Materials Science · Fatigue, creep and time-dependent damage

Sustained loads and creep mechanisms

Explain how defects, diffusion or molecular rearrangement permit continuing deformation under a sustained load. Read a strain–time record, distinguish its rates from its total strain, and test simple models against the actual loading constraint.

Useful background: Elasticity, plasticity and tensile strength; Cracks, fracture and toughness. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

Distinguish sustained-load creep, instantaneous loading strain and stress relaxation.
Explain representative dislocation, diffusion and polymer routes, including the role of accommodation and changing structure.
Calculate interval creep rates, time-dependent elongation and changing true stress with consistent units.
Derive a simple spring–dashpot creep response and recognise which real responses it cannot represent.

A fixed load can produce changing strain

Creep is time-dependent deformation under sustained stress. At a sufficiently large fraction of the melting temperature, diffusion and dislocation processes can contribute strongly in metals and ceramics. This fraction, called homologous temperature, compares absolute temperatures in kelvin; it is not a ratio of Celsius readings. Polymers can creep at ordinary temperatures when molecular relaxation is active.

Imagine a loaded support whose length continues to increase while the applied force is unchanged. A short tensile test may have ended before that slow deformation became appreciable. Creep concerns the response over time; creep rupture is a particular eventual failure endpoint. Appreciable creep does not require that rupture has already occurred.

For a uniaxial record, engineering strain is ε = (L − L₀)/L₀, relative to a stated initial gauge length L₀. A load can first cause a rapid strain and then additional strain during the hold. Here ε_creep(t) = ε(t) − ε(0+) excludes the initial loading strain; 0+ means just after the idealised load application. Actual experiments also need a definition of ramp duration and a correction for thermal expansion or instrument drift.

Holding strain fixed is another experiment. The stress can then decrease with time: stress relaxation. A material can exhibit creep under stress control and relaxation under strain control. The controlled variable and specimen history must accompany the word time-dependent.

Homologous temperature T/T_m is useful for a crystalline material with a meaningful melting reference T_m. The ratio indicates a temperature scale, not a universal onset at one fraction or a complete mechanism identifier. An amorphous polymer’s relevant relaxation processes and glass transition require a different description.

A slow step can enable continuing deformation

In a crystal, a dislocation is a line defect whose motion permits neighbouring regions to slip. Glide moves it along a slip plane and produces shear deformation. Obstacles can arrest that motion. Thermal activation can help a dislocation cross an obstacle; an edge component can also climb out of its original slip plane by absorbing or emitting vacancies, which are unoccupied lattice sites.

Climb can allow a pinned dislocation to bypass an obstacle and resume glide. Much of the accumulated strain may come from the glide step even when slower vacancy transport controls how often the sequence repeats. Naming the rate-controlling step is therefore different from naming the motion that supplies most of the strain.

Diffusional creep is another route. A nonhydrostatic stress can give different grain surfaces different chemical potentials, biasing atomic transport through the lattice or along boundaries. Grains can change shape as matter is redistributed. Neighbouring grains must accommodate the shape changes, including by relative sliding at their boundaries. Boundary sliding without accommodation would create incompatible displacements or openings.

Transport distance and boundary area then matter. In a diffusion-dominated regime, reducing grain size can increase creep rate, even though a finer grain structure may strengthen a metal against another deformation process. That comparison depends on the actual mechanism and state. A single small-grain observation is not enough to identify the controlling route.

In polymers, bond distortion can provide a fast response while chain-segment rotations and rearrangements evolve over longer times. Temperature, crosslinks, entanglements and absorbed species affect mobility. Some delayed strain can recover after unloading; flow, damage or rearrangement beyond reversible constraints can leave residual deformation. Room-temperature creep is therefore possible, without assigning every polymer the same recovery or rate law.

The curve is not a universal law

Primary, approximately steady secondary and accelerating tertiary stages are common descriptions, not mandatory features of every test. Changes in section, damage and microstructure can affect the rate. Report stress convention, temperature, time and specimen state.

