Thermo-Mechanical Coupling Analysis of Inserts Supporting Run-Flat Tires under Zero-Pressure Conditions

Abstract

The inserts supporting run-flat tire (ISRFT) is mainly used in military off-road vehicles, which need to maintain high mobility after a blowout. Regulations show that the ISRFT can be driven safely for at least 100 km at a speed of 30 km/h to 40 km/h under zero-pressure conditions. However, the ISRFT generates serious heat during zero-pressure driving, which accelerates the aging of the tire rubber and degrades its performance. In order to study the thermo-mechanical coupling characteristics of the ISRFT, a three-dimensional finite element model verified by bench tests was established. Then, the stress–strain, energy loss and heat generation of the ISRFT were analyzed by the sequential thermo-mechanical coupling method to obtain the steady-state temperature field (SSTF). Finally, four kinds of honeycomb inserts bodies were designed based on the tangent method, and the SSTF of the honeycomb and the original ISRFT were compared. The results indicated that the high-temperature region of the ISRFT is concentrated in the shoulder area. For every 1 km/h increase in velocity, the temperature at the shoulder of the tire increases by approximately 1.6 °C. The SSTF of the honeycomb ISRFT is more uniformly distributed, and the maximum temperature of the shoulder decreases by about 30 °C, but the maximum temperature of the tread increases by about 40 °C. This study provides methodological guidance for investigating the temperature and mechanical characteristics of the ISRFT under zero-pressure conditions.

1. Introduction

In order to address the safety concerns associated with traditional pneumatic tires, which can be exacerbated by a blowout or puncture, safety tires have emerged [1,2]. The inserts supporting run-flat tire (the ISRFT) is a kind of safety tire based on the structure of pneumatic tires, and its application is mainly for military armored vehicles and military off-road vehicles [3]. In the event of emergency safety cases such as a blowout or puncture, the ISRFT is required to support the vehicle in continuing to drive for distances of at least 100 km and at speeds of 30–40 km/h [4]. However, under zero-pressure continuous driving conditions, the rolling resistance of the ISRFT increases dramatically, reaching approximately 500% of that observed under standard pressure conditions [5]. This generates substantial heat and high tire temperatures, which cause deterioration and separation of the tread and sidewall rubber from the cords and tire performance degradation. The inserts body of the ISRFT studied in this paper is a three-piece structure, each with a radian angle of 120°, which is positioned and assembled on the rim through bolt holes. The assembly model of the ISRFT is shown in Figure 1.

In the field of thermal–mechanical coupled characterization, the derivation of thermal properties from the stress–strain of the material is a wide area of research. Čukanović et al. used different types of shape functions to analyze the thermal buckling of a functional gradient plate based on the von Karmen nonlinear theory [6]. Radaković et al. introduced a new analytically integrable shape function based on high-order shear deformation to investigate the thermal buckling characteristics of functional gradient plates on an elastic foundation, and the validity of the newly proposed shape function was proved by comparison [7]. Chen et al. proposed a coupled thermal–mechanical–abrasive analytical method for prediction of the temperature and material wear of solid rubber tires. Compared with the thermo-mechanical coupling method, the proposed method improves the temperature prediction accuracy by 27.65% [8].

In contrast to conventional pneumatic tires, when the ISRFT is subjected to zero-pressure continuous driving conditions, it is necessary to consider not only the hysteresis heat generation of the tire itself and the convective heat transfer between the inner and outer surfaces of the tire, but also the hysteresis heat generation of the inserts body and the heat transfer with the inner wall of the tire [9]. In the research papers on tire temperature fields, the calculation method for the internal unit heat generation rate of the tire based on the theory of rubber hysteresis loss using Fourier series is widely used [10]. Additionally, the convective heat transfer relation equation for smooth convex rotating surfaces is commonly applied to rolling tires [11]. The above thermal boundary conditions and the sequential thermo-mechanical coupling analysis method are used in this paper to obtain the SSTF of the ISRFT.

In the sequential thermo-mechanical coupling analysis method, the mechanical characteristics must be analyzed first. Based on the results of the analysis, the thermal analysis can then be performed [12]. To investigate the mechanical characteristics of the ISRFT under zero-pressure conditions, Zang et al. conducted bench tests to obtain the radial and lateral stiffness of the ISRFT under varying tire pressures. The results demonstrated that the average radial stiffness increased by 283.34% and the average lateral stiffness increased by 9.92% compared to the rated tire pressure condition when the radial load exceeded 6000 N under zero-pressure conditions [13]. Additionally, an analysis of the grounding stress distribution of the ISRFT under zero-pressure conditions revealed that the grounding stress distribution is concentrated at the endpoints and centers of the contact patch [14].

In this paper, a sequential thermal coupled analysis method is applied to investigate the SSTF of the ISRFT under zero pressure and different speed conditions and comparatively explore the SSTF of the honeycomb ISRFT. The new contributions and information lie in the following: (1) The thermal boundary conditions between the inserts body and the inner wall of the tire were established. (2) The thermal boundary conditions between the inserts body and the rim and between the inserts body and the air were also established. In addition, different from the inflation condition, the stress–strain variation of tires under zero-pressure conditions is more complex and requires higher terms for Fourier series fitting. In this paper, the highest Fourier series to accurately fit the stress–strain of the ISRFT under zero-pressure conditions is summarized in Section 4.

