Dynamic sliding and rolling friction models for linear viscoelastic contact pairs

Technical Summary: Dynamic Sliding and Rolling Friction Models for Linear Viscoelastic Contact Pairs

Problem Statement
Friction modeling in engineering applications, ranging from servo systems to wheel-rail interactions, traditionally distinguishes between sliding and rolling regimes. Sliding friction is typically modeled using finite-dimensional systems of ordinary differential equations (ODEs), often based on the assumption that material advection is absent at the contact interface (e.g., a soft block sliding on a rigid plane). Conversely, rolling friction is formulated using partial differential equations (PDEs) to account for the advection of forces and deformations through the contact patch, as pioneered by Kalker.

However, this dichotomy becomes problematic when dealing with contact pairs where both bodies are deformable or possess significantly different elastic properties, particularly in the presence of viscoelasticity (common in polymers like tires). In such scenarios, the assumption that sliding friction can be purely lumped (ODE-based) breaks down because material particles and internal state variables are advected at different velocities depending on the rheological properties of the contacting bodies. Existing models often struggle to unify these dynamics or correctly handle the kinematic coupling in viscoelastic contact pairs.

Methodology
The paper proposes a unified theoretical framework that combines linear viscoelastic rheologies with nonlinear dynamic friction models. The methodology proceeds as follows:

1. Rheological Modeling: The contact interface is modeled using "bristle-like" elements. The viscoelastic behavior of these elements is described using Generalized Kelvin-Voigt (GKV) models, represented as stable linear state-space systems. This allows for the description of complex memory and hysteresis effects across a broad frequency range.
2. Viscoelasto-Kinematic Derivation: The authors derive a set of governing equations by coupling the rheological state-space representations with kinematic relationships. Crucially, they replace standard time derivatives with material (Eulerian) derivatives to account for the transport of internal state variables ($\zeta$) and bristle deformations ($z$) through the contact area.
3. PDE Formulation: By integrating the constitutive laws with the kinematic transport equations, the authors derive a system of semilinear hyperbolic partial differential equations (PDEs). These equations govern the evolution of the tangential contact force ($f$) and the internal state variables ($\zeta$) over the contact domain.
4. Mathematical Analysis: The resulting PDE system is analyzed to prove its hyperbolic character. The authors demonstrate that the system can be transformed into Riemann variables, decoupling the equations into transport equations along characteristic directions determined by the elastic partitioning of the contact pair.
5. Boundary and Initial Conditions: Appropriate boundary conditions (BCs) are formulated based on the "leading," "neutral," and "trailing" edges of the contact area, ensuring the dissipativity and well-posedness of the problem.

Key Contributions
The paper makes three primary contributions to the field of contact mechanics and friction modeling:

• Systematic Extension to Distributed Frameworks: The work extends dynamic viscoelastic friction models from lumped (ODE) formulations to distributed (PDE) frameworks. It demonstrates that even in sliding contact, if the substrate is compliant or both bodies are deformable, the governing equations naturally become PDEs due to material advection.
• Derivation of Viscoelasto-Kinematic Equations: The authors derive a specific class of equations that govern the coupled evolution of frictional forces, material deformation, and internal state variables for both sliding and rolling contact pairs. These equations explicitly account for the different transport velocities of the two bodies.
• Unification of Sliding and Rolling Dynamics: The theory reveals that sliding and rolling friction share a common mathematical structure: both are described by linear or semilinear hyperbolic PDE systems. The distinction between ODE-based sliding models and PDE-based rolling models is shown to be rheology-dependent rather than intrinsic to the type of motion.

Results
The proposed theory is illustrated through representative examples of sliding and rolling contact, utilizing an extension of the FrBD model coupled with GKV elements:

• Sliding Contact: In the case of a block sliding over a viscoelastic substrate, the model recovers the classical ODE behavior only in the limit where the substrate is rigid (no advection). When the substrate is viscoelastic, the model predicts spatially varying friction forces and bristle deformations. The results show that the magnitude of the friction force depends on the ratio of stiffnesses between the bodies and the number of dissipative branches in the rheological model. Higher-order rheological models introduce spatial relaxation, attenuating the friction force along the contact length.
• Rolling Contact: For rolling contact, the model retains the full PDE character regardless of the stiffness ratio, as material advection is inherent to the rolling motion. The analysis confirms that the hyperbolic nature of the system is preserved, and the characteristic velocities are determined by the principal elastic interaction modes.
• Transient and Steady-State Behavior: Simulations demonstrate that the system exhibits distinct transient behaviors depending on the relaxation times of the viscoelastic materials. In steady-state sliding, the friction force is strictly bounded by the product of the friction coefficient and normal load, with the bound approached as the stiffness ratio favors the rigid body.

Significance and Claims
The paper claims that its framework provides a general theoretical setting applicable to a broad class of viscoelastic frictional systems. The primary significance lies in resolving the conceptual ambiguity between sliding and rolling friction models. By showing that both phenomena can be described by the same class of hyperbolic PDEs, the work suggests that the traditional separation of these problems into ODE and PDE domains is an artifact of simplifying assumptions (specifically, the assumption of a rigid substrate or decoupled kinematics) rather than a fundamental physical distinction.

The authors emphasize that their approach correctly handles the kinematics of material particles in elastically dissimilar pairs, a scenario where previous brush-type models have introduced conceptual difficulties regarding slip and spin definitions. The theory is presented as a rigorous extension of previous work on elastic contact pairs, now incorporating the necessary complexity to model viscoelastic materials found in practical applications like tires. The paper concludes that the distributed formulation is not merely an alternative but a necessary description for accurate modeling of viscoelastic contact, offering a unified perspective that bridges the gap between sliding and rolling dynamics.