1 Introduction

Understanding and minimising driving losses are essential factors in reducing global warming. Depending on the vehicle velocity and ambient temperature, truck-tyre rolling resistance represents up to 70% of the total driving resistance at lower speed levels (Hyttinen et al. 2022; Surcel and Michaelsen 2010). The operating temperature of the tyre is a vital parameter for rolling resistance. This is because of the strong temperature dependency of filler-reinforced rubber, which makes up most of the tyre structure. Although this temperature dependency is well known, the temperature dependency of rolling resistance has long been neglected. Only a few publications have discussed its importance (Hyttinen et al. 2022; Ejsmont et al. 2018; Oswald and Browne 1981; Greiner 2019; Bode 2020; Ejsmont et al. 2022). Nearly all test standards focus on testing tyres in rather simplified conditions. For example, ISO 28580 (ISO 28580:2009 2009) describes a testing procedure where the tyre is tested at an ambient temperature of +25 °C, and the tyre rolling resistance must be stabilised. Thereby, this standard neglects the effects of temperature changes on rolling resistance. There exist several thermal or thermo-mechanical tyre models, such as Farroni et al. (Farroni et al. 2020) and Février and Fandard (Février and Fandard 2008), focusing on performance aspects (tyre friction, vehicle handling, safety, etc.). However, these models are not developed for analysing transient rolling resistance for energy-consumption analyses. Hyttinen et al. (Hyttinen et al. 2023), Greiner et al. (Greiner et al. 2018), Sandberg et al. (Nielsen and Sandberg 2002), and Mars and Luchini (Mars and Luchini 1999) have all created simulation models for transient rolling resistance; only the Hyttinen et al. model considers the effects of ambient temperature on rolling resistance. This model is a phenomenological model that effectively captures the transient rolling resistance. However, the model does not describe the underlying physics of the tyre, and there is a need for such a semi-physical tyre model that can be used in quick parameter studies.

Rubber exhibits two interesting behaviours that make simulations and testing difficult. These are the Fletcher–Gent effect (Payne effect) (Hentschke 2017) and the Mullins effect (Diani et al. 2009). They have similarities, but the Fletcher–Gent effect recovers directly, whereas the Mullins effect recovers only partly over a prolonged time (Diani et al. 2009). The Fletcher–Gent effect can be attributed to the filler–filler and filler–polymer interaction since it is nearly non-existent for unfilled rubber (Fröhlich et al. 2005).

The Fletcher–Gent effect can be described as increasing stiffness at low strain amplitudes, decreasing stiffness at higher strain-amplitude levels, and a strain-amplitude dependency of the loss modulus (Fröhlich et al. 2005; Bergström 2015; Österlöf et al. 2016). This effect is important to consider in rolling resistance calculations (Hyttinen et al. 2022). The Fletcher–Gent effect makes linear viscoelastic models insufficient for rolling resistance calculations at varying axle-load levels. The Mullins effect is the phenomenon of strain softening during the first few loading cycles. This effect is less critical for rolling resistance simulations since it is no longer present when a tyre has been rolling for a certain time. However, it recovers slightly over an extended period, or more considerably with high-temperature conditioning (Diani et al. 2009).

Some semi-physical tyre models have been used to simulate rolling resistance; e.g., Ydrefors (Ydrefors 2022), Davari et al. (Davari et al. 2017), and Greiner et al. (Greiner et al. 2018). These studies have carried out rolling resistance simulations using a small-strain linear viscoelastic model enhanced with a frictional network that creates a Fletcher–Gent-like material response. Davari et al.’s model also includes the effect of adhesive forces on rolling resistance through bristle deformation. However, the models by Ydrefors and Davari et al. do not consider any transient rolling resistance. A similar material modelling approach with Prony-series viscoelasticity and perfectly plastic networks has also been utilised in finite-element simulations (Hyttinen et al. 2022). Rubber also has a strain-amplitude-dependent viscous flow, which the Prony series does not include. Extensions of the Prony series exist, such as the nonlinear viscoelastic Bergström–Boyce model (Dal and Kaliske 2009; Bergström and Boyce 1998). This model considers the Fletcher–Gent effect and viscoelasticity at varying strain amplitudes. There is a clear need for such a physical tyre model that accounts for transient rolling resistance and tyre-temperature dependency, which could be used in parameter studies. This article aims to present a semi-physical thermodynamic tyre model that can capture a tyre’s warming and cooling behaviour and the transient rolling resistance at varying vehicle speeds. The transient rolling resistance is achieved with a temperature-dependent viscous-flow model.

This article is structured as follows: first, the basics of rolling resistance are introduced. Then, the thermodynamic simulation model is presented. The next section provides a definition of the nonlinear viscoelastic model, the temperature-dependent viscous flow and different simulation parameters. Finally, the results and conclusions are presented.

2 Rolling resistance

Rolling resistance is defined as a force opposing the motion of a tyre. It is affected by different parameters such as road roughness (Taryma et al. 2014), road stiffness (Wong 2001), road wetness (Hyttinen et al. 2022; Ejsmont et al. 2015), axle loads (Surcel and Bonsi 2015), vehicle speed (Nakajima 2019) and ambient and tyre temperature (Hyttinen et al. 2022). Of the aforementioned factors, the ambient and tyre temperature most significantly contribute to the magnitude of rolling resistance. Unfortunately, current semi-physical tyre models designed to simulate rolling resistance do not capture these effects. The importance of temperature relates to the fact that rolling resistance, for the most part, is caused by the viscoelasticity (80–95%) in tyre rubber (Aldhufairi and Olatunbosun 2018), and the viscoelasticity has a strong temperature dependency (Hyttinen et al. 2022). At low temperatures, the tyre has a larger hysteresis area, and conversely, the energy dissipation is lower at higher temperatures (Fig. 1).

