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]. Xu et al. [
8] proposed a non-linear rheological model to offer better accuracy in simulating the material’s relaxation 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 Persson’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 Persson’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 Persson’s friction model are evaluated.
Section 5 provides a summary of the conclusions.
2. Measured Viscoelastic Modulus
When a sinusoidal force is applied to a purely elastic material, the stress and deformation occur concurrently, so that both quantities are in phase. On the contrary, when the same sinusoidal force is applied to a purely viscous fluid, the deformation will lag the stress by one-quarter cycle (
radians). This means that when the deformation is maximal, the force is minimal and vice versa. In the case of a viscoelastic material, the phase lag between the force and the deformation lies somewhere between zero and a quarter cycle. Mathematically, a sinusoidal strain can be written using Euler’s formula, as shown in Equation (
1), where
is the amplitude of the strain and
is the frequency:
Similarly, the stress can be written as:
where
is the amplitude of the stress and
is the phase angle. In
Figure 2 the stress and strain phasors are shown as they rotate at a frequency
, with the strain lagging behind the stress by an angle
.
The following relationships can be deduced from
Figure 2:
With the above definitions, the dynamic or complex modulus will have a real and an imaginary part. The real or storage modulus is defined as the ratio between the real part of the stress and the strain:
The imaginary or loss modulus is defined as the ratio between the imaginary part of the stress and the strain:
Equations (
1)–(
6) lead to the broadly used complex or dynamic modulus formula [
12]:
Finally, the tangent modulus, also known as the loss tangent is defined as:
By definition, the modulus of a material is considered as the overall resistance of the material to an applied deformation. Due to its viscoelastic nature, the rubber modulus is split into elastic (storage),
, and viscous (loss),
, components, denoting the ability of the material to store and dissipate energy as heat, respectively. The real and imaginary components of a viscoelastic modulus are collectively referred to as the material’s complex modulus. Equally important is the loss tangent or
, which is defined as the ratio of energy loss to energy storage. The area near the peak value denotes high energy loss and is considered as the area where the tyre is designed to be operating most of the time. While spring–dashpot-based analytical models are useful for representing the physics of viscoelastic materials, the experimentally measured complex modulus as a function of excitation frequency is often used instead. In the case of rubber, the complex modulus is shown to be strongly linked to friction [
13,
14].
Figure 3 shows the typical complex modulus of a rubber used in tyre manufacturing.
Such moduli of rubber materials are obtained using oscillatory strain in tension or shear form, while monitoring the resulting stress [
15]. When rubber is dynamically stretched and released, the returned energy is less than the energy put into the rubber in the first place. This viscoelastic effect can only be tested dynamically. Dynamic mechanical analysis (DMA) instruments measure the viscoelastic modulus in response to applied oscillating strain. The stress response to the deformation is recorded as a function of time or frequency. The rubber sample is oscillated at different frequencies and the same process is repeated at various temperatures. Results are then shifted according to the time–temperature superposition principle [
16], forming a master curve and covering a wide range of frequencies at a chosen reference temperature. The literature was scrutinised and DMA data for different viscoelastic materials were gathered [
4,
11,
13,
15,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27]. While spring–dashpot networks can be used to reproduce DMA data, the required structure of the network is sometimes difficult to decide a priori, and the demand on parameter identification can be significant. In the following sections it is shown that the DMA responses of different materials can be approximated successfully using our proposed formulations with a limited set of parameters.
4. Results
Results show that the empirical formulations presented herein were able to model all the viscoelastic moduli that were examined with correlation levels similar to those presented in the previous section. More specifically, the two proposed approaches were compared against each other and against other methods from the literature, such as the zero/pole fitting presented in [
9] and the fractal derivative generalised Maxwell Model [
30]. Specific performance attributes of the models that are addressed include: (a) versatility, (b) number of parameters required, (c) computational efficiency and (d) integration with Persson’s friction model [
31]. Persson’s model was selected as the “carrier” friction model, as it has been heavily relied upon in the literature for the prediction of rubber friction [
32,
33,
34,
35,
36,
37,
38]. It was also used because it is a particularly computationally intensive model that will benefit from efficient implementations of the viscoelastic material model. Finally, the effects on model parameters of five SBR (styrene-butadiene rubber) compounds from the study presented in [
20] are discussed and the results are linked to the CB (carbon black) content. In line with the findings in [
20], the varying levels of CB in the rubber do not affect the peak values of the viscoelastic modulus, but do influence the gradient of the linear sections.
4.1. Versatility
Versatility of a model is interpreted as the ability to successfully fit the moduli of different materials. The performances of the models in this respect were demonstrated in the previous section, particularly in
Figure 5,
Figure 6,
Figure 7 and
Figure 8 and
Table 3 and
Table 4. Due to its piece-wise nature, the polynomial method is inherently more flexible in terms of fitting any of the three commonly occurring elements of the complex modulus (storage modulus, loss modulus and loss tangent). It is also more likely to fit more accurately a wider range of materials, albeit with increased pre-processing time and an increase in the number of fitting parameters. The LBEM method, on the other hand, requires less pre-processing, does not depend on manual selection of the relevant data segments and relies on fewer parameters than the polynomial method, as described below.
