Dynamic Tire Wear Model

• Dynamic tire wear model is a quantitative framework that characterizes degradation through microscale wear physics and continuous damage accumulation.
• It integrates probabilistic scission kinetics and erosion dynamics with online parameter identification to predict tire performance under variable conditions.
• The approach couples material chemistry calibration with vehicle-level latent-state models, supporting real-time control and adaptive maintenance strategies.

A dynamic tire wear model quantitatively characterizes the evolution of tire performance and material loss under operationally varying mechanical, chemical, and environmental conditions. Contemporary developments incorporate microscale wear physics, online parameter identification, and macroscopic or latent-state degradation modeling across both materials and vehicle-level applications.

1. Physical Basis: Continuous Damage Accumulation in Elastomeric Wear

The dominant wear process for elastomeric (tire) materials under mild frictional sliding is not classical crack propagation but rather the gradual, distributed accumulation of damage by stress-activated chain scission throughout the network, extending far below the surface. The primary state variable is a local, depth-resolved damage fraction $\phi(z,N)$, where $z$ is the depth from the sliding surface and $N$ is the number of sliding cycles. The total worn volume after $N$ cycles is

$$V(N) = A \cdot \ell \cdot \left\langle \phi \right\rangle = A \cdot \ell \int_0^\infty \phi(z,N) \, dz$$

where $A$ is the nominal contact area, $\ell = N L$ is the total sliding distance ($L$ is the stroke length), and $\left\langle \phi \right\rangle$ is the through-thickness average damage (Taisne et al., 22 Jun 2025).

Stress-activated chain scission at contacting asperities follows an Arrhenius-type rate law:

$$k_\text{off}(W) = k_0 \exp\left(\frac{W}{k_B T}\right)$$

with $W$ the elastic strand energy, $k_0$ the zero-load rate, $k_B T$ the thermal energy. For a strand of energy $W$, the mean breakage cycle count $n(W)$ is

$$n(W) \approx \frac{1}{k_0 T_\text{load}} \exp\left(-\frac{W}{k_B T}\right)$$

where $T_\text{load}$ is the dwell time per asperity slip. The threshold energy for scission after $N$ cycles is

$W_N \approx W_0 - k_B T \ln N$

with $W_0$ the zero-load activation barrier. Under a uniform strand energy density $p_0$, the increment in broken chains per area is

$N_\text{chain}(N) \approx p_0 k_B T \ln N$

Translating to the damage fraction via the crosslink density $v_0$ gives logarithmic-in-$N$ damage growth. Subsurface damage decays exponentially with depth:

$$\phi(z) = \phi_\text{max} \exp\left(-\frac{z}{\lambda_s}\right)$$

where the characteristic depth $\lambda_s$ is set by the asperity size $a$ (typically $a \sim 10~\mu \mathrm{m}$, $\lambda_s \approx 2$–$7~\mu \mathrm{m}$).

This wear scenario implies that tire degradation evolves as a slow, probabilistically triggered, fatigue-like process, subject to both surface accumulation and material removal via erosion as the subsurface damage field crosses a detachment threshold (Taisne et al., 22 Jun 2025).

2. Mathematical Structure and Dimensionless Parameters

The dynamic progression of the local damage field (accumulation plus erosion) is given by

$$\frac{\partial \phi}{\partial N} + v_e \frac{\partial \phi}{\partial z} = F(z, \phi)$$

with $v_e$ the erosion speed and $F$ an accumulation source term. In the absence of erosion ($v_e = 0$), pure accumulation gives

$\phi(N) \propto \frac{A_R}{A} p_0 k_B T \ln N$

where $A_R$ is the real contact area (which scales with nominal pressure $P$ and Young’s modulus $E_Y$ by macro-scaling $A_R / A \sim P / E_Y$).

For the model’s practical implementation, key dimensionless and calibration parameters include:

Parameter	Physical Definition	Typical Value
$\Pi$	Dimensionless pressure $P / E_Y$	Load- and material-dependent
$N$	Cycle count	Number of tread revolutions
$a_0$	Pre-logarithmic factor ($A_R / A) p_0 k_B T$	$\sim 10^{17}$ chains$\cdot$m$^{-2}$ per $\ln N$
$k$	Archard-like wear coefficient	$10^{-6}$–$10^{-5}$ (per cycle)
$\phi_M$	Detachment (depercolation) threshold	$1$ ("monolayer" of broken network strands)
$\lambda_s$	Damage penetration length (asperity-scale)	$2$–$7~\mu\mathrm{m}$

This framework naturally introduces multi-scale coupling: contact mechanics, molecular bond energetics, and network chemistry all set measurable wear rates and timescales (Taisne et al., 22 Jun 2025).

3. Implementation Algorithm for Dynamic Wear Simulation

The continuous damage model prescribes a minimal cycle-based update algorithm for dynamic tire wear computations:

1. Local Damage Increment per Cycle For each contact patch $i$ at cycle $N$:

$$\Delta \phi_i(z) = \frac{A_{R,i}}{A} p_0 k_B T / N \cdot \exp\left(-\frac{z}{\lambda_{s,i}}\right)$$

2. Damage Field Update Aggregate increments:

$$\phi(z, N+1) = \phi(z, N) + \sum_i \Delta \phi_i(z)$$

1. Surface Erosion Step If $\phi(0,N) > \phi_M$:
   • Remove a surface slice $\Delta h$ such that $\phi(0,N) \rightarrow 0$
   • Shift the damage field deeper: $\phi(z) \rightarrow \phi(z + \Delta h)$
2. Worn Volume Accumulation Track macroscopic loss: $\Delta V = N L \Delta h A$

The model robustly captures logarithmic damage scaling, depth-dependence set by contact microgeometry, and the influence of material parameters tunable by chemistry or processing (Taisne et al., 22 Jun 2025).

