Rubber Creep Model and Its Influence on Mounting Stiffness

Abstract

The mount of the engine will creep under the action of long-term load. Creep will change its original structure, resulting in changes in static and dynamic characteristics and fatigue life. In order to solve the problem of mounting rubber creep, the creep characteristics were studied in this paper. In order to study the influence of creep characteristics on engine mounting stiffness, a theoretical model of mount creep was established based on the creep mechanism of rubber. The effects of different loads on the creep characteristics of engine mount were studied. The static viscoelastic parameters and creep analysis of rubber mounting were analyzed numerically. The creep variation law of suspension under different loads is obtained. By analyzing the static and dynamic characteristics of no creep, 2.98 mm creep and 3.83 mm creep of engine mount, the creep characteristics and the variation law of mount stiffness of rubber mount were revealed. The results show that the static stiffness of suspension increases with the increase of creep. When the frequency is constant and the creep increases, the dynamic stiffness of the suspension increases obviously. In this paper, the creep characteristics of rubber mount are analyzed, and the results of the analysis provide a design method for rubber mount design.

1. Introduction

Research on rubber creep has achieved many achievements at home and abroad, and the research fields involve important fields such as vehicles, aerospace, ships, and materials [1,2]. For the creep of the vehicle engine mount, its own mechanical properties play a vital role in the stability of the vehicle during driving, because rubber creep will seriously affect the static stiffness and fatigue life of the engine suspension, affecting the stability and comfort of the vehicle during driving [3,4]. At present, the research on rubber creep mainly involves rubber material formulation, algorithm research, experimental testing, rubber super-viscoelastic constitutive model, finite element simulation, secondary developments, and so on. The addition of different additives to the rubber material will change the important properties of the rubber material, such as hardness, elasticity and viscosity, thereby affecting the creep performance of the rubber. In terms of rubber ingredients, Mostafa et al. [5] studied the effect of carbon black dosage on creep and relaxation behavior of styrene-butadiene rubber and nitrile rubber. The results show that creep increases with the increase of carbon black dosage and the decrease of stress level, while the relative modulus and initial relaxation rate decrease with the increase of carbon black dosage. Cui et al. [6] performed elastic stress strain tests and creep tests on gluten-free rubber and two particle-reinforced rubber composites. Performed at three different temperatures, the stress-strain curve of the rubber conforms to the classical rubber elasticity theory, and the creep behavior can be well captured using the Miller–Norton formula. The results show that the creep resistance of the filler in the rubber material is enhanced, which is to some extent consistent with the expected load transfer from the matrix to the particles; Cheriet et al. [7] proposed the modification of asphalt with different contents of natural rubber and studied the effect of natural rubber on the creep properties of asphalt concrete. The results of the study showed that the creep properties of the different mixtures were improved. In the research of rubber creep algorithm, Paimushin et al. [8] propose a technique derived from studying the genetic properties of rubber shear creep, which is based on the finite element method and the integral equation of the genetic viscoelastic theory, and has a Koltunov–Rzhanitsyn genetic nucleus. In terms of rubber creep simulation analysis, Oman et al. [9] conducted an experimental evaluation of stress relaxation and creep of filled rubber and studied in detail the influence of different experimental protocols with loading order as the main variable on creep and relaxation process. Based on the assumptions that creep and stress relaxation are the result of the same viscoelastic mechanism, and stress relaxation can be considered creep at the time of stress reduction, the experimental data show that these assumptions are correct; Yamaguchi et al. [10] studied the stress relaxation and creep behavior of rubber in a state of natural stress, and a method based on the Boltzmann superposition principle was used to compare creep flexibility and the measured values recovered after release from a constant range of loads maintained at different times. Using Boltzmann’s principle of superposition, they concluded that within a certain load time range, the recovery data can also be reduced to a single recovery curve for any given waigara stress. Pascual-Francisco et al. [11] determined the tensile permanent deformation behavior of silicone rubber under different stress intensities, realized the digital image correlation of temperature in the improved creep experimental setup, and applied tensile strengths of 98.11 kPa, 96.2 kPa, 98.1 kPa, 196.2 kPa, at temperatures of 20 °C, 40 °C and 60 °C, respectively, and the recovery of creep was determined. The results show that with the increase of temperature, creep flexibility increases, while tensile strength increases and creep flexibility decreases. Robert et al. proposed an engineering method to evaluate time-dependent rubber vibration-isolation components with creep and stress relaxation, introducing a time-dependent damage function in the superelastic model. The results showed that the predicted values were well matched with the experimental data [12].