The remainder of this paper is structured as follows. Section 2 provides the theory of the SSTF of the ISRFT, including the theory of rubber hysteresis loss and the Fourier series fitting of stress–strain to obtain the formula for calculating the unit heat generation rate. Section 3 establishes an experimentally validated nonlinear finite element model. Section 4 carries out the numerical simulation and analysis of the SSTF through the steady-state rolling results of the ISRFT under zero-pressure conditions, combined with the theory of rubber deformation and heat generation. Four honeycomb structures were also designed as inserts bodies based on the tangent method, and the SSTF results were compared between the honeycombs and the original ISRFT. Finally, Section 5 concludes the paper and summarizes the key findings.

2. Theory of the SSTF of the ISRFT

When a tire turns and brakes, relative slip is generated between the tire and the road surface, and sliding friction generates more heat; while a tire is rolling in the steady state, almost no sliding friction is generated, and in comparison, the heat generated by rolling friction is much less than that emitted by hysteresis losses, which can be ignored when carrying out thermo-mechanical coupling analyses of a tire [15]. In Abaqus, the linear velocity of the tire is initially specified, and then the angular velocity is adjusted to achieve a desired traction, steady-state, or braking simulation. Consequently, it is essential to debug the angular velocity value to ensure that it is under a steady-state rolling condition.

The key to the finite element analysis of the tire SSTF is to determine the numerical magnitude of the heat generation rate. However, the viscoelastic constitutive equations are difficult to obtain due to a certain degree of inadequacy in both theoretical studies and experimental methods of viscoelastic mechanics. In this case, the current finite element analysis of the tire SSTF usually adopts a decoupled research method to deal with the thermo-mechanical coupling problem. In other words, the force field situation, including stress and strain, is first analyzed, and then the strain energy amplitude is calculated according to the relevant equations and multiplied by the loss coefficient of the rubber material to obtain the final heat generation rate. The solution flowchart for the tire SSTF is shown in Figure 2.

2.1. Theory of Hysteresis Loss in Viscoelastic Materials

Rubber, the constituent material of a tire, is classically a viscoelastic material, with properties intermediate between those of an ideal elastomer and an ideal viscous body. Therefore, when a cyclic load is applied to the rubber material of a tire, it exhibits dynamic viscoelastic properties, and the strain lags behind the stress. When subject to a sinusoidally varying stress–strain, the strain always lags behind the stress by a phase difference, resulting in a hysteresis phenomenon, as shown in Figure 3a.

If the stress–strain is strictly sinusoidal, the respective expressions are [12]

$$\epsilon \left(t\right)={\epsilon }_0\sin \left(\omega t\right)$$

(1)

$$\sigma \left(t\right)={\sigma }_0\sin \left(\omega t+\delta \right)$$

(2)

where σ denotes stress, ε denotes strain, ω denotes angular frequency, t denotes time, δ denotes hysteresis loss angle and σ₀, ε₀ denote stress and strain amplitude, respectively.

The phase difference that exists between the stress–strain results in a hysteresis loop in the stress–strain curve of the material, as shown in Figure 3b. The magnitude of the hysteresis loss in the rubber unit material is calculated from the area of the ring with the following expression:

$$E={\displaystyle \int \sigma \left(t\right)d\epsilon \left(t\right)=\pi \cdot {\sigma }_0\cdot {\epsilon }_0\cdot \sin \delta }$$

(3)

where E′ and E″ are the elastic storage modulus and elastic loss modulus:

$$\left\{\begin{array}{l}E\prime ={\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}\cdot \cos \delta \\ E″={\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}\cdot \sin \delta \end{array}\right.$$

(4)

The ratio of loss modulus to storage modulus is the loss factor:

$${\displaystyle \frac{E″}{E\prime }}=\tan \delta$$

(5)

2.2. Fourier Series Fit Process

However, during the process of tire rolling, the rubber material is not subjected to a strictly sinusoidal form of varying stress–strain. It is necessary to adjust Equations (1)–(3) accordingly. Figure 4a depicts the stress profile of the sidewall in the circumferential direction. Figure 4b illustrates the time history of strain amplitude. Figure 4a,b were created by the authors using data from references [16,17], respectively. It can be observed from Figure 4 that during the rolling process of the tire, the maximum value of stress–strain occurs at the time of contact with the ground, and the value of stress remains almost constant the remainder of the time.

The change rule of the strain is consistent with the stress and will not be described here. In order to obtain the maximum value of circumferential stress–strain on the tire, Wei et al. employed the Fourier series for fitting. However, they did not specify the number of terms [18]. Shida et al. concluded that a more accurate fitting could be achieved with a series comprising more than 15 terms [16]. Therefore, the stress–strain can be expanded through the Fourier series, that is [19]

$$\sigma (\theta )=a_0^{\sigma }+{\displaystyle \sum _{n=1}^N\left\{a_n^{\sigma }\cos (n\theta )+b_n^{\sigma }\sin (n\theta )\right\}}$$

(6)

where a₀ denotes the Fourier coefficients, n denotes the number of Fourier expansion stages and n denotes the total number of Fourier expansion stages. The above equation can be simplified to obtain

$$\sigma (\theta )=a_0^{\sigma }+{\displaystyle \sum _{n=1}^N\left\{A_n^{\sigma }\sin (n\theta +{\varphi }_n^{\sigma })\right\}}$$

(7)

where $A_n^{\sigma }$ denotes the maximum stress value, $A_n=\sqrt{a_n^2+b_n^2}$, ${\varphi }_n=\arctan \left({\displaystyle \frac{b_n}{a_n}}\right)$.

