Across injection-moulded gear trains in automotive seat adjusters, window lift drives, and HVAC actuators, the accumulation of viscoelastic strain in the involute flank after 1 × 10⁶ load applications is routinely quantified as an increase in total profile deviation F_α measured under ISO 1328-1:2013 on a Klingelnberg P 26 gear measuring centre or equivalent CNC gear metrology system. Creep modulus E_c(t), defined as the ratio of constant applied stress σ₀ to total time-dependent strain ε(t), is not a single data sheet property but a function of stress level, temperature, moisture equilibrium, moulded-in orientation, and accumulated duty time. For polymeric gear teeth, the relevant strain accumulates under intermittent flank loading rather than under continuous axial stress, yet the ISO 899-1:2017 tensile creep modulus at 1000 h and 20 MPa is frequently the only available material parameter. The transfer from coupon-level creep data to flank form deviation after exactly one million cycles therefore requires a duty-cycle integration of per-mesh contact time, transverse contact ratio, tooth root stress concentration, load sharing ratio, and the crystalline morphology present in the moulded tooth. In a back-to-back gear test rig configured to VDI 2736 Part 1:2014 with a torque of 1.5 N·m and a pinion speed of 1200 min⁻¹, a single tooth is loaded for approximately 1–5 ms per mesh, depending on tooth stiffness, tip relief, and transverse contact ratio. One million cycles under these operating conditions therefore corresponds to a cumulative loaded time on the order of 5.6 h to 16.7 h, which places the relevant creep modulus at short-term or transition-region values rather than at the long-term plateau commonly reported on material datasheets. Consequently, data sheet creep moduli at 1000 h can under-predict the effective stiffness of a polymer tooth in cyclic operation while still failing to capture the irreversible component of tooth bending creep that accumulates as flank profile error. The viscoelastic strain history of the tooth can be described with the Findley power law ε(t) = (σ₀ / E₀) + σ₀ K tⁿ, where E₀ is the initial elastic modulus, K is the creep compliance coefficient, and n is a dimensionless strain-hardening exponent typically between 0.05 and 0.35 for unfilled acetal copolymer at room temperature. The exponent n is not typically supplied on a material datasheet, and its use with injection-moulded gears is further complicated by the fact that the tooth root experiences multiaxial bending rather than uniaxial tension.

Is Backlash Growth Governed by Creep Modulus or Cyclic Softening?

Backlash enlargement after one million cycles is commonly attributed to abrasive wear, but tooth thickness reduction due to creep compliance can produce a geometrically similar shift. The distinction becomes visible when tooth thickness is measured over a defined measurement circle before and after testing, and when the load flank and non-load flank are compared by coordinate measurement. A pure wear process reduces tooth thickness without necessarily increasing profile form error, whereas creep bending produces a progressive top-root tilting of the loaded flank and a rise in F_α without equivalent material removal. The influence of creep modulus on tooth thickness is not instantaneous but cumulative: during each mesh, the tooth bends by an amount inversely proportional to the effective stiffness of the polymer, and if the recovery time between successive engagements is shorter than the longest relaxation times of the polymer, a residual deflection accumulates cycle after cycle. This accumulation is analogous to cyclic creep and cannot be captured by a static creep modulus alone. Single-tooth stiffness c′ is conventionally expressed in N/(mm·µm) and is inversely related to tooth deflection under a given normal load. If the effective modulus of a tooth decreases from 2800 MPa to 1800 MPa while the tooth geometry remains linearly elastic, the corresponding tooth deflection can increase by approximately 55%. That increase moves the load centre toward the addendum, raises sliding velocity at the tip, and increases the instantaneous transmission error under torque. In a zero-torque measurement, however, backlash growth is not directly proportional to creep modulus because the elastic component of tooth deflection is not part of zero-torque backlash; only the accumulated plastic strain, permanent set, wear loss, and pitch deviations remain after unloading. Published data for the isolated contribution of creep modulus to zero-torque backlash after 1 × 10⁶ cycles in POM gears are limited, and the existing comparative studies emphasise that creep modulus effects are entangled with wear, thermal expansion, and moulded-in residual stress.

Conditioning at 23 °C and 50 % RH for 168 h in accordance with ISO 291:2008 stabilises the moisture equilibrium of polyamide 66 but only partially relaxes moulded-in residual stress. In dry-as-moulded PA66 gear teeth, the tensile modulus per ISO 527-2:2012 can be as high as 3100 MPa, but after moisture uptake of approximately 2.5 wt% the modulus may fall to 1200–1600 MPa, and the ISO 899-1 tensile creep modulus at 1000 h may fall below 700 MPa at 23 °C. The corresponding increase in tooth deflection under constant torque is amplified in tooth geometries with low tooth counts because the tooth thickness at the root is smaller and the root stress is higher. For a module 1.0 mm gear with 20 teeth, a 10% drop in effective modulus can shift the pitch-line load distribution toward the addendum, increasing sliding velocity and local flank temperature. This interplay between moisture-induced plasticisation and creep modulus is a known source of gear accuracy drift in HVAC actuators that are stored in high-humidity environments before service. In such actuators, the gear teeth may be moulded to ISO 1328-1 accuracy grade Q8 or better at assembly, but after moisture conditioning and one million cycles under torque, conditioned PA66 gears often show greater total profile deviation change than dry PA66 gears. The scatter in published data for this specific configuration is considerable because the crystallinity of PA66 depends on mould temperature, and the moisture uptake through a tooth flank is diffusion-limited and geometry-dependent.

