A Hyper-Elastic Creep Approach and Characterization Analysis for Rubber Vibration Systems

Article A Hyper-Elastic Creep Approach and Characterization Analysis for Rubber Vibration Systems

Dingxin Leng 1, Kai Xu 1, Liping Qin 2, Yong Ma 2 and Guijie Liu 1,*

1 Department of Mechanical and Electrical Engineering, Ocean University of China, Qingdao 266024, China; [email protected] (D.L.); user_ [email protected] (K.X.) 2 Seventh thirteen Institute of China Shipbuilding Industry Corporation, Zhengzhou 450009, China; [email protected] (L.Q.); [email protected] (Y.M.) * Correspondence: [email protected]; Tel./Fax: +86-0532-66781131

Received: 22 March 2019; Accepted: 15 May 2019; Published: 4 June 2019

Abstract: Rubber materials are extensively utilized for vibration mitigation. Creep is one of the most important physical properties in rubber engineering applications, which may induce failure issues. The purpose of this paper is to provide an engineering approach to evaluate creep performance of rubber systems. Using a combination of hyper-elastic strain energy potential and time-dependent creep damage function, new creep constitutive models were developed. Three diﬀerent time-decay creep functions were provided and compared. The developed constitutive model was incorporated with ﬁnite element analysis by user subroutine and its engineering potential for predicting the creep response of rubber vibration devices was validated. Quasi-static and creep experiments were conducted to verify numerical solutions. The time-dependent, temperature-related, and loading-induced creep behaviors (e.g., stress distribution, creep rate, and creep degree) were explored. Additionally, the time–temperature superposition principle was shown. The present work may enlighten the understanding of the creep mechanism of rubbers and provide a theoretical basis for engineering applications.

Keywords: ﬁnite element analysis; creep behavior; rubber; vibration system; hyper-elasticity; creep damage

1. Introduction Vibration mitigation is an essential design requirement in several industries, such as aerospace, rocket-engine, and automotive [1]. Passive damping technology often utilizes viscoelastic materials to decrease the vibratory level transmitted and the vibration field generated. Traditional viscoelastic materials are rubbers, which are widely utilized for years of service [2]. In practice, when a constant load is applied to a rubber material, its deformation is not a constant; it gradually increases with time, which is known as creep. The creep presents a time-dependent characteristic, which induces the dimensional instability of rubber products over their expected lifetime and may finally lead to early failure and deteriorate the vibration mitigation performance. Hence, it is important to accurately predict the creep behaviors of rubbers so that the fracture failure due to the creep effect can be prevented. Creep is typically classified in three stages, as shown in Figure1[ 3]. The first one is primary creep or transient creep, which is related to the physical rearrangement of polymer chains (e.g., bond stretching/bending and crosslinking between chains of rubber materials) [4]. In the primary creep stage, the strain rate is initially high and reduces with time. When the strain rate diminishes to a minimum value, the secondary stage begins and an obvious time-dependent behavior is presented, that is, the stain increases remarkably after an important length of elapsing time. The third stage is termed as

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that is, the stain increases remarkably after an important length of elapsing time. The third stage is the creeptermed failure as the stage,creep failure in which stage, the in creepwhich resistancethe creep resistance is weakened is weaken anded fracture and fracture inside inside the rubber the is presented.rubber Inis presented. engineering In applications,engineering application creep analysiss, creep practically analysis practically considers considers the ﬁrst twothe first creep two stages. creep stages.

FigureFigure 1. 1.Strain Strain as as a a functionfunction of time time for for the the three three creep creep stages stages [3]. [3].