Fig. 1
figure 1

Illustration of the rubber hysteresis at lower and higher temperatures

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Viscoelasticity does not affect the contact pressure when the tyre is standing still, and the contact pressure is identical at the leading and trailing edges of the contact surface. However, when the tyre is in a rolling motion, the rubber is compressed at the leading edge of the contact patch and expanded at the trailing edge (Fig. 2). Due to rubber viscoelasticity, rubber produces a larger reaction force during the loading cycle (compression) than during the unloading cycle (expansion). As a result, the centre of contact pressure shifts in front of the tyre-rotation axis, creating a resistance \(\left ( F_{r} \right )\) that must be balanced with a driving force to attain constant travelling velocity (Fig. 2). The axle spindle experiences torque only if a tyre is placed on a torque-driven axle because the resulting force created by the rolling resistance and the axle load goes through the axle centre for a free-rolling tyre. In reality, there are always some bearing losses that must be transferred through the tyre. Depending on the definition of rolling resistance, this might or might not be included in the parasitic losses during rolling resistance measurements (Hyttinen et al. 2022; ISO 28580:2009 2009).

Fig. 2
figure 2

Schematic illustration of the tyre contact pressure and reaction forces

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The viscoelasticity converts mechanical energy into heat, increasing the tyre temperature. An increased temperature causes reduced rolling resistance through two different mechanisms. First, a higher temperature inside the tyre cavity increases tyre inflation pressure and stiffens the tyre. This stiffening reduces deformations at different tyre parts, such as sidewalls. Furthermore, lower rubber deformation causes lower strain-induced heat; i.e., lower rolling resistance. The second effect is the decrease of the dissipated energy at higher temperature levels, which reduces the contact pressure shift at the contact patch. Therefore, a transient tyre rolling resistance model should include a viscoelastic constitutive model with temperature dependency. In reality, the deformation state at the contact patch is vastly more complex than described here, but the above explanation provides a basic understanding of the mechanism that creates the majority of rolling resistance.

The model outlined in the following sections is based on the pressure-shift principle and the change in the viscoelastic properties at varying tyre temperatures. The remaining part of the rolling resistance is caused by, e.g., frictional effects at the contact surface (stick–slip) and aerodynamic windage losses (Nakajima 2019). However, in this article, rolling resistance is idealised to originate entirely from viscoelasticity, and the frictional and aerodynamic effects are not considered because the majority of rolling resistance is caused by viscoelasticity (Nakajima 2019).

Rolling resistance can be solved from a moment balance at the wheel centre:

$$ M_{r} = F_{r} r_{l} - F_{c} e=0, $$
(1)

where \(r_{l}\) is the loaded radius of the tyre, and \(e\) is the distance of the shift of the sum of the contact forces \(\left ( F_{c} \right )\) from the axle centre (Fig. 2). A common way to represent the magnitude of the rolling resistance is to normalise it with the axle load \(\left ( F_{z} \right )\). This normalised rolling resistance is called the rolling resistance coefficient \(\left ( c_{rr} \right )\):

$$ c_{rr} \left ( t \right ) = \frac{F_{r}}{F_{z}}. $$
(2)

The coefficient is usually multiplied by a thousand for easier readability. Its unit is kg/ton and is used as a standard unit in the tyre industry to compare different tyres. Combining Eq. (1) and Eq. (2), rolling resistance can be calculated as a ratio between \(e\) and \(r_{l}\):

$$ c_{rr} =1000 \frac{F_{r}}{F_{z}} =1000 \frac{F_{c} e}{F_{z} r_{l}} =1000 \frac{e}{r_{l}}, F_{c} = F_{z}. $$
(3)

3 Simulation workflow

This section describes the thermodynamic simulation model, and the simulation flow of the whole tyre model is illustrated in Fig. 3. The model assumes that the tyre consists of only one rubber compound with a uniform temperature distribution, and the rolling resistance can be calculated entirely from the contact pressure at varying speeds and temperature levels. A new tyre temperature and rolling resistance are calculated for every time step.

Fig. 3
figure 3

Simulation workflow

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The tyre is represented with a 2D circle (Fig. 4) where the tyre-contact patch is divided into multiple \(\left ( i \right )\) vertical segments acting as the tyre-contact patch. The segments can only compress and expand in the vertical direction. The deformations are described through a kinematic relation, which will be explained later. Strain levels for the viscoelastic model are calculated from these segments.

Fig. 4
figure 4

Parameters defining the 2D tyre geometry

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The simulations start by applying a tyre load \(\left ( F_{z} \right )\) to the rotation centre of the tyre. With the viscoelastic model, the stresses, i.e., contact pressure, are calculated from the deformation of the tyre. The deformation of the tyre gives a contact length \(\left ( L_{c} \right )\). The sum of contact forces \(\left ( F_{c} \right )\) is calculated from the stresses of the different segments \(\left ( \sigma _{i} \right )\), the segment spacing \(\left ( \Delta x= \frac{L_{c}}{i_{max}} \right )\) and the tyre width \(\left ( w_{t} \right )\):

$$ F_{c} = \sum _{1}^{i} \Delta x \sigma _{i} w_{t}. $$
(4)

The applied load must be equal to the sum of the contact forces. Thereby, the deformation \(\left ( z \right )\) of the tyre is iterated with a search algorithm until the residual \(\left ( R \right )\) between the applied load and the sum of the contact forces is within the error tolerance \(\left ( e_{tol} \right )\):

$$ R= \left \vert F_{z} - F_{c} \right \vert < e_{tol}. $$
(5)

It is worth nothing that contact forces changing their sign are not considered in the calculations since a sign switch of contact forces would mean that rubber is pulled down to the road. Therefore, only forces with the same sign are used in the calculations.

