case, generally more than 10,000 cycles will be required for the material to reach failure. In high-cycle fatigue situations, a diagram known as an S-N curve is used to show the magnitude of the cyclic stress ((S)) vs. the cycles to failure ((N)) displayed on a logarithmic scale. These curves allow an engineer to quickly estimate the lifespan of a material from the stresses that it will be under.

## Miner’s Rule

Miner’s rule states that the sum of the ratio of the number of cycles at one stress level (( n_i )) to the number of cycles to failure at that level (( N_i )) will equal a constant (( C )) that is generally found through experiments to be between 0.7 and 2.2. However, (C) is often assumed to be 1 for design estimations.

$$sum_{i=1}^{k} {n_i over N_i} = C$$

However, it must be kept in mind that Miner’s rule should only be used for approximation in the early stages of design because it does not take into account the high dependence of fatigue life on probability, and S-N curves are often averaged to account for scatter in this respect. Additionally, although Miner’s Rule is independent of the order that the stresses are applied in, the total fatigue life for said material does depend on the stress order. For example, low-stress cycles immediately followed by high-stress cycles typically cause much more damage than would be accounted for by Miner’s rule, and the opposite may also be true because high stress cycles followed by low stress cycles may introduce compressive residual stress into the system. Residual stress generally serves to strengthen a material against applied stresses.

## Paris’ Law

Paris’ Law describes the rate of growth of the crack per stress cycle ((da over dN) ) in terms of the cyclical stress concentration factor ((Delta K )). In this equation, C and m are constants related to both the material and geometry and can generally be found in tables. (N) is the number of stress cycles and (a) is the length of the crack.

$${da over dN} = C{(Delta K)^m}$$

Paris’ Law is used to create fatigue crack growth rate graphs. These graphs can be separated into three separate sections. Region 1 is known as the threshold region, where both crack growth and stresses are low. Once ∆K has reached the threshold level, the graph enters Region 2. This is the area of the graph that is described by Paris’ Law and is the dominate type of crack growth — relatively slow growing for a wide range of stresses. Once the stress reaches a critical point, the graph enters Region 3, at which point the cracks grow rapidly with a minuscule increase in stress. Generally, Region 3 leads almost immediately to failure.

Fatigue crack formation and propagation are complicated processes that can never be fully predicted in real-life stress situations. However, they can be closely approximated through the aforementioned techniques and planned for in advance. Additionally, it is always important when considering fatigue life to use conservative approximations, as there is much less harm in being too careful than in not being careful enough.

**Is your piping system feeling the pressure and fatigue?**

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