Tyres and degradation models

This article specifies a race-engineering tyre model for Formula One: combined-slip force generation, transient build-up, thermal coupling, wear kinetics, circuit energy characterisation, and stint optimisation. Symbols follow motorsport literature; parameters are given as calibrated ranges suitable for lap-time simulation and strategy work.

Force generation (nonlinear, combined slip)

Baseline lateral force (Magic Formula representation): $F_{y}=D\sin \!{\big (}C\arctan \!{\big (}B\alpha -E(B\alpha -\arctan(B\alpha )){\big )}{\big )},$ with $D=\mu F_{z}$; $B,C,E$ shape parameters; slip angle $\alpha$; vertical load $F_{z}$; peak friction $\mu$.

Combined-slip admissible set (anisotropic ellipse): $\left({\frac {F_{x}}{\mu _{x}F_{z}}}\right)^{n}+\left({\frac {F_{y}}{\mu _{y}F_{z}}}\right)^{n}\leq 1,\quad n\in [1.6,2.2].$

Load-sensitivity (decreasing peak friction with load): $\mu (F_{z})=\mu _{0}\left({\frac {F_{z}}{F_{z0}}}\right)^{a_{\mu }},\quad a_{\mu }\in [-0.05,-0.02].$

Camber thrust (additive near-linear term for moderate camber $\gamma$): $F_{y,\gamma }=C_{\gamma }\,\gamma \,F_{z},\qquad F_{y}(\alpha ,\gamma )\approx F_{y}(\alpha ,0)+F_{y,\gamma }.$

Indicative calibrated ranges (per front/rear tyre, race trim; to be refined per event):

Parameter	Typical range	Notes
Peak lateral friction $\mu$	1.60–1.95	Effective (includes compound & road); varies with temperature and wear
Cornering stiffness $C_{\alpha }$	80–140 kN/rad	Increases with pressure & load to a limit; front < rear asymmetry common
Camber thrust coeff. $C_{\gamma }$	2–6 kN/rad	Increases with vertical load; risk of shoulder over-temp if too high
Stiffness factor $B$	8–15 1/rad	Magic-Formula fit; car- and compound-dependent
Shape factor $C$	1.2–1.6	Controls curvature to peak
Curvature $E$	0.0–0.3	Lower E → sharper peak; sensitive to wear state

Transient build-up and rolling losses

Relaxation-length dynamics (distance form): ${\frac {dF_{y}}{ds}}+{\frac {1}{\lambda }}F_{y}={\frac {C_{\alpha }}{\lambda }}\,\alpha ,\qquad \lambda \in [0.3,0.8]\ \mathrm {m} .$

Rolling resistance (per wheel) for lap-sim: $F_{\mathrm {rr} }=C_{\mathrm {rr} }F_{z},\qquad C_{\mathrm {rr} }\in [0.012,0.018],$ with temperature/wear sensitivity modelled as $C_{\mathrm {rr} }=C_{\mathrm {rr0} }+k_{T}(T_{c}-T^{\ast })+k_{w}w.$

Thermal model (two-node surface/carcass)

Surface $T_{s}$ and carcass $T_{c}$ temperatures: $C_{s}{\frac {dT_{s}}{dt}}=\eta _{s}\,\mu \,F_{z}\,v\,\gamma _{s}-h_{s}A_{s}(T_{s}-T_{\infty })-k_{sc}(T_{s}-T_{c}),$ $C_{c}{\frac {dT_{c}}{dt}}=k_{sc}(T_{s}-T_{c})-h_{c}A_{c}(T_{c}-T_{\infty }).$

Temperature modifiers (Gaussian around target carcass temperature $T^{\ast }$): $\phi (T_{c})=\exp \!{\big (}-k_{\phi }(T_{c}-T^{\ast })^{2}{\big )},\qquad \psi (T_{c})=\exp \!{\big (}-k_{\psi }(T_{c}-T^{\ast })^{2}{\big )},$ used to scale $\mu$ and $C_{\alpha }$ respectively.

Wear and degradation kinetics

Wear state $w\in [0,1]$ (0 fresh, 1 end-of-life). Sliding-power law: ${\frac {dw}{dt}}=k_{w}\!\left(\left|F_{x}v_{s,x}\right|+\left|F_{y}v_{s,y}\right|\right).$

Performance decay: $\mu (w,T_{c})=\mu _{0}\left(1-a_{w}w\right)\phi (T_{c}),\qquad C_{\alpha }(w,T_{c})=C_{\alpha 0}\left(1-b_{w}w\right)\psi (T_{c}),$ where $a_{w},b_{w}\in [0.15,0.45]$ give typical end-of-stint losses.

Failure/artefact modes: thermal fade (high $T_{c}$), graining (low $T_{s}$ and high shear; reversible), blistering (subsurface; irreversible), flat-spot (lock-up; step change in $w$), pick-up (reversible after cleaning laps).