Substituting the hysteresis loss factor δ, we obtain

$$\sigma (\theta )=a_0^{\sigma }+{\displaystyle \sum _{n=1}^N\left\{A_n^{\sigma }\sin (n\theta +{\varphi }_n^{\sigma }+\delta )\right\}}$$

(8)

Therefore, after adjusting Equations (2) to (8), the hysteresis loss of the rubber unit material should also be adjusted from Equation (3) to (9):

$$E_d={\displaystyle \sum _{n=1}^N\left[\pi \cdot n\cdot A_n^{\sigma }\cdot A_n^{\epsilon }\cdot \sin ({\varphi }_n^{\sigma }-{\varphi }_n^{\epsilon }+\delta )\right]}$$

(9)

where $A_n^{\epsilon }$ denotes the maximum strain value, which is simplified to

$$E_d={\displaystyle \sum _{n=1}^N\left[\pi \cdot n\cdot A_n^{\sigma }\cdot A_n^{\epsilon }\cdot \tan \delta \right]}$$

(10)

Since the 3D finite element model of the tire consists of multiple rubber cells, each unit has six components of stress and strain. Therefore, the hysteresis loss energy of all the rubber units is summed up to obtain the hysteresis loss energy of the whole tire:

$$E_T={\displaystyle \sum _{i=1}^{Nele}{\displaystyle \sum _{m=1}^6{\displaystyle \sum _{n=1}^N\left[\pi \cdot n\cdot V_i\cdot A_{n,i,m}^{\sigma }\cdot A_{n,i,m}^{\epsilon }\cdot \tan {\delta }_i\right]}}}$$

(11)

where Nele denotes the cell number, i denotes the cell sequence number, m denotes the number of stress–strain vectors and V denotes the cell volume. The derivation of Equations (6)–(9) above references the dissertation by Wu. The cumulative form of Equations (10) and (11) has been changed.

3. Simulations and Verifications

3.1. Finite Element Modelling of the ISRFT

The specification of the tire is 37 × 12.5R16.5, its outer diameter is 939.8 mm, the width of the tire section is 317.5 mm, the height of the tire section is 260.35 mm and the diameter of the rim is 419.1 mm. The inserts body is a solid-like circular structure, which is mounted on the rim with a width of 110 mm, and the difference between the inner and outer radius is 130 mm. The dimensions of the tire geometry model were derived from actual measured data values, with the tread pattern excluded. The tread grooves are about 8.5 mm thick and 16.2 mm wide. The finite element modelling process of the ISRFT is shown in Figure 5 below.

Firstly, a 2D semi-cross-section of the tire was drawn, after which it was imported into HyperMesh 2022 to define the various parts of the tire and draw the mesh of each part separately. The ply mesh type was defined as SFMGAX1, while the remaining parts were defined as CGAX3H and CGAX4H. The mesh of the other half of the cross-section was generated by reflecting. Finally, the 2D tire model was imported into Abaqus 2022. The rim was defined as an axisymmetric analytical rigid body in the part, and the assembly was completed in the 2D model. The assembled 2D model was generated into a 3D model by the *Symmetric model generation revolve command; the exploded view and the 3D model view are shown in Figure 5 above. The finite element model of the ISRFT has a total of 72,360 elements and 84,672 nodes, of which the tire has 39,528 elements and 47,952 nodes. The inserts body has 32,832 elements and 36,720 nodes. Upon generation of a 3D model in Abaqus, the mesh type underwent a transformation from SFMGAX1, CGAX3/4H to SFM3D4R, C3D6/8H.

It is necessary to create surfaces and features that will facilitate subsequent boundary conditions and load setting after assembly. The steps of this process are as follows. First, create the ground surface: choose “geometry” for the surface type, select “individual” for the regions for the surface, and select the face in contact with the tire to create the surface. Next, create the tire outer surface: choose “mesh” for the surface type, select “by angle” for the regions for the surface, and select the tread and sidewall to create the surface. Subsequently, the inner wall of the tire, the outer surface of the inserts body, the inner surface of the insert support and the rim surface are created, and the options in the process are consistent with the tire outer surface. Finally, select the center point of the tire to create feature RP-1 and the center point of the ground to create feature RP-2.

Based on the surfaces and features created above, the contact, boundary conditions and loads are set for the finite element model. The contacts include (1) the tread–ground contact. The contact properties select tangential behavior and normal behavior. The friction formula in the tangential behavior selects the penalty function, the direction is isotropic, and the friction coefficient is set to 0.8. The shear stress limit is set to “No limit”, the elastic slip is specified as maximum elastic slip, and the fraction of characteristic surface dimension is set to default, “0.005”. In the normal behavior, a hard contact is set, and the constraint execution mode is default, allowing separation after contact. Another contact is (2) the inserts body–tire inner wall: the friction coefficient is set to 0.6 and the rest of the settings are as above. The friction coefficients were derived from tire test bench measurements and have a negligible impact on the result of static simulations.

