1 Introduction

Wear performance is an important index of many rubber products, especially for the tire. It directly affects the safety and life of products. Many scholars have conducted a lot of theoretical and experimental researches on the wear of rubber in order to better understand the wear performance [1]. Also, evaluation system of wear characteristics for rubber composites at room temperature is established. However, there still exists a big difference between the actual wear and normal wear of tires and other rubber products. The reason is the hysteresis heat which causes high temperature in the rolling process of tire and leads to wear. Previous wear evaluation system cannot accurately reflect the high temperature abrasion performance and characteristics [2–4]. Therefore, it is necessary to deeply study wear performance and mechanism of rubber composites at the high temperature.

Akron and DIN abrasion tests were carried out to test wear performance of the rubber composite with different filler (nano-carbon black, alumina, clay and so on), different base material (natural rubber, Styrene Butadiene rubber, silicone rubber, Ethylene Propylene Diene Monomer, etc.), and different test conditions (pressure, surface severity) [5, 6]. Atomic Force Microscope (AFM) and Scanning Electron Microscope (SEM) were employed to investigate the wear surface and further reveal the wear mechanism. The surface abrasion pattern of rubber composite is not always generated. For instance, the wear surface does not produce abrasion pattern at low loads; instead, the wear pattern appears when the load exceeds beyond a critical value. A series of abrasion pattern are observed which are perpendicular to the sliding direction and mutually parallel [7–9].

The above research works only focused on examining the wear surface and did not explore the relationship and laws between the morphology and wear performance.

Fractal geometry is an effective tool to study the surface morphology of rubber composites. The basic theory of fractal geometry is an important branch of nonlinear science, which is widely used to describe and distinguish complicated morphology in nature since its introduction by Mandelbrot in the 1970s. The wear surface of the rubber composites has characteristics of self-similarity and self-affinity. Both local and overall characteristics of rubber surface can be described by multifractal characteristics [10–12].

Fractal theory is mainly applied to study the scaling properties of an irregular shape, a physical quantity or a natural phenomenon. This scaling property can be expressed mathematically by the concept of self-similarity or self-affinity. Despite magnification of the object, this behavior suggests that a fractal object appears statistically similar [13].

Rubber wear surface is similar to wear surface of the cylinder liner and the machined surface. Similarly, it has the fractal characteristics of randomness, self-similarity, self-affinity and multi-scale. Therefore, Fractal theory and method can be used to describe and evaluate the morphology characteristics of rubber wear surface [14].

Tire rubber wear testing is mainly performed at room temperature, which is different from its actual usability temperature. In the actual rolling process, tire temperature continuously rises to 60–80 °C due to the periodic stress and strain hysteresis of rubber. Therefore, in current work, a new wear testing instrument is designed for a more realistic study.

The morphology of rubber wear surface is also analyzed at different temperatures by studying the homogeneity and irregularity. Wear surface morphology of the rubber composites is viewed through an Olympus LEXT OLS4100 3D measuring laser microscope.

The characteristic parameters of multifractal spectrum are then applied to describe the complicated and irregular rubber wear. And spatial phenomenon is defined. The relation between characteristic parameters and wear is established and the fractal characterization of wear rubber surface is realized.

2 Materials and Methods

Fractality can be described briefly as a rough or discrete geometry figure that can be subdivided into infinite parts, and each part maintains certain similarity to the overall structure. Fractal dimension of a straight line, area, volume is respectively 1, 2, 3 in Euclidean geometry. Contrary to the Euclidean geometry, the dimensions of fractal geometry are all fractional. If the topology dimension of an image is Dt, the dimensions of fractal geometry is D, then Dt < D < Dt + 1.

In the box-counting method of multifractal theory, the fractal image is covered by square box with side length of ε, and nij represents the number of pixels of the fractal in the box of (i, j), \( \sum {n_{ij} } \) represents the number of pixels of all fractal numbers. The fractal image probability measure Pij(ε) in every ε × ε box is defined as,

$$ P_{ij} \left( \varepsilon \right) = \frac{{n_{ij} }}{{\sum {n_{ij} } }} $$
(1)

In a scale-free, self-similar area, the probability measure Pij(ε) can be written as,

$$ P_{ij} \left( \varepsilon \right) \propto \varepsilon_{ij}^{{\alpha_{ij} }} $$
(2)

where αij is the dimension of the fractal image in a small square box, which is called the local fractal dimension also the singularity exponent,

$$ \alpha = \mathop {\lim }\limits_{\varepsilon \to 0} \frac{\log P\left( \varepsilon \right)}{\log \varepsilon } $$
(3)

Then, Δα = αmax − αmin defines the disorder of probability measures in overall fractal area, as well as the complexity of the surface. A larger Δα represents a lower surface uniformity of rubber.

