4.1 Introduction: When Stress-Life Fails

The S–N method (Chapter 3) assumes stresses remain elastic throughout the fatigue life. This assumption breaks down when:

1. Local plasticity occurs at stress concentrations (notches, fillets, threads) even if the nominal stress is elastic.
2. Low-cycle fatigue (LCF) dominates: lives below $10^4$–$10^5$ cycles where plastic strain per cycle is significant.
3. Thermal cycling or creep-fatigue interaction produces large inelastic strains.
4. Complex multiaxial loading creates non-proportional strain paths.

The strain-life (ε–N) approach tracks the local strain amplitude $\varepsilon_a$ at the critical point (usually the notch root). It unifies HCF and LCF through the Coffin–Manson–Basquin equation, which adds an elastic strain term (Basquin) and a plastic strain term (Coffin–Manson).

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4.2 Elastic vs Plastic Strain in Cyclic Loading

Under cyclic loading, the stress-strain response at a material point traces a hysteresis loop in $\sigma$–$\varepsilon$ space. The loop width in the strain direction is $2\varepsilon_a$; in the stress direction is $2\sigma_a$.

4.2.1 Total strain decomposition

$$\varepsilon_{\mathrm{total}} = \varepsilon_e + \varepsilon_p$$

• Elastic strain $\varepsilon_e = \sigma / E$ — recoverable, linear.
• Plastic strain $\varepsilon_p$ — permanent, governed by plasticity theory.

4.2.2 Cyclic hardening and softening

Materials may harden (stress amplitude increases with cycles), soften (stress decreases), or stabilize (stable hysteresis loop after ~10–100 cycles). The stabilized loop properties ($K'$, $n'$) are used in analysis.

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4.3 Coffin–Manson–Basquin Equation

The fundamental strain-life relation combines elastic and plastic contributions:

$$\boxed{\varepsilon_a = \underbrace{\frac{\sigma_f'}{E}(2N_f)^b}_{\text{elastic (Basquin)}} + \underbrace{\varepsilon_f'(2N_f)^c}_{\text{plastic (Coffin–Manson)}}}$$

4.3.1 Transition life

The transition fatigue life $N_t$ is where elastic and plastic strain amplitudes are equal:

$$N_t = \left(\frac{\varepsilon_f' E}{\sigma_f'}\right)^{1/(b-c)}.$$

For typical steels: $N_t \approx 10^3$–$10^4$ cycles. Below $N_t$, plastic strain dominates (LCF); above $N_t$, elastic strain dominates (HCF).

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4.4 Ramberg–Osgood Cyclic Stress-Strain Relation

The cyclic stress-strain curve describes the stabilized hysteresis loop mid-loop (from tension peak to compression peak):

$$\boxed{\varepsilon_a = \frac{\sigma_a}{E} + \left(\frac{\sigma_a}{K'}\right)^{1/n'}}$$

where:

• $K'$ is the cyclic strength coefficient (MPa).
• $n'$ is the cyclic strain hardening exponent ($0.10$–$0.20$ for metals).

4.4.1 Relation to monotonic properties

$K' \approx 1.9 S_y$ and $n' \approx 0.15$–$0.20$ for many steels (first approximation). The cyclic curve may lie above (cyclic hardening) or below (cyclic softening) the monotonic curve.

4.4.2 Hysteresis loop shape

Masing's rule: the compressive unloading path is a mirror of the tensile loading path scaled by a factor of 2, giving a symmetric hysteresis loop for stabilized cyclic loading.

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4.5 Neuber's Rule for Notch Analysis

Neuber's rule (1961) relates the nominal (net-section) stress and strain to the local (notch root) stress and strain:

$$\boxed{K_t^2 \sigma_{\mathrm{nom}} \varepsilon_{\mathrm{nom}} = \sigma \varepsilon}$$

where $K_t$ is the elastic stress concentration factor, and $(\sigma, \varepsilon)$ are the local notch-root values.

4.5.2 Modified Neuber for mean stress

For non-zero mean stress, use $\sigma_{\mathrm{nom}}$ and $\varepsilon_{\mathrm{nom}}$ including mean components, or apply Neuber to the stress and strain ranges separately.