Then, the constraint conditions are defined. (1) First, the rigid body constraints are defined, with the rim selected for the analytical surface and RP-1 selected for the reference point; then, the constraints are bound, with the rim selected for the main surface and the inner surface of the inserts body with the tire toe area selected for the secondary surface, and RP-1 is fixed completely. (2) The reference point is selected as RP-2 by the rigid body command. The analytical surface is the ground, and the constraints are created. Then, the five degrees of freedom of RP-2 are constrained. One degree of freedom is reserved for the Z-axis. (3) The Embedded Region command is then employed to embed the carcass ply and belt ply into the corresponding mesh.

Finally, a rated load (12,250 N) in the positive direction of the Z-axis is applied to RP-2, causing the ground to move in the positive direction of the Z-axis and contact the tire.

To ensure the accuracy of the finite element model, mesh sensitivity was analyzed. By adjusting the mesh size, a zero-pressure rated load (12,250 N) simulation was carried out when the number of elements in the tire model was 39,528, 50,508 and 61,488. The deflection results were 104.2 mm, 104.04 mm and 103.96 mm, respectively. The results are shown in Figure 6.

Figure 6 illustrates that when the number of meshes is increased from 39,528 to 61,488, the result of tire deflection only changes by 0.238 mm. This indicates that the simulation results are not affected by the number of finite elements, suggesting that the results converge and the mesh quality is high.

Tire rubber material has nonlinear characteristics and can usually be defined with the generalized Mooney–Rivlin model. The strain energy density function model is [20]

$$W=C_{10}(I_1-3)+C_{01}(I_2-3)+{\displaystyle \frac{1}{D_1}}{(J-1)}^2$$

(12)

The Mooney–Rivlin material parameters and thermo-physical properties of the ISRFT are shown in

Table 1. Rubber material parameters of the ISRFT [21].

Component	C01 (MPa)	C10 (MPa)	D01 (MPa)	Density (kg/m3)	Conductivity (W/(m·K))	Specific Heat (J/(kg·K))	tanδ
Tread	0.5792	0.1448	0.01381	1112	0.341	1.406	0.182
Belt	1.0848	0.2712	0.00737	1144	0.285	1.060	0.11
Sidewall	0.524	0.131	0.01527	1110	0.311	1.372	0.087
Bead filler	−1.6905	8.0598	0.00157	1025	0.323	1.450	0.049
Carcass	0.6159	0.154	0.01299	1058	0.291	1.283	0.082

. The density distribution is uniform, the material type is isotropic, and the specific heat type is constant volume. The material parameters and thermo-physical properties of the inserts body and tire skeleton sections are shown in

Table 2. Inserts body and tire skeleton material parameters [14].

Component	Young’s Modulus (MPa)	Poisson’s Ratio	Density (kg/m3)	Conductivity (W/(m·K))	Specific Heat (J/(kg·K))	tanδ
Inserts body	600	0.48	1200	0.24	1.334	0.445
Bead	166,000	0.3	7800	0.385	1.580	0.143
Carcass ply	10,549	0.4	-
Belt ply	205,351	0.3

below. Specific details are consistent with the above.

The data, including deflection, length, width and area of the grounding imprints, are extracted and compared with the test results to verify the accuracy of the nonlinear finite element model. The simulation model of the ISRFT presented in this paper is based on the ISRFT utilized for the bench tests. It is consistent in terms of quality and dimensions.

3.2. Bench Test of the ISRFT

The test rig comprises an air pump, vertical loading cylinders, a force–displacement sensor test device, a control cabinet, an analog ground platform and reinforcing screws. Prior to commencing the test, the tire was fixed to the test rig via a solid shaft passing through the flange of the tire and left- and right-side mounting pads. The replacement of the left- and right-side mounting pads allows the tire to have different camber angles, with the tire maintained at 0° camber throughout the test. Prior to the acquisition of tire ground impressions, the simulated ground platform was cleared of debris and contaminants, and a suitable size of copy paper was placed on it. The radial loading test of the ISRFT is depicted below.

Because this paper investigates the SSTF distribution and related change rules of the ISRFT under zero-pressure continuous driving conditions, the tire was deflated in the test, and the tire pressure was measured to be 0 kPa using a tire pressure gauge. The ISRFT was initially lowered to make contact with the analog ground platform. Thereafter, the air pressure of the vertical loading cylinder, as illustrated in Figure 7, was adjusted via a knob to ensure that the load was distributed as evenly as possible. When the radial load is minimal, the augmentation of the cylinder air pressure precipitates a swift increase in load, resulting in a paucity of data points when the load is below 2000 N. The knob is meticulously calibrated to ensure a uniform increase in radial load, which ultimately attains the desired value. Upon completion of the action, the valve is opened to deflate the ISRFT to its initial position.

