1. Introduction
Rubber elastomers composed of long chains, macromolecules, and mesh-crosslinked structures have been commonly used in the automotive, aeronautical, and electronic industries. As a commonly used reinforcing agent for rubber products, carbon black has endowed natural rubber with better mechanical properties and thermo-elasticity. For carbon black-filled rubbers, temperature has a great influence on its hyper-elastic behavior. Due to the wide range of temperatures in a variety of applications [
1,
2], it is of great necessity to consider the impact of temperature on the hyper-elastic properties of carbon black-filled rubbers.
Filled rubber is usually used as damping and shock-absorbing components in the automotive and aerospace industries. Since the self-healing nature of elastic components in high-temperature environments or which are subjected to cyclic loading makes the temperature rise, the mechanical response of elastic components will be severely affected due to thermal coupling. [
3] Although the mechanical responses of filled and unfilled rubber have been characterized at room temperature and high temperatures [
4,
5,
6,
7,
8], the effects of temperature on the mechanical response of rubber materials in a certain deformation range, such as a 150% strain, have rarely been studied. From the point of view of tire applications and industrial formulations, it is necessary to carry out tests on the hyper-elastic mechanical properties of carbon black-filled rubber specimens at different temperatures in a wide range of deformations (150% strain) [
9]. Meanwhile, both the filler-rubber matrix and filler–filler interactions at different temperatures also have a significant effect on the thermal behavior of rubber components [
10].
The hyper-elastic mechanical behavior of rubber materials depends not only on the temperature but also the filler content. It is necessary to develop a temperature-dependent model to predict the hyper-elastic behavior of rubber based on the existing hyper-elastic constitutive model. The model is able to clearly describe and expose the temperature characteristics of rubber components with different carbon black contents [
11]. Several thermodynamic models have been proposed to evaluate the effect of temperature on the mechanical properties of filled rubber [
3,
12,
13]. However, the effect of temperature on the mechanical behavior of rubber materials in a larger deformation range (150% strain) is rarely studied. Meanwhile, due to the temperature correlation of different constitutive models having rarely been studied, the hyper-elastic behavior of rubber at different temperatures under a constant elongation strain can be accurately described by different constitutive models. This is helpful to select a fitted constitutive model that can better characterize the hyper-elastic behavior of rubber specimens at various temperatures.
This paper aims to systematically investigate the influence of temperature on the hyper-elastic mechanical behavior of filled rubbers.
Section 2 introduces the commonly used hyper-elastic constitutive model for carbon black-filled rubbers.
Section 3 displays the materials and the experimental setup.
Section 4 shows different hyper-elastic stress–strain curves of unfilled and filled rubbers with various CB contents at different temperatures.
Section 5 not only reveals the relationship between the Yeoh model parameters and the ambient temperature but extends the Yeoh model to an explicit temperature-dependent model. The evaluation results show that the model can accurately reveal the effect of temperature on the hyper-elastic behavior of tire rubber. Combining the relationship between the parameters of the Yeoh constitutive model and the ambient temperature, an explicit temperature-dependent Yeoh constitutive model was developed and has been applied to the FEA. Finally,
Section 6 comes to the conclusion.
4. Discussion
From the uniaxial tensile experimental data at different temperatures, a preliminary study on the temperature dependence of rubber between the Yeoh model, Ogden model, and Arruda–Boyce model was carried out on the C20 rubber specimen. The smaller the residual sum of squares (
RSS), the closer the fit is to 1. In order to obtain the fitting ability of the hyper-elastic intrinsic model more quickly, the residual sum of squares (
RSS) was calculated to evaluate the fitting ability of the Arruda–Boyce model, Ogden model, and Yeoh model.
where
is the experimental value;
is the average of the test values;
is the model fit value;
N is the number of experimental data points involved in the fit. The smaller the
RSS, the larger the
R2, indicating a better overall fit of the model.
