Rolling Tires on the Flat Road: Thermo-Investigation with Changing Conditions through Numerical Simulation

Abstract

A crucial material comprising a pneumatic tire is rubber. In general, the tire, or more specifically, the hysteresis effects brought on by the deformation of the part made of rubber during the procedure, heat up the part. In addition, the tire temperature depends on several factors, including the inflation pressure, automobile loading, car speed, road tire, the environmental conditions, and the tire geometry. This work focuses on using simulations to calculate the temperature and generated heat flow distributions of a rolling tire with constant velocity using the finite element method. For the sake of simplicity, it is assumed that the only components of the tire are rubber, body-ply, bead wire, and the rim. While the other components are believed to be made of a linear elastic material, the nonlinear mechanical behavior of the rubber is characterized by a Mooney–Rivlin model. Investigations are conducted into the combined effects of vehicle loads and inflation pressure. Hysteresis energy loss is used as a bridge to link the strain energy density to the heat source in rolling tires, and their temperature and heat flow distributions may be determined by steady-state thermal analysis. Thanks to the state-of-the-art computing method, the time required for connected 3D dynamic rolling tire simulations is reduced. The simulation outcomes demonstrate that the maximum temperature in this paper is attained with high weights, high velocities, and low inner inflated pressures. Overall, the maximum temperature is increased with the rise of all three variables. Moreover, the rise of the friction coefficient between the tread and road surface moves the high-temperature area towards the tread/sidewall connection area.

1. Introduction

Tire temperature is one of the main causes of tire fatigue and affects tire integrality and effectiveness [1,2,3]. Tire temperature increases are caused by two factors: friction between the tread and the road, and internal heat production from the continuous deformation of the rubber. The finite element method (FEM) has been utilized in many studies to study tire stress and temperature [4,5,6,7,8,9,10,11], which are used to assess static properties, stiffness, resonance frequencies, and vibration modes due to a very high computational load. Hölscher et al. [5] concluded that finite element models can accurately simulate both the overall deformation and the frictional behavior in the tire/ground contact area. Yanjin et al. [7] used simulations of cord–rubber composites with a nonlinear boundary condition in tire–rim and tire–road contacts to examine the impact of the belt cord angle on the performance of tires under various rolling states. They calculated the ideal belt cord angle for a particular radial tire using rebar components for the cord layers. In the finite element model, it was found that the strain energy density of the belt decreased as the belt chord angle rose.

Much research has been conducted in order to determine how the composition of the rubber material affects the temperature rise within the substance [12,13]. Most heat generation studies assert that rubber composites with negligible heat buildup have limited toughness [12]. It has been claimed that the thermal energy loss in rubber is caused by interfacial phenomena rather than the filler network [13]. Many researchers have predicted the temperature of tires via simulation. A numerical procedure used to handle the temperature distribution of a rolling tire during a steady-state simulation was suggested by Park et al. [14]. Using a mechanical solver, the strain and velocity of the tire were assessed during the procedure. The energy dissipation rate of the viscoelastic rubber was then assessed. Last but not least, a thermal solver was used to determine the temperature distribution of the tire. This strategy, however, has a drawback; indeed, it can only be used for steady-state analysis.

Using ANSYS 14.0 software, the static and dynamic behavior of radial tires on highways for civilian emergency vehicles or armored army vehicles was studied by Giurgiu et al. [15]. Their research provided greater insight into the modeling and simulation of the static and dynamic behavior of radial tires for civil emergency vehicles or military armored vehicles when their findings were compared to the imprint of the tire on the road surface. De Gregoriis et al. [16] provided a completely predicted high-fidelity numerical technique for the simulation of a vehicle tire rolling with a constant angular speed over a bumpy road surface. Moreover, a finite detail technique was created by Nallusamy et al. [11] to examine the efficiency of a regular-state rolling tire on a road by altering the tread patterns. Additionally, Johnson and Chen carried out cyclic loading analysis by employing a thermo-mechanical coupled issue [17] and Kan et al. [18] did the same by using Abaqus. They concluded that stress and thermal studies are valid for significant strains and displacements.