Many domestic scholars also have more research results in rubber creep. In terms of rubber material formulation, Chen Xue et al. [13] studied the creep recovery performance and temperature influence of rubber doped with different graphenes and analyzed and evaluated the creep and recovery test results of composite rubber by using a four-parameter model and distribution function method. The results show that the doping of graphene can improve the creep resistance of rubber. Cui [14] studied the effects of carbon black and additives on rubber creep and concluded that as the diameter of carbon black increases, the creep of rubber decreases, and different additives have different effects on creep. In terms of rubber creep simulation analysis and test, Hao et al. [15] fitted the constitutive parameters of rubber materials through tests, established linear and nonlinear viscoelastic creep models, performed creep tests on rubber samples, and concluded that viscoelasticity is limited creep. Ying et al. [16] obtained different samples of polypropylene material within a certain frequency and pressure range, then applied the same load to these samples, and after a certain period of time, observed the creep of these samples. Studies have shown that when the frequency is maintained at a certain time, the greater the pressure of molding, the smaller the creep variable of the polypropylene material sample, and when the molding process keeps the pressure unchanged, the greater the fatigue of the molding, the greater the creep variable. Bin et al. [17] conducted a dynamic test on the rubber shock absorber and studied the influence of rubber creep on the shock absorber. The results show that the greater the creep, the greater the damping and dynamic stiffness of the rubber shock absorber. They concluded that rubber creep will affect the vibration isolation effect of the shock absorber and proposed a plan for the improvement of rubber creep. Xiao [18] introduced different constitutive models of rubber materials, combined with a new rubber material of polyurethane; through rubber test data, the constitutive parameters of the material were fitted by simulation software, and creep analysis was performed on the material. In terms of rubber creep constitutive model and secondary development, Ya [19] conducted creep analysis on the water-lubricated bearing on the basis of considering the creep of the rubber bearing material, combined with the Kelvin creep model and the three-component solid creep model, and obtained the changes of the lubrication film pressure and film thickness of the water-lubricated shaft under the influence of rubber creep. Ji [20] carried out finite element simulation analysis on the vertical stiffness characteristics and creep variables of the rubber axial spring and concluded that it provides a guarantee for the development and design of the rubber axial spring. Zhang [21] used secondary development technology to realize Sburger and Kburger models and analyzed the influence of different function changes on creep by combining relevant calculation model parameters. In order to solve the creep problem of mounting rubber, the creep characteristics of mounting rubber were studied. In order to study the influence of creep characteristics on engine mounting stiffness, a theoretical model of engine mounting creep was established based on the creep mechanism of rubber. The effects of different loads on the creep characteristics of engine mount were studied. The static viscoelastic parameters and creep analysis of rubber bearing were analyzed numerically. The creep laws of suspension under different loads are obtained. The creep characteristics of rubber mounting and the variation law of mounting stiffness were revealed. In this paper, the creep characteristics of the rubber bracket are analyzed and a design method is provided for the design of the rubber bracket. At the beginning of the design, the creep characteristics of rubber were studied systematically. The relation between creep and mounting stiffness is explained systematically. The research topic is innovative.

2. Rubber Creep Mechanism and Theoretical Model

2.1. Creep Mechanism of Rubber

Creep is the external manifestation of viscoelastic properties of rubber materials, and the creep property of rubber is an inherent property of suspension. The mechanism of creep is that rubber polymers intertwine with each other into a mesh structure, and the molecules are rearranged under the action of an external load, leading to molecular mesh reconstruction. Under the external load, the polymer cross-linking bond is damaged, resulting in the deformed rubber being unable to fully recover to the original state after unloading. Rubber creep is divided into three stages, as shown in Figure 1: instantaneous elastic deformation stage, delayed line deformation stage, and plastic deformation stage [7]. When the external force is removed, the deformation of the first two stages can be completely recovered, while the deformation of the third stage is irreversible, resulting in permanent deformation. The creep properties of rubber materials reflect the structural stability of rubber components and have a potential influence on their mechanical properties.