When POM Copolymer Teeth Operate Above 60 °C Under Sustained Torque

At elevated temperatures below the heat deflection temperature of POM copolymer, the creep modulus decays more rapidly with time under load than at room temperature. The time-temperature equivalence principle allows creep data at multiple temperatures to be shifted along the log-time axis to construct a master curve; however, the shift factors are themselves stress-dependent for semicrystalline polymers, and gear teeth near the root experience multiaxial bending and shear. In acetal copolymer, manufacturer technical bulletins using ISO 899-1 show that at 80 °C and 10 MPa, the tensile creep modulus at 1000 h can be less than 50% of the value at 23 °C. When a gear pair is run in a closed-loop test stand at an air temperature of 80 °C, the flank temperature can exceed 100 °C because of hysteretic heating and sliding friction under poor lubrication. At those temperatures, the tooth stiffness drops during the loaded phase of each mesh, producing a larger alternating deflection and a measurable increase in total profile deviation. This is not a classical wear mechanism, because material loss may be minimal; it is a viscoelastic ratcheting of the tooth flank under cyclic bending. The relevant failure criterion in VDI 2736 Part 1:2014 is therefore not only wear rate but also tooth deformation under load and the resulting increase in transmission error. When polyoxymethylene is selected for a continuously loaded gear running near 60 °C, the design should incorporate tip relief and full fillet radius to reduce the load concentration that accelerates local creep, and the allowable root stress should be derated by a temperature factor consistent with the creep modulus retention curve provided by the polymer supplier.

To separate plastic deformation from wear in a one-million-cycle test, tooth thickness is measured at three axial depths using a gear measuring centre, and the difference between the load flank and the non-load flank is mapped. In a typical protocol, total profile deviation F_α, total helix deviation F_β, and total cumulative pitch deviation F_p are recorded before testing, after 10⁴ cycles, after 10⁵ cycles, and after 10⁶ cycles. The tooth thickness at the pitch circle is evaluated with a double-flank gear rolling tester against a master gear, or with a coordinate measuring machine equipped with a rotary table and scanning probe. In unfilled POM copolymer tested at 1.5 N·m and 1200 min⁻¹, the total profile deviation increase after 1 × 10⁶ cycles is often reported in the range of 8–20 µm for a module 1.0 mm gear in VDI 2736 comparative studies, but published data for exactly this configuration is limited and the result is sensitive to moulding anisotropy, gate placement, and pre-test annealing. The same measurement protocol applied to a gear running at 10 N·m would be dominated by wear and thermal damage, making it impossible to isolate the creep modulus contribution. Therefore, creep modulus effects on gear tooth accuracy are best resolved at moderate torque levels where tooth bending is the dominant deformation mode and wear is not yet catastrophic.

Creep Modulus Measurement, Data Sheet Values, and Their Misapplication to Gear Teeth

Creep modulus is normally measured on ISO 527-2 tensile bars under constant stress in accordance with ISO 899-1, and the reported values are axial, homogeneous, and free of orientation gradients. An injection-moulded gear tooth, by contrast, has a core-skin morphology: the skin layer is highly oriented but rapidly cooled, while the core is more slowly cooled and may contain spherulites or flow-induced crystallinity. The effective creep modulus in the tooth root is therefore not equal to the isotropic coupon value, and the difference can amount to 15–30% depending on gate location, packing pressure, and mould temperature. Cross-polarised light microscopy of thin sections from a POM gear tooth often reveals a radially oriented skin around the tooth profile and an isotropic core, which means the load-bearing flank is orthotropic and the creep response is direction-sensitive. Finite element models of polymer gears increasingly assign elastic constants using orientation tensors from injection moulding simulation and then scale the elastic modulus by a creep compliance factor from ISO 899-1 data. However, the accuracy of this approach after 1 × 10⁶ cycles is limited by the absence of cyclic creep data at the relevant stress levels and by the stress relaxation that occurs during the unloaded part of each mesh. In addition, the creep modulus measured in flexure per ISO 899-2:2019 is more representative of gear tooth bending than tensile creep, but even flexural creep data are generated at constant load and constant temperature, which do not replicate the short-duration, high-frequency loading of a gear tooth. When the creep modulus is used in a gear design formula, the selected value should be taken at the cumulative loaded time corresponding to the required number of cycles and at the maximum flank temperature, not at 23 °C and 1000 h, unless a conservative stiffness reduction factor is applied.

