High-Precision Normal Stress Measurement Methods for Tire–Road Contact and Its Spatial and Frequency Domain Distribution Characteristics

Abstract

This study investigates measurement methods for and the distribution characteristics of normal stress within tire–road contact areas. A novel measurement method, integrating 3D scanning technology with bearing area curve (BAC) analysis, is proposed. This method quantifies the rubber penetration depth and calculates contact stress based on rubber deformation. The key innovation of this method lies in this integrated methodology for high-precision stress mapping. In the spatial domain, stress distribution is characterized by the percentage of area occupied by different stress intervals, while in the frequency domain, stress levels are analyzed at various frequencies. The results demonstrate that as the Mean Profile Depth (MPD) of the road texture increases, the areas under stress greater than 1.0 MPa increase, while the areas under stress less than 0.8 MPa decrease. However, when the MPD exceeds 0.7 mm, this effect becomes less pronounced. Higher loads and harder rubber reduce the proportion of areas under lower stress and increase the proportion under higher stress. Low-frequency (<800 1/m) stress components increase with an MPD up to 0.7 mm, beyond which they exhibit diminished sensitivity. Stress at the same frequency is not significantly affected by load variation but increases markedly with increasing rubber hardness. This research provides crucial insights into contact stress distribution, establishing a foundation for analyzing road friction and optimizing surface texture design oriented towards high-friction pavements.

1. Introduction

During travel, vehicles require sufficient acceleration, deceleration, and steering forces, with friction between the tire and road surface being the key source of these forces [1]. From the perspective of friction formation mechanisms, the stress during tire–road contact affects the compression of local structures on the contact surface, thereby determining the friction coefficient between them [2]. Additionally, stress concentration exacerbates issues such as the wear of tires and road surface macro/microtexture and the fatigue failure of tread materials [3,4,5]. Therefore, understanding and evaluating stress distribution in the interaction process are significant for ensuring vehicle safety, improving road material performance, and optimizing tire design.

To understand tire–road interactions, researchers have proposed intelligent tire monitoring systems using piezoelectric, electromagnetic, and triboelectric effects to obtain lateral, longitudinal, and vertical forces and friction coefficients [6,7,8,9,10]. In addition, previous studies have developed devices to monitor tire–road contact forces and moments in dynamic equilibrium, with their cores comprising three resistive strain gauges placed on the inner side of the rim to reflect the tire–road contact forces through rim deformation [11,12]. To obtain stress distributions on a more complex surface without affecting tire damage or deformation, researchers have developed a non-contact load measurement method based on image recognition to acquire the tire deflection under load, combined with tire pressure information and a tire weighing formula [13,14]; although these studies can achieve their purpose of obtaining resultant forces or moments, they cannot reflect the distribution pattern of contact stress and cannot meet the demand for revealing friction or road skid resistance formation mechanisms.

To obtain the tire–road contact stress, previous studies have developed devices based on sensor arrays capable of measuring in three directions, to obtain the longitudinal, vertical, and lateral traction [15,16,17]; in these testing devices, the size of individual sensors in the sensor array is typically larger than 15 mm, and the devices must be placed between the tire and road surface; thus, the measured contact stress is not suitable for evaluating the influence of road texture and cannot be used to explain road skid resistance formation mechanisms. The finite element method has also been used to obtain contact stress, wherein the road was simplified as a surface without roughness, and this method was used to establish a three-dimensional (3D) tire–road model to study the effects of friction forces on the tire–road contact stress distribution under different rolling conditions such as free rolling, braking/acceleration, and turning [18,19,20]. Liu et al. proposed a stress prediction model based on finite element models using machine-learning methods, greatly facilitating the acquisition of contact stress under different load, tire inflation pressure, and slip rate conditions [21]. Tire stress distribution models ignoring road roughness are sufficient to be used in the structural design of flexible pavements, even when loads are typically simplified into dual circular uniform distributions.

