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): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_y = D \sin\!\big( C \arctan\!\big( B \alpha - E ( B \alpha - \arctan(B \alpha) ) \big) \big), } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\mu F_z} ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B,C,E} shape parameters; slip angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} ; vertical load Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_z} ; peak friction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} .

Combined-slip admissible set (anisotropic ellipse): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{F_x}{\mu_x F_z}\right)^{n} + \left(\frac{F_y}{\mu_y F_z}\right)^{n} \le 1,\quad n \in [1.6,2.2]. }

Load-sensitivity (decreasing peak friction with load): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} ): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu}	1.60–1.95	Effective (includes compound & road); varies with temperature and wear
Cornering stiffness Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\alpha}	80–140 kN/rad	Increases with pressure & load to a limit; front < rear asymmetry common
Camber thrust coeff. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\gamma}	2–6 kN/rad	Increases with vertical load; risk of shoulder over-temp if too high
Stiffness factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}	8–15 1/rad	Magic-Formula fit; car- and compound-dependent
Shape factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}	1.2–1.6	Controls curvature to peak
Curvature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E}	0.0–0.3	Lower E → sharper peak; sensitive to wear state

Transient build-up and rolling losses

Relaxation-length dynamics (distance form): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\mathrm{rr}} = C_{\mathrm{rr}} F_z,\qquad C_{\mathrm{rr}} \in [0.012,0.018], } with temperature/wear sensitivity modelled as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{\mathrm{rr}} = C_{\mathrm{rr0}} + k_T (T_c - T^{\ast}) + k_w w. }

Thermal model (two-node surface/carcass)

Surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_s} and carcass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} temperatures: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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), } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{\ast}} ): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\alpha} respectively.

Wear and degradation kinetics

Wear state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \in [0,1]} (0 fresh, 1 end-of-life). Sliding-power law: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dw}{dt}=k_w\!\left(\left|F_x v_{s,x}\right|+\left|F_y v_{s,y}\right|\right). }

Performance decay: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(w,T_c)=\mu_0 \left(1-a_w w\right)\phi(T_c),\qquad C_\alpha(w,T_c)=C_{\alpha0}\left(1-b_w w\right)\psi(T_c), } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_w,b_w \in [0.15,0.45]} give typical end-of-stint losses.

Failure/artefact modes: thermal fade (high Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} ), graining (low Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_s} and high shear; reversible), blistering (subsurface; irreversible), flat-spot (lock-up; step change in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{\ast}} , and traffic factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_k} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t_k = \alpha_1 w_k + \alpha_2 (T_{c,k}-T^{\ast})^2 + \alpha_3 Q_k. }

Cumulative stint time to lap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{\mathrm{stint}}(N)=\sum_{k=1}^{N}\!\big(t_0+\Delta t_k\big). }

Under linear degradation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t_k \approx d k} and pit loss Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} , the continuous optimum stint length is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^{\ast} \approx \sqrt{\frac{2P}{d}}, } useful for first-order stop count decisions. The undercut condition at lap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_2} ; safety-car or VSC resets change optimal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^{\ast}} by reducing the opportunity cost of a stop.

Data acquisition and online identification

Typical online state vector and measurements for estimation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).