The strain of the \(i\)th tyre segments in the vertical direction is related to the tyre-profile thickness \(\left ( L_{0} \right )\) and the vertical deformation of the segment \(\left ( \delta _{i} \right )\):

$$ \varepsilon _{i} =\ln \left ( \frac{\delta _{i}}{L_{0}} \right ). $$
(6)

The following equation gives the segment deformation for all the segments:

$$ \delta _{i} =z+ \sqrt{r_{u}^{2} - L_{i}^{2}} - r_{u}, $$
(7)

where \(r_{u}\) is the unloaded tyre radius, and \(L_{i} \)is the longitudinal length from the axle centre to the segment. The following equation gives the distance from the axle centre:

$$ L_{i} =- \frac{L_{c}}{2} +i \Delta x, $$
(8)

where \(L_{c}\) is the total length of the contact patch. This length is calculated from the unloaded tyre radius and the loaded tyre radius \(\left ( r_{l} = r_{u} -z \right )\) with the circle chord equation:

$$ L_{c} =2 \sqrt{r_{u}^{2} - r_{l}^{2}}. $$
(9)

The angle for the different segments can be calculated with the following equation:

$$ \varphi _{i} = \tan ^{-1} \left ( \frac{L_{i}}{r_{u}} \right ). $$
(10)

The instantaneous time signal \(\left ( t_{v\_i} \right )\) for different segments in the viscoelastic model is given in terms of angular speed \(\left ( \omega \right )\) of the segment angle:

$$ t_{v\_i} = \frac{\varphi _{i}}{\omega}, \omega = \frac{v}{r_{l}}, $$
(11)

where \(v\) denotes the vehicle speed. Note that this segment time signal differs from the absolute time signal\(\left ( t \right )\) for the whole-tyre model. The segment time is a sub-time for the absolute time describing how quickly different segments deform at different speed levels. For every simulation step, the segment time starts from zero.

The contact forces for different segments \(\left ( q_{xy\_i} \right )\) are calculated from the segment stresses \(\left ( \sigma _{i} \right )\), tyre width \(\left ( w_{t} \right )\) and segment spacing \(\left ( \Delta x \right )\):

$$ q_{xy\_i} = \sigma _{i} w_{t} \Delta x. $$
(12)

The rolling resistance force can be solved from the separate segment contact forces with the following equation:

$$ F_{r} = \frac{\sum _{1}^{i} q_{xy\_i} L_{i}}{r_{l}}. $$
(13)

The viscoelastic part of the tyre is temperature dependent. Thereby, the tyre temperature must be solved for every simulation step. The derivation of the tyre-temperature evolution starts with the energy-balance equation, where the energy flow stored in the tyre \(\left ( \dot{Q}_{st} \right )\) is equal to the energy flow out \(\left ( \dot{Q}_{out} \right )\) of the tyre subtracted from the energy flowing into the tyre \(\left ( \dot{Q}_{in} \right )\):

$$ \dot{Q}_{st} = \dot{Q}_{in} - \dot{Q}_{out}. $$
(14)

The rate of energy stored in the material \(\left ( \dot{Q}_{st} \right )\) is defined as the product of the specific heat capacity \(\left ( c_{t} \right )\), the tyre’s temperature gradient \(\left ( \frac{d T_{t}}{dt} \right )\) and mass \(\left ( m_{t} \right )\):

$$ \dot{Q}_{st} = m_{t} c_{t} \frac{d T_{t}}{dt}. $$
(15)

\(c_{t}\) is modelled with a linear function and changes with tyre temperature \(\left ( T_{t} \right )\). The higher the active filler concentration, the more the heat capacity increases (Mandal et al. 2014). The following equation depicts the change in the rubber’s heat capacity:

$$ c_{t} = T_{t} c_{t\_t} + c_{t\_0}, $$
(16)

where \(c_{t\_t}\) is a linear constant describing the increase in heat capacity with increased temperatures, and \(c_{t\_0}\) is a constant describing the heat capacity at a temperature of zero Kelvin.

The heat flowing into the tyre, \(\dot{Q}_{in}\), is assumed to come from the heat dissipation of the viscoelastic model at a certain speed \(\left ( \dot{Q}_{in} \left ( v \right ) = F_{r} \left ( T_{t} \right ) v \right ) \). This viscoelasticity is temperature dependent and explained in a later section. At low slip levels, most heat should come from rubber viscoelasticity. In the future, the model could be expanded to consider frictional effects with a bristle model, similarly as Davari et al. (Davari et al. 2017) have done previously. The frictional effects become increasingly important with high slip levels.

The outflowing energy is taken into account with convective heat transfer to the surroundings \(\left ( \dot{Q}_{out\_t-a} \right ) \), convective heat flow between the tyre and road \(\left ( \dot{Q}_{out\_t-r} \right )\) and radiative heat transfer between the tyre and surroundings \(\left ( \dot{Q}_{out\_r} \right ) \). The convective heat transfer is proportional to the temperature difference between the tyre and ambient temperature:

$$ \dot{Q}_{out\_t-a} \left ( v \right ) = h_{a} \left ( v \right ) A_{t} \left ( T_{t} - T_{a} \right ), $$
(17)

where \(h_{a}\) is the heat-transfer coefficient and \(A_{t}\) is the outer area of the tyre. It is the thermal power per unit area \(\left ( q \right )\) divided by the temperature difference between the surroundings and the surface temperature \(\left ( h= \frac{q}{\Delta T} \right )\). In this model, the heat-transfer coefficient consists of natural convection \(\left ( h_{free} \right )\) and a forced convection coefficient \(\left ( h_{forced} \right )\) that is speed dependent. The total convective heat-transfer coefficient is described with a linear equation with speed dependency \(\left ( h \left ( v \right ) =v h_{forced} + h_{free} \right )\) (Namjoo and Golbakhshi 2014). Several other approximations do exist for the heat-transfer coefficient. For more heat-transfer coefficient approximations, see the article by Qin and Hiller (Qin and Hiller 2013). If the outer area of the tyre is not known beforehand, it can be roughly approximated with \(A_{t} =2 A_{s} + A_{tr} - A_{cp} =2\pi \left ( r_{u}^{2} - r_{l}^{2} \right ) +2\pi r_{u} w_{t} - L_{c} w_{t}\), where \(A_{s}\) and \(A_{tr}\) are the areas of the sidewall and the tread. \(A_{cp}\) is the contact-patch area that is calculated as a product of the contact-patch length \(\left ( L_{c} \right )\) and width of the tyre \(\left ( w_{t} \right )\). Alternatively, the tyre-contact-patch area could be calculated from the vertical tyre load \(\left ( F_{z} \right )\) and inflation pressure \(\left ( p_{t} \right )\) of the tyre \(\left ( A_{cp} = \frac{F_{z}}{p_{t}} \right )\).

