Rate, direction and temperature effects

The deformation mechanism can change

A faster test gives thermally activated processes less time to act. Temperature changes mobility and may change phases or damage mechanisms. Orientation changes which crystal or fibre directions carry load. These are physical conditions, not just reporting details.

Start with the same deformation endpoint. Engineering axial strain is e = u/L₀, where u is the extension between specimen marks and L₀ their original separation. Its rate r = de/dt describes how quickly this ratio changes; the dot notation ė means the same rate. Since strain is dimensionless, r has units s⁻¹. A rate 0.001 s⁻¹ means an increase of 0.001 in strain ratio per second, or 0.1 percentage point per second, when it is constant.

For a fixed reference gauge, r = (du/dt)/L₀. At L₀ = 50.0 mm, a constant specimen-gauge extension speed 0.0500 mm/s gives r = 0.00100 s⁻¹. Reaching e = 0.0200 then takes 0.0200/0.00100 = 20.0 s and adds 1.00 mm. At r = 0.100 s⁻¹ the same endpoint takes 0.200 s, with gauge speed 5.00 mm/s. The two tests leave very different times for a competing process to act.

A testing-machine crosshead can move because the grips, frame and material outside the selected gauge deform. Its speed is not automatically du/dt for the specimen marks. Nor does a constant speed give the same engineering strain rate for different L₀. Deformation and relative strain explains the gauge; Elastic response and elastic modulus distinguishes its deformation from load-train movement.

In a polymer, fast bond distortion and slower segmental rearrangements can both contribute. A rapid excursion may end before much rearrangement occurs, while a longer load permits more time-dependent deformation. This makes the retained rapid-test/creep example plausible without assigning all polymers one relaxation time. Viscoelasticity, temperature and time develops the separate creep and relaxation models.

In a crystalline metal, dislocations encounter obstacles to motion. A thermally activated step is one whose occurrence depends on fluctuations that help cross an energy barrier; applied stress can assist that passage. At a fixed temperature and comparable internal state, requiring the same plastic deformation more quickly can demand more stress when such steps control the response. Temperature changes their rates and can also enable recovery, in which defect arrangements reorganise.

That mechanism does not prove “faster means stronger”. In some solute-containing alloys, atoms can diffuse toward temporarily arrested dislocations and increase resistance to their motion. The time available for this interaction can give greater flow resistance at a slower rate in a particular temperature/rate window: negative rate sensitivity. Other windows and mechanisms can give another trend. A tensile observation must establish its sign rather than assume it.

Rotate the material direction, not just the drawing

Slip in a crystal occurs on particular planes and in particular directions. An axial pull has different shear components on differently oriented planes. Changing the crystal orientation relative to the pull therefore changes which slip systems are favourably loaded and how much axial stress is needed to activate them. This is distinct from changing the density of obstacles to motion; Plastic deformation and yield behaviour introduces the moving-dislocation mechanism.

A polycrystal contains many grains. If their orientations are distributed with a preference, called crystallographic texture, averaging over them need not erase direction dependence. Rolling can establish such a texture and a processing history. An aligned fibre or molecular structure can produce another directional path. Direction and size can change observed behaviour prepares those architectural distinctions; there is no claim that all directional differences share one microscopic cause.

For a rolled sheet, call the rolling direction RD and the in-plane direction at right angles to it TD, the transverse direction. A coupon cut and pulled along RD samples a different material direction from one cut and pulled along TD. Rotating both a coupon and its loading axis together on a page does not change that relative angle.

Anisotropy is a direction-dependent material response under stated conditions. It is not established just because a longer or thinner specimen stretches farther, or a larger section carries more force. Compare the same response definition with dimensions and conditions controlled or normalised. Stress F/A₀ separates force from original area; gauge strain separates extension from length. Loading mode also matters: uniaxial tension, bending and shear impose different fields.

