Creep Stress Relaxation

Creep Stress Relaxation (structures/materials/creep-stress-relaxation)

Use when the task is the fixed-total-strain stress decay of a preloaded metallic element (a bolted joint preload, a spring preload or an interference-fit fastener) held at elevated temperature: the relaxed stress after the hold from the closed-form integral of the relaxation ODE d(sigma)/dt = -E * eps_dot under Norton power-law creep, the retained-preload fraction, the time to relax to a target fraction of the preload, and the retained-preload margin check against a required retained fraction with the PASS/FAIL verdict. This leaf implements the deterministic closed-form algebra in pure Python, stdlib only. It pairs with structures/materials/creep-rupture, which owns the constant-stress vein: a preload held at fixed total strain relaxes here, while a part under a sustained stress with accumulated creep strain, rupture-life and design-life margin questions stays with creep-rupture. The fixed-strain decay this leaf computes has no function in the constant-stress sibling, whose creep accumulation eps_c = eps_dot * t cannot answer how sigma falls while the total strain is held. Adjacent fences: structures/thermal-structures/thermal-stress-analysis owns the constrained-expansion load side of the elevated-temperature environment, and the structures/fatigue pack owns cyclic endurance and creep-fatigue interaction. This leaf does NOT cover rupture life, the Larson-Miller parameter, the Monkman-Grant cross-check, time to 1 percent creep strain or accumulated creep strain at a sustained stress (all constant-stress products of creep-rupture), statistically based tensile design values (structures/materials/mmpsd-allowables), or the room-temperature elastic-plastic curve (ramberg-osgood).

Domain quick reference

• Fixed-strain relaxation ODE: with the total strain held constant the elastic strain unloads one-for-one into creep strain, d(sigma)/dt = -E * eps_dot, so under the Norton law eps_dot = A * sigma^n * exp(-Q / (R * T)) the stress obeys d(sigma)/dt = -K * sigma^n with the relaxation rate coefficient K = E * A * exp(-Q / (R * T)) in Pa^(1-n) / s (A in 1/s per Pa^n, n the stress exponent, Q the activation energy in J/mol, R = 8.314 J/mol/K, E the high-temperature elastic modulus in Pa, T in K).
• Power-law branch (n != 1): the separable ODE integrates in closed form to sigma(t) = [sigma_0^(1-n) + (n - 1) * K * t]^(1 / (1 - n)); sigma(0) = sigma_0 and for n > 1 the stress decays monotonically with the long-time tail sigma ~ [(n - 1) * K * t]^(1/(1-n)) toward zero, never reaching it in finite time.
• Linear branch (n = 1): sigma(t) = sigma_0 * exp(-K * t), the n -> 1 limit of the power branch (Newtonian viscous relaxation).
• Retained fraction: f(t) = sigma(t) / sigma_0 in (0, 1] for n >= 1.
• Time to a retained fraction f: t(f) = sigma_0^(1-n) * (f^(1-n) - 1) / ((n - 1) * K) for n != 1 and t(f) = -ln(f) / K for n = 1; smaller f needs longer t.
• Retained-preload margin: margin = retained / required - 1 with the verdict PASS when the margin is >= 0 (the available / required - 1 convention shared with the creep-rupture margin checks).
• Units: stress in Pa (MPa times 1e6), temperature in K (Celsius plus 273.15), time in seconds (hold times in hours convert by 3600). All functions raise ValueError on non-positive stress, temperature or required fraction, on negative time, on a fraction outside (0, 1], on a stress exponent below 1, or on an unknown material name.

Workflow

1. Fix the operating point: the initial preload stress sigma_0 in Pa (convert MPa by multiplying by 1e6), the metal temperature T in K (Celsius plus 273.15) and the hold time in seconds (hours times 3600). Select the material: pass the registered name "inconel-718" (the default, reference-only typicals A = 2.0e-47, n = 7.0, Q = 360000 J/mol, E = 2.1e11 Pa shared with the creep-rupture sibling, plus the modulus this ODE needs) or a dict of overrides on top of the default alloy with any subset of keys norton_a, norton_n, norton_q and elastic_modulus.
2. Compute the relaxed stress after the hold with relaxed_stress(sigma_0, time_s, temp_k, material): the closed-form integral of the fixed-strain relaxation ODE, the power-law branch for n != 1 and the exponential branch for n = 1. A zero hold time returns sigma_0 exactly.
3. Read the retained-preload fraction with retained_fraction(sigma_0, time_s, temp_k, material): sigma(t) / sigma_0, the preload-retention fraction in (0, 1].
4. Find the time for the preload to relax to a target fraction of its initial value with time_to_relaxed_fraction(fraction, sigma_0, temp_k, material): the inverted closed form in seconds, for example the time to 80 percent of the preload for a fastener joint.
5. Run the retained-preload margin check with preload_margin(sigma_0, time_s, temp_k, required_fraction, material): it returns the retained fraction at the hold time, the margin retained / required - 1 and the verdict PASS when the margin is

   = 0 else FAIL. PASS means the joint holds the required retention through the design hold; FAIL means re-torque, a higher initial preload or a cooler operating point is needed.