Tire life is crucial to preserving the health and durability of a vehicle. Temperature is the primary determinant of tire life since it impairs the construction of the tire and the rubber [18,19]. Monitoring tire stress, deformation, and temperature distribution at various ambient temperatures is essential to preventing temperature-related problems. Schuring [20] and Schuring et al. [21] studied the effects of ambient temperature on tire rolling loss. They came to the conclusion that the temperature of the ambient air has a direct impact on the heat that is dissipated. The inflation pressure is a crucial property that affects the temperature of the tire. The temperature distribution of a steady-state rolling tire was also predicted by Tang et al. [22]. As the inflation pressure rose, the high-temperature region diminished and shifted to the shoulder position. Sokolov [9], Ebbott et al. [10], and Zang et al. [23] used the finite element method to study the thermal state of a pneumatic tire. They produced the same temperature distribution in the tire as Tang et al. [22]. The fundamentals of tire rolling resistance on a road were studied by Hall and Moreland [24]. They discovered that the highest temperatures are recorded near the tire shoulder, which is also where the majority of heat generation occurs, and where the tire is thickest.

The road is another element that influences tire temperature. The direct measurements made by a flatbed sensor device of three-dimensional (3D) tire pavement contact stresses were emphasized by De Beer et al. [25]. Wang et al. [26] improved the three-dimensional tire-pavement interaction model to forecast the tire-pavement contact stress distributions. Douglas et al. [27] developed a mechanism to evaluate the contact stresses for various tire types, wheel weights, inflation pressures, and contact stresses under full-scale tires.

This work suggests a new technique for numerical modeling in order to forecast the internal tire temperature growth at various speeds, inflation pressures, and loads. There were two main elements to our process. To accurately assess the impact of the tire on the road, structural analysis of a rolling tire in contact with the road was first carried out with transient analysis. The hysteresis energy losses were then used to calculate the heat-generating rate. With the use of a dynamic tire rolling simulation, the total strain energy was obtained. The hysteresis energy losses were determined by using hysteresis data obtained from a dynamic mechanical study and utilized to estimate the temperature distribution via a steady-state thermal analysis.

2. Simulation Methodology

The calculations in this paper employed a standard passenger tire size (185/60R15). Figure 1 depicts the size and two-dimensional cross-section of the tire.

Some fair assumptions and simplifications were made for the simulations in order to simplify the situation:

• It was supposed that the tire was made of steel bead wires, rubber, cord plies, and rim.
• The rubber was considered homogenous, isotropic, and hyperelastic over the whole temperature range.
• It was assumed that the steel bead wire, rim, and cord ply were homogenous, isotropic, and elastic materials.
• It was assumed that the road and the rim were rigid.
• It was assumed that the friction coefficient between the road and the tire was constant at 0.1, while neglecting the fluctuation in the amount of friction between the road and tire. Then, the friction coefficient was changed to investigate the effect of this variable on the distribution of temperature within the rubber.

Table 1. Material properties used in the simulation [22].

Material	Rubber	Body-Ply	Bead Wire
Density (kg/m3)	1400	1200	6500
Poisson’s ratio	-	0.3	0.3
Young’s modulus (Mpa)	-	500	207
Mooney–Rivlin constant (Mpa)	C10 = 8.061	-	-
	C01 = 1.805	-	-
Thermal conductivity (W/m·°C)	0.293	0.293	60.5
Hysteresis	0.1	-	-

provides a summary of the material attributes [22]. These values had been previously obtained by experiments in the literature, and were provided by Kenda Rubber Industrial Corporation. Bead wires were considered to have stainless steel qualities in terms of their mechanical and thermal properties. The inner volume of the tire was applied with the equivalent inflated pressure and heat transfer properties in order to represent the filled air response. Moreover, the reinforcements were also modeled using 3D elements with structural steel properties to achieve a sense of their real behavior in terms of their structural and heating responses. The mesh for the simulation was constructed using structural mesh and using ANSYS ICEM. The overall simulation mesh is shown in Figure 2. The software package employed for the simulation work was provided by ANSYS, Inc, which includes ANSYS Transient Mechanical and ANSYS Steady-state thermal version 19.1 [28].