The heat flow through radiation is proportional to the difference between the fourth power of the tyre temperature and the ambient temperature:

$$ \dot{Q}_{out\_r} \left ( v \right ) = S e_{e} A_{t} \left ( T_{t}^{4} - T_{a}^{4} \right ), $$
(18)

where \(S\) is the Stefan–Boltzmann constant \(\left ( S= 5.670374419 \frac{1}{10^{8}} \frac{\mathrm{J}}{\mathrm{s}} \frac{1}{\mathrm{m}^{2} \ \mathrm{K}^{4}} \ \right )\), \(A_{t}\) is the outer area of the tyre, and \(e_{e}\) is the emissivity. The emissivity describes how efficiently a body emits thermal energy through radiation. Black tyre rubber emits thermal energy efficiently, which is why it has a high emissivity value \(\left ( \approx 0.95-0.97 \right ) \). A perfect emitter would have an emissivity value of 1.

The heat flow between the tyre and the road \(\left ( \dot{Q}_{out\_cp} \right )\) through the contact-patch area \(\left ( A_{cp} = L_{c} w_{t} \right )\) is also considered with convective heat transfer:

$$ \dot{Q}_{out\_t-r} \left ( v \right ) = h_{r} \left ( v \right ) A_{cp} \left ( T_{t} - T_{r} \right ), $$
(19)

where \(T_{r}\) is the road temperature. Introducing \(\dot{Q}_{st},\ \dot{Q}_{in}\) and \(\dot{Q}_{out}\) into Eq. (14) gives the final temperature evolution equation:

$$ \frac{d T_{t}}{dt} = \frac{F_{r} v- h_{a} A_{t} \left ( T_{t} - T_{a} \right ) - h_{r} A_{cp} \left ( T_{t} - T_{r} \right ) - S e_{e} A_{t} \left ( T_{t}^{4} - T_{a}^{4} \right )}{m_{t} c_{t}}. $$
(20)

The tyre-temperature gradient is time integrated using Euler integration over time steps \(\left ( \Delta t \right )\):

$$ T_{t \_ t +\Delta t} = T_{t \_ t} + \frac{d T_{t}}{dt} \Delta t. $$
(21)

Some general remarks on how the tyre structure or tyre parameters influence the rolling resistance coefficient can be made using the temperature-gradient equation. The equation shows that tyre temperature highly depends on the ambient temperature, heat-transfer coefficients, and the current tyre temperature. Higher mass or heat capacity does not affect the stabilised tyre temperature but makes the temperature evolution, heating or cooling, slower. A slower heat-up means that the transient part of the rolling resistance lasts longer. A tyre with higher rolling resistance and otherwise identical to a tyre with lower rolling resistance will reach higher tyre temperatures. This higher temperature is because \(\dot{Q}_{out}\), the tyre geometry and mass are similar for both tyres, but \(\dot{Q}_{in}\) is larger for the tyre with higher rolling resistance.

The temperature-gradient equation also gives insights into how rolling resistance could be reduced. For example, by increasing the surrounding temperature of the tyre or reducing the convective heat-transfer coefficient, the tyre temperature would increase, while rolling resistance would be lower for the same tyre. The rolling resistance would decrease because of lower viscoelasticity at higher tyre temperatures, as shown earlier by Hyttinen et al. (Hyttinen et al. 2022). Tyres could be coloured so that the emissivity would be lower, resulting in lower radiative heat transfer and lower rolling resistance. The temperature rate equation also suggests, for example, that a tyre with a smaller outer area should yield a lower rolling resistance presuming that viscoelasticity becomes lower at higher tyre temperatures. Additionally, lower speed levels might be beneficial from a rolling resistance perspective to reduce cooling effects. Furthermore, rubber hysteresis for filler-reinforced rubber is generally lower at lower strain rates.

4 Constitutive model

This section briefly introduces rubber models and describes the theory of the temperature-dependent nonlinear viscoelastic model that is used in the tyre model. In addition, the model parameters are given a physical interpretation.

4.1 Hyperelasticity

Hyperelastic models are based on the assumption that materials store energy upon deformation and that there exists a hyperelastic strain energy-density function \(\left ( W \right )\) describing the stresses. The Helmholtz free-energy function describes the hyperelastic energy density with the following function:

$$ W = e_{0} - T \eta _{0}, $$
(22)

where \(e_{0}\) is the internal energy of the material, \(T\) is the absolute temperature, and \(\eta _{0}\) is the material entropy. The first part describes the energetic elasticity, whereas the latter describes the entropic elasticity. Polymer materials generally have entropic elasticity, which many micromechanical models describe using statistical mechanics. The basic idea is that a polymer macromolecule consists of multiple randomly organised jointed chains (monomers). Upon deformation, the polymer chains are straightened, causing a decrease in possible microstates of a material; i.e., decrease in entropy (Holzapfel 2000). Only one microstate can be occupied when the polymer chain is fully straightened. Removing the load causes the polymer to recoil to its original maximum entropy state. One way to represent entropy is through the following equation:

$$ \eta _{0} =N k_{B} \ln \Omega \left ( r \right ) +c, $$
(23)

where \(N\) is the number of separate freely jointed chains (monomers) in the reference polymer molecule chain and \(k_{B}\) is Boltzmann’s constant. \(\Omega \left ( r \right )\) is a probability distribution of the distance between polymer ends (Bergström 2015). One approximation of the probability function by Kuhn and Grün (Bergström 2015) is based on the Langevin function \(\left ( \mathcal{L} \right ) \).