4. Data-Driven, Vehicle-Level Degradation Models and Latent-State Approaches

At the vehicle performance scale, dynamic tire wear is modeled by latent degradation state estimation frameworks.

In Formula 1 applications, a Bayesian state-space model captures evolving tire degradation as a latent variable $\alpha_t$ (tire pace) with dynamics:

• Observation:

$$y_t = \alpha_t + \gamma \, \mathrm{fuel}_t + \varepsilon_t,\quad \varepsilon_t \sim N(0, \sigma^2_\varepsilon)$$

where $y_t$ is lap time and $\mathrm{fuel}_t$ is fuel mass.

• Latent Process:

$$\alpha_{t+1} = (1 - I_{\mathrm{pit},t})(\alpha_t + \nu) + I_{\mathrm{pit},t}\, \alpha_{\mathrm{reset}} + \eta_t$$

where $I_{\mathrm{pit},t}$ indicates a pit stop, $\nu$ is the lap-wise degradation increment, and $\alpha_{\mathrm{reset}}$ is the new-tire pace (Cappello et al., 29 Nov 2025).

Extensions such as compound-specific wear ($\nu[c]$), time-varying degradation ($\nu_t$ with autoregressive $\beta[c]$), and asymmetric error modeling (skewed-$t$) are also introduced for predictive robustness. Bayesian inference provides parameter uncertainty and actionable forecasts for real-time race strategy (Cappello et al., 29 Nov 2025).

Performance evaluation (e.g., RMSPE, CRPS) confirms improved predictive power versus ARIMA-style baselines and enables interpretable quantification of degradation rates and uncertainty intervals per stint and compound.

5. Online Model Identification and Adaptive Control in High-Performance Vehicles

For real-time control and safety in autonomous or racing vehicles, tire wear is modeled as a time-varying parameter identification problem. Core methodologies employ online learning—specifically, Extreme Learning Machines (ELM)—to non-parametrically model the lateral tire force characteristic $F_y(\alpha)$:

$$F_y(\alpha; \Theta) = D \sin\bigl[C\,\arctan\bigl(B\alpha - E(B\alpha - \arctan(B\alpha))\bigr)\bigr]$$

where the canonical "Magic Formula" parameter vector $\Theta = \{B,\, C,\, D,\, E\}$ is not static but is represented by time-varying ELM weights $\beta_{f,r}(t)$ for front and rear tires. The complete parameter set, including rolling and aero-drag constants ($C_r, C_d$), is updated online using observed vehicle states, control inputs, and measured outputs via stochastic gradient descent with momentum (Kalaria et al., 2023).

This implicit, data-driven wear tracking supports integration with nonlinear model predictive control (MPC), in which updated tire properties directly inform safety-constrained trajectory optimization.

6. Material Chemistry, Toughness–Wear Tradeoff, and Calibration

Varying the polymer-filler network prestretch $\lambda_0$ tunes a critical antagonism: higher fracture energy ($G_c$) can be achieved at the expense of accelerated wear by lowering the average activation energy for bond rupture $W_0$, manifesting as an increased fatigue-wear coefficient $k$ (Taisne et al., 22 Jun 2025). This coupling mandates careful calibration for tire applications demanding both resilience to catastrophic failure and operational durability.

Empirical calibration parameters include:

• $$G_{c,\mathrm{DN}} = 400~\mathrm{J}\,\mathrm{m}^{-2}$$ (double-network, DNE)
• $$G_{c,\mathrm{TN}} = 2400~\mathrm{J}\,\mathrm{m}^{-2}$$ (triple-network, TNE)
• $k_{\mathrm{DN}} \approx 0.9 \times 10^{-6}$, $k_{\mathrm{TN}} \approx 6 \times 10^{-6}$ (per cycle)

Modulation of these through network design enables constrained optimization of wear and toughness for practical tire formulations.

7. Model Limitations, Assumptions, and Integration Across Scales

Within material-centric approaches, explicit camber and normal-load variability are neglected, and wear is characterized solely via accumulated mechanochemical damage—not via explicit state variables for "wear" in the system state vector (Kalaria et al., 2023). In vehicle-level models, tire performance evolution is considered latent and inferred statistically, but does not directly encode physical microstructural degradation.

A plausible implication is that full-scale predictive frameworks may require hierarchical coupling: mechanochemical field evolution (micro-scale), latent state degradation (macro-scale), and online adaptation in response to operational data. Robust integration mandates physically motivated calibration and closed-loop data assimilation at all levels.

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These dynamic tire wear modeling approaches—rooted in continuous damage mechanics, probabilistic scission kinetics, and adaptive system identification—provide a comprehensive set of algorithms and physical insights for predicting, controlling, and mitigating tire wear across disciplines and use cases (Taisne et al., 22 Jun 2025, Cappello et al., 29 Nov 2025, Kalaria et al., 2023).