The creep rate of rubber changes with the increase in time. In the initial state, the creep rate of rubber is faster under transient loading, then shows a slow increase trend, and finally tends to a stable state.

2.2. Viscoelastic Model of Rubber

In the viscoelastic constitutive model of rubber, the viscoelastic model is simplified into two parts: spring and sticky pot, and then various models are composed by different series and parallel connections. As for the viscoelastic constitutive models of rubber, there are the Maxwell model, Kelvin model, three-element solid model, Burgers four-element model, and the generalized Maxwell model and Kelvin model. The above viscoelastic constitutive models are described below.

2.2.1. Maxwell Model

Maxwell’s model consists of a simple linear spring connected in series with a Newton sticky pot, as shown in Figure 2 [10,12]. For a linear spring, when the external force is loaded, it can produce an instantaneous deformation response, and when the external force is removed, the deformation can be fully recovered. Therefore, the spring follows Hooke’s law, and its stress–strain relationship is

$$\sigma =E\epsilon$$

(1)

where $\sigma$, $\epsilon$ represent the normal stress and the normal strain respectively, and E is the elastic modulus of tension and compression. The sticky pot is contrary to the spring; when the external force is withdrawn, due to internal energy consumption, its deformation cannot be fully recovered, leading to permanent deformation, and its stress–strain relationship follows Newton’s law of viscosity

$$\sigma =\eta \dot{\epsilon }$$

(2)

where $\eta$ represents the coefficient of viscosity; $\dot{\epsilon }$ represents strain; $\epsilon$ derivative with respect to time being the strain rate. According to Equations (1) and (2), its constitutive equation can be obtained as follows

$$\dot{\epsilon }=\frac{\dot{\sigma }}{E}+\frac{\sigma }{\eta }$$

(3)

where $\dot{\sigma }$ represents the derivative of stress $\sigma$ with respect to time, namely stress rate.

In Maxwell’s model, when stimulated by the outside world, its instantaneous response is determined by the spring, and the sticky pot cannot generate an instantaneous response, as shown in Figure 3. Therefore, at the initial state t = 0, the stress–strain relation corresponding to the model is

$${\epsilon }_0=\frac{{\sigma }_0}{E}$$

(4)

When the excitation is constant stress, that is, with increasing time, the Maxwell model produces a creep response. It can be obtained from Maxwell constitutive equation

$$\dot{\epsilon }=\frac{\sigma }{\eta }$$

(5)

By integrating the formula, the strain state under constant stress loading is obtained.

$$\epsilon \left(t\right)=\frac{{\sigma }_0}{\eta }t+\frac{{\sigma }_0}{E}$$

(6)

Rubber material is a cross-linked polymer; under constant external excitation, its deformation will eventually tend to a stable state with the increase of time. Figure 3 shows that under constant stress loading, the model can deform infinitely with time, which represents the ideal creep response of a viscous body. Therefore, the Maxwell model cannot describe the creep performance of rubber materials.

2.2.2. Kelvin Model

The Kelvin model is composed of a spring and a sticky pot in parallel, as shown in Figure 4. The sticky pot cannot produce instantaneous strain or instantaneous stress; the initial stress is produced by the spring. The total stress generated by the model is the sum of the stresses of each part, and its constitutive equation is

$$\sigma =E\epsilon +\eta \dot{\epsilon }$$

(7)

At the initial state, constant stress is applied to the model, which can be obtained by integrating the above equation

$$\epsilon =\frac{{\sigma }_0}{E}\left(1-e^{-\frac{t}{\tau }}\right)$$

(8)

When $t\to \infty$, $\tau =\frac{\eta }{E}$, the strain of the model tends to a horizontal asymptote $\epsilon =\frac{{\sigma }_0}{E}$, so the Kelvin model can be used to describe the creep characteristics of viscoelastic bodies, as shown in Figure 5. For the rubber material, there are still some shortcomings. Under the initial state of constant stress, the corresponding elastic strain will be generated, which cannot be ignored. Therefore, the model has some shortcomings in describing the creep of rubber material.