If the polymer is only considered to consist of rigid, freely jointed chains that cannot stretch, the free-energy function is entirely described by the entropic part of the free-energy function. As there is a temperature in the entropy equation, it might seem intuitive that tyre rubber would also increase its stiffness proportionally with temperature, as the function suggests. However, this does not usually occur for tyre rubber because it is compounded with reinforcing fillers. On the contrary, the stiffness often decreases with increasing temperature when enough filler is added to the rubber matrix. This is because adding fillers to the rubber increases the effects of filler–filler and filler–polymer contribution to the stiffness, causing a decrease in stiffness with increasing temperature (Reoroji and Vol 2013).

The hyperelastic functions are often based on one to two deviatoric strain invariants \(\left ( \overline{I}_{1} - \overline{I}_{2} \right ) \). Stable hyperelastic models should satisfy the Drucker stability criterion \(\left ( d \boldsymbol{\sigma}: d \boldsymbol{\varepsilon} \geq 0 \right )\) (Bergström 2015). This criterion means that the internal energy should only increase with increasing stress increments. Higher-order polynomial models like Yeoh do not necessarily fulfil this criterion with all the possible parameter combinations (Österlöf et al. 2015). Choosing a robust simulation model that creates a convex stress–strain space with all parameter combinations is beneficial for parameter studies. One example of a model that fulfils this requirement is the Arruda–Boyce micromechanical hyperelastic model, also called the eight-chain model. The Arruda–Boyce model underestimates biaxial stresses because it uses only the first strain-invariant, which is more sensitive to uniaxial than biaxial strains (Bergström 2015; Melly et al. 2021). However, this is not an important issue to consider for a tyre 2D tyre model where the tyre segments experience only uniaxial strains.

4.2 Nonlinear viscoelasticity

Hyperelastic models alone are not suitable for rolling resistance simulations, although they have been used in some finite-element simulations to calculate rolling resistance (Daesa and Rodkwan 2018). Hyperelastic models do not have any material dissipation, but they work well as components of more complicated viscoelastic models, where viscous-flow elements accompany hyperelastic models. The most common way to model rubber viscoelasticity is by using linear viscoelastic theory. One of the most widely used linear viscoelastic models is the Prony series (Mashadi et al. 2019). Linear viscoelastic models are effective simulation models when the required simulation strain range is narrow. However, they have some drawbacks at large strains when simulating filler-reinforced rubber, such as strain-magnitude-independent stiffness and energy dissipation; i.e., they do not capture the Fletcher–Gent effect without modifications. There are some examples where the Prony series is modified to consider this effect by mapping different parameters to different strain levels (Austrell and Olsson 2012). However, this method makes the material model calibration and parameter studies cumbersome. Therefore, some other technique should be used. The Prony series cannot decouple the connection between the storage and loss modulus (Bergström 2015). This is why predictions of the storage and loss moduli might not always fit the experimental data simultaneously, even when multiple networks are stacked in parallel.

The nonlinear viscoelastic Bergström–Boyce model consists of the Arruda–Boyce hyperelastic model, and the Bergström–Boyce flow element is a micromechanically inspired model (Bergström and Boyce 2001). It is an extended version of the Prony series, which partly fixes the inadequacies of the Prony series. It adds strain amplitude and strain-rate dependency to the viscous flow. The model can be further enhanced by stacking multiple parallel Bergström–Boyce models together to reach better stress–strain estimations for a specific rubber (Rafei et al. 2019). However, using multiple highly nonlinear Bergström–Boyce flow elements brings a new problem: the number of material parameters increases, and the parameters are not necessarily unique. A similar stress–strain estimation can be achieved with multiple parameter sets, making the parameters challenging to tune. The tyre model used in this article does not necessarily need to capture a specific material’s stress–strain response perfectly. It is more important that the model remains understandable, that the parameters have a clear meaning, and that they capture the main properties of rolling resistance at varying temperatures. This kind of rolling resistance model should be able to be done with only one Bergström–Boyce flow element combined with the Arruda–Boyce hyperelastic model. Additionally, tyres are made of multiple rubber materials, further promoting the idea of using only one Bergström–Boyce flow element and not focusing on simulating one rubber material precisely. Therefore, the Bergström–Boyce model should be able to capture the crucial mechanical properties for rolling resistance simulation; i.e., speed- and amplitude-dependent energy dissipation and stiffness. This model assumes that the elastic stresses originate from straightening cross-linked polymer chains that create a polymer matrix. The viscoelastic relaxation is motivated by the relaxation of free polymer chains in a rubber matrix (Bergström and Boyce 1998). During deformation, the free polymer chains deform with the polymer matrix and then relax by gliding through the polymer matrix (Fig. 5).

Fig. 5
figure 5

Illustration of a cross-linked rubber matrix that gives the equilibrium response of the viscoelastic model and one free polymer chain, causing viscoelastic overstress. (a) The polymer network is undeformed, and the free chain is tangled into the polymer network. (b) The network is deformed quickly, and the free chain initially stretches, causing overstress. (c) Over a prolonged period, the free chain glides through the network, and the material relaxes

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Figure 6 presents the whole nonlinear viscoelastic model. Network A describes the fully recoverable equilibrium response; i.e., the energy stored in the material or the cross-linked polymer matrix. The equilibrium response is described with a micromechanical Arruda–Boyce model. Network B depicts the viscous strain-dependent relaxation behaviour; i.e., nonrecoverable response. The viscoelastic spring consists of a similar Arruda–Boyce model with a different shear modulus and a nonlinear viscous damper element. Over a prolonged period, network B relaxes until the stresses are fully described by network A. The structure is similar to the Prony series, with one hyperelastic model and one viscoelastic network.