In a common metallic primary stage, increasing resistance to further dislocation motion can reduce the rate. Recovery processes can oppose that hardening. An interval with approximately balanced structural evolution may have a nearly steady rate. These descriptions connect slope with evolving state; a straight interval is not proof of a unique atomic mechanism.

Later acceleration can involve necking or loss of effective load-bearing area, cavities, cracks or another structural change. Under constant force, a smaller section raises true stress and can promote further deformation: a feedback between geometry and response. Under other controls, microstructural damage can still accelerate strain. The curve alone does not separate these causes.

The invented record below begins at 0+ with 0.40% strain. Its initial interval rates decline, its 2–6 h interval has a common average rate, and its later interval rates rise. The figure uses straight joins between chosen points and displays the actual interval averages below; no unmeasured smooth instantaneous-rate curve or rupture time is implied.

Original illustrative engineering-strain observations during a sustained-load test at stated constant temperature. Strain is relative to the unloaded initial gauge length. No material identity or rupture endpoint is supplied.
Time after loading (h)	Total strain ε (%)
0+	0.40
1	0.70
2	0.85
4	1.05
6	1.25
7	1.45
8	1.85

Original chosen total strains at 0+, 1, 2, 4, 6, 7 and 8 h are 0.40, 0.70, 0.85, 1.05, 1.25, 1.45 and 1.85%. Interval-average rates are 0.30, 0.15, 0.10, 0.10, 0.20 and 0.40%/h. Straight joins do not assert the instantaneous rate between observations. No rupture event is supplied.

Slope and accumulated strain answer different questions

Over an interval from t₁ to t₂, the average strain rate is (ε₂ − ε₁)/(t₂ − t₁). Strain is dimensionless; rate has units of inverse time. A percentage must be divided by 100 before it is used as a strain fraction. For example, 0.10%/h is 0.001/h, not 0.10/h.

Strain rate is the slope of a strain–time curve, whereas accumulated creep strain is its change from the defined initial loading value. In an interval of approximately constant rate, Δε_creep ≈ rate × duration. In a variable-rate record, add the strain changes or integrate the rate over the actual history. A global average can conceal early slowing and later acceleration.

To obtain elongation from engineering strain, use ΔL = L₀ε with the same reference L₀. Compare like quantities: the total change from the unloaded state includes the initial loading strain, while additional elongation during the hold excludes it.

The table supports observed deformation through 8 h. It supplies neither an eventual rupture time nor a safe allowable strain. Estimating a time to an additional strain under a constant-rate assumption is another conditional calculation, requiring that assumption to remain appropriate over the interval.

Constant force is not constant true stress

Engineering stress uses the reference area A₀: σ_eng = F/A₀. True axial stress uses the current section area A: σ_true = F/A. For a uniform section these are section averages; a necked or damaged body also has a nonuniform local state.

With fixed force, a shrinking area increases F/A even when F/A₀ stays the same. Holding true stress fixed instead requires adjusting force as the area changes. A strain–time comparison that silently switches between these controls has changed the experiment.

The mechanism and rate also depend on temperature and material state. Grain structure, particles, phase changes, cavities and polymer mobility may evolve during a hold. Compare measurements under defined histories before attributing a rate change to one stress exponent or one atomic pathway. The advanced high-temperature lesson develops conditioned stress/temperature rate fits and an evolving microstructure case.

A simple model explains one time-dependent response

An ideal Maxwell model connects an elastic spring and a viscous dashpot in series. It is a constitutive analogue: the elements share the same uniaxial stress σ and their strains add. The spring has ε_s = σ/E with modulus E in pascals. The dashpot has dε_v/dt = σ/η, with viscosity parameter η in Pa·s. Their physical pictures do not mean that a polymer contains literal tiny springs and pistons.