The rated load of the tire is 12,250 N, and a radial load of 120% of the rated load, i.e., 14,700 N, is applied to the tire. The deflection, length and width of grounding imprints data were collected. For the bench tests, six tests were conducted under test conditions with a rated load of 12,250 N, and four tests were conducted with a load of 14,700 N. When the load reached the desired value, the results of the deflection were all consistent, indicating that the tests were reproducible. The comparison results of the simulation and test are shown in Figure 8 and Figure 9.

In Figure 8, when the radial load is less than about 6000 N, the inserts body is not involved in the load-bearing action. In the test and simulation, the deflection increases rapidly with the increase in the radial load. When the radial load is greater than about 6000 N, the inserts body is involved in load bearing, the radial stiffness of the ISRFT increases, and the deflection decreases abruptly with the rate of increase of the radial load. When the radial load is 2000–4000 N, there is a certain deviation between the test deflection and the simulated deflection, and the test curve shows obvious fluctuation, with a maximum error of about 21.55% when the load is 2300 N. This may be caused by external factors such as the tread pattern and load loading speed. In a previous study [13], the test curve fluctuated between 3000 N and 5000 N, with the largest error in deflection of 15.48% when the load was 4000 N. The ISRFT is rotated at a certain angle during each test to avoid permanent deformation from repeated loading at the same position. Therefore, the grounding position of the ISRFT will change each time, resulting in different forms and degrees of tread warping, which may be the main reason for the fluctuation of the curve. When the radial load is 6000–14,700 N, the error between the simulated deflection and the test deflection is no more than 1 mm, less than 2%. Compared with the maximum error of 8.4% (6000–14,700 N) in the previous study, the model developed in this paper has higher accuracy under zero-pressure conditions.

Prior to the test, the copy paper was placed on the platform, and then the requisite radial force was applied to the tire in order to create an impression on the copy paper. As the radial force of the tire is gradually increased from 0 to the required radial force for the test, the grounding imprints on the copy paper include the entire process. However, the grounding imprints shown in the simulation results were only the result at that value of radial force. Therefore, there are differences in details between the two. Figure 9 and

Table 3. Comparison of length, width and area of grounding imprints.

Radial Load (N)	Test Width (mm)	Simulation Width (mm)	Test Length (mm)	Simulation Length (mm)	Test Area (mm2)	Simulation Area (mm2)	Area Error (%)
9800	235	216.579	580	561.139	136,300	121,530	10.84
12,250	235	225.463	580	562.015	136,300	126,713	7.03
14,700	235	235.875	580	560.513	136,300	132,211	3

compare the length, width and area of the grounding imprints obtained from the simulation and the test.

The shape of the contact patch is rectangular, and with the increase in the radial load, the color of the imprints at the four endpoints and the center of the rectangle deepens, and the trend of the test and simulation is consistent. Under different radial loads, the error of the imprint length between the simulation and test is 3.36% at the maximum and 3.1% at the minimum, which has good accuracy and stability. The maximum error of the imprint width is 7.84% and the minimum is 0.8%; the error decreases with the increase in the radial load, and the maximum error is within the acceptable range. The maximum error of the imprint area is 10.84% and the minimum is 3%; the error decreases with the increase in the radial load. In a study by Genovese et al., the simulated and tested tire grounding area error was 6.53% [22]. Therefore, combining the simulation and test comparison results of deflection and grounding imprints, it is concluded that the nonlinear finite element model of the ISRFT has high accuracy under zero-pressure conditions.

4. Numerical Simulation Analysis

4.1. Steady-State Rolling Simulation

Before conducting the analysis of the SSTF, it is necessary to extract the stress–strain values of the ISRFT under the steady-state rolling conditions and then to combine it with the rubber material loss factor through the Fourier series fitting to calculate the heat generation rate of each unit.

The relevant technical indicator report shows that the maximum velocity of the off-road vehicle equipped with this type of tire is 131 km/h, and the maximum average velocity on the off-road surface at the automotive proving ground is 40.8 km/h [3]. T/COS 015-2023 “Specification for aluminum-based support body of deflating protection wheels” issued by the China Ordnance Society stipulates that the ISRFT needs to be driven safely at a speed of 30~40 km/h for no more than 100 km [4]. Therefore, combining the above two studies and considering the high mobility required for the ISRFT in special scenarios, the minimum driving speed of 30 km/h and the maximum driving speed of 80 km/h are set, and the middle speed of 55 km/h between the two speeds is taken as a transition. Therefore, under the zero-pressure rated load conditions, the steady-state rolling simulations of the three velocity conditions of 30 km/h, 55 km/h and 80 km/h are carried out, which basically cover the full velocity range under the zero-pressure rated load conditions. The steady-state rolling simulation results of ISRFT under zero-pressure and three speed conditions are shown in Figure 10.

In the legend, the unit of velocity V is mm/s, and ω is the angular velocity of convergence. In the steady-state rolling simulation, the velocity at the center of the tire is a given velocity, the velocity in the tire grounding area is 0, and the velocity at the highest point of the tire is twice the given velocity. The velocity distribution cloud images are highly consistent under the three velocity conditions, which indicates that the simulation results have good convergence and high reliability.

Under steady-state rolling conditions, the effective rolling radius of pneumatic tires is slightly larger than the static load radius. However, the difference is numerically small [23,24]. This is evidenced by the rolling test of the ISRFT, where the numerical difference between the effective rolling radius and the static load radius is almost nonexistent after the inserts body is involved in carrying the load [5]. Therefore, by verifying the results of the static simulation, the effectiveness of the steady-state rolling tire can be verified to some extent.