3. Test Method for Suspension Creep

Rubber creep test according to the national standard GB/T19242-2003—“vulcanized rubber under compression or shear determination of creep”. The rubber creep test adopts the sample with a diameter of 29.0 ± 0.5 mm and height of 12.5 mm ± 0.5 mm. The loading mode is vertical compression. Before the test, the sample and its device shall be adjusted in the following ways:

(1) the compression device is required to keep two steel plates parallel, the roughness of two steel plates ≤ 0.2 μm.

(2) The lower pressure plate shall be fixed in the middle of the bottom of the box, and the distance between the lower pressure plate and the thermostat shall be kept above 40 mm.

(3) It is required that the sample should be placed in the thermostat in, the compression device, so that all components reach thermal balance.

(4) The sample was compressed to (25 ± 2)% of the initial strain at the speed of 25 mm/min, and then the load was unloaded. After the static sample returned to the initial state, the operation was performed again five times. The test load is 1000 N, and the direction is vertical downward. During the test, all the load should be loaded on the sample within 6 s, and then the deformation of the sample should be measured at different times. In the compression process, the initial thickness ${\delta }_1$ should be measured after the sample is compressed to 10 min ± 0.2 min, and the final thickness ${\delta }_2$ of the sample can be measured according to the recommended termination time (such as 1 h, 2 h, 4 h, etc.). Then, the calculation formula of compressive creep increment $\Delta \epsilon$ is

$$\Delta \epsilon ={\epsilon }_2-{\epsilon }_1=\frac{{\delta }_1-{\delta }_2}{{\delta }_0}$$

(9)

$${\epsilon }_1=\frac{{\delta }_0-{\delta }_1}{{\delta }_0}$$

(10)

where ${\epsilon }_1$ represents the compressive strain of the compressed sample measured at 10 min; ${\epsilon }_2$ denotes the strain of the compressed sample at the end of the test; ${\delta }_0$ represents the initial thickness of the compressed sample, in mm; ${\delta }_1$ denotes the thickness of the compressed sample measured at 10 min (in mm); ${\delta }_2$ represents the final thickness of the compressed sample at the end of the test. In order to avoid the accidental test results and improve the accuracy of the test, the above rubber creep test should be carried out independently on more than three rubber samples under the same conditions. The test results show that the static stiffness corresponding to no creep, 2.98 mm creep and 3.83 mm creep are 210 N/mm, 262 N/mm and 281 N/mm respectively.

4. Influence of Rubber Creep on Static and Dynamic Characteristics of Mount

4.1. Influence of Rubber Creep on Static Characteristics of Engine-Mounted Main Spring

The static characteristics of the suspended rubber main spring were simulated and analyzed based on different creep models, and the geometric models without creep, 2.98 mm creep and 3.83 mm creep were established. The static characteristics of the suspended rubber main spring were analyzed based on the creep. Considering that there would be errors in solving different models, the three 3D models with different creep were divided into grid cells of the same size to eliminate the errors in the calculation results caused by grid reasons, and then the divided grid models were imported into ABAQUS for simulation calculation respectively. Simulation analysis steps: Just set a static analysis step, fix the lower end of the main spring, set coupling constraints in the middle loading position of the main spring, concentrate the six degrees of freedom of all nodes at the loading position to a single point, and apply a load of 1000 N in the -Y direction, as shown in Figure 6.

The resulting data were extracted and the corresponding force–displacement curve-was drawn, as shown in Figure 7. Generally, the static stiffness of the suspended main spring can be obtained by the slope of the linear section of the curve. Therefore, the static stiffness corresponding to the non-creep, 2.98 mm creep and 3.83 mm creep are 201.8 N/mm, 253.3 N/mm and 269.6 N/mm, respectively, after calculation. The error between numerical simulation results and experimental results is small. The reliability of the numerical model is verified. It can be used for simulation calculation.

When the creep increases from 0 to 2.98 mm, the static stiffness of the suspended main spring increases by 51.5 N/mm, while when the creep increases from 2.98 mm to 3.83 mm the static stiffness of the suspended main spring increases by 16.3 N/mm. The variation of the static stiffness of the suspended main spring is greatly affected by the creep. As can be seen from the stress cloud diagram of the suspended main spring in Figure 8, for different degrees of creep, the shape variable of the suspended main spring under the same load changes very significantly. As the load increases, the size of the suspended main spring is compressed less in the same time.

