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The S-N curve eastimate the service life of materials above 103 (often >104) cycles. The corresponding stress level is usually below 2/3 of yielding stress. The "low-cycle fatigue" model, on the other hand, is made for 104 cycles and below. The stress level usually steps into plastic range.
Strain-Life Equation

When plastic strain occurs, the service life of material decreases, often no more than 104, a.k.a., in low-cycle fatigue range. The research of low-cycle fatigue was traditioanlly done for pressure vessels, power machinery that are exposed to a heat source/sink which induces thermal expension (thermal stress) to the structure. The low-cycle fatigue is usually presented as the plastic strain in log scale against cycles to failure N also in log scale.

To add Fig 12-13, low-cycle fatigue

The result of low-cycle fatigue is near a straight line for common metal meterials such as steel and is often referred as Coffin-Manson relation:

 where is the amplitude of plastic strain. is fatigue dutility coefficient defined by the strain intercept at 2N = 1. For common metal materials, . N is the number of strain cycles to failure and 2N is the number of strain reversals to failure. c if called fatigue ductility exponent. For common metal materials, -0.7 < c < -0.5. A smaller c value results in a longer fatigue life.
Bridge between High and Low Cycles

The Coffine-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, describe high-cycle low strain behavior

 where is the amplitude of alterning stress is the amplitude of elastic strain E is Young's modulus is fatigue strength coefficient and is approximately equal to the monotonic true fracture stress . N is the number of strain cycles to failure and 2N is the number of strain reversals to failure. b if called fatigue strength exponent. For common metal materials, -0.12 < b < -0.05. A smaller c value results in a longer fatigue life.
Universal Strain-Life Equation

Manson proposed a simplified formula known as the mothod of universal slopes

 where is the amplitude of alterning stress. is the true strain at fracture intension. E is Young's modulus. N is the number of strain cycles to failure.