Mechanical Behavior · Creep

Creep and deformation mechanism maps: which mechanism wins?

Start with the creep curve: drag the stress up and watch the steady slope and the rupture time change. Then click the map to see which mechanism sets that slope.

What do the three stages of a creep curve show?

Creep is slow, permanent stretching under a constant load, below the yield stress. It matters above about 0.3 to 0.4 of the melting temperature Tm (in kelvin). T/Tm is called the homologous temperature. The steady (secondary) rate follows the power law ε̇ = A (σ/G)n exp(−Q/RT), where G is the shear modulus and Q the activation energy for diffusion.

Strain against time at constant load. Dashed: the steady slope. The time axis rescales to the rupture time.
Steady rate against stress (log-log) at your temperature. The slope is the stress exponent n.

Load and temperature

Try it: in nickel at T/Tm = 0.6, raise the stress from 30 to 60 MPa. The steady rate rises about 80 times and the rupture time falls from 54 h to about 41 min. The local n reads 5.8 at 30 MPa, above the tabulated 4.6, because diffusion along dislocation cores feeds climb here.

Which creep mechanism wins? The deformation mechanism map

Four mechanisms run at the same time. Each colour is where one of them gives the fastest strain rate. Dislocation glide: dislocations slide on slip planes. Power-law creep: dislocations glide and climb past obstacles. Coble creep: atoms diffuse along grain boundaries. Nabarro-Herring creep: atoms diffuse through the grains. Rate equations and data are from Frost and Ashby (1982).

Dislocation glidePower-law creepCoble creepNabarro-Herring creepThin lines: shear strain rate, 1/sclimb fed by core diffusion (left) or lattice diffusion (right)
Move over the map to read it. Click to pick a point.

Material and microstructure

Try it: set T/Tm = 0.6 and 20 MPa in nickel, then slide the grain size from 100 µm down to 10 µm. The point moves from power-law creep into Coble creep, and the strain rate rises about 130 times.

How do atoms move in each mechanism?

Under tension, atoms leave the grain faces with low normal stress (top and bottom here) and add onto the faces being pulled apart (left and right), so the grain gets longer. They can go through the grain (Nabarro-Herring) or along its boundary (Coble). In power-law creep, a dislocation stuck at a particle climbs: it absorbs vacancies, steps up to a new slip plane and glides on. The number of moving dots shows each path's share at your settings.

Nabarro-Herring: through the grain. Blue: faces with low normal stress. Red: faces under tension.
Coble: along the grain boundary (violet).
Climb: glide, stop, climb, glide.

Which path carries the atoms?

Try it: in nickel with 100 µm grains, raise T/Tm from 0.6 to 0.9. Boundary diffusion has the lower activation energy, so the lattice path gains: the Coble share falls from 99% to about 22%.

How far can you extrapolate a creep test? The Larson-Miller parameter

Power-plant parts are often designed for 100,000 h (about 11 years) or more; lab tests last hours to months. The Larson-Miller parameter P = T (C + log10 tr), with T in kelvin and rupture time tr in hours, assumes that the same stress gives the same P at every temperature. Here three short "tests" come from the model above, and P carries them down to a service temperature. Rupture times use the Monkman-Grant rule: ε̇min × tr = constant.

Rupture life against temperature at your stress. Dots: tests. Coral: Larson-Miller prediction. Grey: what the model gives. The strip under the axis shows the dominant mechanism.

Tests and service

Try it: with nickel at 30 MPa, C = 20 predicts 3.8 years at service, 66 times the model's 510 h. Press "Fit C": C becomes 11.2 and the prediction (846 h) is within 2 times of the model. Then lower the service T/Tm to 0.5: climb there is fed by core diffusion and the error grows to about 4 times.