To obtain more refined contact stress, an Xsensor pressure mapping system based on flexible sensors with low stiffness that can be deformed with road roughness was developed [22]. A deformable pressure film was also used to test the stress distribution between tires and the road with different roughness levels [23]. The mapping system can provide limited precision in stress measurement due to the dispersion of local stress resulting from the stiffness of flexible sensors, and stress values tend to be underestimated. Srirangam used the CT scanning of road specimens to establish models considering the influence of the road surface texture on contact [24]. Liu et al., Peng et al., and Zhu et al. used point cloud data obtained by laser scanners to construct road surface roughness models and simulate road contact with tires, but these numerical models are typically difficult to validate due to a lack of real, detailed contact information [25,26,27]. Theoretically, Persson proposed a contact theory considering 3D surface multi-scale roughness, providing analytical solutions for rubber and road surfaces with multi-scale roughness [28]. Dubois first suggested roughly dividing surface roughness into two scales using the watershed algorithm, identifying large-scale stress before obtaining smaller-scale stress; introduced a higher-precision surface texture design; captured detailed contact information; and then used the convolution of force and Green’s function to obtain the tire’s dynamic response [29]. However, theoretical methods have many assumptions with unknown differences with respect to reality.

In summary, placing flexible sensors within the contact patch or using numerical simulations are the primary methodologies for investigating tire–road contact stresses. However, the geometric configuration and dimensions of the contacting surfaces significantly affect the stress concentration and distribution. These methods cannot obtain stress distributions matching road roughness scales due to the insufficient consideration of road roughness and its effects. To enhance the precision of contact stress acquisition methods, this study proposes a high-precision methodology for measuring normal stresses specifically in rubber–road contact scenarios, and after comparison and verification with existing methods, it explores the impact of road texture characteristics on stress distribution patterns. This research would advance the understanding of tire–road interaction by providing a more accurate and detailed method to quantify contact stresses. Industrially, the proposed method offers a valuable tool for pavement engineers and tire manufacturers to optimize surface texture designs aimed at high-friction pavements, supporting the development of safer and more durable road infrastructure and tires.

2. Tire–Road Stress Distribution Measurement and Its Spatiotemporal Characterization

2.1. Measurement Principle

2.1.1. Assumptions

The tire–road contact normal stress testing in this study is primarily based on the following assumptions:

Assumption 1. 

Rubber is modeled as a series of independent springs (Winkler foundation model). The displacement at any point on the rubber surface is proportional to the normal stress applied at that specific point and independent of stresses acting elsewhere on the surface. Additionally, due to the large spacing between rough peaks of the road surface, deformations of the rubber surface caused by individual peaks can be treated as independent [30]. Thus, strain at any point can be derived from its displacement, and subsequently, stress at that point can be determined.

Assumption 2. 

Rubber is assumed to behave as a linear elastic material. While rubber exhibits obvious viscoelastic characteristics across most temperature ranges, previous studies have applied linear elastic assumptions in numerical simulations of rubber–road contact, with results showing a good correlation with experimentally measured data [31]. Therefore, the normal stress at any point can be calculated as the product of the materials’ elastic modulus and strain.

Assumption 3. 

The contact interface between the rubber block and road surface is as shown in Figure 1f, where the lowest points of the road texture that are in contact with and enveloped by rubber are assumed to lie on a common horizontal plane. The Indentor method in the ROSANNE project also assumed similar contact conditions when solving for the enveloping profile, in which coated putty applied to surfaces was used to visualize the contact, and the validity of this assumption was confirmed [32].

Under these three assumptions, the normal stress at each point within the contact area can be obtained by multiplying the material’s elastic modulus by the corresponding strain. Since the rubber layer thickness and its compression at every point across the contact area are already known, the strain of rubber at each point on the surface of the rubber block within the contact area can be determined. Specifically, the sum of normal forces acting within the contact region must equal the total applied load, and A refers to the projected area of the actual contact area on the horizontal plane; thus, the stress at each point across the contact area can be determined.

2.1.2. Test Methods

Based on the aforementioned assumptions, the pivotal step in determining the normal stress is the quantification of the rubber penetration depth into the road texture. The testing process for this penetration depth is divided into two phases (Figure 1).

First, obtain the bearing area ratio curve of the surface using a 3D scanner (Creaform, Lévis, QC, Canada) to capture an m × n road surface texture point cloud elevation matrix Z, assuming a point cloud resolution of res, where the vertical projection side lengths of the road surface point cloud are m × res and n × res, respectively, with a projection area of A (Figure 1a); define a plane at a depth d below the highest point of the surface, where the projection area of points in the upper portion of matrix Z that lie above this plane is denoted as A_d (Figure 1b). The ratio of A_d to A represents the proportion P of the bearing area to the total area at a given height d. By traversing the surface elevation data from the highest to the lowest point, the relationship between P and the varying h values constitutes the bearing area ratio curve (Figure 1c).