2.1. Dynamic Rolling Analysis

By carrying out transient simulations, the computing time required to arrive at the steady solution would increase in comparison to the steady-state procedure, since the behavior of a rolling tire is entirely transitory. As a result, a successful approach was developed to mimic rolling conditions. The dynamic rolling analysis was performed over two steps.

Step 1:

Loading Analysis

The tire was simulated under different pressures and loadings with static-state mechanical simulations. The displacements were then obtained under a non-rotating state. Different loadings (1, 2, 3, 4, 5, 6 kN) were applied to the rim, and an inflation pressure of 32 psi was placed at the interior surfaces of the tire. The loading force was applied by a remote force to the inner rim surface with a scope to the center point of the rim. This was performed with the aim of simulating the effect of the shaft. The boundary conditions for the loading analysis are shown in Figure 3a.

Step 2:

Rolling Analysis

The boundary conditions for the upward displacement of the road were established using the displacement data from Step 1. Different speeds were added to the road component to simulate the rolling effect, and this speed was applied constantly to the road. The rim was considered to have a constant displacement in any direction that was not free rolling around the x-axis. Following the investigation, the overall strain energy density of the tire was discovered. This step was performed via transient analysis, so it took longer than Step 1. The boundary conditions applied in the rolling analysis are demonstrated in Figure 3b. The simulation time was 1s for 10 time steps.

Introducing two steps into the rolling analysis aims to speed up convergence. If loading analysis was utilized in the dynamic rolling analysis, the loading and rolling analyses should also be linked and used, in addition to the damping ratio. As a result, obtaining the converged solution would take longer. Although the displacement control approach does not produce “real” dynamic simulation results, the quantity of computer resources required for the simulation is substantially decreased; thus, these results are adequate for the present applications. When comparing the static and dynamic rolling models for the reference case, the achieved results showed that the average energy density varied at around approximately 4%. Hence, the simplified model is able to ensure a low computational cost with acceptable error.

2.2. Steady-State Thermal Analysis

All rubber components were heated using the heat generation rate (H_G) presented in Section 3.2, which served as the primary heat source. The ambient air, all tire parts, and the road were all assumed to have sink temperatures of 25 °C. The air within the tire was regulated to a temperature of 38 °C. Figure 4 demonstrates the different boundary conditions of the different surfaces. These values were taken from the literature [14] and are summarized in

Table 2. Heat transfer conditions utilized for thermal analysis in the thermal study [14].

Location	Heat Transfer Coefficient (W/m2·K)	Sink Temperature (°C)
Tread/road	12,000	25
Tread/air	16.18	25
Sidewall/air	16.18	25
Body-ply/cavity-air	5.9	38
Liner/rim	88,000	25

.

3. Conceptual Modeling of Rubber Materials

3.1. Model for Hyperelastic Material by Mooney–Rivlin Function

The primary component of a tire is rubber. The rubber component has cord plies attached inside to keep the rubber from deforming under pressure. The bead wire is the stiffening component of the tire that limits the inner deformation of the rubber and cord ply. The rubber material is considered to be an isotropic hyper-elastic material. Tires deform repeatedly and rapidly during high-speed driving, and their hyperelastic behavior leads to hysteresis. The heat produced by the hysteresis effect causes tires to heat up. Gehman’s work in Ref. [8] provided a general review of the material properties of tire parts. The Mooney–Rivlin function may be used to explain the mechanical behavior of incompressible rubber. The Mooney–Rivlin material has previously been incorporated into ANSYS Engineering Data.

