Cyclic Strain Hardening Exponent⚠ unverified
Mechanical / Materials · Return the cyclic strain hardening exponent
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| n_prime | nprime | — | 0.2 | Cyclic strain hardening exponent, dimensionless. Default is 0.2 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | n | — | The cyclic strain hardening exponent, dimensionless |
The science & history
Understanding the Parameters
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Cyclic exponent $n'$ — typically $0.1$–$0.2$ for metals. It plays the same role for the stabilised cyclic stress–strain curve that the monotonic strain-hardening exponent $n$ plays for the tensile curve: a larger $n'$ means more hardening as cyclic plastic strain grows.
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Cyclic vs monotonic — many metals cyclically soften (annealed, initially soft) or cyclically harden (cold-worked, initially hard) until a stable hysteresis loop forms; $n'$ describes that stabilised loop, which can differ substantially from the monotonic curve.
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Link to fatigue exponents $b$, $c$ — in the strain-life model, the elastic strain amplitude follows Basquin ($\propto (2N)^{b}$) and the plastic amplitude follows Coffin–Manson ($\propto (2N)^{c}$). Compatibility of the two with the cyclic stress–strain curve forces $n' = b/c$.
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Pass-through caveat — since the calculator returns whatever $n'$ you supply, its value is only as good as your material data (from a cyclic test or handbook), not something it derives.
Derivation (Approaching a Proof)
The stabilised cyclic stress–strain curve is written as a power law between stress amplitude and plastic strain amplitude:
$$\sigma_a = K'\,(\varepsilon_{p,a})^{n'}.$$
The strain-life model separately gives, versus reversals $2N$,
$$\sigma_a = \sigma_f'\,(2N)^{b} \quad(\text{Basquin}), \qquad \varepsilon_{p,a} = \varepsilon_f'\,(2N)^{c} \quad(\text{Coffin–Manson}).$$
For these three relations to be mutually consistent, substitute the last two into the first: $\sigma_f'(2N)^{b} = K'\,[\varepsilon_f'(2N)^{c}]^{n'} = K'\varepsilon_f'^{\,n'}(2N)^{c\,n'}$. Matching the powers of $(2N)$ requires $b = c\,n'$, i.e.
$$n' = \frac{b}{c}.$$
So the cyclic hardening exponent is not independent — it is fixed by the two fatigue-life exponents. The registry card, lacking $b$ and $c$, just echoes a supplied $n'$.
Dimensional check. $n'$ is a dimensionless exponent; as the ratio $b/c$ of two dimensionless strain-life exponents, it is likewise dimensionless.
History and Development
The cyclic stress–strain curve and its exponent $n'$ come from the strain-life (local strain) fatigue methodology developed by Coffin, Manson, Morrow, and Landgraf in the 1950s–1960s for low-cycle fatigue — where plastic strains dominate and the stress-life S-N approach breaks down. The compatibility relation $n' = b/c$ ties the cyclic hardening behaviour to the fatigue-life exponents, unifying the cyclic stress–strain curve with the strain-life equation.
Related Concepts: Fatigue Strength Coefficient, Fatigue Life cycles, S-N Curve, Endurance Limit Unmodified, Marin Endurance Limit
Notes: Pass-through — registry returns the input $n'$ unchanged (empty equation, default 0.2). Physically $n' = b/c$ (Basquin/Coffin–Manson exponents); typical $0.1$–$0.2$. Describes the stabilised cyclic curve (cyclic hardening/softening), which can differ from the monotonic $n$.