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4.6 Mean Stress Corrections in Strain-Life

Mean stress affects strain-life similarly to stress-life, but corrections are applied to the elastic term (which is mean-stress sensitive) rather than the plastic term.

4.6.1 Morrow correction

Replace $\sigma_f'$ with $(\sigma_f' - \sigma_m)$ in the elastic term:

$$\varepsilon_a = \frac{\sigma_f' - \sigma_m}{E}(2N_f)^b + \varepsilon_f'(2N_f)^c.$$

Tensile mean stress ($\sigma_m > 0$) reduces the effective fatigue strength, shortening life. Compressive mean stress extends life.

4.6.2 Smith–Watson–Topper (SWT) parameter

For combined mean and alternating stress:

$$\boxed{\sigma_{\max} \varepsilon_a = \frac{(\sigma_f')^2}{E}(2N_f)^{2b} + \sigma_f' \varepsilon_f' (2N_f)^{b+c}}$$

where $\sigma_{\max} = \sigma_a + |\sigma_m|$. SWT is widely regarded as the most accurate mean-stress correction for strain-life, especially for tensile mean stress.

4.6.3 Walker correction

$$\sigma_{\max}^{1-m} \varepsilon_a = f(2N_f)$$

with material-dependent exponent $m \approx 0.5$–$0.7$. More flexible than SWT for aluminum alloys.

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4.7 Low-Cycle vs High-Cycle Fatigue: Selection Guide

Rule of thumb: If the local notch-root stress exceeds yield on the first cycle, use ε–N. If the nominal stress is below yield and life exceeds $10^5$ cycles, use S–N.

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4.8 Multiaxial Strain-Life and Critical Plane Approaches

Under multiaxial loading (shafts with bending + torsion, notched plates), the strain state is tensorial. Approaches:

1. von Mises equivalent strain: $\bar{\varepsilon}_{\mathrm{vm}} = \sqrt{\frac{2}{3}e_{ij}e_{ij}}$ — simple but ignores plane orientation.
2. Critical plane method (Fatemi–Socie): search over all planes for the maximum shear strain range $\Delta\gamma_{\max}/2$ combined with normal stress on that plane.
3. Brown–Miller: $\varepsilon_a = \frac{\Delta\gamma_{\max}}{2\sqrt{3}} + \frac{\Delta\varepsilon_n}{2}$ — combines max shear and normal strain range.

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4.9 Experimental Determination of Strain-Life Constants

From stabilized hysteresis loop tests (strain-controlled, $R = -1$):

1. Cyclic stress-strain: plot $\sigma_a$ vs $\varepsilon_a$ for several strain levels → fit $K'$, $n'$.
2. Strain-life: plot $\varepsilon_a$ vs $N_f$ on log–log → separate elastic and plastic contributions by measuring loop width components.
3. Typical values from monotonic properties (Roessle–Fatemi correlations):

• $\sigma_f' \approx S_{ut} + 345$ MPa (for $S_{ut}$ in MPa)
• $b \approx -0.1$
• $\varepsilon_f' \approx 0.59 \psi$ where $\psi = \%RA$ (reduction in area)
• $c \approx -0.6$

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4.10 Design Procedure: ε–N Fatigue Analysis

1. Identify critical location: notch root, fillet, thread root.
2. Determine nominal loading: $\sigma_{\mathrm{nom}}$, $\varepsilon_{\mathrm{nom}}$ (from FEA or hand analysis).
3. Apply Neuber's rule: solve for local $\sigma$, $\varepsilon$ using $K_t$ and Ramberg–Osgood ($K'$, $n'$).
4. Extract strain amplitude: $\varepsilon_a$ (half the total strain range).
5. Apply mean stress correction: Morrow or SWT.
6. Solve Coffin–Manson for $N_f$: numerical root-finding (Newton on $\log 2N_f$).
7. Compare to required life with safety factor.

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4.11 Summary

The strain-life method is essential for:

• Notched components where local plasticity occurs.
• LCF applications (thermal cycling, seismic, bolt pre-load cycling).
• Accurate life prediction in the transition region ($10^3$–$10^5$ cycles).

The Coffin–Manson–Basquin equation, combined with Neuber's rule and Morrow/SWT mean-stress corrections, provides a self-consistent framework from 1 cycle to $10^7$ cycles. Use the Live Solver in this chapter to iterate $N_f$ numerically.