The engine is mounted in a left-right symmetric structure. While the axial static stiffness is calculated, the radial static stiffness of the suspended main spring is also simulated and analyzed. Figure 9 shows the force–displacement curves of the calculated results of three creep models, namely, non-creep, 2.98 mm creep and 3.83 mm creep. The radial static stiffness values are 133.4 N/mm, 144.2 N/mm and 157.5 N/mm, respectively. From the change of radial static stiffness, it can be concluded that when the creep increases from 0 to 2.98 mm, the stiffness value increases by 10.8 n/mm, and when the creep increases from 2.98 mm to 3.83 mm, the stiffness value increases by 13.3 N/mm.

In the analysis of the axial and radial static stiffness of the main suspension spring of rubber creep, the axial and radial static stiffness of the main suspension spring increase with the increase of the creep. Moreover, axial stiffness is more affected by creep than radial stiffness.

4.2. Influence of Rubber Creep on Engine Mounting Static Characteristics

This article uses a four-cylinder car machine right-transverse engine mount. For the hydraulic mount, the finite element simulation analysis method is used for the computation of the static characteristics of a suspended load in simulation analysis and the incentive and constraint article according to the installation location, suspended in the powertrain and encouraged by the defined direction, mainly load-bearing when the installation is in the axial direction; the load size is 1000 N. According to the division of engine-mounting finite element meshes in Section 2, the engine hydraulic mounting meshes after 2.98 mm creep and 3.83 mm creep are modeled. The partitioned mesh was imported into ADINA software to calculate its axial static stiffness, and the force and displacement data of three different creep models were obtained according to the extraction method of force and displacement in the preceding text, then plotted on the same curve, as shown in Figure 10. The static stiffness of the suspension without creep, 2.98 mm creep and 3.83 mm creep were calculated to be 252 N/mm, 332.8 n/mm and 392.3 n/mm, respectively. Compared with the model without creep, the static stiffness increased by 80.8 N/mm and 140.3 N/mm, respectively, orby 32.1% and 55.7%, respectively. It can be seen from the results that after the creep of the suspension, with the increase of the creep the axial static stiffness value shows a trend of increasing, and the change of the static stiffness and the change of the suspension creep are nonlinear.

The displacement of the mount structures with no creep, 2.98 mm creep and 3.83 mm creep were collected. As shown in Figure 11, it can be seen that after creep occurs, the displacement of the mount first decreases and then increases with the increase of creep. The creep of the mount will reduce the overall size from changing the limit size at the time of installation, causing the inflection point of the nonlinear stiffness of the mount to appear later. As a result, in some limiting conditions, the suspension fails to reach the nonlinear stiffness point in time and cannot provide the required stiffness value of the working condition, which will seriously affect the normal driving state of the vehicle. Therefore, it is necessary to strictly control the creep range of the suspension. At the initial stage of the design model, the influence of the suspended structure and material on creep must be considered within a reasonable range.

4.3. Influence of Rubber Creep on Dynamic Characteristics of Engine-Mounted Main Spring

4.3.1. Frequency Variation Characteristics of Engine-Mounted Main Spring

The main spring of engine suspension is made of pure rubber. The dynamic characteristics of the main spring are described by the constitutive models of hyperelasticity and viscoelasticity. The hyperelastic parameters adopt the Mooney-Rivlin constitutive parameters [22]. The viscoelasticity involved is different from that of creep. In creep analysis, static viscoelasticity is used to describe the creep process, while dynamic viscoelasticity is used to describe the dynamic process of the main spring. In ABAQUS, there are two kinds of viscoelastic domain. One is a kind of viscoelastic time domain—a frequency domain of viscoelasticity; the other is a two-domain viscoelasticity defined through the Prony series, mounting the main spring dynamic characteristic analysis using frequency domain viscoelasticity. The Prony series needs inputs of g_i, k_i and τ_i, three parameters (provided by the enterprise), as shown in

Table 1. Dynamic viscoelastic parameters.

prony	gi	ki	τi
	0.1954	0.0401	0.5737
	0.14	0.0035	0.019
	0.0604	0.004	0.000303

.