3.2. Heat Generation Rate

In this study, the heat produced within a rolling tire was determined using the strain energy density and hysteresis. FEA simulation was used to determine the overall strain energy density ($U_{total}$). The lost strain energy density is derived by multiplying the total strain energy density from the deformation module by the hysteresis [29]:

$$U_{loss}=H\cdot U_{total}$$

(1)

This work considers that the lost strain energy density, which is the energy that is not recovered after deformation, fully contributes to internal heat generation. The following equation states that frequency is defined as velocity divided by the circumference of a rolling tire:

$$f=\frac{V}{2\pi R}$$

(2)

To obtain the heat production rate per unit volume or heat flux (W/m³) for each mesh element, the rolling tire frequency is first multiplied by the lost strain energy density:

$$H_G=U_{loss}\cdot f$$

(3)

4. Simulation and Thermo-Investigation of Rolling Tires on the Flat Road Affected by Some Factors

4.1. Mesh Sensitivity Study

The mesh sensitivity of the mesh used in this study is presented in

Table 3. Summary of the mesh sensitivity.

Mesh No.	No. Perimeter Divisions	Rubber	Body-Ply	Rim	Bead Wire	Total Elements	Tire Displacement (mm)
1	92	18,321	3412	2893	1423	27,471	5.82
2	104	23,657	4562	3718	1885	35,705	6.96
3	120	30,624	5800	4872	2320	45,936	7.70
4	140	39,711	7740	6434	3056	59,997	7.78
5	160	51,955	9902	8334	3921	78,032	7.81

. The mesh size was evaluated by the number of divisions over the perimeter of the tire, the number of elements for each part of the tire and the total elements of the tire. The element count of each part was controlled by three divisions over the three dimensions of the part. It was ensured that each part was increased by a ratio of roughly 1.3 in the mesh count. It was discovered that mesh 3 produces a low deviation in terms of tire displacement compared to mesh 4, of 1.04%. This indicates that mesh 3 is converged enough for a balance between computational cost and accuracy. In addition, all the mesh elements were hexahedral, which contributed to the accuracy of the simulations. Overall, the mesh chosen in this study contained 30,624, 5800, 4872, and 2320 elements in the rubber, body-ply, rim, and bead wire, respectively. The overall mesh had roughly 46,000 elements.

4.2. Loading and Rolling Analysis

This subsection describes the process of carrying out the loading analysis; this was performed using varying loading sizes, ranging from 1 kN to 6 kN and rising by the increment of 1 kN, and with the inflated pressure kept at 32 psi. The displacements were then recorded and compared with the data from a report published by Lin and Hwang [29], who had also previously conducted experiments and simulations with a tire of the same loading as that used in this work. They had also previously validated their results via static-state analysis and only using loading conditions. Moreover, their geometrical model is identical to the model in this study, and uses identical materials. The results are shown in Figure 5.

The visualization shows that the results from the present study are in better agreement with the experimental data than those from Lin and Hwang [29]. Indeed, the maximum deviation of all cases from the experimental data is roughly 4%. Hence, the reliability of the numerical method has been rationally authenticated for mechanical deformation in static loading. Figure 6 displays the contour plots of the von Mises stress on the rubber portion of a steady-state rolling tire at 32 psi of inflation, 3 kN of load, and 80 km/h. It can be observed that the regions near the rim are the principal load-carrying elements. Moreover, it is clear that the magnitude of stress undertaken by the rubber is significantly reduced due to the presence of the rim and bead wires.

4.3. Thermal Analysis

This subsection demonstrates the process of investigating thermal distributions within the tire using different boundary conditions, including inflated pressure, loading, and velocity. The detailed ranges for the parametric study are shown in

Table 4. Ranges for parametric study.

Parameters	Ref. Value	Min	Max	Step
Weight (N)	3000	1000	6000	1000
Pressure (psi)	32	27	35	1
Velocity (km/h)	80	20	140	20

.

4.3.1. Change in Loading

The variation in the strain energy density, the average temperature over the rubber, and the highest temperature on the rubber part are shown in Figure 7a–c, respectively. It was found that all three variables increased when the applied vertical force was increased. The maximum temperature on the tire was about 34.4 °C at F = 6 kN. The temperature distributions in a 2D cut section of half the tire are shown in Figure 8a–c. It can be observed that the highest temperature areas were located in the shoulder and sidewall regions. These are the regions of the tire that experienced the highest level of deformation. Moreover, the bottom parts of the tire that were in contact with the air were also the regions that experienced high temperatures, due to the high level of deformation. The bottom parts in contact with the road should have experienced the highest temperature if the sipes did not exist there, due to the friction between the tire and the road. However, the areas of high temperature slightly decreased in size when the loading increased.

