Generally, the method of the physical experiment is used by the industry to validate their rubber product design improvements. However, there are significant limitations of rubber testing with the experimental method. The experimental methods need additional equipment, floor area, and operators to arrange the testing setups. One method to solve these issues is to use properly validated numerical models of composites. Combining such models with design experiments will make it possible to gain the best operating and design parameters [ 72 ].
Wongwitthayakool obtained the correlation between viscoelastic results got from oscillatory Rubber Processing Analyzer (RPA) as a standard test and the temperature rise measured from a flexometer [ 48 ]. The temperature rise can thus be measured using the RPA data. The study built a relationship between filler characteristics, magnitude of reinforcement, viscoelastic, and temperature rise behaviors. As a standard test, the temperature rise (T) is often measured from the high loading flexometer using the RPA results. tan δ is less useful as an indicator of HBU than E″.
The temperature rise is the result of energy dissipation, explained by the hysteresis that is the mechanical response under cyclic loadings. Its prediction requires the resolution of the mechanical and thermal equations.
Uncoupled algorithm. It consists of deformation, dissipation, and thermal modules. First of all, solve one cycle of the mechanical equation to evaluate the dissipation. Then solve the thermal equation for many cycles on one fixed geometry until the rise of temperature is significant. After that, use the actual temperature to update the mechanical equation.
Fractional step algorithm. The thermomechanical equation is divided into two easier equations that are solved separately (first solve the mechanical equation and then solve the thermal equation).
As a result of the low simulation cost, the uncoupled method is constantly being used to solve the thermomechanical coupling equation that occurs under cyclic loading. It is proved that the numerical results agreed with the experimental data very well. To simulate the appearance of the self-heating during the fatigue process, the finite element method (FEM) often be used [ 19 ].
The heat generation and the hysteresis loss of rubber composites under different frequencies and surrounding temperatures are predicted based on the Maxwell model [ 8 ]. As the loading frequency rises, the loss factor of the rubber r composites rises as well, and the reliance of the frequency weakens as the temperature rises. A chief cause is that the motion of polymer chains cannot follow the change of external loading with high frequency, which brings about much more energy loss and internal friction in rubber composites. With the ambient temperature increases, the thermal movement of the chain segment tends to quicken, and the effect of frequency on the loss factor shows a decreasing tendency.
The amplitude of the Payne effect is tended to rise with the temperature dropping. The hysteresis loss of the rubber material is tended to decrease with the increase of temperature. A method to forecast the hysteresis loss of the rubber material at various temperatures is devised and confirmed in light of the experimental data from the Payne effect and the Kraus model. The predicted hysteresis losses according to this model at different temperatures and strain amplitudes agreed with the experimental results well. In other words, if the relationship between the Payne effect and temperature is given, the hysteresis loss can be predicted in the condition of a known strain amplitude at an arbitrary temperature [ 76 ]. It is not enough to only consider the influence of temperature rise. To acquire more precise simulation data for the low-frequency loading and the large strain amplitude conditions, the dynamic property softening needs to be taken into consideration.
The Kraus model is used to describe the Payne effect. It turns out that hysteresis loss showed an increasing tendency with the strain amplitude and frequency. The energy lost across the entire deformation cycle is frequently calculated using the viscoelastic model, which is related to hysteresis loss with strain amplitude and loss modulus [ 17 ]. NR/CB cylindrical specimens were passed the dynamic compression mode test ( Figure 1 b). A comparison between the simulation and test data of temperature rise was carried out. It turns out that ( Figure 10 ), at the low-frequency condition, the simulation results and the experimental data do not match well [ 75 ].
In Equation (11), ∆ε c is the strain amplitude eigenvalue, E″ m is the maximum loss modulus. E″ ∞ is the loss modulus at higher strain amplitude, which reaches its asymptotic plateau. m is a non-negative phenomenological exponent with the value about 0.5. It is almost independent of frequency, temperature, and filler content. In Equation (12), τ i and E i are the relaxation time and the elastic modulus of the Maxwell element, respectively. The relationship between Δε and E″ can be expressed by the Kraus model, and the relationship between ω and E″ can be expressed by the Maxwell model, as shown in Equations (11) and (12). They are always used to describe the strain amplitude dependence and the frequency dependence behaviors of rubber composites, respectively.
Luo [ 17 76 ] deeply considered the influence of frequency and loading strain amplitude on temperature rise. The relationship among loss modulus, frequency, and strain of the composites is built up through the Kraus model () and Maxwell model ().
In Li’s work [ 74 ], the transient temperature of rubber tires is performed according to the thermo–mechanical coupling approach and viscoelastic theory ( Figure 9 ). They used nature rubber and carbon black N234 to make a solid rubber tire, then tested the tire by a rubber rolling test apparatus. The solid rubber tire was composed of two parts: metal rim and rubber tire. With the rotation cycle from start, the temperature rises rapidly. Owing to thermal balance, a platform emerges at the time of around 1000 s for the simulated results and about 2000 s for the experiment results. There is always a delay between the simulated data and experiment data. Latency increases as the rotational speed and compressive displacement increase.
Saux’s group used a simple phenomenological approach to overcome the difficulties of the classical method based on thermodynamic models. The validity of the method is suitable both on the transient and stabilized temperature fields. This uncoupled approach is still suitable for a temperature rise of less than 20 °C [ 73 ].
