Low cycle fatigue (LCF) usually refers to situations where the stress is high enough for plastic deformation to occur, the accounting of the loading in terms of stress is less useful and the stain in the material offers a simpler and more accurate description. This type of fatigue is normally experienced by components which undergo elastic-plastic or plastic deformation. The reason why strain life is more suitable for low cycle fatigue is that the strains allow the elastic and plastic components to be distinguished from each other, whereas the stress is way less sensitive to the plastic component in the case of high amplitude. The arbitrary classification between high cycle fatigue (HCF) and low cycle fatigue is considered to be roughly 10,000 cycles; however, the exact number of cycles really depends on the properties of the metal.

The research of low cycle fatigue was traditionally done for boilers, steam turbines, gas turbines, pressure vessels, nuclear reactors and other power equipment that are exposed to high temperatures which induce thermal stresses in the components.

**Coffin-Manson Relation**

Low-cycle fatigue is usually characterized by the *Coffin-Manson relation* (published independently by L. F. Coffin in 1954 and S. S. Manson in 1953):

where,

- Δε
_{p}/2 is the plastic strain amplitude - 2
*N*is the number of reversals to failure (_{f}*N*cycles)_{f} - ε
_{f}‘ is an empirical constant known as the*fatigue ductility coefficient*, the failure strain for a single reversal. A fairly good approximation is ε’_{f}= ε_{f}with fracture ductility ε_{f}= ln[100/(100 – RA)], where RA is the reduction of area in %. Other values in estimating ε’_{f}include:

ε’_{f} = 0.35 ε_{f}

ε’_{f} = 0.5 ε_{f} (c = -0.6)

ε’_{f} = 0.75 ε_{f}^{0.75} (c = -0.6)

ε’_{f} = 0.76 ε_{f}^{0.6} (c = -0.6)

ε’_{f} = 0.50 ε_{f}

ε’_{f} = 0.71 ε_{f}

*c*is an empirical constant known as the*fatigue ductility exponent*, commonly ranging from -0.5 to -0.7 for metals in time independent fatigue. Fairly ductile metals (say ε_{f}= 1.0) have value of c = -0.6; for strong metals (say ε_{f}= 0.5) a value of c = -0.5 is more reasonable. Slopes can be considerably steeper in the presence of creep or environmental interactions

**Basquin’s Equation**

The Coffin-Manson formula describes the relationship between plastic strain and fatigue life in the low-cycle high-strain fatigue regime. Basquin’s equation, on the other hand, describes high-cycle low strain behavior in elastic nature:

or

where

- σ
_{a}is the elastic stress amplitude - Δε
_{e}/2 is the elastic strain amplitude - 2
*N*is the number of reversals to failure (_{f}*N*cycles)_{f} - E is modulus of elasticity
- σ’
_{f}is fatigue strength coefficient. A fairly good approximation is σ’_{f}= σ_{f}where σ_{f}is fracture stress with necking corrections. For steels with hardness below 500 HB, σ_{f }= UTS + 50 ksi. Other values in estimating σ’_{f}include:

σ’_{f} = 1.09 σ_{f}

σ’_{f} = 0.92 σ_{f} (b = -0.12)

σ’_{f} = 1.15 σ_{f} (b = -0.12)

- b is the fatigue strength exponent, commonly in the range -0.12 < b < -0.05 with an average of -0.085

The Fatigue+ software provides a material database including extensive fatigue data for carbon steels, low alloy steels, high alloy steels, aluminum alloys, as well as cast metals and welds. The user can edit the database or add materials to the database.

**Morrow Design Rule (Basquin-Coffin-Manson relation)**

The Coffin-Manson formula describes the relationship between plastic strain and fatigue life in the low-cycle high-strain fatigue regime. Basquin’s equation describes high-cycle low strain behavior, as discussed above. Morrow combines elastic strain and plastic strain into a total strain relationship as follows:

The figure below shows the elastic, plastic and total strain fatigue curves. The strain fatigue curves are also called E-N curves as opposed to S-N curves for stress fatigue curves. The elastic strain curve represents Basquin’s equation; the plastic strain curve represents Coffin-Manson’s equation; and the total strain curve represents Morrow’s equation. Notice from the figure that total strain is the sum of elastic strain and plastic strain. The intersection of elastic and plastic curves is denoted as 2N_{t}, which is the demarcation of material behaviors. The first half of the total strain fatigue curve on the left with higher slope represents low cycle fatigue (LCF); the second half on the right with the lower slope pertains to lower strain ranges that impart stresses in the elastic regime, which represents the high cycle fatigue (HCF).

The following chart is an example of the strain fatigue curves generated by the Fatigue+ software.

**Fatigue curve of ASTM A36**

Elastic – Basquin’s equation

Plastic – Coffin-Manson’s equation

Total – Morrow’s equation

### Cyclic Stress-Strain Behavior

Metals are metastable under application of cyclic loads, and their stress-strain response can be drastically altered when subjected to repeated plastic strains. Depending on the material properties, initial state and its test condition, a metal may cyclically harden or cyclically soften.

As illustrated in the following figures, the total strain is controlled as shown in figure (a) in the test, and the stress response is observed as shown in figure (b). The stress required to enforce the strain increases on subsequent reversals, the metal undergoes cyclic hardening. The hardness, yield, and ultimate strength increase.

Cyclic stress-strain behavior in materials is characterized by hysteresis loops in plots of stress versus strain, as shown in figure (c). These loops occur because energy is dissipated as internal friction (heat) in the material, causing a lag in the response (strain) of the material to a forcing function (stress) during cyclic deformation.

In about 20% to 40% of the total fatigue life in eigher hardening or softening materials, the steady-state condition is usually achieved, as illustrated in the following figure. For completely reversed, R = -1, strain-controlled conditions with zero mean strain, the total width of the loop is Δε, or total strain range. The total height of the loop is Δσ, or the total stress range.

At steady-state, the stress-strain power law is

where

σ = cyclically stable stress amplitude

ε_{p} = cyclically stable plastic strain amplitude

K’ = cyclic strength coefficient

n’ = cyclic strain hardening exponent (value varies between 0.10 and 0.20, with an average value very close to 0.15)

Re-arrange the equation gives

The total strain is the sum of the elastic and plastic components, and can be written as

K’ and n’ are usually obtained from a curve fit of the cyclic stress-strain data, however, the following properties may be related:

### Mean Stress Correction

High cycle fatigue is sensitive to mean stresses. In the software Fatigue+, the following options are available for mean stress correction:

For **Basquin** model, select **Soderberg**, **Goodman**, **Gerber**,**Morrow**, **Smith-Watson-Topper**, or **None**. (See High Cycle Fatigue)

For **Basquin-Coffin-Manson** relation, select **Morrow**, **Smith-Watson-Topper**, **Manson-Halford**, or **None**.

In the design code based fatigue models, the mean stress correction is often incorporated into the models in compliance with corresponding design codes.