Strain Life Fatigue

Strain-Life Fatigue Analysis (structures/fatigue/strain-life-fatigue)

Use when the load point is a strain (or a nominal stress at a notch) rather than a nominal elastic stress far from yielding: the Coffin-Manson relation gives the total strain amplitude as the sum of elastic and plastic amplitude curves against reversals to failure, the life at a strain amplitude is found by inverting that relation, and the Neuber rule bridges the nominal elastic stress at a notch to the local elastic-plastic strain when the section yields. This is the low-cycle fatigue (LCF) leaf; it pairs with the high-cycle S-N leaf structures/fatigue/stress-life-curve (the stress-life counterpart above the transition), structures/fatigue/notch-sensitivity (source of the fatigue notch factor k_f input), structures/fatigue/miner-damage and structures/fatigue/load-spectrum-counting (accumulating damage over variable loads), and structures/fatigue/goodman-diagram (mean-stress correction, out of scope here, assumed zero mean).

Domain quick reference

• Coffin-Manson total strain amplitude at 2N_f reversals to failure: eps_a = (sigma_f_prime / E) * (2N_f)^b + eps_f_prime * (2N_f)^c, with fatigue strength coefficient sigma_f_prime, fatigue strength exponent b < 0, fatigue ductility coefficient eps_f_prime, fatigue ductility exponent c < 0, modulus E. The first term is the elastic amplitude, the second the plastic amplitude.

• Life from strain: invert eps_a(2N_f) by bisection on log(2N_f), wide deterministic bracket (reversals_to_failure).

• Strain from life: direct evaluation of the two terms (strain_amplitude).

• Transition life 2N_t: the reversal count where the elastic and plastic amplitudes are equal. A load point with 2N_f below 2N_t is low-cycle (plastic-dominated); above it, high-cycle (elastic-dominated) (regime_classification).

• Ramberg-Osgood cyclic curve: eps = sigma/E + (sigma/K_prime)^(1/n_prime), cyclic strength coefficient K_prime, cyclic strain hardening exponent n_prime.

• Neuber rule at a notch: sigma_loc * eps_loc = (k_f * S)^2 / E, with nominal elastic stress amplitude S, fatigue notch factor k_f (an input here, from notch-sensitivity methods elsewhere), solved with the Ramberg-Osgood curve for the local stress and strain (neuber_local_strain). Fully elastic when the local strain equals sigma_loc/E, plastic otherwise.

• Local strain amplitude equals eps_loc for the fully reversed life, read through the Coffin-Manson curve with zero mean stress assumed (strain_life_point).

• Module material table (representative typicals, reference-only; not reproduced from MMPDS):

  Property	7075-T6 class aluminum	4340 class steel
  sigma_f_prime (MPa)	690	1750
  b	-0.10	-0.08
  eps_f_prime	0.55	0.50
  c	-0.60	-0.70
  E (GPa)	71.7	200
  K_prime (MPa)	900	1800
  n_prime	0.10	0.08

  Values are representative magnitudes from the open fatigue literature, reference-only. Pass any of the seven constants as a property dict to override the defaults for a specific alloy.

Workflow

1. Identify the load point: a fully reversed strain amplitude eps_a, or a nominal elastic stress amplitude S at a notch with fatigue notch factor k_f.
2. For a direct strain input, get the life: reversals_to_failure(eps_a) returns 2N_f; cycles are 2N_f / 2.
3. Categorize the point: regime_classification(eps_a) returns "low-cycle" when 2N_f < 2N_t (transition_reversals) and "high-cycle" otherwise.
4. For a notch load point, bridge the stress to the local strain: neuber_local_strain(k_f, S) returns sigma_loc, eps_loc and a plastic flag from the Neuber identity on the Ramberg-Osgood curve.
5. Read the local strain back through the curve: reversals_to_failure(eps_loc) gives the local notch life.
6. Run the one-call summary strain_life_point(S, k_f) for sigma_loc, eps_loc, reversals and cycles to failure, regime and verdict.
7. Confirm the deterministic checks with the contract test scripts/test_strain_life_fatigue.py.

Worked example

Representative aluminum (default table), fully reversed loading.