The second phase focuses on experimentally determining the contact area bearing load and interpolating to obtain the penetration depth. Specifically, use a loading equipment to apply a controlled pressure (Figure 1d), test the contact area S between the rubber and the road specimen under specified pressure conditions, and calculate the ratio P of the contact area to the total projected area (Figure 1e). As shown in Figure 1f, when the rubber is squeezed against the asphalt pavement, it deforms and envelops parts of the surface. This enveloping phenomenon corresponds to the contact interface between the rubber and the road. By interpolating on the bearing area ratio curve obtained in the first phase, the depth corresponding to the P value, which represents the depth of rubber penetration into the road surface under the given load condition, is determined (Figure 1c). The loading equipment features a vertically movable loading head connected to a rubber specimen (60 mm diameter) bonded to a steel plate. This facilitates the effective load transfer to the pavement specimen with dimensions of 60 × 60 mm² placed on the lower plate. Precise alignment between the loading head and the road specimen is meticulously achieved prior to testing. A servo-hydraulic Universal Testing Machine (UTM) supplied by IPC is employed, capable of delivering the necessary load control and ensuring proper alignment.

2.2. Comparative Verification of Normal Stress on Interface

Pressure film and theoretical solutions are widely employed in contact characteristic studies. Therefore, this paper applies these two methods for comparison.

2.2.1. Test Surface

For the purpose of facilitating comparison, this paper designed two regular surfaces as shown in Figure 2. The first surface featured five hemispheres, each with a diameter of 16 mm. This specific configuration was chosen to simulate the nominal size of typical pavement aggregate particles. The second surface was a wave surface defined by the equation z(x, y) = 2.5cos(0.4x)cos(0.4y), possessing an amplitude of 2.5 mm. This amplitude was selected to correspond to the typical roughness amplitude observed on actual pavement surfaces.

This study selected 0.73 MPa as the test load, which approximated the typical contact pressure experienced between heavy vehicle tires and road surfaces. Figure 2b shows the contact areas between the aforementioned surfaces and the rubber. These contact areas were measured using the experimental setup described in Section 2.1.2.

2.2.2. Stress Testing

The stress values during the contact between the rubber and the road surface exhibit a wide distribution range. Therefore, this study selected Prescale pressure films (Fujifilm Corporation, Tokyo, Japan), with ranges from LW to 4 LW. The corresponding stress intervals for these film types are shown in

Table 1. Prescale pressure film model and the corresponding pressure intervals.

Prescale Pressure Film Model	Testing Pressure Range (MPa)
Low pressure (LW)	2.5~10
Super low pressure (LLW)	0.5~2.5
Ultra super low pressure (LLLW)	0.1~0.6
Extremely low pressure (4LW)	0.05~0.2

. This selection was made to ensure comprehensive coverage of the entire pressure distribution spectrum encountered during contact.

During testing, the pressure load was maintained for a minimum duration of 5 s to ensure the film fully developed color (Figure 3). Recognizing that ambient temperature and relative humidity affect the film’s color development, potentially causing differences between the analyzed stress values and the actual pressures, the environmental conditions were recorded during testing.

After the pressure film develops color indicating contact, the stress values recorded by colors from different types of pressure films require analysis. Initially, pressure film images were captured using an Epson Perfection V330 photo scanner as shown in Figure 4a (Seiko Epson Corporation, Nagano, Japan). Subsequently, based on the temperature and humidity conditions recorded during testing, the appropriate color density–stress value curve (Figure 4b) was chosen. This curve defines the relationship between color density and stress values. Then, the stress results were derived from the film images using this color curve. Analyzing various types of pressure films requires using respective color charts, and this process was completed using the dedicated stress analysis software FPD-8010E (ver. 1.5.0.3). After analysis, the pressure film image visually represents the pressure value through variations in red color intensity. Colors outside the expected range, such as yellow or green are shown (Figure 4c), indicating conditions beyond the normal test scope. After removing these out-of-range pressure data points, effective stress measurements were obtained (Figure 4d).