For example, a coupon with A₀ = 10.0 mm² carrying 2000 N and one with A₀ = 5.00 mm² carrying 1000 N both have engineering stress 200 MPa. The force difference alone does not establish anisotropy. At a common rate, temperature and strain, a properly matched directional stress difference is evidence of directional response, while texture, fibre orientation, damage or other state evidence is needed to identify its mechanism.

Hold the endpoint and change one comparison factor

Use an original fictional rolled-sheet dossier, with all values invented for teaching. Coupons begin from the same verified stock/state and have L₀ = 50.0 mm and A₀ = 10.0 mm². Each is loaded monotonically at its prescribed constant specimen-gauge engineering rate. The recorded engineering stress S is taken at the same e = 0.0200, with uniform deformation independently stipulated up to that reading. Specimen temperature is measured and held at the stated value; preparation and observation conventions match.

The table gives one illustrative representative reading for each condition, not an experimental dataset or a set of population means. Its values are supplied observations; they are not calculated from the mechanism descriptions. Replicate variation and measurement uncertainty have not been supplied. In particular, S is stress at 2.00% total engineering strain, not an assigned 0.2% proof stress, UTS, elastic modulus or service allowable.

At 293 K in RD, raising rate from 0.001 to 0.100 s⁻¹ changes S from 200 to 240 MPa. The observed increment is 40 MPa and the relative change is 40/200 = 20.0%, using the slower value as reference. In TD at the same temperature, 160→184 MPa gives 24 MPa or 15.0%. The rate ratio is 100 in both cases, but the stress changes are different.

At 333 K, the RD rate increment is 154 − 140 = 14 MPa, or 10.0%; the TD increment is 180 − 150 = 30 MPa, or 20.0%. Thus even in this deliberately simple record there is no one additive rate correction that describes every slice. All four chosen pairs have positive rate effects; this feature was stipulated in this fictional table and does not override the possible negative mechanism already discussed.

Direction also changes its ranking here. At the slower rate, RD exceeds TD by 40 MPa at 293 K, but falls 10 MPa below it at 333 K. At the faster rate those differences are 56 MPa and −26 MPa. A rate or direction value has meaning together with the temperature and endpoint; “the stronger direction” is not a complete label for this record.

The first figure displays those paired readings. It does not connect them into measured curves or demonstrate a particular crystal mechanism. Because every original area is 10.0 mm², the 200 MPa reading corresponds to 2000 N and the 240 MPa reading to 2400 N. Dividing either by the 2% strain would give a secant from the origin, not establish the initial elastic modulus.

A changed condition is not an isolated cause

Suppose a report compares RD at 293 K and 0.001 s⁻¹, S = 200 MPa, with TD at 333 K and 0.100 s⁻¹, S = 180 MPa. The second reading is 10.0% lower. Calling that a 10% softening caused by faster loading would change three factors at once: rate, direction and temperature. The arithmetic is valid for the two readings, but the causal label is unsupported.

Use a matched rate comparison instead. In TD at 333 K, the actual supplied pair is 150→180 MPa, a 20.0% rise. A temperature comparison at fixed TD and 0.100 s⁻¹ is 184→180 MPa, a reduction of 4 MPa relative to 184 MPa, or about 2.17%. A direction comparison at 333 K and 0.100 s⁻¹ is RD 154 versus TD 180 MPa. Each answers a different question.

The rate increments themselves change with temperature: RD 40→14 MPa and TD 24→30 MPa. This is an interaction in the descriptive sense that the effect of one factor depends on another. It does not require a statistical interaction test or prove which microscopic mechanism caused the difference; uncertainty and replication are missing.

Fair comparison does not mean every condition can be made physically identical. A different rate can evolve a different defect state or heating history by the time the same strain is reached. Record that history rather than calling every difference an instantaneous intrinsic rate sensitivity. An unmeasured change of temperature or state is an alternative explanation that the labels alone cannot exclude.

Compare a response surface

Data across several rates and temperatures can support a constitutive model. One curve fixes only part of that surface. Interpolation and extrapolation must preserve mechanisms and phase state; a fit can be numerically smooth while being physically inappropriate.