Fig. 6
figure 6

1D representation of the Bergström–Boyce nonlinear viscoelastic model, where the viscous flow depends on the strain magnitude, strain rate and rubber temperature

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The deformation gradient \(\left ( \boldsymbol{F} \right )\) is acting on the whole model. For both networks, the deformation gradient is equal \(\left ( \boldsymbol{F} = \boldsymbol{F}_{A} = \boldsymbol{F}_{B} \right )\), and they experience the same deformation. As the networks are in parallel, they have a separate contribution to the total stress \(\left ( \boldsymbol{\sigma}_{tot} \right )\):

$$ \boldsymbol{\sigma}_{tot} = \boldsymbol{\sigma}_{A} + \boldsymbol{\sigma}_{B}, $$
(24)

where \(\boldsymbol{\sigma}_{A}\) and \(\boldsymbol{\sigma}_{B}\) are the stress tensors from networks A and B. For network B, the deformation gradient is multiplicatively decomposed into elastic \(\left ( \boldsymbol{F}_{B}^{e} \right )\) and viscous \(\left ( \boldsymbol{F}_{B}^{v} \right ) \)deformation in the following way, as shown in Fig. 7:

$$ \boldsymbol{F}_{B} = \boldsymbol{F}_{B}^{e} \boldsymbol{F}_{B}^{v}. $$
(25)
Fig. 7
figure 7

Kinematics of the viscoelastic model with a multiplicative split of the deformation gradient

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The stress response for network A and the elastic part of network B are modelled with the Arruda–Boyce hyperelastic model, also called the eight-chain model. The Arruda–Boyce model is based on statistical mechanics. The structure of the model consists of eight polymer-chain macromolecules located on a unit cell’s diagonal in a principal stretch space. These chain molecules on a diagonal of a unit cell are illustrated in Fig. 8. Upon deformation, the macromolecules are stretched and rotated in the direction of loading in uniaxial tension. In compression, the rotation of the macromolecules occurs perpendicular to the loading direction. As a result, the molecules stretch only and do not compress. The strain energy-density function \(\left ( W \right )\) for the Arruda–Boyce model is given by the following equation (Bergström and Boyce 2001):

$$ W=N k_{B} T \sqrt{n} \left [ \beta \lambda _{chain} - \sqrt{n} ln \left ( \frac{\beta}{\sinh \beta} \right ) \right ], $$
(26)

where \(N\) is the number of the network chains in the cross-linked polymer reference unit, \(k_{B}\) is the Boltzmann constant, \(T\) is the temperature, \(\beta = \mathcal{L}^{-1} \left ( \frac{\lambda _{chain}}{\sqrt{n}} \right )\) and \(n\) is the number of freely jointed rigid links. The Langevin function is given by \(\mathcal{L}= \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}} - \frac{1}{x}\). The equation expresses the Brownian motion of a particle. The closed-form equation for the inverse Langevin function \(\left ( \mathcal{L}^{-1} \right )\) does not exist. However, this inverse function has different approximations, such as by the Bergström approximation (Bergström 2015). Various finite-element software utilise different inverse Langevin approximations with surprisingly large differences. These differences make it problematic to transfer material parameters between different software directly. In this article, the approximation by Bergström is used:

$$ \mathcal{L}^{-1} \approx \left \{ \textstyle\begin{array}{l} 1.31446 \tan \left ( 1.58986x \right ) +0.91209x,\ \mbox{if} \left \vert x \right \vert < 0.84136 \\ \frac{1}{sign \left ( x \right ) -x},\ \mbox{if} 0.84136\leq \left \vert x \right \vert \leq 1. \end{array}\displaystyle \right . $$
(27)
Fig. 8
figure 8

Arruda–Boyce eight-chain hyperelastic model. (\(\mathbf{a}\)) undeformed unit cube, (\(\mathbf{b}\)) cube in tension, and (\(\mathbf{c}\)) the unit cube compressed model. Upon deformation, the eight chains rotate and stretch

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The effective chain stretch \(\left ( \lambda _{chain} \right )\) is given by the following equation for uniaxial incompressible deformation:

$$ \lambda _{chain} = \sqrt{\frac{\lambda ^{2} + \frac{2}{\lambda}}{3}}. $$
(28)

The following equation gives uniaxial stresses for the equilibrium network:

$$ \boldsymbol{\sigma}_{A} = \frac{\mu _{A}}{\lambda _{A_{chain}}} \frac{\mathcal{L}^{-1} \left ( \frac{\lambda _{A_{chain}}}{\lambda _{L}} \right )}{\mathcal{L}^{-1} \left ( \frac{1}{\lambda _{L}} \right )} \left [ \lambda _{A}^{2} - \frac{1}{\lambda _{A}} \right ], $$
(29)

where \(\mu _{A}\) is the initial shear modulus and \(\lambda _{A_{chain}}\) is the effective chain stretch, \(\lambda _{A}\) is the stretch ratio, and \(\lambda _{L}\) is the limiting factor for the chain stretch for network A.