TheThe reliability reliability of of creep creep predictions predictions isis dependentdependent on on the the application application of computational of computational model models.s. Hyper-elasticHyper-elastic models models are are commonly commonly utilized utilized to describe the the nonlinear nonlinear properties properties of rubber of rubber materials materials and rubber-basedand rubber-base devices.d devices. In In mathematical mathematical expression, the the hyper hyper-elasticity-elasticity of rubber of rubber is issued is issued from from the strainthe strain energy energy density, density, which which is is a a function function of principal invariants invariants related related to the to theCauchy Cauchy–Green–Green deformationdeformation tensors tensors and and thethe Jacobian Jacobian matrix matrix of the of deformation the deformation gradient gradient [5]. Widely [5]. applied Widely hyper applied- hyper-elasticelastic models models include include the Mooney the Mooney–Rivlin–Rivlin model [6,7 model], the Biderman [6,7], the model Biderman [8], the model Ogden [model8], the [9 Ogden], modelthe [9 Yeoh], the model Yeoh model [10], the [10 Neo], the–Hookean Neo–Hookean model model[5], and [5 polynomial], and polynomial series [1 series1]. The [11 parameters]. The parameters in these phenomenological models are identified according to the experimental data. Additionally, in these phenomenological models are identified according to the experimental data. Additionally, some hyper-elastic models are based on microscopic responses of polymer chains in the network of somerubber hyper-elastic materials, models e.g., six areor eight based constrained on microscopic-chain model responsess [12,13]. of Hyper polymer-elastic chains models in focus the network on of rubberportraying materials, nonlinear e.g., force six– ordeflection eight constrained-chain responses of rubbers; models however, [12 ,they13]. cannot Hyper-elastic describe modelsthe time- focus on portrayingdependent creep nonlinear responses force–deflection due to the models responses without of referring rubbers; to th however,e elapsed theyloading cannot time [1 describe4]. the time-dependentCreep behavior creep responses is attributed due to to the the models time-related without viscoelasticity referring to of the rubber elapsed materials. loading timeThe [14]. descriptionCreep behavior of viscoelastic is attributed behavior to the can time-related be achieved viscoelasticity by taking into account of rubber the materials. appropriate The amounts description of viscoelasticof elastic and behavior damping can elements be achieved into viscoelastic by taking models. into accountTypical computational the appropriate models, amounts such as of the elastic and dampingMaxwell element elements and into the viscoelastic Kelvin–Voight models. model Typical, are suitable computational for depicting models, linearly such viscoelastic as the Maxwell elementproperties; and the however Kelvin–Voight, long-term model, nonlinear are suitable creep for responses depicting are linearly not accurately viscoelastic predicted. properties; Subsequently, some complex viscoelastic models were proposed. Skrzypek [15] proposed a creep however, long-term nonlinear creep responses are not accurately predicted. Subsequently, some model by modifying Boltzmann’s superposition principle to describe nonlinear creep laws, in which complextime- viscoelasticdependent creep models strain were was stud proposed.ied. Lee [1 Skrzypek6] established [15] a proposed modified viscoelastic a creep model model by in prony modifying Boltzmann’sseries to study superposition creep characteristics principle toof compressed describe nonlinear rubber products creep laws, and a in finite which element time-dependent analysis was creep strainpresented. was studied. Majda Lee [3] [16 developed] established a modified a modified Burger viscoelastic model with model tunable in prony damping series and to stiffness study creep characteristicscoefficients of for compressed calculating rubber creep products deformation and, which a finite was element validated analysis by wasexperimental presented. results. Majda [3] developedAlthough a modified these complex Burger viscoelastic model withmodels tunable can forecast damping creep and nonlinearity, stiffness their coeffi complexitycients for results calculating creepin deformation, time-consuming which calculations was validated and substantial by experimental creep experime results.ntal Although results these must complex be input viscoelasticted to modelsdetermi canne forecast large numbers creep of nonlinearity, model parameters, their complexitywhich limits the results practical in time-consumingapplication. calculations It is noteworthy that a mechanical model for creep analysis with high efficiency and reasonable and substantial creep experimental results must be inputted to determine large numbers of model accuracy is particularly attractive. Recently, Luo proposed an easily implemented creep damage parameters, which limits the practical application. model for predicting long-term creep characteristics of polyisoprene rubbers [17]. However, his work focusedIt is noteworthy on the creep that analysis a mechanical under a model fixed loading for creep level analysis and temperature with high and effi ciencythe validation and reasonable for accuracyvario isus particularly conditions was attractive. not considered. Recently, In Luopractic proposede, rubber an materials easily implemented are commonly creep used damagein various model for predictingconditions long-term(loading level, creep temperature, characteristics humidity, of polyisoprene oxygen aging) rubbers and some [17]. studies However, revealed his work that the focused on theloading creep level analysis and under ambient a fixed temperatures loading level are major and temperature factors which and largely the validation affect the for creep various conditionscharacteristics was not [3]. considered. It is also emphasized In practice, that rubber creep materials performances are commonlyof rubbers under used indifferent various loading conditions (loadingconditions level, temperature,and temperatures humidity, are significant oxygen for aging) engineering and some designers studies [1 revealed8]. Some thatrelated the work loading has level and ambient temperatures are major factors which largely affect the creep characteristics [3]. It is also emphasized that creep performances of rubbers under different loading conditions and temperatures are significant for engineering designers [18]. Some related work has been conducted, e.g., Rivin [18] carried out creep tests of compressed rubber components under different levels of static loading and Polymers 20182019, 1011, x 988 FOR PEER REVIEW 3 of 20 19 been conducted, e.g., Rivin [18] carried out creep tests of compressed rubber components under differentthe correspondence levels of static characteristics loading and were the discussed. correspondence Oman [characteristics19] observed the were influence discussed of test. Oman programs [19] observedand loading the conditionsinfluence of on test the programs creep responses and loading of rubbers conditions and dionff erentthe creep creep responses performances of rubb wereers andcompared. different Wang creep [20 performances] performed awere laboratory compared evaluation. Wang [ on20] theperformed creep viscosity a laboratory and sti evaluationffness of tireon therubber creep under viscosity low and stiffness high temperatures of tire rubber and under the temperature-related low and high temperature creep stabilitys and the was temperature analyzed.- relatedThese studies creep stability are mainly was experimental analyzed. These works, studies in which are mainly creep deformation experimental is qualitativelyworks, in which discussed creep deformationfrom the test results. is qualitatively Nevertheless, discussed due to from observation the test di ffi results.culties, Nevertheless, the stress variation, due to creep observation rate, and difficulties,creep degree the are stressnot fully variation, addressed, creep but ratesuch, informationand creep degree is beneficial are not to explore fully addressed creep mechanism, but such for informationthe engineering is benef designicial of to rubber explore systems. creep mechanism for the engineering design of rubber systems. As mentioned above, although creep behaviors of rubbers have been theoretically studied for decades, it has been solved by the viscoelastic mechanical models. However, However, the viscoelastic model is not not an an optimal optimal solution solution for for engineering engineering application applications.s. Different Diff erentfrom the from previous the previous work, the work, present the studypresent investigates study investigates creep performance creep performance with a modified with a modified hyper-elastic hyper-elastic mechanical mechanical model modelwith a withtime a- dependenttime-dependent creep creepdamage damage function. function. The integration The integration of the ofproposed the proposed model model in the incommercial the commercial finite elementfinite element software software Abaqus Abaqus is provided is provided by utilizing by utilizing user user subroutines. subroutines. Its Its validation validation is is presented presented by by experimental work. work. By By utilizing utilizing the the proposed proposed model model for predicting for predicting creep behaviors creep behaviors of rubber of vibration rubber vibratiosystems,n the system engineerings, the engineering potential of potential the proposed of the model proposed is validated. model The is validated.detailed arrangement The detailed of arrangementthe present work of the is aspresent follows. work The is experimentalas follows. The testing experimental of rubber testing materials of rubber is shown materials in Section is shown2. The inconstitutive Section 2. equation The constitutive and its numerical equation implementation and its numerical in the implementat finite elemention software in the finite areprovided element softwarein Section are3. Theprovided identified in Section parameters 3. The identified of the proposed parameters model of are the determined proposed model by the are experimental determined bydata. the Section experimental4 validates data. the Section developed 4 validates approach the and developed discusses approach the creep and performances discusses of the rubber creep performancesmaterials under of multi-level rubber materials loadings and under temperatures. multi-level For loadings further evaluatingand temperatures. engineering For potential, further evaluatingthe proposed engineering approach is utilized potential, for predictingthe proposed creep approach performances is of utilized rubber vibrationfor predicting systems. creep The performancesmain conclusions of rubber are drawn vibration in Section system5. s. The main conclusions are drawn in Section 5.