From
Figure 2 and
Figure 3, it can be seen that the Arruda–Boyce model shows general “S-shaped” stress–strain characteristics of the hyperelastic behavior of carbon black-filled rubber at different temperatures. However, there are still some obvious deviations between the fitted results and the experimental data. It was difficult for the Arruda–Boyce model to reflect the nonlinear characteristics of the hyper-elastic mechanical behavior of carbon black-filled rubber at different ambient temperatures under the 150% strain. This was consistent with the conclusion summarized in the previous theoretical presentation. Meanwhile, the Arruda–Boyce model cannot reflect the nonlinear characteristics of the hyper-elastic mechanical behavior of carbon black-filled rubbers in small and medium deformations well, showing a more significant error with experimental data. The Ogden model (
N = 3) and Yeoh model also show an “S-shaped” stress–strain curve for the hyper-elastic behavior of carbon black-filled rubber at different temperatures. This indicates that the fitted curves of the two models can reasonably describe the experimental data under the 150% strain.
Table 2 lists the parameters fitting of the Ogden constitutive model (
N = 3) with the experimental data of the C20 rubber specimen at different temperatures.
Figure 4 shows the trend of the parameters of the uniaxial tensile data with temperature. The Ogden constitutive model (
N = 3) fit the experimental data in
Figure 2 and
Figure 3 well.
Figure 4 and
Table 2 show the relationship between the parameters of the Ogden model (
N = 3) and temperature. As can be seen from the diagram, the parameters of the Ogden model have no law with the change of temperature, so it can be said that the parameters have no temperature dependence. Due to the excessive parameters of the Ogden model (
N = 3), the model did not converge easily when fitting to the experimental data, resulting in longer computation times and a low applicability of the model.
From
Table 3 and
Figure 5, it can be seen that the material parameters
, and
vary approximately as a quadratic function with temperature, which indicates that the material parameters are correlated with temperature. As the temperature gradually increased, the rubber gradually softened, and the shear modulus decreased. The reason for this is that the material parameter
indicates the initial shear modulus at small strains. However, the rubber started to “harden” after reaching the turning temperature. The ability of the carbon black-filled rubber to resist the strain increased, and the shear modulus gradually rose. This was consistent with the trend of the experimental data in
Figure 1(b1). The material parameter
indicates the softening phenomenon of the filled rubber in a medium deformation. The larger the material parameter
, the more obvious the softening phenomenon, and when it came to the turning temperature, the filled rubber was the softest, and the material parameter
was the largest. The material parameter
indicates the phenomenon that the material started to harden again in a large deformation, and after reaching the turning temperature, the material parameter
became larger due to the hardening of the filled rubber. Therefore, there was significant dependence between the Yeoh model and temperature, and this temperature dependence can be expressed by numerical fitting using the quadratic function.
In summary, the ArrudaBoyce model cannot reflect the nonlinear characteristics of the hyper-elastic mechanical behavior of carbon black-filled rubbers in small and medium deformations well, showing a more significant error with experimental data. Although the Ogden constitutive model (N = 3) fit the experimental data well, there also existed an irregularity of the model parameters with temperature. Due to the excessive parameters of the Ogden model (N = 3), the model did not converge easily when fitted to the experimental data, resulting in longer computation times and a low applicability of the model. On that basis, it can be concluded that there was significant dependence between the Yeoh model and temperature, and this temperature dependence can be expressed by numerical fitting using the quadratic function.
Using the Yeoh constitutive model, the relationship between the material parameters and temperature can be expressed as:
where
are the temperature-dependent parameters of the Yeoh constitutive model, which can be determined by fitting the Yeoh constitutive model. For different volume fractions of carbon black-filled rubbers, the temperature-dependent parameter values can be obtained by fitting the parameters of the Yeoh constitutive model at different temperatures by Equation (9). The details are shown in
Table 4,
Table 5 and
Table 6 and
Figure 6.
By combining Equations (2) and (9), the Yeoh constitutive model with the explicit temperature parameter can be obtained.
Based on Equation (10) and the parameters in
Table 4,
Table 5 and
Table 6, the stress–strain curves for four different contents of carbon black-filled rubbers at different temperatures were plotted, which can be used to predict the trend of the parameters of the Yeoh constitutive model with explicit temperature parameters.
Figure 7 shows the prediction curves of the model (a1–d1) with local zoom-in plots (a2–d2). From
Figure 7, the predicted results of the Yeoh constitutive model with the explicit temperature parameter are in good agreement with the experimental results. This indicates that the Yeoh constitutive model with the explicit temperature parameter can more accurately describe the nonlinear hyper-elastic mechanical behavior of carbon black-filled rubber at different temperatures under the 150% strain.