Under different velocity conditions, the stress–strain data of the ISRFT are extracted, and the belt ply and carcass ply are ignored. According to the Fourier variation and related formulas derived in Section 2, the maximum value of hysteresis energy loss of the cross-section element is obtained in the circumferential direction, and then the heat generation rate is calculated. The example diagram of the stress data is shown in Figure 11.

In Figure 11a, the radial force applied to the ISRFT is 12,250 N, and the velocity is 80 km/h. The equivalent stress values of the elements located on the red curve are extracted for analysis.

Figure 11b illustrates the Fourier series fitting curve, and the total number of Fourier expansion steps n is taken as 24 to obtain a better fit. In instances where the stress is more extreme and complex than that, it is necessary to increase the value of n to achieve a superior fit. In the dataset of this paper, the value of n was set to a maximum of 30 in order to achieve the optimal fit. Upon exceeding 30, the transition smoothing process results in a reduction in the efficacy of the fit to the stress maximum.

4.2. Thermal Boundary Condition Setting

Before conducting the numerical simulation of the tire SSTF, the following assumptions need to be made:

• The properties of tire viscoelastic materials are not affected by temperature, and all the heat generated in the rolling process of the tire originates from hysteresis loss.
• The steady-state rolling of the tire does not take into account the circumferential temperature gradient, and the temperature is the same in each cross-section.
• When the tire is in thermal equilibrium, the thermo-physical parameters such as thermal conductivity, specific heat capacity and the loss factor of rubber material are regarded as constant.
• The rubber material used in tires is isotropic.

In Section 2, the hysteresis energy loss of the rubber unit material element was obtained through the theoretical derivation of Equation (10), and the heat generation rate of each unit is

$$Q_d={\displaystyle \sum _{m=1}^6\frac{E_d}{T}}={\displaystyle \frac{\omega }{2}}\cdot {\displaystyle \sum _{m=1}^6{\sigma }_m}\cdot {\epsilon }_m\cdot \tan \delta$$

(13)

where T represents the rolling period, ω represents the steady rolling angular velocity of the tire, σ_m represents the stress amplitude and ε_m represents the strain amplitude.

The heat source is the calculated heat generation rate of each unit. The heat transfer equation of the ISRFT under zero-pressure continuous driving conditions is heat conduction and heat convection, and the contact between the inserts body and the tire is heat conduction. The heat convection boundary mainly includes the boundary of the inserts body and the tire bead in contact with the rim, the sidewall boundary and the inner liner boundary in contact with the air, and the tread boundary in contact with the road surface. This is shown in Figure 12.

The boundary conditions of convective heat transfer can be expressed as follows:

$$Q_2=\left(h_c+h_r\right)\left(T_s-T_d\right)$$

(14)

where h_c is the convective heat transfer coefficient on the inner and outer surface of the tire, h_r is the radiative heat transfer coefficient of the inner and outer surface of the tire, T_S is the temperature of the tire surface (°C) and T_d is the air temperature (°C).

Numerous researchers and scholars have shown the convective heat transfer coefficient of various parts of the tire:

$$\begin{gathered} h_T=2.2\cdot v^{0.84} \\ {h}_B=h_C=0.4\cdot h_T \\ {h}_s={\displaystyle \frac{\left(h_T-h_B\right)\cdot \Delta l}{l}} \end{gathered}$$

(15)

where h_T is the convective heat transfer coefficient of the crown, v is the linear velocity of steady rolling, h_B is the convective heat transfer coefficient on the inner surface, h_C is the inner surface convective heat transfer coefficient, h_s is the sidewall convective heat transfer coefficient, l is the sidewall arc length (mm) and Δl is the length of the arc from the crown (mm) [19].

After consulting the relevant data, the convective heat transfer coefficient is modified to determine the convective heat transfer coefficient of each part of the tire, as shown in

Table 4. Convective heat transfer coefficient of each part of tire [19].

Part	Coefficient of Convective Heat Transfer (w/m2k)
Tread	118
Sidewall	90
Rim	168
Inner liner	52

.

4.3. Results of SSTF Distribution

4.3.1. SSTF Distribution of ISRFT

Based on the assumption that there is no temperature gradient along the circumferential direction of the tire in the state of thermal equilibrium, the section temperature of the tire can be regarded as the whole temperature field distribution of the tire, and the model can be simplified to a 2D model. The four-node unit is changed to DC2D4, and the three-node unit is changed to DC2D3. The temperature at the boundary is set to be 22 °C. The results of the SSTF distributions are shown in Figure 13a–c.

It can be seen from the above that the SSTF distribution of the ISRFT has a high consistency under the zero-pressure rated load and different velocity conditions, the highest temperature is located at the shoulder, followed by the tread and sidewall parts, and the temperature of the bead is lower. The Path1, X and Y paths are shown in Figure 14a. The temperature changes on Path1 under three velocity conditions are extracted, as shown in Figure 14b.