For an initially relaxed model loaded rapidly to a constant stress σ₀, the spring immediately contributes σ₀/E. Integrating the constant dashpot rate gives ε_v = σ₀t/η. Thus ε(t) = σ₀/E + σ₀t/η during the hold, with t in seconds when η uses Pa·s. The total strain increases even though stress is fixed.

On ideal instantaneous unloading, the spring strain is removed immediately. The dashpot strain remains in this particular model because its rate becomes zero at zero stress. A real polymer may instead show additional delayed recovery, a limiting compliance, nonlinear flow or damage. The simple Maxwell response cannot describe every such behaviour or the three-stage metallic record above.

Choose the model to answer a defined question, then test its missing behaviour. In the illustrative Maxwell case the rate is constant because E, η and stress are prescribed constants. In a real evolving specimen, the observed curve and additional structural evidence decide whether that simplification is useful.

A spring and dashpot in series share stress and add strains. With E=2.0 GPa, η=10^13 Pa·s, σ=10 MPa and a 1000 s hold, spring strain is 0.50%, dashpot strain is 0.10% and total strain is 0.60%. The ideal spring recovers immediately on unloading; the dashpot retains 0.10% in this model. This analogue is not the three-stage record or a universal polymer model.

Worked examples

Separate loading strain, interval rate and creep elongation

Use the illustrative strain record and L₀ = 50 mm. Find the mean rate from 2 to 6 h, the additional creep strain and elongation by 8 h, and the total elongation at 8 h. What does the later acceleration establish?

From 2 to 6 h, the total-strain change is 1.25% − 0.85% = 0.40% over 4 h. The mean rate is 0.10%/h = 0.001/h ≈ 2.778 × 10^−7 s^−1. The initial loading offset cancels in this interval difference.
By 8 h, additional creep strain is 1.85% − 0.40% = 1.45% = 0.0145. Additional elongation during the hold is 50 × 0.0145 = 0.725 mm.
Total elongation from the unloaded reference is 50 × 0.0185 = 0.925 mm. Its 0.200 mm difference from the additional value is the initial 0.40% loading elongation.
The 6–7 h rate is 0.20%/h and the 7–8 h rate is 0.40%/h. This record accelerates late. Geometry, damage and changing structure are possible contributors; the curve alone does not identify which one caused it or when rupture will occur.

The 2–6 h mean rate is 0.10%/h, about 2.778 × 10^−7 s^−1. By 8 h the hold adds 0.725 mm, while total elongation is 0.925 mm. Acceleration is observed, but its mechanism and rupture time are unresolved.

Track the section as a load is held

An illustrative uniform specimen carries a fixed tensile force of 10,000 N. Its original area is 100 mm² and its current area later becomes 80 mm². Find engineering and current true stress. What force would maintain the original true-stress value at that later area?

Initially, force divided by area is 10,000/100 = 100 N/mm² = 100 MPa. With unchanged reference area, engineering stress remains 100 MPa.
At the later area, true stress is 10,000/80 = 125 MPa, a 25% increase from 100 MPa. Constant force did not preserve constant true stress.
To hold true stress at 100 MPa when A = 80 mm², force must be 100 N/mm² × 80 mm² = 8000 N. That is a different force-control history.
These are section-average arithmetic comparisons for the stated uniform model. They do not identify a creep law, prove that the section is stable or resolve stresses around cavities and cracks.

Engineering stress stays 100 MPa; the later true stress is 125 MPa. Maintaining 100 MPa true stress at 80 mm² would require 8000 N.

A constant-stress model can accumulate strain without a new cycle

An initially relaxed Maxwell model has E = 2.0 GPa and η = 1.0 × 10^13 Pa·s. A rapid load establishes σ₀ = 10 MPa, then holds it for 1000 s. Find initial elastic strain, additional dashpot strain and total strain just before unloading. What remains just after ideal unloading?

