Creep Life: Method

Method: the four parameters, the least-squares fit, the lower bound, the life-fraction rule

Scope

The tool estimates the creep-rupture life of a metal at a stress and a temperature from a time-temperature parameter, and works backward to the stress or temperature that gives a required life. The rupture model is either an equation printed in MIL-HDBK-5J, your own coefficients in the same form, or a least-squares fit of creep-rupture tests that you paste. With times to a creep strain in place of rupture lives, the same fit gives an isostrain creep curve. Every relation is a closed form or comes from two U.S. government documents: MIL-HDBK-5J (sections 9.6.4.2 and 9.9.4, the creep and stress-rupture figures of chapter 6) and NASA Memorandum 3-10-59E by Manson and Mendelson. The papers of Larson and Miller, Orr, Sherby and Dorn, Manson and Succop, Robinson, and Monkman and Grant are cited for the ideas, not reproduced.

The four equations

MIL-HDBK-5J section 9.6.4.2 writes four relations between the time t (hours), the absolute temperature T and X = log(stress), all logarithms to base 10:

Larson–Miller: log t = c + (b₁ + b₂ X + b₃ X² + b₄ X³) / T
Orr–Sherby–Dorn: log t = c + b₁ / T + b₂ X + b₃ X² + b₄ X³
Manson–Succop: log t = c + b₁ T + b₂ X + b₃ X² + b₄ X³
Manson–Haferd: log t = c + (T − T_a)(b₁ + b₂ X + b₃ X² + b₄ X³)

Each says that one combination of time and temperature, the parameter P, depends on the stress alone, so that tests at every temperature fall on one master curve of stress against P:

Larson–Miller: P = T (C + log t), with C = −c
Orr–Sherby–Dorn: P = log t − D / T, with D = b₁; the apparent activation energy is Q = 2.3026 R D
Manson–Succop: P = log t + B T, with B = −b₁
Manson–Haferd: P = (log t − log t_a) / (T − T_a), with log t_a = c; (T_a, log t_a) is the point where the constant-stress lines of log t against T meet

The handbook states that the Larson–Miller relation models most materials adequately and is usually preferred. Its printed equation for Inconel 718 forging is nevertheless of the Manson–Succop form, and the one for HAYNES HR-120 is of the Larson–Miller form. C = 20 is the value usually quoted with the parameter; the least-squares constants of real data sets in the two government documents range from 15 to 24.

Printed equations offered

Inconel 718 forging (MIL-HDBK-5J Figures 6.3.5.1.7(a) to (e), pages 6-61 to 6-65): stress rupture and the times to 0.10, 0.20, 0.50 and 5.00 percent creep, which share the coefficients b₁ to b₄ and differ in c, with standard errors of 0.29, 0.56, 0.41, 0.34 and 0.37 in log hours; the data lie between 1000 and 1300 °F and about 37 to 158 ksi. HAYNES HR-120, annealed (Figures 6.3.10.1.7(a) and (b), pages 6-114 and 6-115): stress rupture from 283 tests between 1100 and 2200 °F, standard error 0.155. The stress ranges are read from the plotted tests of the figures. The handbook cautions that each model "may provide unrealistic predictions for temperatures and stresses beyond those represented", so results outside the range are flagged.

Fitting pasted tests

Each line of the box is one test: stress, temperature, hours. Following NASA Memorandum 3-10-59E and the handbook, the dependent variable is log t and the sum S of the squares of its residuals is minimised. For the Larson–Miller, Orr–Sherby–Dorn and Manson–Succop forms the constant (C, D or B) enters the equation linearly together with the polynomial coefficients, so one linear least-squares solution gives the optimum constant and the master curve at once. For the Manson–Haferd form the equation is linear once T_a is assumed; T_a is searched below the lowest test temperature for the smallest S, as the memorandum does by trial. The memorandum shows that S is often flat over a wide range of T_a as long as log t_a moves with it, so a searched T_a far from another analyst's value need not mean a different curve. When S keeps falling to the edge of the search, the data do not define a convergence point and the page says so.

The master curve is a polynomial of degree 1, 2 or 3 in X. The handbook writes the cubic and warns that such polynomials can behave badly outside the data; a lower degree is safer for a small data set. A constant may also be held at an entered value, for example C = 20, and the page lists all four parameters fitted to the same tests so that their standard errors can be compared.