A constitutive model relates response to deformation, rate, temperature, direction and the relevant internal state or history. A response surface is a way to display several of those dependencies while holding others fixed. The table contains eight points on a 2%-strain slice; it is not a complete stress–strain history at every rate or a unique continuous surface.

Consider only the RD, 293 K pair 200 MPa at r₀ = 0.001 s⁻¹ and 240 MPa at r₁ = 0.100 s⁻¹. Candidate P assumes the power interpolation S_P(r) = 200 MPa × (r/r₀)^m. To fit both endpoints, 100^m = 240/200 = 1.20, so m = ln(1.20)/ln(100) ≈ 0.0395906. The ratios inside ln and the exponent are dimensionless, and ln denotes the natural logarithm.

Candidate L instead assumes stress is linear in rate between these endpoints: S_L(r) = 200 MPa + 40 MPa × (r − r₀)/(r₁ − r₀). Both rules reproduce the two supplied observations. They are interpolation assumptions for one fixed-strain/temperature/direction slice, not complete material laws. No parameter was supplied by the source mechanisms.

At the unmeasured middle decade r = 0.0100 s⁻¹, P gives 200 × 10^m ≈ 219.089 MPa. L gives 200 + 40 × 0.009/0.099 ≈ 203.636 MPa. “Middle decade” means halfway along the logarithmic rate axis, not halfway between 0.001 and 0.100 arithmetically. Two data points cannot choose between these different predictions; the second figure marks each prediction separately from the observed endpoints.

A further matched observation at the missing rate, together with uncertainty and state/history evidence, could discriminate these rules within this slice. Agreement at one extra point would still not validate a full law for another temperature, direction, strain or mechanism. The exponent m from separate constant-rate tests also need not equal an instantaneous rate-jump derivative evaluated at one unchanged internal state.

Do not extend either curve outside the measured bracket merely because its formula remains smooth. Thermal activation, solute interactions, recovery, phase changes, damage or self-heating can alter the governing response. Replacing a missing record with an extrapolated number is not an operating limit or a test-based failure guarantee.

A temperature label does not replace a thermal and mechanical history

Temperature affects both present mobility and the structure present when loading begins. Recovery, precipitation, phase transformation or polymer rearrangement may require time. Two specimens at the same instantaneous temperature can therefore differ if one was heated rapidly and the other held long enough for structural change. Neither a hot-chamber setting nor an equilibrium phase diagram proves that the gauge reached that state.

Rapid deformation can also generate heat through irreversible work, and the time available for heat transfer changes with the test. Check actual specimen temperature when a comparison assumes isothermal conditions. Here it was explicitly controlled in the fictional table; no adiabatic heating, temperature rise or heat-transfer coefficient was calculated.

A useful record includes original gauge/section, material direction, specimen and preparation history, strain definition, rate-control method, measured temperature and its evolution, the selected response endpoint and measurement uncertainty. Specimens, dimensions and test conditions prepares that record. Changing from tension to compression or bending also needs a relevant field model; direction is not a substitute for loading mode.

Keep mechanisms, observations and models separate. A plausible activation mechanism explains why a dependency might occur; a matched record establishes what was observed under its conditions; a fitted rule supplies additional assumptions about unmeasured points. Each can be useful without becoming a universal faster/colder/stronger rule, manufacturing setting, service-life claim or component prescription.

Worked examples

Try the reasoning

1. Can one crystal-direction value represent every direction?

   Show the answer and reasoning

   Not generally.

   Anisotropy requires directional information. A preferred grain or fibre arrangement changes paths relative to the load. Match geometry and conditions before assigning a difference to material direction.

2. Why avoid extrapolating across a phase transition?

   Show the answer and reasoning

   The governing structure and response can change.

   A model for one phase need not describe another. Transformation kinetics and prior heating time can also make equal instantaneous temperatures correspond to different structures.