During the unloaded portion of each gear rotation, the polymer tooth has time to partially recover its viscoelastic strain. The extent of recovery depends on the recovery spectrum and the time between successive mesh contacts. At 1200 min⁻¹, the time between successive tooth engagements for a 20-tooth pinion is approximately 2.5 ms, which is far shorter than the relaxation times of high-molecular-weight acetal chains at room temperature. As a result, the tooth enters the next contact before full recovery has occurred, and a residual deflection accumulates cycle after cycle. This accumulation is analogous to cyclic creep and cannot be captured by a static creep modulus alone. If the same gear operates at 60 min⁻¹, the inter-mesh recovery time is 50 ms, allowing greater recovery and reducing the accumulated tooth deflection for a given number of cycles, although the wear regime may change. This frequency sensitivity is one reason why a polymer gear may exhibit acceptable accuracy after 1 × 10⁶ cycles at low speed but unacceptable transmission error at high speed, even when the total load duration is identical. Consequently, gear designers should not compare creep modulus data across materials without also comparing the loading frequency, inter-mesh recovery time, and the resulting stress-relaxation spectrum of each polymer. Published data for the specific frequency dependence of gear tooth accuracy after one million cycles are limited, but the underlying viscoelastic principles are well documented in polymer mechanics.

Thermal Softening and Stress Relaxation Confound One-Million-Cycle Accuracy Data

Gear tooth accuracy after one million cycles is seldom controlled by creep modulus in isolation. Frictional heating at the flank raises the local temperature, and that temperature rise lowers the yield stress and the creep modulus of the polymer. In dry-running acetal gears, adhesive wear can occur when the flank temperature approaches the melting point, producing material transfer and abrupt changes in profile deviation. In lubricated or greased gears, the thermal load is smaller but migration of grease into the root can plasticise certain polymers. A rigorous test protocol therefore records bulk temperature at the tooth root with an infrared pyrometer and torque at the output shaft with a torque transducer, while the gear pair is run in a thermally controlled chamber conforming to a defined temperature tolerance. When reporting accuracy change after 1 × 10⁶ cycles, the standard practice is to normalise the result to the initial moulded geometry and to separate thermal expansion from permanent creep. For POM copolymer, the coefficient of linear thermal expansion is approximately 110 × 10⁻⁶ K⁻¹ in the flow direction and 130 × 10⁻⁶ K⁻¹ transverse to flow, so a temperature difference of 20 °C across a 1.0 mm tooth can produce a dimensional shift of 2.2–2.6 µm, which is of the same order as the creep-induced profile error being measured. Without compensation for thermal expansion, the creep modulus effect on tooth accuracy may be overestimated. Similarly, stress relaxation during the unloaded intervals of a cycling gear reduces the peak tensile stress at the tooth root and can therefore reduce the rate of irreversible creep accumulation compared with a continuously loaded coupon at the same nominal stress.

Injection moulding process conflicts for creep-resistant gear teeth are most acute with acetal copolymer because the crystallinity that raises short-term stiffness also introduces anisotropic shrinkage that can dominate pre-test accuracy. On a typical production machine with a reciprocating screw diameter of 30 mm, barrel temperature zones from hopper to nozzle set at 180 °C, 190 °C, 200 °C, and 210 °C, and a mould temperature of 90 °C, the achievable peak injection pressure is 1800 bar. Process trials using a 40:1 L/D twin-screw compounding line to prepare glass-fibre-reinforced PEEK have shown that residual fibre length distribution after compounding is a stronger determinant of creep modulus than average fibre length because short fibres act as stress concentrations at the tooth root. The gate location for a gear cavity is preferably a central pin gate or a three-point diaphragm gate to avoid weld lines at the root; a weld line at a tooth root can reduce local strength and creep resistance by up to 50%, although published data for this specific loss in one-million-cycle gear accuracy is limited. Packing pressure and cooling time must be balanced: excessive packing increases tooth shrinkage anisotropy, while insufficient packing leads to microvoids that accelerate creep. The relevant quality criterion is not simply dimensional tolerance at demoulding but the stability of that geometry after annealing and after one million load cycles. Post-mould annealing of POM copolymer at 130 °C for 2 h in circulating air can relieve some frozen-in stress and improve dimensional stability, but it can also cause differential tooth shrinkage if the gear is not supported on a flat surface during the heat cycle.

Operationally, POM copolymer gear teeth must not be run against shafts or housings made from materials that generate acidic degradation products at elevated temperature, because the resulting depolymerisation lowers molecular weight and creep modulus rapidly. Amine-based lubricants should also be avoided on acetal copolymer gears because amines can catalyse depolymerisation at the tooth flank and produce a loss of profile accuracy that is easily mistaken for a creep modulus failure. PA66 gears should avoid continuous exposure to hot aqueous solutions that remove plasticisers or alter crystallinity at the surface, and designs should either allow for a measured increase in backlash or specify a moisture-resistant polyamide variant. The use of ISO 899-1 creep modulus data for gear design is valid only when the operating temperature, moisture uptake, and loading frequency remain within the ranges specified in the standard test report; outside those boundaries, the predicted tooth accuracy after 1 × 10⁶ cycles is not experimentally transferable. Published data isolating creep modulus as the sole variable for injection-moulded gear tooth accuracy after exactly one million cycles are limited, and the available VDI 2736 comparative studies indicate that multi-variable effects of moisture, moulded-in orientation, local flank temperature, and wear dominate the measured scatter in total profile deviation.