2. Experimental Experimental T Testingesting 2.1. Quasi-Static Compression Tests 2.1. Quasi-Static Compression Tests A quasi-static compression test was performed before the creep experiment to analyze nonlinear A quasi-static compression test was performed before the creep experiment to analyze nonlinear hyper-elasticity of the rubber materials. Acrylonitrile-butadiene rubbers (Yi-Ke Rubber Manufacturing hyper-elasticity of the rubber materials. Acrylonitrile-butadiene rubbers (Yi-Ke Rubber Corporation, Qingdao, China) were utilized for preparing the rubber samples. The measurements of Manufacturing Corporation, Qingdao, China) were utilized for preparing the rubber samples. The the studied cylinder-shaped samples were 29.0 mm in diameter and 12.5 mm in thickness, as shown in measurements of the studied cylinder-shaped samples were 29.0 mm in diameter and 12.5 mm in Figure2. The loading was applied along the axial direction of the sample. All the tests were performed thickness, as shown in Figure 2. The loading was applied along the axial direction of the sample. All using a servo-hydraulic testing system (type: WDW-50) in which the load cells with 1% accuracy and the tests were performed using a servo-hydraulic testing system (type: WDW-50) ±in which the load the displacement sensor with 0.01 mm accuracy were equipped. The frequency range of this machine cells with ±1% accuracy and ±the displacement sensor with ±0.01 mm accuracy were equipped. The was 0–20 Hz. The rubber sample was constrained between two plates in the experimental system. The frequency range of this machine was 0–20 Hz. The rubber sample was constrained between two plates experimental setup is shown in Figure3. in the experimental system. The experimental setup is shown in Figure 3.