The distributions of heat flux in a half section of the part of the rubber with loadings of 1 kN, 3 kN, and 5 kN can be observed in Figure 9. It is seen that the heat flux generated comes from the deformation of the rubber part mainly located in the contact regions of the tire and the road. This is due to the friction between these two parts. This friction is created by applying a frictional contact between the two parts. This deformation expands to the sidewall regions, which is also the part that experiences high levels of heat flux. Moreover, the parts between the rubber and the beads are affected by the stress when rolling, so this is also a part in which deformation occurs.

4.3.2. Change in Pressure

The variations in the strain energy density, averaged temperature, and total temperature with an increasing inflated pressure are demonstrated in Figure 10a–c. All three variables are observed to increase linearly when the pressure is increased from 27 psi to 35 psi. The variations in all three variables are lower than the increased rate of changing loading. The maximum temperature ranges between 32.9 °C and 34.5 °C. It is generally observed that the temperature is inversely proportional to the inflated pressure for high-loaded vehicles. However, for the lightweight loads below 3 kN this paper studied, the phenomenon is quite the opposite, despite low levels of fluctuation. Nevertheless, this is still in accordance with the findings of some research found in the literature on tire subjected to lightweight loads and low pressures [22].

Figure 11a–c shows the distributions of temperature on the 2D plane section of half the tire with inflation pressures of 30 psi, 32 psi, and 34 psi. It was discovered that the regions of high temperature were at the same position as those analyzed above. The high-temperature zones slightly expanded as the inflation pressure increased.

Figure 12a–c shows the distributions of the heat flux generated by the strain energy density on a 2D plane segment of half the tire with inflation pressures of 30, 32, and 34 psi. It was found that the distribution of heat flux remains the same with varying pressures. However, the region that experiences the highest heat flux at different pressures becomes more prominent at the contact region between the road and the tire. This is due to the more inflated tire, with the contact region between the tire and the road thus shrinking. This leads to a higher stress being applied to this contact region and it becoming more deformed.

On the other hand, since the relationship between pressure and temperature is proportional, this is quite controversial. It is common knowledge that the temperature decreases as the tire inflation pressure increases due to the lower deformation level. However, reference [22,30] also notes a rising temperature trend as the inflated pressure increases for the tire at a low loading and low pressure. Thus, simulations for the tire with an increasing pressure with a higher loading (i.e., 6 kN) were conducted to observe whether the presented model shows the expected behavior for higher loads.

The variations in the strain energy density, averaged temperature, and total temperature with increasing inflated pressure and 6 kN of loading are demonstrated in Figure 13a–c. As expected, the three variables decrease with the increase in the inflated pressure due to the lower deformation of the tire and the lower contact areas between the tire and the road. The maximum temperature decreased from 36.4 °C at an inflated pressure of 27 psi, and to 33.8 °C at an inflated pressure of 34 psi. Moreover, the trend seen for the variables drop in its rate of decline as the variables converge to their asymptotic values.

Furthermore, it is evident that the rubber in this study behaves similarly to that of earlier studies after evaluating the distributions of temperature and heat flux for the three analyzed pressures of 28 psi, 32 psi, and 34 psi in Figure 14 and Figure 15. However, an exception exists for the high loading and high pressure at 6 kN and 34 psi. The heat flux distribution towards the shoulder part of the rubber in Figure 15b exhibits an abrupt spike. This may be explained by the fact that heavy loads and pressure cause the tire to flex more, and that the weaker part of the rubber is the place to be damaged first.

4.3.3. Change in Velocity

The development of the strain energy density, averaged temperature, and highest temperature with changing velocity are shown in Figure 16a–c, respectively. Firstly, with changing velocity, the strain energy density is increased when velocity increases from 20 to 40 km/h. After that, it decreases from 40 to 80 km/h and rises strongly from 80 to 120 km/h. The strain energy density decreases again when the velocity is higher than 120 km/h, even though the fluctuation is very low (around 0.56%) compared to the two previous variables. In addition, the increasing trend in the averaged and total temperature increases linearly with the increasing velocity. The reason for the fluctuating strain energy density, but the linearly varied temperature, is that the higher the velocity, the higher the frequency of the tire.