6. Read the result with the model assumptions in view: steady-state Norton creep only (primary creep is neglected, the same documented conservative assumption the creep-rupture sibling makes; the stress relaxes monotonically, so the model overestimates the retained preload early and converges as primary creep exhausts), and the power-law tail for n > 1 (sigma ~ t^(-1/(n-1)), strictly positive after any finite hold).
7. Confirm the deterministic checks with the contract test scripts/test_creep_stress_relaxation.py.

Worked example

A 200 MPa bolt preload (sigma_0 = 2.0e8 Pa) at fixed total strain in the default Inconel-718 class alloy held at 700 C (973.15 K), the hot operating point where relaxation of this alloy is meaningful over hours to days. All values are real outputs of the module:

• After 100 h (3.6e5 s): relaxed_stress(2.0e8, 3.6e5, 973.15) = 114410840.82577138 Pa, retained fraction 0.5720542041288569: the preload has relaxed to 57.21 percent of its initial value.
• After 1000 h (3.6e6 s): relaxed_stress(2.0e8, 3.6e6, 973.15) = 78364659.4406817 Pa, retained 0.39182329720340847: 39.18 percent retained, the 100 h to 1000 h drop showing the slowing power-law tail, not a linear bleed.
• Time to relax to 90 percent of the preload at 700 C: time_to_relaxed_fraction(0.9, 2.0e8, 973.15) = 11527.301420325737 s = 3.202 h; to 80 percent: 36800.19442005393 s = 10.222 h; to 50 percent: 823680.8543417541 s = 228.800 h (9.53 days). Monotone: 3.2 h < 10.2 h < 228.8 h.
• Temperature sensitivity at the same 1000 h hold: 650 C (923.15 K) gives 116399871.02150063 Pa (0.5820 retained) and 600 C (873.15 K) gives 169638258.38946512 Pa (0.8482 retained) against 0.3918 at 700 C: a 100 K slip from 700 C to 600 C nearly doubles the retained preload, the exp(-Q / (R * T)) Arrhenius sensitivity.
• Retained-preload margin after 1000 h at 700 C: preload_margin(2.0e8, 3.6e6, 973.15, 0.35) = {"retained_fraction": 0.39182329720340847, "margin": 0.11949513486688135, "verdict": "PASS"} (39.18 percent retained against a 35 percent requirement); preload_margin(2.0e8, 3.6e6, 973.15, 0.45) = {"retained_fraction": 0.39182329720340847, "margin": -0.12928156177020345, "verdict": "FAIL"} (against a 45 percent requirement the joint needs re-torque or a higher initial preload).
• Initial rate identity: at 200 MPa and 700 C the initial creep rate is 1.2140624548073926e-08 1/s, so E * eps_dot = 2549.5311550955244 Pa/s and the stress first falls at about 2.55 kPa per second; the module finite difference over a 1 ms step matches within 1e-6 relative.
• Newtonian branch: with the override dict {"norton_a": 5.0e-16, "norton_n": 1.0, "norton_q": 0.0} (K = E * A = 1.05e-4 1/s), relaxed_stress(2.0e8, 1.0e4, 873.15) = 69987549.82223107 Pa, exactly sigma_0 * exp(-K * t) within 1e-12 relative.
• 600 C sanity: relaxed_stress(2.0e8, 1.0e4, 873.15) = 199844353.157 Pa, retained 0.99922: at 600 C the alloy relaxes only 0.08 percent in 1e4 s, which is why the worked example lives at 700 C.

Pitfalls

• Confusing this leaf with the constant-stress sibling: creep-rupture accumulates creep strain at a sustained stress and reports rupture life and design-life margins; this leaf answers how a preload decays at fixed total strain. The fixed-strain decay question has no function in creep-rupture's constant-stress machinery, and the constant-stress life products are not claimed here.
• Running the relaxation check at a cool operating point: at 600 C the default alloy retains 99.92 percent of the preload after 1e4 s, so a room-temperature or mildly elevated check misses the failure mode entirely; the relaxation check lives at the hot operating point (700 C in the worked example) where the decay is meaningful over hours to days.
• Mixing units in the input chain: stress enters in Pa (convert MPa by 1e6), temperature in K (Celsius plus 273.15) and time in seconds (hold times in hours convert by 3600); a seconds-versus-hours slip misreads the relaxation by 3600x.
• Forgetting the modulus is an input: E is the high-temperature elastic modulus of the material at the operating point, an input constant (2.1e11 Pa for the default alloy); it is never derived from the stress-strain curve here.
• Overriding the material carelessly: a dict override merges over the default alloy, so a partial override (for example only norton_n) silently keeps the other Inconel-718 class constants; quoting the defaults for another alloy is a material-data error.
• Reading the verdict without the steady-state-only assumption: the model overestimates the retained preload early because primary creep is neglected; the margin is conservative in the direction of the retained preload only as the creep exhausts.
• Treating the module material as an alloy database: the Inconel-718 class constants are reference-only typicals shared with the creep-rupture sibling; production preload-retention margins need measured alloy data.