4.2. Number of Parameters
Both formulas make use of parameters that can be linked to the critical sections of the data, such as storage modulus transition regions or the peak values of the loss modulus and loss tangent. The LBEM method requires 10 parameters to be defined, whereas the polynomial method needs a minimum of 16 parameters for the storage and loss modulus to be evaluated.
An important quality of the LBEM approach is that the parameters are directly linked to typifying quantities of the data, such as the loss tangent frequency peak location, the curve’s growth rate and the loss tangent peak value.
4.3. Computational Efficiency
In terms of solution efficiency, both formulas provide a greatly reduced computation time compared with the common data interpolation method, the proposed zero/pole formula of [
39] and the FDGM (fractal derivative generalised Maxwell) model [
9].
Table 5 compares solution times of Persson’s isothermal friction model for a single friction point using a resolution of 200 frequency segments, and several complex modulus implementations. The commonly used data interpolation method was used as a benchmark and had a mean computation time of 0.318 s per friction point. The FDGM model (that utilises a zero/pole formulation) had a computational time of 1.371 s, which is an increase of
with respect to the interpolation method. The proposed LBEM model had a reduced computation time of 0.184 s, a reduction of
with respect to the interpolation method and
with respect to the zero/pole model. On the other hand, the polynomial method resulted in a
or
more efficient solution compared to interpolation and the zero/pole techniques, respectively, and is also
less computationally demanding than the LBEM method. The reduced efficiency of the LBEM approach in comparison to the polynomial method is related to the use of computationally expensive mathematical functions, such as exponentials and the hyperbolic secant.
4.4. Application to Persson’S Friction Model
Figure 9 shows the friction coefficient estimated over a wide range of sliding speeds for the three different materials used previously. Persson’s isothermal friction model has been used for this estimation [
11] with the model parameters provided in
Table 6.
All models gave similar friction results, with the LBEM model providing the best agreement with the interpolated data. A discrepancy was produced by the FDGM model, which is most likely a result of extrapolation at frequency extremes. However, the FDGM model may include an increased number of parameters or "zones"; therefore, there is room for improving its fitting accuracy [
9].
4.5. PHR Content
We used the data published in [
20] to study the effect of different concentrations of CB (carbon black) on the DMA data, and by extension, the effects on the identified parameters from both formulations. Five SBR compounds with different parts phr (per hundred rubber) of carbon black, ranging from 0 to 60 phr, are shown in
Figure 10. Both models were able to fit the curves with a maximum normalised root mean squared error of
for the LBEM formulation and
for the polynomial formulation. From the raw data it can be seen that as the phr of CB increases, so does the storage modulus in the rubbery (low frequency) region. The storage modulus also increases with the phr of CB in the glassy (high frequency) region, albeit at a reduced rate. The peak of the loss tangent is broadly inversely proportional to the concentration of CB.
With regard to the polynomial model,
Figure 11 demonstrates the linear relationship between the vertical shift of the linear section (see
Figure 4a) and the phr concentration of CB in rubber specimens. R-squared values are included in
Table 7.
From
Figure 12 it can be observed that the parameters
,
k,
,
,
and
p of the LBEM model have R-squared values equal to or higher than
, indicating a high level of correlation with the carbon black concentration. As expected from the raw data analysis, the parameter
that controls the vertical shift of the storage modulus shows a high correlation. However, the parameter
k that controls the logistic rate of increase or steepness of the curve also shows a high correlation, and this was not easy to spot from a visual analysis of the data. In addition, as indicated from the visual analysis of raw data, the parameter
p that controls the bell function peak value shows a clear correlation with CB concentration. The loss tangent parameter
that controls the bell function’s vertical shift at high frequencies after the peak, together with
and
that determine the bell’s rate of increase at low frequencies and high frequencies, respectively, also have R-squared values above
.
On the basis of the observed dependency of critical model parameters on CB content, the LBEM model and to a lesser extent the polynomial model, lend themselves to the creation of virtual hypothetical rubber materials to be simulated within friction models, so that the performance of a new tyre with new materials can be predicted.
5. Conclusions
The proposed empirical models are capable of not only accurately reproducing DMA data but also significantly reducing the computational time. This makes the models ideal for integration into advanced friction models, such as [
11,
13,
14], where the viscoelastic complex modulus is evaluated multiple times and over a wide range of frequencies.
With the polynomial method, good predictions of the storage and loss moduli and the loss tangent can be obtained by selecting the piecewise linear/non-linear segments of the curves. Formulation is versatile in that in can fit a broad range of viscoelastic moduli because of the adjustability of the number of segments. The proposed LBEM formula uses parameters that can be easily linked to the typifying quantities of the data, such us the storage moduli in the rubbery and glassy regions or the frequency where the loss tangent is maximum. The fitting quality of both models was quantitatively evaluated against the data using the NRMSE and metrics. The for the polynomial method was above for all cases, whereas the performance of the LBEM formula varied more widely, with varying from to . Comparable results can be obtained with the FDGM model at the expense of computation time. The proposed models are primarily suited to calculations in the frequency domain, but they are not applicable the creep and relaxation tests because the relaxation time cannot be readily deduced from their formulation. With regard to the polynomial model, only a subset of model parameters can be linked to the CB content of the rubber, and by extension, to the chemical composition of the rubber. On the other hand, the LBEM model comes with a smaller parameter set that is linked to material composition; therefore, it can be used for the creation of hypothetical rubber materials to assess/optimise the grip of new tyres ahead of production.