To visualize the relationship between the effect of temperature on the “softness” and “hardness” of the rubber, the correlation between the adhesive stress and the temperature at different constant elongation strains of 0.2, 0.6, 1, and 1.4 was investigated. From
Figure 8, it can be seen that, as the temperature increased, the stress in the rubber specimens (C20, C40, C60) at a constant elongation first decreased with the increase in temperature and then, after reaching the turning temperature, increased again with the increase in temperature. The carbon black-filled rubber sample first became “soft” with the increase in temperature and then gradually became “hard” when it reached the turning temperature. The temperature changed with the number of carbon black-filled masses. The stress transition temperature increased with the increase in the volume fraction of the carbon black. However, for the unfilled carbon black rubber specimen C00, the stress tended to increase roughly linearly with the increasing temperature at different constant elongation strains. The above results were the same as the conclusion of the stress–strain curves measured by the above tests.
There are two reasons for the temperature dependence of the hyper-elasticity of carbon black-filled rubbers. Firstly, due to the gradual increase in temperature, the movement between molecules is more intense. The intermolecular potential energy is reduced, which results in a thermal softening effect of the filled rubber. Secondly, due to the gradual increase in temperature, the conformational entropy of the long-chain molecular system of rubber changes. The thermos-elasticity of the rubber is enhanced, which makes the filled rubber show a thermal hardening effect. The thermal softening effect of the filled rubber plays a major role at lower temperatures. When the test temperature exceeds the turning temperature, the thermal hardening effect of the filled rubber gradually plays a major role. Therefore, the phenomenon of “softening first, then hardening” of the filled rubber occurs with the increase in temperature. Meanwhile, due to the addition of carbon black, the thermal softening effect of the filled rubber increases with the increase in the volume fraction of the carbon black filling, the thermal hardening effect gradually decreases, and the turning temperature becomes higher and higher.
5. Application of the Yeoh Model with Explicit Temperature Parameters in FEA
From Equation (10) and
Table 4,
Table 5 and
Table 6, the temperature-dependent characterization parameters of rubber specimens with four different carbon black-filled mass fractions can be obtained from the model parameters at different temperatures. Then, uniaxial tensile simulations were performed on C60 rubber specimens at 293 K using ABAQUS/CAE. The simulation results were compared with the experimental data. A dumbbell-shaped model with the same properties as the experiment was built using ABAQUS/CAE. Then, the uniaxial tensile model was obtained by adding material properties, building components, setting analysis steps, and dividing meshes, as shown in
Figure 9. The tensile specimen model used the C3D8RH unit. In order to resemble the experimental process as much as possible, this simulation coupled the specimen area with reference points A and B. In addition, the boundary condition that A is completely fixed and the displacement along the Y-axis is applied to the reference point B was set. The tensile process of the universal power tensile tester was simulated.
It can be seen from
Figure 10 that the stress distribution in the middle part of the dumbbell model is more uniform, and the stress concentration region of the specimen is a circular arc from narrow to wide. The stress–strain curve for the simulation was also determined from the uniform deformation region in the middle part of the dumbbell model. From
Figure 11, the simulated uniaxial tensile data are more consistent with the trend of the experimental data. This indicates that the Yeoh constitutive model with apparently included temperature parameters can predict the uniaxial tensile test data at different temperature ranges under the 150% strain. Because the model parameters can be obtained through uniaxial tensile tests, they can be better applied to actual working conditions, with a guarantee of certain accuracy requirements. Therefore, the Yeoh constitutive model with explicit temperature parameters has high engineering applicability.
If the temperature of the rubber specimen is known, the corresponding model parameters can be calculated immediately from the Yeoh constitutive model with explicit temperature parameters. Therefore, the Yeoh constitutive model with explicit temperature parameters can be quickly applied to finite element analysis. The Yeoh constitutive model with explicit temperature parameters provides a more convenient and accurate method for the analysis of other hyper-elastic finite element models. However, the simulation results of the Yeoh constitutive model with explicit temperature parameters still have some deviations from the experimental data. This indicates that there is still room for improvement in the model.