Dynamic sliding and rolling friction models for linear viscoelastic contact pairs

Abstract: This paper considers the sliding and rolling contact between viscoelastic bodies. Combining linear viscoelastic rheologies for bristle-like elements with nonlinear dynamic friction models, it derives a class of viscoelasto-kinematic equations, formulated as a system of partial differential equations (PDEs) governing the evolution of the frictional force, bristle deformations, and internal state variables at the interface between the contacting bodies. The resulting system is analysed mathematically, demonstrating that linear viscoelasticity preserves the hyperbolic character of the PDE systems typically encountered in rolling contact. The proposed theory is illustrated through representative examples of both sliding and rolling contact, highlighting that these two processes, whilst often treated as distinct, may in fact exhibit closely related underlying dynamics. Overall, the framework provides a general theoretical setting applicable to a broad class of viscoelastic frictional systems.

• The paper develops a unified semilinear hyperbolic PDE framework for dynamic sliding and rolling friction, coupling GKV viscoelasticity with Dahl, LuGre, FrBD, and Bouc-Wen laws.
• The analysis proves symmetric hyperbolicity, establishes well-posed solutions, and demonstrates passivity for the FrBDₙ₊₁-GKV model under specified pressure-transport conditions.
• The results show that distributed versus lumped friction dynamics arises from contact kinematics and material advection, with deformable sliding bodies producing spatial force relaxation and rolling force vanishing at zero creepage.

This paper develops a unified theoretical framework for dynamic friction modelling of sliding and rolling contact between viscoelastic bodies, formulated as systems of semilinear hyperbolic partial differential equations (PDEs). Building on the author's prior work on elasto-kinematic equations for elastic contact pairs (2606.09128), it extends the theory to linear viscoelastic rheologies of the Generalised Kelvin-Voigt (GKV) type, coupled with established nonlinear friction laws such as Dahl, LuGre, FrBD, and Bouc-Wen. The central claim is that the classical dichotomy between sliding friction models (finite-dimensional ODEs) and rolling friction models (distributed PDEs) is not intrinsic to the phenomena but is a consequence of the contact kinematics and, specifically, of the advection of material particles through the contact region.

Modelling framework

The paper begins with a general structure for dynamic friction models, in which the tangential force is generated by bristle-like elements whose relative deflection $\bm{z}$, internal viscoelastic states $\bm{\zeta}$, and generated force $\bm{f}$ evolve according to a coupled system of evolution equations. The function $\bm{h}_0$ governing bristle kinematics is instantiated by the two-dimensional Dahl/LuGre, FrBD, and Bouc-Wen formulations, all regularised to be continuous (avoiding non-smooth set-valued laws). The bristle rheology is described by stable linear state-space realisations equivalent to GKV elements, with distinct realisations permitted for the two contacting bodies. Because the forces generated by the two bodies must be equal and opposite, the relative deflection and the contact force admit the constitutive relation $$\bm{f} = \bar{\mathbf{K}}_0\bm{z} + \bar{\mathbf{C}}\bm{\zeta}$$, where the composite stiffness $\bar{\mathbf{K}}_0$ is the harmonic-like combination of the two bodies' micro-stiffnesses.

Viscoelasto-kinematic equations

The key structural contribution lies in the treatment of the material (Eulerian) derivatives. Under a small-displacement-gradient assumption, the paper defines a common interface coordinate $\bm{x}$ as a symmetric pull-back of the two bodies' coordinates, weighted by the matrices $\mathbf{S}_1$ and $\mathbf{S}_2$, which partition the elastic deformation between the bodies and satisfy $\mathbf{S}_1 + \mathbf{S}_2 = \mathbf{I}_2$. In the purely elastic case this recovers the coordinate introduced in the author's earlier elasto-kinematic analysis; in the viscoelastic case an additional contribution $\bm{\zeta}$0 appears. Notably, the paper argues that the bristle force $\bm{\zeta}$1, rather than the deflection $\bm{\zeta}$2, is the natural state variable in the viscoelastic setting: reformulating the dynamics in terms of $\bm{\zeta}$3 introduces higher-order coupling terms that complicate the analysis considerably.