Based on Equation (1), the model of the relationship between heat generation and crosslink density is created, along with heat generation and dynamic lag loss [ 14 ]. The dynamic characteristics are updated as a function of strain and temperature due to the modified Kraus model and iterative solution process. The influence of creep on deformation and dynamic softening on the loss model of rubber materials are taken into consideration [ 10 ]. The numerical results agree with the experimental data well. Therefore, it requires precise experimental dynamic material characteristics to achieve the exact prediction.
Khiêm et al. put forward a physically motivated constitutive model to describe the inelastic behavior of filled rubber composites in multiaxial states of deformation [ 18 ]. Therefore, the rubber networks are separated into two anisotropic damage networks (M and H) and an induced anisotropic elastic network (E). The M network linked polymer chains attached to the filler’s surface in irreversible adsorption, while the H network contains polymer chains attached to a reversible adsorption site ( Figure 11 ). Fillers are regarded as rigid bodies in the damage networks. The Mullins effect explains the deformation caused by permanent damage in the M network, and the hysteresis is caused by the strain-induced network recovery in the H network.
The study also compared the published available test data of filled silicone rubber composites. As seen in Figure 12 , there is a little bit of a misconception about the model with the tested data.
Circular loading rubber composites display a complex response, The main characteristic is stress softening and hysteresis caused by fatigue and dissipative heating. According to the thermodynamic principles, Guo and his coworkers [ 21 77 ] presented a new thermo viscoelastic damage method to predict inelastic fatigue phenomena. Two kinds of dissipative network rearrangements were taken into consideration, containing the unrecoverable rearrangements and the recoverable rearrangements viscoelasticity inducing damage. This is supposing that the recoverable viscoelastic rearrangements are caused by the movement of non-entangled free chains and entangled chains superimposed on a neat elastic perfect rubber network ( Figure 13 ).
They used styrene-butadiene rubber was filled with three different amounts of carbon black prepared dog-bone-shaped specimens. The dynamic stretching mode test ( Figure 1 a) results were compared with the constitutive model. The predictive model capabilities were checked from the comparison between the calculated data and the experiment data. This model can predict the temperature evolution, particularly for a dependence of temperature evolution on filler loading and pre-stretch level, which is in accordance with the result of intrinsic dissipation. However, this model is only useful for flat, thin specimens with a constant cross-section that is subjected to cyclic pre-stretched loads of constant amplitude.
According to the dynamic stretching mode test ( Figure 1 a), the comparison of the stress–strain hysteresis loop at the 250th cycle is shown in Figure 14 a [ 77 ]. The temperature changes tested in the middle region of the sample surface are compared with the simulation results in Figure 14 b. Although there is a slight discrepancy between the model demonstration and the tested results of the stress–strain curves, the temperature evolution is almost predicted by the model.
According to the second principle of thermodynamics, a thermo-visco-hyperelastic constitutive model is established to describe the self-thermal evolution of elastic materials under cyclic loading ( Figure 15 a). The constitutive model regards the stress–strain response as the result of two polymer networks acting in parallel. The model shows that the total of the overall resistance to deformation should balance with an equilibrium state A and a time-dependent deviation with regard to the equilibrium response B. A is the non-linear elastic spring and B is the non-linear elastic spring in series with a viscous dashpot [ 78 ].
This temperature rise will have a great effect on the constitutive stress–strain behavior by producing a thermal softening of the rubber composites. The thermo-mechanical model, which is used to explain temperature-related mechanical behavior, consists of a thermal resistance functioning in series with a mechanical resistance connecting to the stress-free thermal dilatation and the big strain rubber elastic behavior, respectively [ 79 ].
Taking account of the influence of filler on the thermo-mechanical response, introduce a tensile amplification factor into the model ( Figure 15 c) [ 80 ].
They created dog-bone-shaped specimens out of styrene-butadiene rubber that were filled with three different concentrations of carbon black. Then, the samples were tested by the dynamic stretching mode ( Figure 1 a). The predicted temperature evolutions due to the temperature rise of the filled rubbers are presented in Figure 16 . The data showed that the model captures temperature evolution from temperature rise in a satisfactory manner. However, for large volumes of particulate fillers, it is obvious that there is a significant difference between the test results and simulation results. This difference is attributed to the influence of highly adding amounts on the model under large strains.
A three-element model was used to reproduce the stress–strain data with two rubber elastic elements and a viscous element [ 81 ]. The model considers three factors to predict the temperature changes at the adiabatic conditions: viscous dissipation effects, isentropic elastic, and entropic elastic, as shown in Figure 17 81 ].
According to the model, the nominal stresses added to are equal to the total nominal stress. Besides that, the nominal stress is equal to the stretch ratio.
The upshot is thus that, in the case of small deformation, the is entropic elastic effect is dominant, while in the case of large deformation, the entropy elastic effect is dominant. Carbon black-filled styrene-butadiene rubber was selected for the test. The samples were tested by the dynamic stretching mode test. Figure 18 summarizes the relationship between the predicted and experimental temperature changes and the strain under different loading conditions. The calculated temperature changes are lower than the experimental ones. The author thought that there are two reasons. The first reason is the restriction of measuring and testing technique. It is rather difficult to record precise figures at very small deformation due to noise. There is a second reason for the discrepancy is the estimation shortage of the heat conduction. Actually, there is thermal convection and conduction with the surrounding air inside the sample and between the sample holder.
However, in order to use this technique on rubber composites, some questions need to be further investigated. For example, a quantitative interpretation of the temperature changes of different rubber composites under different loading modes is required. Heat conduction in the rubber composites and heat convection from the surface of the rubber composites to the outside air should be taken into consideration.
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