Verification

• Confirm relaxed_stress(2.0e8, 3.6e5, 973.15) returns 114410840.82577138 Pa and relaxed_stress(2.0e8, 3.6e6, 973.15) returns 78364659.4406817 Pa, with retained fractions 0.5720542041288569 and 0.39182329720340847; the 600 C and 650 C 1000 h cases return 169638258.38946512 and 116399871.02150063 Pa.
• Confirm time_to_relaxed_fraction(0.9, 2.0e8, 973.15) returns 11527.301420325737 s, 0.8 -> 36800.19442005393 s and 0.5 -> 823680.8543417541 s, strictly ordered, and the round trip: relaxing for the time computed for a fraction f and reading the retained fraction recovers f within 1e-9 for f in {0.3, 0.5, 0.8, 0.95}.
• Confirm preload_margin(2.0e8, 3.6e6, 973.15, 0.35) returns margin 0.11949513486688135 with verdict PASS and the 0.45 requirement returns margin -0.12928156177020345 with verdict FAIL; a required fraction equal to the achieved retention gives margin 0.0 and PASS (never assert at exact equality at the boundary).
• Confirm the decay identities: a zero hold returns sigma_0 within 1e-6 relative; the 1 ms finite difference matches -E * A * sigma_0^n * exp(-Q / (R * T)) = -2549.5311550955244 Pa/s within 1e-6 relative; the n = 1 branch matches sigma_0 * exp(-K * t) within 1e-12 relative and the n = 1 + 1e-9 power branch matches the exponential within 1e-6 relative.
• Confirm monotone decay over the grid [0, 1e3, 1e4, 3.6e5, 3.6e6, 3.6e7] s at 700 C (strictly falling, every value positive, the 1e4 h retained fraction 0.26709127676828587 in (0.2, 0.4)), that the retained fraction after 1000 h falls as T rises over [873.15, 923.15, 973.15, 1023.15] K, and that for n > 1 a higher preload relaxes to a lower retained fraction at fixed T and t (100 MPa 0.757, 200 MPa 0.392, 400 MPa 0.196), the closed-form consequence of the sigma_0-independent power-law tail.
• Confirm the material handling: the dict override reproducing the default constants gives identical numbers within 1e-12 relative, and the shared constant set is proven by the default alloy Norton rate at 300 MPa and 600 C equaling the creep-rupture sibling anchor 1.2698242552930268e-09 1/s within 1e-6 relative.
• Confirm every non-positive stress, temperature or required fraction, every fraction outside (0, 1], every negative hold time, every stress exponent below 1 and every unknown material name raises ValueError.
• Run the contract test offline: python3 scripts/test_creep_stress_relaxation.py (42 tests, deterministic, passes under both the system python3 and the pyenv 3.13.12 interpreter).

Related leaves

• structures/materials/creep-rupture: the constant-stress owner in this pack; sustained-stress creep rate, accumulated creep strain, rupture life and design-life margin questions route there, and this leaf consumes the shared Inconel-718 class Norton constants internally.
• structures/thermal-structures/thermal-stress-analysis: the constrained-expansion load side of the same elevated-temperature environment (owned by that leaf, not here).
• structures/materials/ramberg-osgood: the room-temperature elastic-plastic curve beneath the creep regime.
• structures/materials/fracture-toughness and structures/materials/crack-tip-plasticity-correction: the crack-driven failure modes that compete with preload decay in hot fasteners.
• structures/materials/material-selection: temperature limits and alloy family screening before the preload-retention check.
• structures/fatigue/stress-life-curve and structures/fatigue/strain- life-fatigue: cyclic life methods, the non-creep route for the same part (creep-fatigue interaction stays with the fatigue pack).

Behavior contract (gate 3)

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

python3 scripts/test_creep_stress_relaxation.py

The test covers the worked-example anchors (100 h relaxed stress 114410840.82577138 Pa retained 0.5721, 1000 h 78364659.4406817 Pa retained 0.3918, the 600 C and 650 C cases, the 0.9/0.8/0.5 time-to-fraction anchors 11527.30 / 36800.19 / 823680.85 s), the round trips recovering each target fraction, the margin verdicts at the 0.35 PASS and 0.45 FAIL requirements and the equality boundary, the zero hold time and initial-rate identities, the monotone decay grid and the 1e4 h power-law tail, the n = 1 exponential branch and the n -> 1 limit, temperature and preload monotonicity, material dict overrides and the shared-constant parity with creep-rupture, and ValueError rejection of non-physical inputs and unknown materials.

Compliance

• Standards referenced, not reproduced: MMPDS documents elevated- temperature creep and relaxation design practice for metallic airframe materials; FAR-25 frames the elevated-temperature part and its strength demonstration. The equations above are standard engineering methodology, summary-only per standards-map.yaml, and the material constants are reference-only typicals, not a reproduced data table.
• compliance: STANDARDS-REF, gated: false.