Dimension function f(α) is defined as a subset of singularity exponent α,

$$ f(\alpha ) = - \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{\log N_{\alpha } (\varepsilon )}}{\log \varepsilon } $$
(4)

The complexity and irregularity of the wear surface is mainly shown by Δf(α) = f(αmin) − f(αmax). When Δf(α) > 0, the multifractal spectrum curve is left-hook-shaped, so the subset of maximum probability numbers are less than the minimal ones. Thus, most of the wear rubber surface is comprised of sharp peaks with the relatively lower height. When Δf(α) < 0, the multifractal spectrum curve is right-hook-shaped, so the subset of maximum probability numbers is larger than the minimal ones. Hence, most of the wear rubber surface contains very high blunt peaks.

Because of the unique one to one correspondence between α and f(α), there is a dimension spectrum to describe multifractal properties, called the multifractal spectrum.

The sum of q square of Pij(ε) can be expressed in partition function χq(ε), and the partition function τ(q) is related to q as,

$$ \tau \left( q \right) = \frac{{\ln \chi_{q} \left( \varepsilon \right)}}{\ln \varepsilon }\left( {\varepsilon \to 0} \right) $$
(5)

where τ(q) is called mass index; q is called weighting factor.

Where mass index τ(q) can be written in the form,

$$ \alpha = \frac{d\tau \left( q \right)}{dq} $$
(6)

From the Legendre conversion of τ(q) and q, f(α) is written as,

$$ f\left( \alpha \right) = \alpha q - \tau \left( q \right) $$
(7)

In this paper, multifractal spectrum of the wear rubber surface is obtained through previous processes. The gray level images were covered by small squares with the 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 of the wear surface length, and the corresponding probability Pij(ε) with weighting factor q = [− 20, 20] are then calculated using a Matlab program written for fractal dimension and multifractal spectrum analysis.

The experimental setup includes rubber mixer (type XSM-500), roll mill (type BL-6175-BL), plate vulcanizing apparatus (type HS-100T-FTMO-2PT), 3D measuring laser microscope (type LEXT OLS4100) and the new type wear testing instrument.

The new type wear testing instrument is designed for simulating the high temperature rolling friction process of the tire rubber. The new wear testing instrument is composed of an adjustable-speed motor, a speed controller, a sandglass support, a sand funnel, a grinding wheel and support, a counterbalance, a load lever with scale, heating wires with voltage regulator, thermocouples and temperature controller, a slip ring, an adjusting wheel and plate, a screw, a lock nut and adjusting nut, and a base.

This newly designed instrument can process rubber wear testing under different temperatures, loads, speeds, and angles. It can provide a more realistic and accurate description of the tire wear process and different aspects in practical situations.

The tread formula of all-steel radial tire is following: Natural rubber 100 parts, N330 carbon black 37.3 parts, White carbon black 15 parts, Zinc oxide 3.6 parts, Stearic acid 2 parts, Plasticizer A 2 parts, Antioxidant RD 1.5 parts, Antioxidant 6PPD 2 parts, Silane coupling agent 3 parts, N-(oxidiethylene)-2-benzothiazolyl sulfonamide 1.5 parts, Sulfur 1 part.

Rubber samples are prepared by the means of plate vulcanizing apparatus. To prepare rubber strip specimen, rubber compound is placed in the rheometer mold cavity for 15 min at temperature 150 °C, and pressure of 15 MPa.

The strips are then convolved and attached to a special rubber wheel before drying them in an oven for firm bonding.

Next, the wear testing is performed on pre-polished strips for one hour using a new testing instrument, and then strip samples are cleared for weighing using a balance with the accuracy of 0.001 g.

The imaging process is carried out with OLS4100 3D measuring laser microscope which is equipped with a dual confocal scanning system. The dual confocal scanning can remove the light spots in a fuzzy image and captures planes with same height. Due to these particular qualities, this microscope can be used to capture precise images from samples of materials with different reflective characteristics. Also, the dual confocal scanning generated by round light holes guarantees consistency in different directions.

With the OLS4100 laser microscope, a precise 3D image of the wear surface with a resolution of 10 nm in z-axis and an x–y plane resolution of 0.12 μm can be obtained in a noncontact manner by a laser of short-wavelength i.e. 405 nm.

The steps followed in gray scale image acquisition are as under:

  1. 1.

    The microscope control software is started to finish the initialization of system automatically.

  2. 2.

    The Z axis coarse adjusting knob is unlocked. The 20 times objective lens and zoom 1× is selected in visible light mode. The coarse image is adjusted based on the Z axis.

  3. 3.

    The image to a clear state is tuned and adjusted to fine focus. Magnification is switched to 20 times before starting observation and measurement.

  4. 4.

    Laser mode is activated and fine scanning mode is selected. Because sample is uneven, the highest position (upper limit) and the lowest position of the sample (the lower limit) is properly set.