3. A new 40.0 mm specimen gauge has measured extension speed 0.0800 mm/s. On the engineering definition used here, find strain rate and time to 3.00% strain if the speed stays constant. What if 0.0800 mm/s is only a machine setting?

   Show the answer and reasoning

   r = 0.00200 s⁻¹ and t = 15.0 s for the stated specimen-gauge speed. A crosshead setting alone does not establish that gauge rate.

   Extension to 3.00% is 0.0300 × 40.0 = 1.20 mm, and 1.20/0.0800 = 15.0 s. Deformation elsewhere in the load train can make the machine and gauge motions differ.

4. At r = 0.001 s⁻¹, compare RD and TD at 293 K and 333 K. Is one direction always higher in the supplied record?

   Show the answer and reasoning

   RD exceeds TD by 40 MPa at 293 K, but TD exceeds RD by 10 MPa at 333 K. No single directional ordering describes both slices.

   Use 200−160 and 140−150. The response is stress at the same total 2% strain, not a universal strength endpoint. This descriptive reversal does not identify its microstructural cause without further evidence.

5. A learner applies the 293 K RD rate increase of 40 MPa to the 333 K RD slow value 140 MPa and predicts 180 MPa at the faster rate. Compare with the supplied reading.

   Show the answer and reasoning

   The actual 333 K RD faster-rate reading is 154 MPa. The reused additive correction overpredicts it by 26 MPa.

   At 333 K the matched increment is 14 MPa, not 40 MPa. The effect of changing rate depends on the temperature slice in this record; neither increment is an independent universal material constant.

6. Two new coupons with matched rate, temperature and 2% strain give 2200 N over 10.0 mm² in RD and 1100 N over 5.00 mm² in TD. Does the force difference alone establish different directional stress at that endpoint?

   Show the answer and reasoning

   No. Both endpoint engineering stresses are 220 MPa.

   Divide each force by its own original area. Equal endpoint stress also does not establish complete isotropy: other strains, loading modes, moduli or states have not been compared.

7. The fictional table has positive rate effects in all four pairs. Does that disprove a separately observed negative rate effect in another alloy or temperature/rate window?

   Show the answer and reasoning

   No. The table defines one conditional fictional record; the competing mechanisms and states need not match.

   Thermally assisted glide, solute/dislocation interactions, recovery and thermal evolution can compete. Check endpoint, state, rate definition, temperature and repeat evidence rather than imposing a universal sign.

8. Suppose an additional matched RD, 293 K reading at 0.0100 s⁻¹ were 204 MPa with measurement uncertainty far smaller than the roughly 15 MPa gap between P and L. Which interpolation is more consistent with that new point, and what remains unestablished?

   Show the answer and reasoning

   L’s 203.636 MPa is much closer than P’s 219.089 MPa. That point would support L over P there, while leaving other conditions and extrapolation unvalidated.

   A new observation can discriminate the stated candidates. It would not prove the stress is globally linear in rate, establish an operating envelope or validate temperature/direction/history dependence elsewhere.

9. A report divides the RD, 293 K slower endpoint 200 MPa by total strain 0.0200, obtains 10 GPa and calls it Young’s modulus. What is missing?

   Show the answer and reasoning

   The quotient is a secant from the origin. A justified initial reversible linear interval and appropriate specimen-gauge strain evidence are missing.

   The table supplies one 2%-strain reading, not the initial slope, unloading response, proof construction or full tensile maximum. A conditional stress value must retain its endpoint instead of becoming an unspecified material property.

What to carry forward

• Attach rate, orientation and temperature to data.
• Extend models only with physical justification.
• Define engineering strain rate from the actual specimen gauge, not an unexplained machine speed.
• Time-dependent mechanisms can compete; a positive or negative rate effect belongs to its window and state.
• Pair conditions before assigning a difference to rate, temperature or material direction.
• A shared endpoint bracket does not select a unique interpolation or a complete constitutive law.
• Preserve specimen temperature and structural history; a smooth extrapolation is not an operating prescription.