The elastic part of network B is also modelled with Arruda–Boyce. The shear modulus is given as a fraction \(\left ( s \right )\) in relation to network A with parameter \(s\ \left ( \mu _{B} = \mu _{A} s \right )\). The parameter \(\lambda _{L}\) is the same for both networks. The stress over the viscous and the elastic elements at network B must be the same since they are in series. Therefore, the creep deformation over the viscous element \(\left ( \dot{\varepsilon}_{B}^{v} \right )\) must be solved numerically before the stress over the spring element can be calculated. The stress for network B is calculated after the viscous creep is solved, according to the following equation:

$$ \boldsymbol{\sigma}_{B} = \frac{\mu _{B}}{\lambda _{B_{chain}}^{e}} \frac{\mathcal{L}^{-1} \left ( \frac{\lambda _{B_{chain}}^{e}}{\lambda _{L}} \right )}{\mathcal{L}^{-1} \left ( \frac{1}{\lambda _{L}} \right )} \left [ \lambda _{B}^{e 2} - \frac{1}{\lambda _{B}^{e}} \right ], $$
(30)

where \(\lambda _{B_{chain}}^{e}\) is the effective chain stretch and \(\lambda _{B}^{e}\) is the stretch ratio for the elastic part of network B.

The equation describing the viscous creep rate for network B is the following:

$$ \dot{\varepsilon}_{B}^{v} = \dot{\varepsilon}_{0} \left ( \lambda _{B_{chain}}^{v} -1+\xi \right )^{C} \left [ R \left ( \frac{\tau}{\tau _{base}} - \tau _{cut} \right ) \right ]^{m}, $$
(31)

where \(\tau \) is the effective shear stress driving the viscous flow, \(\lambda _{B_{chain}}^{v}\) is the effective chain stretch for the flow element, \(R \left ( x \right ) = \frac{x+ \left \vert x \right \vert}{2}\) is the linear ramp function (Fig. 9) with a unity gradient when the argument is larger than its absolute value and \(\tau _{cut}\) is the cut-off stress for the viscous flow. This gives a limit when viscous flow does not occur if the driving stress \(\left ( \tau \right )\) normalised over the flow resistance \(\left ( \tau _{base} \right )\) is lower than the \(\tau _{cut}\) value. Below the cut-off stress, the model does not experience any viscous flow as the equation receives a zero multiplier. \(\dot{\varepsilon}_{0}\) is a dimensional-consistency constant \(\left ( \frac{1}{s} \right )\). Analysing the viscous flow equation, it can be seen that the first part, \(\left ( \overline{\lambda}_{B}^{v} -1+\xi \right )^{C}\), is dependent on the viscoelastic effective chain stretch \(\left ( \overline{\lambda}_{B}^{v} \right )\), which gives an amplitude dependency for the viscous flow. The parameter \(\xi \) is used merely to improve numerical stability with zero strain levels to prevent the singularity of the model since parameter \(C\) is zero or below zero. Therefore, the magnitude of parameter \(\xi \) is usually low \(\left ( \xi \approx 0.01 \right )\) (Bergström and Boyce 2001). The second part of the equation describes the strain-rate dependency with the parameter \(m\). The higher the values for the \(\dot{\varepsilon}_{B}^{v}\) function become, the faster the viscoelastic relaxation occurs.

Fig. 9
figure 9

Ramp function

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The viscous flow reduces into the Prony series flow model with one viscoelastic network when parameters \(\tau _{cut} =0,\ m=1\) and \(C=0\):

$$ \dot{\varepsilon}_{B}^{v} = \dot{\varepsilon}_{0} \frac{\tau}{\tau _{base}}. $$
(32)

As can be seen from the previous equation, there is no dependency on the strain amplitude for a linear viscoelastic material model. The linear viscoelastic model is analogous to a viscous damper with a constant geometry valve of the orifice that limits fluid flow. On the other hand, a nonlinear viscoelastic flow element can be thought of as a variable valve in a fluid damper that changes its geometry with different shaft positions or speeds.

The final part of the flow-rate equation \(\left ( \vartheta T_{t}^{\alpha} \right )\) gives the temperature dependency of the viscous flow:

$$ \dot{\varepsilon}_{b}^{v} = \dot{\varepsilon}_{0} \left ( \overline{\lambda}_{B}^{v} -1+\xi \right )^{C} \left ( R \left ( \frac{\tau}{\tau _{base}} - \tau _{cut} \right ) \right )^{m} \vartheta T_{t}^{\alpha}, $$
(33)

where \(T_{t}\) is the tyre temperature, \(\vartheta \) is the magnitude scaling constant, and \(\alpha \) is the temperature-sensitivity scaling factor. The viscous creep function must be scaled upwards at increasing temperatures so that there is faster relaxation at higher temperatures.

A similar kind of temperature scaling could also be added to the shear modulus of network A. Depending on whether the filler–filler and filler–polymer contributions to the elasticity or the entropic elasticity dominate, the stiffness increases or decreases with increasing temperature. The temperature in Eq. (26) is not used, meaning that the shear modulus is considered to be constant at all temperatures.

4.3 Interpretation of model parameters

The model has many parameters. This section aims to interpret the parameters and give them a physical meaning. These parameters can be summarised as follows:

  • \(\mu _{A}\) gives the vertical stiffness of the tyre, as illustrated in Fig. 10. It is used to tune the vertical stiffness of the tyre at low deformation levels. A higher value gives a higher base stiffness of the tyre.

    Fig. 10
    figure 10

    Arruda–Boyce hyperelastic model when parameter \(\mu _{A}\) is varied

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  • \(\lambda _{L}\) defines the strain stiffening at high deformation levels (the maximum stretch of a polymer chain). Thereby, it is used to tune the stiffening of the tyre at higher axle loads. A lower value of \(\lambda _{L}\) makes the tyre stiffer at high strain levels (Fig. 11).

    Fig. 11
    figure 11

    Arruda–Boyce hyperelastic model when the parameter \(\lambda _{L}\) is varied

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  • \(\tau _{base}\) gives the viscous-flow resistance of the model. Higher flow resistance reduces the model’s viscous flow, causing a slower relaxation of the viscous model.

  • The exponent \(m\) adjusts the strain-rate sensitivity. Higher values make the model less strain-rate dependent.

  • The parameter \(s\) defines the relative stiffness of network B elasticity. The ratio between \(s\) and \(\tau _{base}\) describes the base relaxation behaviour.