To obtain the effective stress distribution across the entire contact area, stress measurements were initially conducted on partial areas using individual specification films (Figure 5a–d). The Iterative Closest Points (ICP) algorithm was then employed to precisely align and merge these partial stress distributions, ensuring a comprehensive and accurate representation of the stress over the entire contact area (Figure 5e). The ICP algorithm iteratively minimized the distances between corresponding stress patterns and boundary conditions, refining the alignment and merging process to address any discrepancies. This meticulous approach enhanced the reliability and validity of the stress analysis, providing valuable insights for further engineering and design considerations.

2.2.3. Analytical Solution Based on Hertz Contact Theory

Contacts satisfying the following assumptions are called Hertz contacts: (1) Small deformation occurs within the contact area. (2) The contact area forms an elliptical shape. (3) The contacting objects can be treated as elastic half-spaces, where only distributed normal pressure acts on the contact interface. Hertz theory provides results that closely approximate reality under these assumptions, particularly when the surface profiles near the contact interface can be approximated by second-order paraboloids, and the contact area is significantly smaller than both the overall dimensions of the objects and their relative radii of curvature. Therefore, the stress distribution for a single hemisphere surface is shown by Equation (1).

$$P={\left({\displaystyle \frac{6F{E}^{*2}}{{\pi }^3{R}^2}}\right)}^{1/3}{\left[1-r^2{\left({\displaystyle \frac{3FR}{{4E}^{*}}}\right)}^{-2/3}\right]}^{1/2}, r\le a$$

(1)

where

• P—Normal stress on the interface;
• F—Force acting on the contact medium;
• E*—Equivalent elastic modulus;
• R—Radius of the sphere;
• r—Position within the contact area, expressed in polar coordinates;
• a—Radius of the areas where stress acts.

Regarding the five-hemisphere and wave surfaces shown in Figure 2, the distance between peaks is sufficiently large compared to the contact area on each peak, thus allowing the deformations to be considered independent. The radius of the five hemispheres is 8 mm, and the radius of the wave surface is represented by the Gaussian curvature radius, which depends on the two principal curvature radii of each small peak. Numerical analysis results indicate that the Gaussian curvature radius of small peaks on the wave surface is approximately 2.5 mm. The elastic modulus of the rubber was taken as 6.8 MPa based on previous studies, and the stress distributions were calculated accordingly.

2.2.4. Comparison of the Stress Results

Comparing the stress distribution obtained from the pressure film method (Figure 6a,d) and the method proposed in this paper (Figure 6b,e) reveals that the stress from both methods reaches maximum values at the surface vertices; the differences in maximum stress are approximately 0.4 MPa for the five-hemisphere surface and 0.1 MPa for the wave surface, respectively. The relative difference in stress values across the surface is consistently below 5%. Therefore, these findings indicate that the stress test results obtained using the method introduced in this paper are reliable.

Compared to the contact area determined by the method proposed in this study and the pressure film test results, the contact area derived from the analytical solution is significantly larger (Figure 6c,f). This discrepancy arises because Hertz contact theory posits that stress acts within areas of radius not exceeding a, and for contact between rigid spheres and elastic half-spaces, a cannot exceed the radius R of the sphere, necessitating the fulfillment of the following relationship (Equations (2) and (3)).

$$a={\left({\displaystyle \frac{3FR}{{4E}^{*}}}\right)}^{1/3}$$

(2)

$$R\ge \sqrt{{\displaystyle \frac{3F}{{4E}^{*}}}}$$

(3)

On the surface composed of five hemispheres, the stress area is larger because the radius is smaller than the allowed minimum value typically associated with concentrated stress, necessitating a larger area to distribute the load. Similarly, on the wave surface, where the radius is also small, stress overlap occurs due to the proximity of the surface vertices. Given that the aggregate diameter on road surfaces typically ranges between 4.75 and 9.5 mm, its interaction with rubber more closely resembles contact with a wave surface and rubber. This suggests that the elastic half-space solution under Hertz contact theory is not well-suited for studying the stress distribution between rubber and the road surface.

2.3. Stress Characterization Based on Spatial and Frequency Domain

Spatial domain stress information refers to the numerical magnitude of stress at each position on the contact interface, which is essential for understanding performance related to the interaction between tread rubber and the road surface. For example, on wet road surfaces, higher stress within the contact area facilitates water drainage, enabling more direct tire–road contact, and thus enhancing friction. Additionally, higher stress is closely related to tire wear mechanisms.