Main spring: the dynamic stiffness of the simulation analysis to set up for two-step analysis. One is a static general analysis step, and the other is a viscosity analysis step, in order to accurately calculate the main spring dynamic stiffness, loading in a way consistent with the actual working condition. The first step was analysis, with the main spring axial preload set to 1000 N; in the second step, the analysis applied a sine along the main spring axial displacement. Figure 12 shows the sinusoidal displacement loading function.

$$x=A\sin \left(\omega t\right)$$

(11)

$$\omega =2\pi f$$

(12)

In the formula, the frequency f is taken as 1–49 Hz, and the frequency is taken once every interval. A represents the vibration amplitude, and the value is 1. Under the condition of no creep, 2.98 m creep and 3.83 mm creep, the dynamic stiffness simulation analysis of the suspended main spring is carried out, and the corresponding dynamic stiffness value of each frequency point is obtained, in N/m. The calculation results of dynamic stiffness of three kinds of creep suspension main springs are drawn into a curve, as shown in Figure 13. According to the curve, when the creep is constant, the dynamic stiffness of the suspension main springs increases with the increase of frequency, and when the frequency is 1–15 Hz, the increase rate of dynamic stiffness is relatively large, and then the dynamic stiffness increases slowly. When the frequency is constant, the dynamic stiffness of the suspended main spring increases with the increase of creep, and when the creep increases from 0 to 2.98 mm, the dynamic stiffness value increases greatly. For example, when the frequency is 15 Hz, the corresponding dynamic stiffness of the suspended main spring without creep, 2.98 mm creep and 3.83 mm creep are 287.2 N/mm and 322.0 N/mm, respectively. When the creep variable increases from 0 mm to 2.98 mm, the dynamic stiffness increases by 12.1%, and when the creep variable increases from 0 mm to 3.83 mm, the dynamic stiffness increases by 15.0%.

4.3.2. Amplitude Variation Characteristics of Engine-Mounted Main Spring

The amplitude variation characteristic describes the variation of the dynamic stiffness of the suspension main spring under the action of different vibration amplitudes. In the actual working environment, the suspension will be subjected to different vibration amplitude excitation loads, making it necessary to study the amplitude variation characteristics of the suspension main spring. In this paper, three vibration amplitudes of 0.1 mm, 0.5 mm and 1 mm are defined. The loading frequency is 1–49 Hz, and the frequency is taken once every interval. After simulation analysis, the dynamic stiffness curves of the suspension main spring under three kinds of creep, namely, non-creep, 2.98 mm creep and 3.83 mm creep, are obtained, as shown in Figure 14.

From the dynamic stiffness curves of the suspended main spring under three creep states, the same change law can be obtained. At the same frequency, the dynamic stiffness decreases with the increase of the vibration amplitude, and the smaller the vibration amplitude, the greater the increase of the dynamic stiffness. The material of engine suspension main spring is rubber, its damping angle itself is very small, and the creep change does not change the viscoelastic damping effect of rubber; creep has little influence on the damping angle of the suspension main spring; this paper does not consider the influence of creep on the damping angle of the suspension main spring.

When the frequency is 11 Hz and the vibration amplitude is 0.1 mm, the corresponding force–displacement values under three creep states, namely, hysteresis loops, are shown in Figure 15: no creep, 2.98 mm creep and 3.83 mm creep. The area enclosed by the hysteresis loop represents the energy dissipated by the suspension main spring after a period of movement. The larger the ellipse area, the greater the damping coefficient of the main spring, that is, the greater the work done by the damping force. As can be seen from Figure 15, with the increase of creep, the inclination angle of the hysteresis loop is larger, and the slope of the corresponding curve is also larger.

5. Conclusions

The static and dynamic characteristics of an engine mount with no creep, 2.98 mm creep and 3.83 mm creep are analyzed, and the three numerical analysis results are compared. The results show that the static stiffness of suspension increases with the increase of creep. When the frequency is constant and the creep increases, the dynamic stiffness of the suspension increases obviously. The increase of creep leads to the backward shift of the peak frequency of dynamic stiffness, while the change of vibration amplitude has little influence on the dynamic stiffness of suspension, and has no influence on the frequency points of the minimum and maximum values of dynamic stiffness. The influence of creep on static stiffness is slightly larger than that on dynamic stiffness.