10mm

Figure 2. RubberRubber samples samples for for quasi quasi-static-static compression loading.

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(a) (b) (c) (a) (b) (c) FigureFigure 3.3. ExperimentalExperimental setup:setup: ((aa)) Quasi-static Quasi-static compression compression of of rubber rubber sample, sample, (b )(b creep) creep compression compression of Figure 3. Experimental setup: (a) Quasi-static compression of rubber sample, (b) creep compression rubberof rubber sample, sample, and and (c) ( creepc) creep compression compression of rubberof rubber device. device. of rubber sample, and (c) creep compression of rubber device. InIn the the load-deflection load-deflection compression compression tests, tests, the specimens the specimens were compressed were compressed at 25 mm at/min 25 according mm/min In the load-deflection compression tests, the specimens were compressed at 25 mm/min toaccording the standard to the test standard method test [21 ].method The compression [21]. The compression was applied was without applied interruption without interruption up to 39% relative up to according to the standard test method [21]. The compression was applied without interruption up to deflection39% relative for axialdeflection compression. for axial Four compression. loading–unloading Four loading cycles–unloading were performed cycles in were tested performed specimens. in 39% relative deflection for axial compression. Four loading–unloading cycles were performed in Thetested temperatures specimens. for The the temperatures two tests were for maintained the two tests at 23 were1 ◦ Cmaintained and 55 1 at◦C, 23 respectively ± 1 °C and and 55 ± dry 1 °C, air tested specimens. The temperatures for the two tests were± maintained± at 23 ± 1 °C and 55 ± 1 °C, withrespec relativetively and humidity dry air was with less relative than 30%. humidity The load-deflection was less than responses30%. The l ofoad the-deflection rubber samples responses in the of respectively and dry air with relative humidity was less than 30%. The load-deflection responses of multiplethe rubber loading samples and in unloading the multiple cycles loading are shown and unloading in Figure 4cycles. are shown in Figure 4. the rubber samples in the multiple loading and unloading cycles are shown in Figure 4.