When considering the temperature distribution on the cross-section of half the tire in Figure 17a–c, it has been discovered that due to air convection at the inner surface, the temperature there is higher than at the outer surface. Moreover, the maximum temperature of the tire rises along with the expansion of the high-temperature regions when increasing the velocity. With a velocity of more than 80 km/h, the highest temperature on the tire is much higher than that on the rim.

By observing the distributions of heat flux in the parts of the rubber with velocities of 40, 80, and 120 km/h in Figure 18, it can be seen that with the low velocity, the highest heat flux regions are the rubber–bead wire contact zones. The deformation caused by friction is trivial. When increasing the velocity, both the bead wire–rubber and rubber–road contact regions are deformed more severely. Overall, increasing loading, pressure or velocity increases the temperature of the tire.

4.3.4. Change in Friction Coefficient between the Road and the Rubber

In this subsection, the three distinct types of road conditions that affect the amount of friction between the rubber and the surface are considered: 0.1 (icy road), 0.2 (slippery road), and 0.5 (unsurfaced road). In addition, all other variables are kept as reference values.

When considering the distributions of temperature and heat flux in the rubber part in Figure 19 and Figure 20, it can be seen that the slippery and icy road distributions are almost identical. The only difference is at the contact location between the rubber and the bead wire. However, when the friction is increased to the level of the unsurfaced road, a region of high heat flux appears in the sidewall region, and this level of high deformation spreads out to the rubber/air contact. This region of high deformation increases the local temperature of the rubber part near this region and expands it to the inner part of the rubber. It can be concluded that the tire, while on an unsurfaced road, experiences high-temperature distributions at the sidewall region, the inner surface of the tread, and the connection area between these two parts. This is due to the greater amount of contact between the tire and the road, inducing a higher level of deformation at the weak region of the tire (the shoulder region). This phenomenon is similar to the tire on the road, showing a lower friction coefficient, but under high loading and inflated pressure.

5. Conclusions

The temperature rise in a tire during normal running conditions is investigated in this work, with the help of a finite element simulation. In addition, a parametric study of boundary conditions is carried out to analyze the impact of changing conditions on the varying temperature distribution inside the tire. The mesh is composed of hexahedral elements. The simulation methodology is divided into three steps: the static loading analysis, the transient rolling investigation step, and the steady-state thermal inspection. The division of the stages ensures that the computational cost is reduced effectively, while the error is kept at an acceptable level in the fully investigated model. The hexahedral mesh, transient simulation and mesh sensitivity study guarantee the reliability of the simulation outcomes. The loading analysis results are then differentiated from the experimental results of a previous study. When changing the applied load, the maximum temperature is 34.5 °C. The maximum temperature values for changing the inflation pressure and velocity are 33.7 °C and 36.5 °C, respectively. While the tire is under a low inflated pressure, the studied temperature properties of the tire rise with the increasing pressure. However, when the tire is under higher loadings, the investigated temperature properties of the tire scale inversely with the pressure. The maximum tire temperature at the loading of 6 kN peaks at 36.4 °C with an inflated pressure of 27 psi. In addition, the changing friction coefficient increases the maximum temperature from 33.0 °C at a friction coefficient of 0.1 to 42.3 °C at a friction coefficient of 0.5. Moreover, the maximum temperature is increased with the rise in all three variables. However, the regions of high-temperature decrease when the three variables are enlarged. Further, when the tire is under high loading and pressure, or in the case of an increased friction coefficient, the high temperature and deformation area are moved toward the sidewall/tread connection area due to a higher level of deformation at the weaker area of the tire.

In car tires, siping is another crucial factor in diffusing the heat from the rubber part. Siping varies for different types of car tires, and the effect of the sipes depends on the road type. Future studies should concentrate on the effects of the road and rubber siping on the temperature distribution of the rubber part and, thus, optimize the siping geometry in order to decrease the total heat of the rubber.