  • \(C\) makes the viscous flow strain dependent. This can be useful for parameter studies with varying axle loads or analysing the effect of lift axles on rolling resistance. The parameter \(C\) is either zero – i.e., no strain-amplitude dependency – or \(C<0\). The lower the values are for this parameter, the quicker the viscous flow is at low strain levels. For a polymer material, the value for \(C\) is often close to -1 (Bergström and Boyce 2000).

  • \(\xi \) and \(\tau _{cut}\) are the least meaningful parameters for the rolling resistance model. \(\xi \) can be considered to be a low number (Bergström and Boyce 2001). \(\tau _{cut}\) gives a permanent set to the model, which is not required for rolling resistance simulations. Thereby, \(\tau _{cut}\) can be set to zero.

  • \(\alpha \) gives the sensitivity of the temperature change, and the parameter \(\vartheta \) gives the magnitude of the temperature scaling.

5 Simulation results

In this section, parameter settings and simulation results are presented. The model has not been parametrised using measurement data. Thereby, some assumptions have been made regarding the simulation parameters. The road and wind temperatures and convective cooling parameters are identical in the model, but they might differ significantly in real life. Also, heat-transfer coefficients should depend on road wetness (Browne and Wickliffe 1980). These additions are left for future work. One way to measure heat-transfer coefficients could be to use thin-film heat-flux sensors (Assaad et al. 2008). The convective heat-transfer coefficient parameters are extracted from the measurements by Browne and Wickliffe (Browne and Wickliffe 1980). The temperature variation for the specific heat capacity of 10% nanosilica-filled rubber is extracted from the publication by Arunava et al. (Mandal et al. 2014).

The chosen parameters represent a 295/80 R22.5 tyre. The model is calibrated so that the tyre deforms vertically by approximately 35 mm with an applied vertical load of 30 kN. The parameters are chosen so that the rolling resistance – i.e., damping – is set to be increasing with vehicle speeds between 0–100 km/h.

The required time step depends on the magnitude of the transient rolling resistance gradient or how quickly and often the simulated speed changes. The steeper the transient rolling resistance, the smaller the time step must be. Here, the simulations are conducted with \(\Delta t =200 s \). The simulation parameters are presented in Tables 1 and 2.

Table 1 Simulation parameters for the tyre geometry and cooling
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Table 2 Viscoelastic parameters
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During the simulations, rolling resistance is simulated at four consecutive prolonged speed steps (80 – 40 – 10 – 60 km/h) similar to previous measurements performed by Hyttinen et al. (Hyttinen et al. 2023). Figures 12 and 13 show the transient rolling resistance and the inverse relationship between tyre temperature and rolling resistance. Initially, the tyre warms at a speed level of 80 km/h until a stable rolling resistance has been reached. When the speed decreases (80 – 40 km/h and 40 – 10 km/h), the rolling resistance reduces, and the tyre starts to cool. When the tyre cools, the rolling resistance increases. When the speed increases at the end of the simulation (10 – 60 km/h), rolling resistance increases considerably, and the tyre warms until a stable temperature level has been reached. Rolling resistance has different stabilised temperature and rolling resistance values due to different speed-dependent cooling and rubber hysteresis levels.

Fig. 12
figure 12

Rolling resistance simulation at four different speed levels (80 – 40 – 10 – 60 km/h) at +25 °C ambient temperature

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Fig. 13
figure 13

Tyre temperature simulation at four different speed levels (80 – 40 – 10 – 60 km/h) at +25 °C ambient temperature

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The transient part of the rolling resistance becomes larger at lower temperatures (Fig. 14), as pointed out by Hyttinen et al. (Hyttinen et al. 2022). Figure 15 shows the tyre temperature with different ambient temperatures using the same speed steps. Hyttinen et al. (Hyttinen et al. 2022) show that the difference between the tyre and ambient temperatures increases with decreasing ambient temperatures, which the model captures effectively.

Fig. 14
figure 14

Rolling resistance during four prolonged speed steps at four different ambient temperatures

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Fig. 15
figure 15

Tyre temperature during four prolonged speed steps at four different ambient temperatures

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Figures 16–18 show how rolling resistance is affected when, for example, tyre area, mass and forced convective cooling vary while other parameters are kept constant. The results suggest that there is a high potential to reduce rolling resistance by affecting the forced convection, i.e., reducing the airflow around the tyre.

Fig. 16
figure 16

Tyre rolling resistance with two different outer tyre areas during four speed steps

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Fig. 17
figure 17

Rolling resistance compared with two different tyre masses during four speed steps

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Fig. 18
figure 18

Rolling resistance with two different forced convection parameters

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6 Conclusions

This article presents a semi-physical thermodynamic tyre model that captures transient rolling resistance and tyre temperature. The 2D tyre is modelled using a Bergström–Boyce nonlinear viscoelastic model, and the rolling resistance is solved from the simulated contact stresses. The viscoelastic model has a temperature-dependent viscous flow. Furthermore, a new temperature is solved from the heat-flow equation for every time step. From the insights gained during this work, the following can be concluded:

  • The proposed semi-physical simulation model captures the shape of the transient rolling resistance at varying speeds and ambient temperatures.

  • Rolling resistance and tyre temperature can be simulated with the same model, and both are dependent on each other.

  • The model gives insights into how multiple tyre and cooling parameters affect tyre temperature and rolling resistance.

  • The cooling parameters of the tyre have a considerable effect on the tyre temperature and, thereby, the simulated rolling resistance.

Future work could include parametrising the model with experimental data. It could also be of interest to expand the model to take into account the effect of tyre slip, tyre pressure, and water pumping to further study their effect on rolling resistance. The model should also be suitable to be used in combination with a bristle tyre model having a variable contact pressure. In addition to the longitudinal tyre segments, the model could have multiple lateral segments so that the model can be used to simulate the effect of camber on rolling resistance.