(a) (b) (a) (b) Figure 4. Experimental compression force-deflection history for the rubber sample. (a) 23°C and (b) FigureFigure 4.4. ExperimentalExperimental compressioncompression force-deflectionforce-deflection history for the rubber sample.sample. ( a) 23°C 23 ◦C and and (b (b) ) 55°C. 5555°C.◦C. Figure 4 shows a representation of the Mullins effect [22], and a permanent set is presented upon FigureFigure4 4 shows shows a a representation representation of of the the Mullins Mullins e effectffect [[2222],], andand aa permanentpermanent setset isis presentedpresented uponupon unloading which denotes viscoelastic effects such as hysteresis. Due to the Mullins effect, the unloadingunloading whichwhich denotes denotes viscoelastic viscoelastic eff ects effects such such as hysteresis. as hysteresi Dues. to Due the Mullinsto the Mullinseffect, the effect, “loading the “loading softening” of rubber samples is clearly indicated, especially in the first and second loops, softening”“loading softening of rubber” of samples rubber issamples clearly is indicated, clearly indicated, especially especially in the first in andthe first second and loops, second i.e., loops, the i.e., the force value from the second loading is lower than that from the first loading at a given forcei.e., the value force from value the second from the loading second is loadinglower than is lower that from than the that first from loading the first at a givenloading deformation, at a given deformation, approximately 16.7%. The stable force-deformation loop is presented in the third cycle. approximatelydeformation, approxim 16.7%. Theately stable 16.7%. force-deformation The stable force-deformation loop is presented loop is in presented the third in cycle. the third Hence, cycle. in Hence, in the following creep tests, before experimental data was recorded, several loading– theHence, following in the creep following tests, before creep experimental tests, before data experimental was recorded, data severalwas recorded, loading–unloading several loading cycles– unloading cycles were conducted to diminish the Mullins effect. Additionally, compared with Figure wereunloading conducted cycles to were diminish conducted the Mullins to diminish effect. the Additionally, Mullins effect. compared Additionally, with compared Figure4a,b, with with Figure the 4a,b, with the temperature increasing, the slope of the loading curve of the rubber sample reduces, temperature4a,b, with the increasing, temperature the slopeincreasing, of the the loading slope curve of the of loading the rubber curve sample of the reduces, rubber sample which indicates reduces, which indicates that the equivalent stiffness of rubber is weakened. thatwhich the indicates equivalent that sti theffness equivalent of rubber stiffness is weakened. of rubber is weakened. 2.2. Creep Compression Tests 2.2. Creep Compression Tests Creep tests of rubber materials under multiple levels of loading and temperature were Creep tests of rubber materials under multiple levels of loading and temperature were conducted. The sample had a diameter of 12.5 mm and a height of 29.0 mm and three levels of applied conducted. The sample had a diameter of 12.5 mm and a height of 29.0 mm and three levels of applied loading (e.g., 1.5 kN, 2.0 kN, and 2.5 kN) and two levels of temperature (e.g., 23 ± 1 °C and 55 ± 1 °C) loading (e.g., 1.5 kN, 2.0 kN, and 2.5 kN) and two levels of temperature (e.g., 23 ± 1 °C and 55 ± 1 °C)

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2.2. Creep Compression Tests Creep tests of rubber materials under multiple levels of loading and temperature were conducted. The sample had a diameter of 12.5 mm and a height of 29.0 mm and three levels of applied loading Polymers 2018, 10, x FOR PEER REVIEW 5 of 20 Polymers(e.g., 1.5 2018 kN,, 10, 2.0 x FOR kN, PEER and REVIEW 2.5 kN) and two levels of temperature (e.g., 23 1 C and 55 1 C)5 wereof 20 ± ◦ ± ◦ assumed along the vertical direction. All the creep tests lasted 48 h. The measurement of deformation were assumed along the vertical direction. All the creep tests lasted 48 h. The measurement of werewas performed assumed alon usingg athe non-contact vertical direction laser extensor. All the displacement creep tests sensor lasted operated 48 h. The at measurementthe measurement of deformation was performed using a non-contact laser extensor displacement sensor operated at the deformationaccuracy of 0.1%.was performed The creep experimentalusing a non-contact setup islaser shown extensor in Figure displacement3b. sensor operated at the measurement accuracy of 0.1%. The creep experimental setup is shown in Figure 3b. measurementThe time-deformation accuracy of 0.1%. curves The of creep rubbers experimental during the setup creep is tests shown are shownin Figure in 3 Figureb. 5. TheThe timetime--deformationdeformation curvescurves ofof rubbersrubbers duringduring thethe creepcreep teststests areare shownshown inin FigureFigure 55..

