# Empirical Models for the Viscoelastic Complex Modulus with an Application to Rubber Friction
## 1 Introduction  
Viscoelastic materials exhibit both viscous and elastic characteristics when stressed so that their behaviour is in-between that of a purely viscous liquid and a perfectly elastic solid. When a viscoelastic material is deformed, part of the deformation energy dissipates, and the rest is stored as reversible elastic energy. The properties of viscoelastic materials such as rubber can vary widely according to temperature, frequency and chemical composition [1]. Rubber is a highly deformable material employed in a wide range of products, from everyday necessities, e.g., shoe soles, to industrial applications--notably, vehicle tyres. Such applications demand large deformations, vibration damping [2] and enhanced gripping characteristics (e.g., tyres and conveyor belts). With safety-critical components such as tyres, it is important to be able to predict friction for different types of rubber; however, the non-linear nature of rubber poses significant modelling challenges [3].  
The response of a rubber material can be obtained from rheological models. Linear viscoelastic materials can be represented by combinations of springs and dashpots. The most common ones are the Maxwell and Voigt-Kelvin models [4] which use springs and dashpots connected in series or in parallel, respectively (see Figure 1). The standard linear solid (SLS) is an upgraded rheological model in which an extra spring is added in parallel with the Maxwell unit to allow for an accurate representation of a larger group of polymers [5]. Additionally, the Weichert model offers a relaxation spread over a longer period by using a single spring in parallel with several Maxwell units [6, 7]. [PERSON] et al. [8] proposed a non-linear rheological model to offer better accuracy in simulating the material'srelaxation modulus and viscoelastic responses. A logistic-type function approximation was proposed by [8] to simulate the viscoelastic response of spring-dashpot networks. Another recent study employed a combination of a generalised Maxwell model and a relative fraction derivative model to reproduce viscoelastic material behaviour [9].  
Both models proposed herein describe rubber's viscoelastic response in the frequency domain. As shown in [10], rubber friction strongly depends on the rubber's frequency response, and this is accounted for in [PERSON]'s friction model [11]. The first model uses a group of polynomials to capture the viscoelastic frequency response with great accuracy, whilst the other method uses a generic empirical formulation with few parameters to expedite the parameter identification process and to offer a compact set of equations. Thereafter, the proposed formulations are applied to [PERSON]'s friction model without thermal effects [11] to evaluate their computational efficiency and accuracy. The paper is structured as follows: In Section 2, the measured moduli used as the benchmark are presented. In Section 3, the polynomial equation method and the empirical model are described. In Section 4, the ability of both models to fit measured data and their performances as part of [PERSON]'s friction model are evaluated. Section 5 provides a summary of the conclusions.