Hyperbolic structure and boundary conditions

The governing PDE system for the augmented state $\bm{\zeta}$4 is shown to be symmetric hyperbolic: the advection matrices $\bm{\zeta}$5 and $\bm{\zeta}$6 are symmetric by construction, being linear combinations of the symmetric matrices $\bm{\zeta}$7 and $\bm{\zeta}$8. This establishes that linear viscoelasticity does not destroy the hyperbolic character typical of rolling contact systems — a result the author extends to time-varying contact domains via a level-preserving diffeomorphic mapping onto a cylindrical domain. A rotation into Riemann variables, with rotation angle determined solely by the reciprocal elastic behaviour of the pair, diagonalises the advection operators and recasts the system as an interconnection of four vector-valued transport equations with principal velocities given by eigenvalue-weighted combinations of the two bodies' transport velocities. This makes explicit that the elastic partition matrices govern not only the static load sharing but also the characteristic propagation structure of the friction dynamics.

Boundary conditions are prescribed on the viscoelasto-kinematic leading edges, enforcing vanishing principal stress components at inflow. A noteworthy remark is that these conditions are generally incompatible with the zero-deflection conditions imposed at the leading edge by lower-order models such as LuGre — a concrete point of tension with established practice. Well-posedness is proved for time-invariant domains and velocities via $\bm{\zeta}$9-semigroup theory (Bardos, Pazy), yielding unique mild solutions in $\bm{f}$0 and classical solutions under additional regularity. Passivity of the FrBD$\bm{f}$1-GKV model is established with an explicit quadratic storage function, under the condition $\bm{f}$2 on the pressure-velocity field, which is automatically satisfied for constant pressure and spin-free transport.

Applications to sliding and rolling contact

The theory is illustrated on one-dimensional line contact problems with isotropic elastic partition, parameterised by the scalar $\bm{f}$3 quantifying the share of elastic compliance attributed to the substrate.

Sliding contact. For a block sliding over a viscoelastic substrate, the substrate states are advected while the block's bristles are not; the governing system is therefore an interconnection of hyperbolic PDEs with nonlocal ODEs. In the purely elastic steady state, a closed-form solution shows that the normalised longitudinal force satisfies $\bm{f}$4, whose second term is strictly positive: the friction force magnitude is always strictly below $\bm{f}$5, approaching that bound only as $\bm{f}$6, where the classical spatially uniform ODE-type behaviour of finite-dimensional sliding models is recovered. This result directly substantiates the paper's thesis: the distributed character of sliding friction emerges whenever both bodies are deformable. Numerical studies with rubber-like parameters show that additional dissipative branches attenuate the force through spatial relaxation, while longer relaxation times increase the steady-state force. In transient sliding, the block's bristle deformation at the leading edge need not vanish — it decays exponentially — a distinction from steady conditions that becomes relevant in velocity-reversal scenarios.

Rolling contact. With advection present in both bodies, the system retains its full PDE character for any positive rolling and translational velocities. The elastic steady-state force depends on three arguments — relative slip, rolling velocity, and $\bm{f}$7 — because the rigid relative velocity is the kinematic difference between rolling and translational speeds. A sharp qualitative contrast with sliding emerges: the longitudinal rolling force vanishes identically at zero creepage, whereas the sliding force tends to a nonzero limit as the sliding velocity approaches zero. Transient rolling simulations under a slip step input confirm trends consistent with the steady analysis.

Limitations and open questions

Several restrictions are stated explicitly by the author. The mathematical analysis (hyperbolicity proof aside) assumes time-invariant contact areas and velocities; the well-posedness theorem requires $\bm{f}$8 and Lipschitz continuity of the nonlinearity, and the passivity result relies on the divergence condition on the pressure-transport field, which may fail for general pressure distributions or spin. The applications are confined to one-dimensional line contact without spin and with decoupled longitudinal/lateral dynamics; combined slip-spin conditions, two-dimensional non-uniform pressure distributions, nonlinear viscoelasticity, and viscoplasticity are all deferred. No experimental validation is presented — the numerical parameters are inspired by, but not fitted to, rubber-like material data — so the quantitative fidelity of the framework for tyres and elastomeric components remains an open empirical question.

Conclusion

The paper derives a class of viscoelasto-kinematic PDEs that unifies dynamic sliding and rolling friction models for viscoelastic contact pairs, proves their symmetric hyperbolicity and well-posedness, and establishes passivity for the FrBD$\bm{f}$9-GKV class. Its principal conceptual contribution is the demonstration that lumped versus distributed friction dynamics is a kinematic — not constitutive — distinction, governed by whether material particles are advected through the contact region. The analytical steady-state results for sliding and rolling contact, and the incompatibility identified between classical leading-edge conditions and the correct viscoelastic boundary conditions, are concrete findings of immediate relevance to brush-type tyre and wheel-rail modelling.