  5. 5.

    After the ‘get’ button is pressed, OLS4100 automatically starts taking photos. It took about 10 s to finish photo shooting process photo. Then, software automatically popped up pictures of samples and scale is generated.

  6. 6.

    Saving image.

The imaging principle of the microscope ensures the coaxial of light and excludes the interference of other light sources.

The gray value of the treat gray image has relatively fixed height. The 2D gray level images of the rubber wear surface captured by OLS4100 are shown in Fig. 1. These images record the morphology of the rubber wear surface at the temperature of 25, 60 and 80 °C, then converted to black-and-white binary images according to the gray level.

Fig. 1
figure 1

2D microscope images of wear rubber surface

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3 Results and Discussion

Fractal dimension of the wear rubber surface at different temperatures is calculated by a Matlab program. The geometric arithmetic average deviation Ra of 39 positions at three different temperatures is measured by OLS4100 3D measuring laser microscope. The fitting curve between the geometric arithmetic average deviation Ra and the corresponding fractal dimensions illustrate the roughness of the wear surface of rubber composite materials.

Figure 2 depicts that the smaller is the fractal dimension, the poor is the homogeneity of the rubber wear surface and vice versa.

Fig. 2
figure 2

Ra–D curve at different temperatures

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The multifractal spectrum q–α, q–f(α) and α–f(α) is created from black-and-white binary images using a Matlab program, shown in Fig. 3.

Fig. 3
figure 3

Multifractal curves. a q–α curves, b q–f(α) curves, c α–f(α) curves

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From Fig. 3a, b, it can be seen that the rate of change of α and f(α) approaches 0 as the weighting factor q is close to − 20 or 20. Hence, q = [− 20, 20] can meet the multifractal requirements of the gray level images. Moreover, α decreases steadily with the increase of q, which illustrates that the figure is multifractal. From Fig. 3c, it can be found that the multifractal spectrums are convex functions of α while the α–f(α) curves are left-hook-shaped.

Multifractal spectrum width Δα gives the size of probability distribution of the entire fractal structure. In other words, multifractal spectrum width Δα reflects the size of the probability distribution range of gray values. High uneven gray value probability distributions result in wider multifractal spectrum curve α–f(α) and increased Δα, due to more dispersive surface height distributions However, more uniform gray value probability distributions result in narrower multifractal spectrum curve α–f(α) and smaller Δα, due to more concentrated surface height distributions. if Δα = 0, the probability distribution is uniform, and the surface height is constant. Δα represents the smoothness of rubber wear surface. The larger is Δα, the poor is the distribution uniformity, and the high is rubber surface wear. Δα is maximum at 80 °C as shown in Fig. 3c, and therefore the rubber wear surface is highly coarse. The second remarkably high value of Δα is obtained at 60 °C, followed by the value at 25 °C. Because Δf(α) > 0, the multifractal spectrum curve is left-hook-shaped, the subset of maximum probability numbers is less than the minimal ones. Thus, most of the wear surface of rubber is comprised of relatively lower spikes.

In addition, different temperatures process curves with different opening areas corresponds to different Δα.

The key values of α and f(α) are extracted from α–f(α) curves using a Matlab program and the results are displayed in Table 1.

Table 1 Key values of α–f(α) multifractal spectrum
Full size table

And, the 3D microscope gray images of wear rubber surfaces are shown in Fig. 4. It can be seen that, the values in Table 1 match up with the surface morphology characteristics in Fig. 4. As temperature increases, the corresponding Δα and Δf(α) increase, resulting in rough wear surface with lower height homogeneity and higher complexity.

Fig. 4
figure 4

3D microscope gray images of rubber wear surface

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From the multifractal spectrum curve shown in Fig. 3c, Δα is equal to 0.238, 0.281 and 0.283 at the temperature of 25, 60 and 80 °C, respectively. This implies that at 25 °C, rubber surface wear is highly fine. This analytical result is consistent with the 3D morphology measured by 3D laser microscope shown in Fig. 4.

4 Conclusions

The rubber wear surface has obvious fractal characteristics. Spectrum width Δα and spectrum change Δf(α) are obtained from multifractal spectrum of gray image using multifractal theory, and they are used to estimate the height uniformity and roughness of rubber wear surface. Comparing with the traditionally and empirically quantitative surface wear analysis method, the multifractal spectrum analysis is more convenient and accurate for rubber wear investigation.

With increase in temperature, the geometric arithmetic average deviation Ra increases, and fractal dimensions decrease. Moreover, the multifractal Δα and Δf(α) increase, indicating the homogeneity of wear is worsen on the rubber surface with higher complexity.

This research established a comprehensive description of the height homogeneity and complexity of wear rubber surface under different temperatures based on multifractal theory. It achieved an effective method for the study of surface morphology of rubber composite materials.