Different road surfaces exhibit distinct surface pavement textures, causing variations in the actual contact areas. However, the nominal contact area, which is the bottom area of the rubber block when it contacts the road surface, remains constant. To facilitate investigation of the pavement texture effects on stress distribution, this study divided stress into specific intervals and introduced the stress area percentage. This metric is defined as the ratio of the area corresponding to a specified stress interval to the nominal contact area, serving as an evaluation index for spatial domain stress distribution.

From the frequency domain perspective, energy dissipation within tread rubber is widely considered as an important cause of friction, tire–road noise, and tire rolling resistance. The amount of energy dissipation is determined by the frequency and magnitude of stress acting on the rubber. In characterizing stress distribution in the frequency domain, this study adopts frequency domain analysis methods for pavement texture from the ISO/TS 13473-4 standards [33], using the stress level to measure the stress amplitude for different frequency components. The stress level can be calculated using Equation (4).

$$L_{\mathrm{pres},\lambda }=10\lg {\displaystyle \frac{Z_{\mathrm{p},\lambda }}{{a^2}_{\mathrm{ref}}}}$$

(4)

where

λ—Center wavelength of 1/3 octave band (m);

L_pres,λ—Stress level corresponding to 1/3 octave band with λ as the center wavelength (dB);

a_ref—Reference value of stress amplitude, 10⁻⁶ (Pa);

Z_p,λ—Energy within 1/3 octave band with λ as the center wavelength (Pa²·m).

This definition yields the frequency and magnitude of the excitation force generated by the road texture on the rubber per unit time, assuming a relative speed of 1 m/s between the tire/rubber slides on the road surface. When the sliding speed of the rubber block increases to v m/s, the frequency of the excitation force acting on the rubber becomes v times the original, with the stress level unchanged.

3. Relationship Between Pavement Texture and Stress Distribution

This study investigated factors influencing stress distribution by examining the contact area between rubber blocks and 44 road surfaces exhibiting significantly different surface morphologies. This experiments were conducted under two rubber hardness conditions with 55 and 68 HSA, four normal loads of 0.18, 0.35, 0.54, and 0.73 MPa, and a constant temperature of 25 °C. The selection of the 68 HSA value was based on its close approximation to the hardness of a typical tire under static conditions at normal temperature, which is typically standardized at 25 °C. However, during rolling, the viscoelastic nature of rubber induces heat accumulation, leading to a temperature increase. Consequently, elevated temperatures can reduce rubber hardness below its static value. Analysis suggests that the hardness may decrease to approximately 55 HSA [34]. Furthermore, 55 HSA rubber was selected as it represents the standard material property used in the British Pendulum Tester, which is employed to evaluate pavement–tire friction under standardized conditions.

Based on bearing area ratio curves obtained from pavement texture data and the measured contact areas, the stress distribution within the contact area under corresponding conditions was determined according to the principles and measurement method introduced in Section 2.1.1 and Section 2.1.2.

Stress characterization followed the methods introduced in Section 2.3. To ensure the representativeness of stress distribution, 10 pavement profiles were extracted from the road surface point cloud data, with the average taken as the stress frequency domain analysis result for each surface.

3.1. Influence of Vertical Load

3.1.1. Spatial Domain Stress Distribution

Figure 7 illustrates the relationship between stress distribution in the spatial domain and pavement texture characteristics. The vertical axis denotes the stress area percentage, which refers to the ratio of the area corresponding to a specific stress interval to the nominal contact area. The stress intervals are uniformly set at 0.2 MPa increments according to the stress magnitude. The horizontal axis shows the Mean Profile Depth (MPD), which quantifies the average depth of the pavement texture, calculated according to the ISO 13473–1 standard [33].

Figure 7 also illustrates how the vertical load affects the trend of stress area percentages across different stress intervals as the MPD varies. Here, the vertical axis represents the stress area percentage, while the horizontal axis shows the MPD. Various colored and shaped markers are used to distinguish different loading conditions, with rubber hardness consistently at 55 HSA.

As Figure 7 illustrates, the relationship between the stress area percentage and the MPD is influenced by the stress magnitude. At lower stress, as the MPD increases, the stress area percentage corresponding to that specific stress interval gradually decreases. Conversely, at higher stress, the stress area percentage gradually increases with rising MPD. This indicates that the trend of stress area percentage with MPD reverses as the stress increases, although a transitional phase precedes the clear emergence of this reversal.