The combination of Figure 13 and Figure 14b reveals the presence of four peaks in temperature on Path 1. The first peak is observed at the tread area, which is attributed to the temperature fluctuations induced by tread grooves. The temperature is lower at the grooves and higher at the areas without grooves. The second peak emerges at the shoulder of the tire, where the temperature difference among the three velocity conditions is the most pronounced. In other words, the temperature in this area is more sensitive to changes in velocity. This is because the hysteresis loss generated at the shoulder is the largest, and the thickness of the rubber here is the largest, which is not conducive to heat dissipation. As illustrated in Figure 14b, the temperature at the shoulder of the tire exhibits a linear pattern with the change in velocity. For every 1 km/h increase in velocity, the temperature at the shoulder of the tire increases by approximately 1.6 °C. The third and fourth peaks appear sequentially at the transition position of the sidewall of the tire due to severe bending of the sidewall area. The temperature of the sidewall area varies less with velocity, indicating that the temperature of the sidewall area is mainly determined by the hysteresis loss caused by rubber deformation and is not sensitive to velocity changes.

4.3.2. SSTF Distribution of the Inserts Body

In comparison to tires, the inserts body has a relatively high elastic modulus and small deformation, resulting in less hysteresis loss. Therefore, under steady-state rolling conditions, the temperature of the inserts body should be lower than that of the tire. Figure 15 presents the temperature distribution of the inserts body under three velocity conditions, as extracted from the X and Y paths in Figure 14a.

Under zero-pressure rated loads and steady-state rolling conditions, the temperature change of the inserts body is less affected by velocity compared to the tire. This is due to the material composition of the inserts body, which is polyurethane (PU) with a high elastic modulus and high hardness. This results in a very small strain, which in turn leads to a very small hysteresis loss. Therefore, reducing the stress–strain value of tires under zero-pressure rated load conditions can reduce the heat generation and improve endurance.

4.4. Honeycomb Structure Optimization

As illustrated in Figure 13, the temperature of the tire is considerably higher than that of the inserts body. The high temperature accelerates the deterioration of tire rubber under zero-pressure driving conditions. To enhance the zero-pressure endurance capability of the ISRFT, some scholars have optimized the design and altered the materials of the inserts body [25]. The honeycomb structure is well-suited for use in the structural design of non-pneumatic safety tires (NPTs) due to its excellent cushioning, vibration damping and heat coupling properties [1,2,26,27]. In our previous study, a honeycomb structure design method based on the tangent method was proposed [28]. In this paper, four multi-level self-similar honeycomb inserts body models are designed based on the method, which is schematically shown below.

In Figure 16, O represents the coordinate origin, r represents the inner diameter of the honeycomb inserts body, R represents the outer diameter, D represents the wall thickness, θ denotes the angle occupied by a honeycomb cell with respect to the coordinate origin, d_i denotes the diameters of the cell elements of each layer, and the number of basal circles n can be derived from the following equation:

$$n={\displaystyle \frac{2\pi }{\theta }}$$

(16)

The diameter of the base circle is known from the following geometric relationship:

$$d_1=2\cdot r\cdot \sin {\displaystyle \frac{\theta }{2}}$$

(17)

Based on the geometric relationship of tangency, the formula for calculating the diameter of a circle of neighboring layers is derived:

$$d_{i+1,i-1}=d_i\cdot \left(\tan {\left({\displaystyle \frac{\pi }{n}}\right)}^2+{\displaystyle \frac{1}{\cos \left(\frac{\pi }{n}\right)}}\pm \sqrt{{\left(\tan {\left({\displaystyle \frac{\pi }{n}}\right)}^2+{\displaystyle \frac{1}{\cos \left(\frac{\pi }{n}\right)}}\right)}^2-1}\right)$$

(18)

The structural parameters of the honeycomb inserts body are presented in

Table 5. Structural parameters of honeycomb inserts body.

n	d0 (mm)	d1 (mm)	d2 (mm)	d3 (mm)	d4 (mm)	d5 (mm)	d6 (mm)
24	43.21	54.24	68.08	85.46	107.28	/	/
30	36.33	43.57	52.25	62.66	75.15	90.12	/
36	31.28	36.39	42.33	49.25	57.30	66.66	/
45	25.72	29.03	32.76	36.98	41.73	47.11	53.17

below.

The three-dimensional models of the multi-level self-similar honeycomb inserts body are shown in Figure 17, where n refers to the number of honeycomb cells in a circle in the circumferential direction. The mass of the honeycomb inserts body in Figure 17a–d is 8.64 kg, 12.13 kg, 13.02 kg and 14.87 kg. The mass of the original inserts body is 29.62 kg. Compared to the original inserts body, the mass of the honeycomb inserts body has decreased by 70.83%, 59.36%, 56.04% and 43.88%, respectively.

4.4.1. SSTF Distribution of Honeycomb ISRFTs

Figure 13 and Figure 14b demonstrate that the change in velocity affects the temperature value of the ISRFT, but not its SSTF distribution pattern. At a higher velocity, the temperature value fluctuates more significantly, and the SSTF distribution pattern is easier to summarize. Therefore, under zero-pressure rated load and V = 80 km/h conditions, the SSTF of four types of honeycomb ISRFTs is simulated. The element type is changed to DC3D8R/DC3D6. The temperature at the boundary is set to 22 °C. The SSTF distribution of the tire is displayed using a 2D half-section diagram. However, the SSTF distribution of the honeycomb inserts body cannot be displayed using a 2D section, so a 3D model of the honeycomb cell is used for display. As an illustrative example, consider the cross-sectional view of a honeycomb ISRFT with N = 36, as depicted in Figure 18.