The statistics are the number of tests n, the number of fitted constants p, the standard error of estimate s_e = √(S / (n − p)), R² and the standard deviation of log t. A searched T_a counts as a fitted constant. The handbook gives 0.1 to 0.2 as typical standard errors. The scatter band shown is ± 2 s_e in log t, a factor on life.

Lower-bound life

For a fit, the lower bound is the one-sided prediction bound for one further test: log t_low = log t − t_{γ, n − p} s_e √(1 + h), with Student's t at the confidence γ and the leverage h = x′(A′A)⁻¹x of the service point in the fit, which grows as the point moves away from the tests. For a Manson–Haferd fit with a searched T_a the leverage is taken with T_a fixed at its optimum. For a printed equation only the standard error is known, and the bound is log t − z s_e with z from the normal distribution; the handbook prints an inverse matrix for the Inconel 718 equation from which the uncertainty of the coefficients could be added, and the tool does not use it. This bound is not the handbook's tolerance limit (a stated fraction of the population at 95 percent confidence), which also needs the variance between heats.

Allowable stress and temperature, design factor

The equation is solved for the highest stress at which the life reaches the required life at the service temperature, searching from the top of the stress range of the model downward to half of its lowest stress, and for the highest temperature at which it does at the service stress. A factor on stress reads every life at the stress times the factor and divides the allowable stress by it; a factor on life divides every life by it. Because log t falls steeply with stress, a small factor on stress is worth a large factor on life; the page shows whichever is chosen. The factors are the analyst's, not code values.

Life fraction over a history

Robinson's rule adds the fractions of life used at each condition, D = Σ t_i / t_r,i, and expects rupture when D reaches 1. The remaining life at the service condition is (1 − D) t_r. The rule is linear and ignores the order of the blocks, which real materials do not always do. The history is pasted one block per line (stress in the stress unit of the model, temperature, hours).

Time-temperature equivalence

At one stress, two exposures use the same fraction of the life when they reach the same value of the parameter. For the Larson–Miller form T₁ (C + log t₁) = T₂ (C + log t₂); the other forms follow from their parameters in the same way. The result depends on the constant of the model only, not on the master curve. It is the usual basis for shortening a test by raising its temperature, and it holds only while the same mechanism governs at both temperatures.

Monkman–Grant relation and Norton's law

Monkman and Grant found that the rupture life and the minimum creep rate of a material are tied by t_r ε̇_min^m = C_MG, with m near 1. The page gives the life for an entered rate and the rate that corresponds to the model life. Norton's law, ε̇ = B σⁿ, gives the secondary creep rate at one temperature; the page multiplies it by the exposure for the creep strain and divides a strain limit by it for the hours to that strain. The constants m, C_MG, B and n are material constants that must come from tests; the defaults are illustrative numbers, not properties of the alloy in the model dropdown.

Units

The service stress is entered in any stress unit and converted to the stress unit of the model. Temperatures are plain numbers in the unit chosen in the form and are converted to the absolute scale of the equation (degrees Rankine or kelvin) with 459.67 and 273.15. The two source documents add 460 to degrees Fahrenheit; the difference moves a fitted constant in its fourth figure.

Data sets offered

Alloy 325 (MOD): the 41 simulated tests of MIL-HDBK-5J Table 9.9.4.1 (page 9-240), the handbook's own example of a creep-rupture test plan, for which its Figure 9.9.4.1(c) uses a Larson–Miller constant of 19; the fit here gives 19.4. 17-22A(S) steel and 18-8 stainless steel: Tables I and II of NASA Memorandum 3-10-59E, whose worked examples (C = 23.8 and 15.3, D = 41,470 and 32,900, T_a = 650 °F and −300 °F) the fit reproduces. The 18-8 points are readings of isothermal curves, not single tests, so their scatter understates that of tests.

Limits

The parameters are empirical correlations. They do not know about changes of microstructure, oxidation, section size or notches, and the handbook does not recommend extrapolating its equations beyond their data. Multiaxial stress, cyclic loading, creep-fatigue interaction, relaxation and primary or tertiary creep strain are outside the tool. A mean curve fitted to a few heats is not a design allowable.