Use consistent pascal units: E = 2.0 × 10^9 Pa and σ₀ = 1.0 × 10^7 Pa. Initial spring strain is σ₀/E = 0.005 = 0.50%.
The dashpot rate is σ₀/η = 1.0 × 10^−6 s^−1. In 1000 s it adds 0.001 = 0.10% strain. This follows from constant stress and a constant η, not from a real material fit.
Just before unloading, total strain is 0.005 + 0.001 = 0.006 = 0.60%. The two contributions add because the model elements are in series.
Ideal unloading removes the spring contribution. The model retains 0.001 strain, or 0.10%, in its dashpot. Delayed recovery in an actual specimen would show a behaviour absent from this two-element model.

The model starts at 0.50% strain, adds 0.10% during the hold and reaches 0.60%. Immediately after ideal unloading it retains 0.10%; that is a model result, not a universal polymer recovery rule.

Try the reasoning

In the illustrative record, calculate the mean rate from 0+ to 1 h and from 1 to 2 h. Does the comparison establish vacancy-controlled climb?
Show the answer and reasoning
The rates are 0.30%/h and 0.15%/h. The average rate halves, but the mechanism is not uniquely identified.
A changing slope describes the response. Several evolving processes can reduce it; suitable temperature/stress dependence and structural evidence would help distinguish mechanisms.
A crystalline specimen is at 600 K and has a stated melting reference 1200 K. Calculate its homologous temperature. Does the ratio alone prove a particular creep rate or rule out creep in a room-temperature polymer?
Show the answer and reasoning
T/T_m = 0.50. It gives neither a particular creep rate nor a conclusion for an unrelated polymer.
Both temperatures are absolute. Mechanisms, stress, structure and observation duration remain inputs; polymer relaxation is not determined by this crystalline melting ratio.
A sustained-load record rises from 0.20% to 0.35% strain over 30 minutes. Find the mean rate in s^−1. A report divides 0.15 by 1800 and calls that strain per second. Identify its error.
Show the answer and reasoning
The change is 0.0015 strain over 1800 s, giving about 8.333 × 10^−7 s^−1. The report is 100 times too large.
The difference 0.15 is in percentage points. Divide by 100 before converting it to dimensionless strain.
Keep E, stress and hold duration from the Maxwell worked case, but double η. Predict and calculate the initial strain, added hold strain and strain retained after ideal unloading.
Show the answer and reasoning
Initial strain remains 0.50%. The added hold strain halves to 0.05%, total strain becomes 0.55%, and the model retains 0.05% after unloading.
E controls the spring contribution; the dashpot rate is inversely proportional to η. This model comparison changes no real specimen’s molecular architecture or damage state.
A specimen is stretched to a fixed total strain and its stress falls during the hold. A learner calls the graph a constant-stress creep curve. What experiment is actually described, and what must remain visible in its report?
Show the answer and reasoning
It describes stress relaxation under strain control. The imposed strain, load ramp, temperature, duration and material state belong with the measured stress history.
A falling force at fixed displacement is not the same boundary condition as a rising strain at fixed stress. The distinction matters even if the same mobility processes contribute.
A finer-grained specimen is stronger in a short tensile test yet creeps faster in a particular sustained high-temperature test. Explain one compatible mechanism and why neither test invalidates the other.
Show the answer and reasoning
Boundaries can impede dislocation glide in the tensile comparison while shorter diffusion distances and boundary processes promote deformation in a diffusion-dominated creep regime.
The mechanisms and timescales differ. This is a plausible interpretation requiring matched material state and supporting evidence, not a conclusion that finer grains always accelerate every kind of creep.

What to carry forward

Creep rate, accumulated creep strain and rupture time are different quantities.
Defect motion, transport, accommodation and molecular rearrangement supply distinct routes to time-dependent deformation.
A strain–time slope describes behaviour; it does not by itself identify the microscopic route.
Keep initial loading strain, percent/fraction units, current area and the controlled variable explicit.
A simple time-dependent model is useful only for the response and conditions it actually represents.

References

Teaching resources and research supporting this lesson’s explanations and examples.