Notably, when the MPD exceeds 0.7 mm, the stress area percentages for all stress intervals stabilize and exhibit minimal further change. This reversal in the relationship between stress area percentage and MPD may be related to the contact area: on surfaces with a smaller MPD, the actual contact area between the rubber and the road surface is larger. Since the total stress must balance the applied load, the stress values distributed across these contact areas are correspondingly smaller, resulting in larger areas associated with lower stress intervals, and vice versa.

The influence of vertical load on the trends of area percentage with MPD varies across different stress intervals. For low-stress intervals (0–0.2 MPa), under low-load conditions (0.18 MPa), the area percentage values are higher and decrease significantly with increasing MPD, stabilizing once the MPD exceeds approximately 0.7 mm. As load gradually increases, this decreasing trend progressively weakens. When the load reaches 0.7 MPa, the area percentage remains virtually constant regardless of increasing MPD.

Within the 0.2–0.4 MPa stress interval, under all load conditions investigated in this study, the area percentage values are relatively similar and consistently exhibit a relatively clear decreasing trend with increasing MPD, with comparable rates of decrease.

Examining the 0.6–0.8 MPa stress interval reveals that the area percentage under high-load conditions is significantly higher than under low-load conditions. As the load increases from 0.18 to 0.73 MPa, the trend of area percentage change with MPD transitions from increasing to decreasing, representing the reversal of the trend. This reversal occurs at different stress intervals depending on the load conditions. For the 0.18 MPa load, it appears at 0.6–0.8 MPa; under 0.35 and 0.54 MPa load conditions, it appears at the 0.8–1.0 MPa interval; and under the 0.73 MPa load condition, it begins to emerge when the stress interval reaches 1.0–1.2 MPa.

With larger loads, the stress interval at which the reversal in area percentage change with MPD occurs shifts to higher stress. This suggests that on surfaces with an MPD of less than 0.7 mm, as the load increases, the area corresponding to lower stress increases more rapidly, and the rubber–road contact tends to generate a greater proportion of intermediate-range stress (0.6–0.8 MPa) to bear the load.

For stress intervals above 0.8 MPa, the area percentage under low-load conditions is minimal and remains almost unchanged with varying MPD. However, under loads exceeding 0.35 MPa, and when the MPD does not exceed 0.7 mm, the stress area percentage increases significantly with rising MPD. Given that the nominal contact stress between vehicle tires and road surfaces typically exceeds 0.35 MPa, the area subjected to high stress increases substantially as the MPD rises to 0.7 mm, after which it stabilizes. That is, heavier vehicle loads result in a larger percentage of the contact area experiencing high stress.

3.1.2. Frequency Domain Stress Distribution

Figure 8 illustrates the trends of stress levels across different frequency ranges with changing MPD. The scatter plot reveals distinct turning points within the trend. Furthermore, it demonstrates the influence of vertical load on the behavior of stress levels corresponding to each frequency range as MPD changes. Different colored and shaped markers represent different load conditions, with rubber hardness consistently maintained at HSA55.

Overall, when the MPD is less than 0.7 mm, the stress levels across different frequencies gradually increase with rising MPD; when the MPD exceeds approximately 0.7 mm, the stress levels across all frequencies decrease. Therefore, at an MPD of about 0.7 mm, the excitation force generated by the road texture reaches its maximum amplitude across various frequency ranges.

The stress levels at each frequency under different load conditions are tightly clustered, making it difficult to discern the specific impact of load on the stress level. Based on this observation, this study concludes that the overall trends of the stress level with MPD across different frequencies are minimally affected by load. Additionally, when the center frequency of the one-third octave band is less than 800 1/m, under all tested load conditions, stress levels first increase and then stabilize with increasing MPD (MPD < 0.7 mm); for one-third octave bands with higher center frequencies, stress levels initially increase with rising MPD, but when the MPD exceeds approximately 0.7 mm, the stress levels drop sharply compared to those on surfaces with a smaller MPD before stabilizing. In summary, under the different load conditions investigated, when the road surface MPD reaches 0.7 mm, the stress levels across almost nearly all frequency ranges attain their maximum values.