What to remember

Steady rate sets the lifeRupture time is roughly a constant divided by the steady creep rate (Monkman-Grant). Halve the rate and the life doubles.
Stress exponent names the mechanismn near 4 to 5 means dislocation climb in pure metals. n = 1 means diffusional creep. Measure n before you pick a model.
Small grains creep faster at low stressCoble creep scales as 1/d3 and Nabarro-Herring as 1/d2. Coarse grains slow both; single crystals remove the grain-boundary path.
Extrapolate only within one mechanismLarson-Miller works while the test and the service point sit in the same field of the map. Check the map before you trust the number.
More detail: the equations, the data and the limits of the model

All rates are shear rates, as in Frost and Ashby. σs is the shear stress, G = μ the shear modulus at temperature T, k Boltzmann's constant. Tensile values use σ = √3 σs and ε̇ = γ̇/√3.

Modulus:

μ(T) = μ0 [1 + (T − 300)/Tm · (Tm/μ0)(dμ/dT)]

Dislocation glide past discrete obstacles (eq. 2.9), with obstacle strength τ̂ = μ b / l:

γ̇ = γ̇0 exp[−(ΔF/kT)(1 − σs/τ̂)]

Power-law creep (eqs. 2.20 and 2.21), with power-law breakdown (α′ = 1000, chapter 4):

γ̇ = A (Deff μ b / kT) [sinh(α′ σs/μ)/α′]n
Deff = Dv + (10 ac/b2)(σs/μ)2 Dc

Diffusional flow (eqs. 2.29 and 2.30). The first term is Nabarro-Herring, the second Coble:

γ̇ = (42 σs Ω / kT d2) [Dv + (π δ/d) Db]

Adding them (section 3.2): net rate = the larger of glide and power-law creep, plus diffusional flow. A field boundary is where two rates are equal.

Data from Frost and Ashby, chapter 4, Table 4.1 (obstacle spacings from the chapter 4 text):

QuantityNickelAluminium
Atomic volume Ω (m3)1.09 × 10−291.66 × 10−29
Burgers vector b (m)2.49 × 10−102.86 × 10−10
Melting point Tm (K)1726933
Shear modulus at 300 K, μ0 (GPa)78.925.4
(Tm/μ0)(dμ/dT)−0.64−0.50
Lattice diffusion D0v (m2/s)1.9 × 10−41.7 × 10−4
Qv (kJ/mol)284142
Boundary diffusion δD0b (m3/s)3.5 × 10−155.0 × 10−14
Qb (kJ/mol)11584
Core diffusion acD0c (m4/s)3.1 × 10−237.0 × 10−25
Qc (kJ/mol)17082
Power-law exponent n4.64.4
Dorn constant A3.0 × 1063.4 × 106
Glide: ΔF/μ0b3, γ̇0 (1/s)0.5, 1060.5, 106
Obstacle spacing l (m)4 × 10−8 work-hardened, 2 × 10−7 annealed

μ0 is Frost and Ashby's shear modulus for a dislocation in the real crystal, worked out from single-crystal elastic constants. It is not the same quantity as the polycrystal shear modulus used in the strengthening lab (79 GPa for Ni and 26 GPa for Al, Hansen 2004), though for these two metals the numbers are close. It is the value their creep constants were fitted with.

Limits. Pure metals only. Solutes and precipitates slow creep by orders of magnitude, so alloys need their own data. Equation 2.9 has no backward jumps, so at zero stress it still gives a small rate; this page counts glide only above 0.2 τ̂. Not included: lattice-resistance glide (small in FCC metals), grain-boundary sliding, Harper-Dorn creep, threshold stresses and recrystallisation. The primary and tertiary stages are drawn with fixed proportions (primary strain 0.25 × the Monkman-Grant constant); only the steady rate and the rupture time are computed. The loading strain uses E = 2.6 μ.

On this site: Diffusion and kinetics for real alloys (the D = D0 exp(−Q/RT) behind every rate here) · Strengthening mechanisms · Reading a tensile curve · Dislocations and Burgers vectors (climb) · EBSD: KAM and GND density after creep · Grain boundaries and the CSL · Strain-rate sensitivity by nanoindentation · Fatigue and S-N curves

Questions people ask

What is creep in materials?