Compounds, operating windows, prescriptions

Pirelli supplies five slick compounds (C1–C5). Three are nominated per event (H/M/S labels). Indicative (modelling) windows—replace with event notes when available:

Compound	Nominal grip (rel.)	Wear rate (rel.)	Indicative carcass window (°C)	Typical use
C1 (hard)	Low	Low	95–115	High-energy/abrasive; long stints; hot ambient
C2	Low–Med	Low–Med	95–115	Long stints; high-load corners
C3	Med	Med	90–110	Baseline race compound on many tracks
C4	Med–High	Med–High	85–105	Qualifying bias; cooler ambient; short stints
C5 (soft)	High	High	85–100	Street circuits; low energy; peak grip runs

Prescriptions (event bulletins): minimum starting pressures (front/rear), maximum static camber (per axle), blanket rules (where allowed). These override generic models for compliance and must be enforced in setup.

Circuit energy characterisation and degradation rates

Use a lateral/longitudinal energy index and texture/abrasion rating to forecast wear and window risk. Pirelli’s track severity scale (1–5) maps well to model priors:

Circuit	Track energy (1–5)	Dominant load	Surface & notes	Expected stint shape (S/M/H)
Monaco	1	Low lateral; traction-limited	Smooth street; low energy	Very shallow degradation; thermal management critical at low speeds
Monza	2	Braking/traction; low lateral	Smooth; long straights	Low deg; front-tyre warm-up critical; flat-spot risk
Bahrain	3–4	Traction + braking; heat	Abrasive; hot ambient	Medium–high deg; rear thermal fade risk
Barcelona (Catalunya)	4	Sustained lateral	Medium-abrasive; long corners	Medium–high deg; front-left graining if cool
Silverstone	5	Very high lateral	Fast, high-energy	High deg; carcass temperature control is limiting
Suzuka	5	Mixed; long-radius lateral	Smooth-medium; esse complex	High deg; front-limited early, rear-limited late
Zandvoort	4	Banked lateral loading	Fresh abrasive after resurfacing	Medium–high deg; camber window tight

Example calibrated degradation slopes (race runs, dry; representative order-of-magnitude for modelling, per compound on “medium” energy tracks):

Compound	Linear deg $d$ (s/lap)	Nonlinear term at end-of-stint (extra s over last 5 laps)
C1	0.010–0.020	0.2–0.6
C2	0.015–0.030	0.3–0.8
C3	0.020–0.040	0.5–1.2
C4	0.030–0.060	0.8–1.8
C5	0.040–0.080	1.0–2.5

Strategy modelling and optimal stints

Per-lap loss including wear, thermal offset from target $T^{\ast }$, and traffic factor $Q_{k}$: $\Delta t_{k}=\alpha _{1}w_{k}+\alpha _{2}(T_{c,k}-T^{\ast })^{2}+\alpha _{3}Q_{k}.$

Cumulative stint time to lap $N$: $T_{\mathrm {stint} }(N)=\sum _{k=1}^{N}\!{\big (}t_{0}+\Delta t_{k}{\big )}.$

Under linear degradation $\Delta t_{k}\approx dk$ and pit loss $P$, the continuous optimum stint length is $N^{\ast }\approx {\sqrt {\frac {2P}{d}}},$ useful for first-order stop count decisions. The undercut condition at lap $n$: ${\big (}t(n)-t(n+1){\big )}_{\mathrm {old} }\;>\;P-{\big (}t(n)-t(n+1){\big )}_{\mathrm {new} }.$

Fuel burn-off and DRS availability couple into $d$ and $\alpha _{2}$; safety-car or VSC resets change optimal $N^{\ast }$ by reducing the opportunity cost of a stop.

Data acquisition and online identification

Typical online state vector and measurements for estimation: $x=\{C_{\alpha },\mu ,\lambda ,T_{s},T_{c},w\},\qquad y=\{F_{x},F_{y},\omega ,{\tfrac {d\psi }{dt}},a_{y},T_{\mathrm {IR} }\}.$ Teams use EKF/UKF observers to update $x$ in real time from wheel speed, brake temps, steering, IMU, and IR cameras; stint-end mass and vibration signatures cross-check wear $w$.

Validation workflow (practical)

1. Identify $B,C,D,E,C_{\alpha },\lambda$ on the tyre rig and from clean track segments.
2. Fit $C_{s},C_{c},h_{s}A_{s},h_{c}A_{c},k_{sc}$ from controlled long-runs.
3. Calibrate $k_{w}$ against abrasion metrics and compare to stint mass loss.
4. Close the loop in lap-sim; verify predicted vs observed stint shape, undercut thresholds, and pit δ.
5. Replace generic windows with event-specific Pirelli prescriptions upon bulletin release.

See also

• Race strategy modelling
• Lap time and delta analysis
• Chassis and suspension design
• Aerodynamics in Formula One

References

• FIA Formula One Technical Regulations (Tyres & Wheels; current season).
• Pirelli Motorsport, Event Technical Notes (pressures, camber, nominations, blanket rules).
• Pacejka, H., Tire and Vehicle Dynamics, Elsevier.
• Milliken, W. & Milliken, D., Race Car Vehicle Dynamics, SAE.
• Dixon, J., Tires, Suspension, and Handling, SAE.
• Selected SAE papers on tyre thermal/wear modelling and combined slip in racing applications.
• Bosch, Automotive Handbook (combined slip, relaxation length, rolling resistance).