3.2. Influence of Rubber Hardness

3.2.1. Spatial Domain Stress Distribution

Figure 9 illustrates the influence of rubber hardness on the spatial domain stress distribution. The vertical axis represents the stress area percentage, while the horizontal axis indicates the MPD. Different colored and shaped markers represent various rubber hardness values, with the load magnitude consistently maintained at 0.53 MPa.

This study primarily investigates the influence of rubber hardness from three distinct perspectives: stress distribution uniformity, the stress area percentage, and the variation in this area percentage with MPD. Initially, regarding stress exceeding 1.2 MPa, the analysis reveals that hard rubber consistently exhibits a stress contact area across all road surfaces examined in this study. In contrast, for soft rubber, the contact area with stress exceeding 1.2 MPa can be observed only on a few surfaces. Additionally, for stress less than 1.0 MPa, the area percentage between hard rubber and the road surface is smaller compared to that for soft rubber and the road surface. Conversely, for stress above 1.4 MPa, the area percentage between soft rubber and the road surface is smaller than that between hard rubber and the road surface. These observations suggest that harder rubber produces more uniform and higher rubber–road stress.

As rubber hardness increases, the trends in the area percentages corresponding to each specific stress interval evolve differently in relation to the MPD. For stress lower than 0.8 MPa, the area percentages for both soft and hard rubber demonstrate a decreasing trend with increasing MPD, eventually stabilizing when the MPD exceeds 0.7 mm. When the stress value exceeds 0.8 MPa, a transformation in the trend becomes evident. Specifically, the area percentage corresponding to stress between soft rubber and the road surface exhibits an increasing trend with rising MPD when the MPD is less than 0.7 mm, followed by a slight decrease or stabilization. Similarly, the area percentage corresponding to stress between hard rubber and the road surface clearly demonstrates this transformation for stress greater than 1.4 MPa.

3.2.2. Frequency Domain Stress Distribution

Figure 10 represents the trends of stress levels corresponding to each one-third octave band with different center frequencies, as the MPD changes under varying rubber hardness conditions. Different colored and shaped markers denote distinct rubber hardness values, with a constant load at 0.53 MPa.

The data reveals that harder rubber produces higher stress levels across all one-third octave bands, regardless of their center frequencies. This phenomenon may be attributed to the fact that harder rubber results in a smaller contact area with the road surface, resulting in higher stress in the contact area. However, this variation in stress distribution due to rubber hardness does not alter the fundamental effect of MPD on stress levels. Specifically, the energy loss remains greatest when the MPD is approximately 0.7 mm.

4. Discussion

In this study, the rubber–road contact stress was measured under static loading conditions. The results characterize the frequency and magnitude of the excitation force generated by the road texture acting on the rubber per unit time, corresponding to a relative velocity of 1 m/s. From the perspective of the friction formation mechanism, the energy dissipated by the rubber directly determines the high-speed skid resistance performance of the pavement. Equation (5) shows the calculation method for the energy dissipation power per unit volume of the rubber [35].

$$\overline{\mathrm{p}}=-{\displaystyle \frac{1}{2}}\omega {{\sigma }_0}^2\mathrm{Im}\left({\displaystyle \frac{1}{\widehat{G}\left(\omega \right)}}\right)$$

(5)

where

• $\overline{ \mathrm{P} }$—Dissipated energy power of stress σ₀ in unit volume.
• ω—Frequency of stress σ₀, corresponding to the center wavelength.
• |$\widehat{G}$*(ω)|—Complex modulus of rubber, which is a function of frequency ω.

At each center frequency of stress levels acting on the tread rubber, a higher stress causes a greater energy loss per unit volume of the rubber. If the area affected by the excitation force on the rubber is directly related to the penetration depth, then when the road’s MPD reaches 0.7 mm, the friction value per unit volume between the rubber and the road surface reaches its maximum. Furthermore, even with varying loads, an MPD of 0.7 mm consistently corresponds to the peak rubber–road friction.

It should be noted that harder rubber generates higher stress but not necessarily a larger energy dissipation per unit volume. This is because the viscoelastic characteristics of the rubber change with hardness. However, this principle helps to explain why when the same type of rubber is used at lower temperatures or higher speeds, the increased rubber hardness generates larger stress amplitudes across different frequencies, making the rubber more susceptible to wear.