Creep is slow, time-dependent plastic strain under a constant stress that is below the yield stress. In metals it becomes important above about 0.3 Tm for pure metals and about 0.4 Tm for alloys, because it needs diffusion.

What are the three stages of creep?

Primary: the rate falls as dislocations tangle. Secondary (steady state): hardening and recovery balance, so the rate is constant. Tertiary: the rate rises as cavities and necking reduce the load-bearing area, until rupture.

What is the difference between Nabarro-Herring and Coble creep?

Both move atoms by diffusion from grain faces with low normal stress to faces under tension, and both have n = 1. Nabarro-Herring goes through the grain (lattice diffusion, rate ∝ 1/d2). Coble goes along the grain boundaries (rate ∝ 1/d3) and wins at lower temperature and in fine grains.

What does the stress exponent n in power-law creep tell you?

n = d log ε̇ / d log σ at fixed temperature. In pure metals n of 4 to 5 points to dislocation climb; n = 1 points to diffusional creep. A rising n at high stress (above σs/μ of about 10−3) is power-law breakdown.

How do you read a deformation mechanism map?

Find your stress (divided by the shear modulus) and your temperature (divided by Tm). The field you land in is the mechanism that gives the fastest strain rate there. The contour through the point gives the strain rate. Maps are drawn for one grain size.

How do you calculate the Larson-Miller parameter?

P = T (C + log10 tr), with T in kelvin and tr in hours. C = 20 was proposed for steels by Larson and Miller (1952). Fit C to your own data, and do not carry P across a change of mechanism.

Why are turbine blades made as single crystals?

In nickel-base superalloy blades, removing grain boundaries removes Coble creep, grain-boundary sliding and the boundaries where creep cavities form. That leaves the slower dislocation creep, which the alloy's precipitates then slow further.

References

Show the 9 references
  1. H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford (1982). Chapter 2 (eqs. 2.9, 2.20, 2.21, 2.29, 2.30; ΔF = 0.5 μb3 and γ̇0 = 106/s), section 3.2 (adding the rates), chapter 4 and Table 4.1 (Ni and Al data, obstacle spacings, α′ ≈ 103), Figs. 4.1 to 4.4 (Ni) and 4.13 to 4.14 (Al). Full text: defmech.engineering.dartmouth.edu.
  2. M. F. Ashby, A first report on deformation-mechanism maps, Acta Metallurgica 20, 887 to 897 (1972). doi:10.1016/0001-6160(72)90082-X
  3. F. R. N. Nabarro, Deformation of crystals by the motion of single ions, in Report of a Conference on the Strength of Solids, Physical Society, London (1948), pp. 75 to 90.
  4. C. Herring, Diffusional viscosity of a polycrystalline solid, Journal of Applied Physics 21, 437 to 445 (1950). doi:10.1063/1.1699681
  5. R. L. Coble, A model for boundary diffusion controlled creep in polycrystalline materials, Journal of Applied Physics 34, 1679 to 1682 (1963). doi:10.1063/1.1702656
  6. J. Weertman, Theory of steady-state creep based on dislocation climb, Journal of Applied Physics 26, 1213 to 1217 (1955). doi:10.1063/1.1721875
  7. F. R. Larson and J. Miller, A time-temperature relationship for rupture and creep stresses, Transactions of the ASME 74, 765 to 775 (1952).
  8. F. C. Monkman and N. J. Grant, An empirical relationship between rupture life and minimum creep rate in creep-rupture tests, Proceedings ASTM 56, 593 to 620 (1956).
  9. M. E. Kassner, Fundamentals of Creep in Metals and Alloys, 3rd ed., Butterworth-Heinemann (2015), chapters 1 to 3 and 5.
Cite this page: Tripathy, Manisha. “Creep and Deformation Mechanism Maps.” untethered atom, 2026, https://untetheredatom.com/mechanical-behavior/creep-and-deformation-maps.
BibTeX 
@misc{tripathy2026creepmaps,
  author = {Tripathy, Manisha},
  title  = {Creep and Deformation Mechanism Maps},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/mechanical-behavior/creep-and-deformation-maps}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.