• eps_a = 0.01: reversals_to_failure(0.01) = 2135 reversals (1068 cycles). Since 2135 < 2N_t = 3266 reversals, regime_classification returns "low-cycle". The 0.01 life sits in the low-cycle band (between 1e3 and 1e5 reversals).
• eps_a = 0.002: reversals_to_failure(0.002) = 8.11e6 reversals, well above both the 0.01 life and 2N_t, so the point is "high-cycle" (elastic-dominated).
• Transition: 2N_t = 3266 reversals (1633 cycles); at that point eps_e = eps_p = 4.285e-3, and the total amplitude is 8.569e-3.
• Monotonicity: over the strain ladder 0.004, 0.006, 0.008, 0.012, 0.015 the predicted lives fall strictly: 5.63e4, 1.01e4, 3.99e3, 1.34e3, 788 reversals.
• Neuber rule, k_f = 2.5, S = 200 MPa: neuber_local_strain(2.5, 200e6) returns sigma_loc = 459.1 MPa, eps_loc = 7.60e-3, plastic flag True. The local strain exceeds the nominal elastic value S/E = 2.79e-3, and the Neuber identity holds: sigma_loc * eps_loc = 3.487e6 Pa = (k_f * S)^2 / E. Reading the local strain through the curve gives 4664 reversals, just above 2N_t, so this notch point is categorized high-cycle even though the notch root yields; the summary notes that.
• Same notch at S = 300 MPa: strain_life_point(300e6, 2.5) returns sigma_loc = 546.0 MPa, eps_loc = 1.437e-2, plastic flag True, reversals to failure 870 (435 cycles), regime "low-cycle", verdict "low-cycle plastic-dominated fatigue life".
• Steel contrast: for the 4340 entry 2N_t = 682 reversals and reversals_to_failure(0.01) = 752 reversals, showing the shorter ductile transition of the higher strength steel.

Pitfalls

• Reading a high-cycle life off the elastic term alone: the total amplitude is the SUM of the elastic and plastic Coffin-Manson terms, and below the transition the plastic term dominates; a plastic-only or elastic-only estimate misses the life by orders of magnitude.
• Applying a mean-stress correction that is not modeled: this leaf assumes fully reversed (zero mean) loading; mean-stress effects belong to the goodman-diagram leaf, so a nonzero-mean load point needs that correction before the life here means anything.
• Treating a local yield as high-cycle automatically: the Neuber point at S = 200 MPa yields at the notch root (plastic flag True) yet lands at 4664 reversals, just above 2N_t, and is categorized high-cycle - the regime string follows the local strain life, not the yield flag.
• Confusing reversals with cycles: reversals_to_failure returns 2N_f, and cycles are 2N_f / 2 (2135 reversals is 1068 cycles); quoting reversals as cycles doubles the reported life.
• Using k_f below unity or a notch stress beyond the model: k_f < 1 raises ValueError, and the Neuber identity sigma_loc * eps_loc = (k_f * S)^2 / E only holds when the Ramberg-Osgood solution is the one used to bridge.
• Forgetting the material table is representative: the 7075-T6 and 4340 constants are typicals from the open literature (reference-only), so an alloy-specific analysis must pass the property dict override rather than quote the defaults.

Verification

• reversals_to_failure(0.01) returns 2135.47 reversals (between 1e3 and 1e5) and regime_classification(0.01) == "low-cycle".
• reversals_to_failure(0.002) returns 8.11e6 reversals and the regime is "high-cycle"; predicted life falls monotonically as eps_a rises over five ladder points.
• At 2N_t = 3266 the elastic and plastic amplitudes agree to 1e-6 relative.
• neuber_local_strain(2.5, 200e6): sigma_loc * eps_loc equals (k_f * S)^2 / E to 1e-9 relative, eps_loc > S/E, plastic flag True; with k_f = 1.0 and S = 30 MPa the flag is False and sigma_loc = S.
• Round trip: strain_amplitude(reversals_to_failure(eps_a)) recovers eps_a for the ladder points.
• ValueError rejection: non-positive or non-finite strain, stress, modulus or reversals, k_f < 1, unknown material names, n_prime outside (0, 1), K_prime <= 0.
• Run the contract test offline: python3 scripts/test_strain_life_fatigue.py (34 tests, deterministic).

Related leaves

• structures/fatigue/stress-life-curve: the high-cycle S-N counterpart for loads far below yield.
• structures/fatigue/notch-sensitivity: fatigue notch factor k_f from k_t and material sensitivity, the input the Neuber bridge needs.
• structures/fatigue/goodman-diagram: mean-stress correction, assumed zero mean in this leaf.
• structures/fatigue/miner-damage and structures/fatigue/load-spectrum-counting: damage accumulation over the variable amplitude spectrum once each point life is known.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_strain_life_fatigue.py

The test covers the Coffin-Manson amplitude and its monotonic fall with life, the 0.01 and 0.002 worked-example lives and their regime strings, the transition life with elastic-plastic equality at 2N_t, monotonic life over five strain points, the Ramberg-Osgood curve values and bounds, the Neuber anchor (local stress and strain, plastic flag, product identity to 1e-9 relative, elastic limit with k_f = 1), the property-dict override path, and ValueError rejection of non-positive, non-finite, out-of-range and unknown inputs.

Compliance

• Standards referenced by name, not reproduced: FAR 25 (damage tolerance and fatigue evaluation practice), CS-25 (EASA counterpart), MMPDS (material allowables context). The material constants in this leaf are representative typicals stated in this document, not MMPDS table values, and the equations are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.