Under a zero-pressure rated load, the deformation of the honeycomb inserts body is greater than that of the original inserts body. This results in a stress concentration at the original tire shoulder being dispersed to the tread. Consequently, there is a significant increase in the temperature of the honeycomb inserts body, with the highest temperature position being transferred from the tire shoulder to the tread. The highest temperature of the tire is slightly reduced, and the SSTF distribution of the tire is more uniform. Figure 18f illustrates that the SSTF distribution along Path1 on the 2D half section of the four types of honeycomb ISRFTs exhibits remarkable consistency. Figure 18g depicts the temperature changes along Path1 between the honeycomb ISRFT and the original ISRFT. It can be observed that due to the reduced structural stiffness of the inserts body, all four peak positions have undergone a slight shift towards the negative x-axis direction. The temperature at the tread area has increased by approximately 38 °C, while the temperature at the shoulder area has decreased by approximately 25.6 °C. The highest temperature along Path1 has decreased by approximately 16.5 °C. The temperature changes at the sidewall and shoulder areas are relatively small.

4.4.2. SSTF Distribution of the Honeycomb Inserts Body

The cellular structures of the honeycomb inserts body are extracted as shown in Figure 19a. Two paths, Path2 and Path3, are plotted in Figure 19b–e, and the SSTF results of the cellular structures on these two paths are extracted. Figure 19f illustrates that the temperature of the cellular structure with N = 24 along the Path2 is significantly higher than the other three groups.

The temperature of N = 30 is nearly identical to that of N = 36, while the temperature of N = 45 is the lowest. On Path2, the temperature distribution pattern of the four cellular structures is consistent, with the temperature at the center slightly lower than on both sides. This is due to the presence of slight warping at the center of the tread, which results in a stress–strain value at the center of Path2 that is not as large as the value in direct contact with the tire on both sides. The SSTF distribution of the inserts body along the Y path in Figure 15b also exhibits a similar pattern. Figure 19g illustrates that there are differences in the SSTF distribution of the four cellular structures on Path3. When the normalized distance is greater than 0.3, the temperature of N = 24 is significantly higher than the other three groups. The temperature of N = 30 is nearly identical to that of N = 36, and the temperature of N = 45 is the lowest. When the normalized distance is less than 0.3, the rate of temperature decrease is highest on the first surface and then decreases as the normalized distance increases.

5. Conclusions

In this paper, the steady-state rolling analysis, energy loss analysis and heat conduction analysis of the ISRFT are carried out by using the thermo-mechanical sequential coupling method under zero-pressure rated load and three velocity conditions. The SSTF distribution of the ISRFT under the action of thermo-mechanical coupling is obtained. The SSTF results of the ISRFT were as anticipated, and the honeycomb ISRFT did not effectively dissipate heat, as evidenced by the absence of a notable reduction in the maximum temperature on the outer surface of the inserts body. This indicates that the internal structure of the inserts body exerts a relatively limited influence on the maximum temperature of the outer surface. Consequently, it may be advisable to consider redesigning the outer surface of the inserts body to enhance heat dissipation. After a comparative analysis, the following conclusions are drawn:

• Under zero-pressure rated load conditions, for each 1 km/h increase in the operating speed of the ISRFT, the shoulder temperature rises by approximately 1.6 °C, the tread temperature by approximately 0.67 °C and the sidewall temperature by approximately 0.37 °C. The temperature rise of the bead and other components is relatively minor and less sensitive to speed.
• Compared to the original the ISRFT, the maximum temperature of the shoulder of the honeycomb ISRFT decreases by about 30 °C, and the SSTF distribution is more uniform. However, the maximum temperature of the tread increases by about 40 °C.
• Compared to the original inserts body, the four groups of the honeycomb inserts body exhibited a reduction in mass of 70.83%, 59.36%, 56.04% and 43.88%, respectively. This resulted in a notable enhancement in lightweight performance. The temperature of the honeycomb inserts body increased by 60.4 °C, 55.5 °C, 54.6 °C and 50 °C, respectively.

This paper is limited by space and only discusses the thermal–mechanical coupling results of the ISRFT under different speed conditions. On the basis of this paper, subsequent studies can include radial load, tire pressure, camber angle and other tire operating parameters in the research scope, and other materials or advanced composite materials can be selected to be applied to the inserts body. In addition, the orthogonal method can be considered to design the test to determine the test parameters and obtain more test output values accordingly.

With the use of the honeycomb inserts body, the constituent materials are reduced, and the manufacturing cost will be significantly reduced. This directly leads to a reduction in the amount of waste rubber produced, which is beneficial for the advancement of sustainable development and the maintenance of ecological health. In addition, the mechanical characteristics, grounding characteristics, longitudinal forces, lateral forces and aligning torque of the ISRFT under zero-pressure conditions will be changed. The study of the mechanical characteristics and thermal–mechanical coupling characteristics of the honeycomb ISRFT is a direction that can be expected in future research.