The UK Transport Research Laboratory (TRL) tested friction coefficients on multiple road sections under wet, high-speed conditions and reported that changes in the friction coefficient with road texture depth are limited. Specifically, when the estimated mean texture depth SMTD of asphalt pavement is less than 0.8 mm, the friction coefficient under high speed increases with increasing SMTD. However, when SMTD exceeds 0.8 mm, the friction coefficient is no longer significantly affected by SMTD. Additionally, the rate at which friction decreases with increasing speed also ceases to diminish as SMTD increases beyond 0.8 mm [36].

Based on existing research correlating the Mean Profile Depth (MPD) and SMTD, an SMTD of approximately 0.8 mm corresponds to an MPD of approximately 0.7 mm. Consequently, the conclusion derived from stress frequency domain analysis aligns with the findings from the UK TRL’s friction coefficient testing at high speeds. It should be noted that the relationship between the road texture and friction coefficient tested by the TRL under wet conditions is also influenced by the tire tread pattern, which serves as a drainage function.

Furthermore, the conclusion of this study is based on test data obtained under dry conditions, whereas the relationship established by the TRL was derived from tests under wet, high-speed conditions. When a high-speed rolling tire contacts a wet road surface, accumulated water fills the microtexture due to insufficient drainage, reducing the pavement texture, contributing to contact compared to dry conditions. That is, the effective contact area decreases. Nevertheless, theoretical analyses suggest that larger surface texture wavelengths influence the depth of rubber penetration into the road surface, and the stress acquisition approach in this study is based on the penetration depth. Therefore, even under wet conditions, it is likely that the conclusion that an MPD of 0.7 mm generates maximum energy loss and consequently maximum friction remains valid. Certainly, subsequent studies should incorporate actual testing under wet conditions to enable more comprehensive analysis.

Previous studies used laser scanned point cloud data for road roughness modeling and FEM-based tire contact simulations [25,26,27]. However, obtaining the stress distributed on the pavement texture remains problematic because defining the real interaction boundary is challenging. Therefore, further improvements are needed regarding model accuracy, boundary conditions, and the load cases considered.

5. Conclusions

This study innovatively proposes a high-precision tire–road normal stress testing method and characterizes stress distribution through spatial–frequency domain analysis, thoroughly examining the influence of different road textures on contact stress distribution characteristics. The following main conclusions are drawn:

• The normal stress testing method is based on the penetration depth of rubber blocks on the road surface, and the relationship between rubber deformation and stress. Pressure film testing verified the measurement accuracy of this method with maximum measurement deviation below 5%, confirming its reliability and applicability.
• Mean Profile Depth (MPD) significantly impacts the stress spatial distribution. Research demonstrates that when the MPD is less than 0.7 mm, increasing MPD decreases the proportion of the area under stress less than 0.8 MPa while increasing the proportion under stress greater than 1.0 MPa. However, when the MPD exceeds 0.7 mm, the stress distribution stabilizes and exhibits minimal further change with increasing MPD.
• In the frequency domain analysis, stress levels at frequencies below 800 1/m increase significantly with MPD and stabilize when the MPD exceeds 0.7 mm. This finding verifies that an MPD of approximately 0.7 mm represents a critical threshold for optimizing rubber energy loss, thereby maximizing friction under wet, high-speed conditions.
• Both load and the rubber hardness significantly influence the stress distribution. Increasing load decreases the proportion of the area under low stress while increasing the proportion under high stress. Similarly, increasing rubber hardness increases the proportion of area under high stress and significantly elevates stress levels across various frequencies.

This research expands the technical methods of contact mechanics testing, providing novel solutions for complex contact problems. It establishes a theoretical foundation for optimizing road surface textures aimed at high-friction pavements and supporting the development of safer and more durable road infrastructure and tires. Despite these significant contributions, certain limitations warrant further exploration. For instance, this research primarily relies on results from static load conditions, whereas the dynamic responses of tire–pavement contact during actual driving may produce different stress distributions. Moreover, considering the nonlinear viscoelastic characteristics of rubber and temperature effects, future research should deepen the analysis of tire–pavement contact stress under dynamic loads. Furthermore, extending the investigation to encompass a broader range of tire compounds, such as those with varying stiffness, materials, or patterns, would provide deeper insights into how tire composition influences stress transfer. Similarly, exploring interactions properties with different pavement surfaces is crucial to refine testing methods and models, ultimately yielding more comprehensive and accurate conclusions based on actual driving conditions and material combinations.