((aa)) ((bb)) Figure 5. a b FigureFigure 5.5. ExperimentalExperimentalExperimental creepcreep historyhistory forfor rubberrubber materialsmaterials materials... (( (aa))) 2323 23 °C°C◦C andand and ((b (b)) ) 5555 55 °C.°C.◦C. In Figure5, it is shown that, in the initial creep stage, the deformation largely increases, which is InIn FigureFigure 55,, itit isis shownshown that,that, inin thethe initialinitial creepcreep stage,stage, thethe deformationdeformation largelylargely increases,increases, whichwhich assumed to be accelerated creep. In the long-term creep stage, the deformation gently increases with isis assumedassumed toto bebe acceleratedaccelerated creepcreep.. IInn thethe longlong--termterm creepcreep stage,stage, thethe deformationdeformation gentlygently increasesincreases elapsed time, which is termed as stable creep. Compared with Figure5a,b, the creep deformation withwith elapsedelapsed time,time, whichwhich isis termedtermed asas stablestable creep.creep. ComparedCompared withwith FiFiguregure 55aa,,b,b, thethe creepcreep deformationdeformation increases with the increase in temperature and at a ﬁxed temperature, the slope in the long-term creep increasesincreases withwith thethe increaseincrease inin temperaturetemperature andand atat aa fixedfixed temperature,temperature, thethe slopeslope inin thethe longlong--termterm creepcreep stage approximately keeps a constant in diﬀerent levels of applied loadings, which demonstrates that stagestage approximatelyapproximately keepskeeps aa constantconstant inin differentdifferent levelslevels ofof appliedapplied loadings,loadings, whichwhich demonstratesdemonstrates thatthat the creep characterization is less sensitive to loading levels than temperature. thethe creepcreep charactcharacterizationerization isis lessless sensitivesensitive toto loadingloading levelslevels thanthan temperature.temperature. Additionally, creep tests of the rubber vibration system were conducted. At room temperature Additionally,Additionally, creepcreep testtestss ofof thethe rubberrubber vibrationvibration systemsystem werewere conducted.conducted. AtAt roomroom temperaturetemperature (e.g., 23 1 C), 2.0 kN loading was applied along the normal direction for 48 h. The test setup was ((e.ge.g..,, 2323 ±± 11 °C),°C),◦ 2.02.0 kNkN loadingloading waswas appliedapplied alongalong thethe normalnormal directiondirection forfor 4848 hh.. TheThe testtest setupsetup wwasas shown in Figure3c. The time-deformation curve in creep test at 23 C is presented in Figure6. shownshown inin FigureFigure 3c.3c. TheThe timetime--deformationdeformation curvecurve inin creepcreep testtest atat 2323 °C°C◦ isis presentedpresented inin FigureFigure 66..

FigureFigureFigure 6.6. 6. TimeTimeTime-deformation--deformationdeformation curvecurve curve ofof of rubberrubber rubber vibrationvibration vibration systemsystem system inin in creepcreep creep test.test. 3. Numerical Simulation 3.3. NumericalNumerical SSimulationimulation 3.1. Constitutive Model 3.1.3.1. ConstitutiveConstitutive MModelodel In the analysis of experimental data, the rubber materials in creep tests showed nonlinear elasticity In the analysis of experimental data, the rubber materials in creep tests showed nonlinear in theIn initialthe analysis load/deﬂection of experimental characteristics data, the and rubber time-dependent materials in creep creep behaviors tests showed in the nonlinear long-term elasticityelasticity inin thethe initialinitial load/deflectionload/deflection characteristicscharacteristics andand timetime--dependentdependent creepcreep behaviorsbehaviors inin thethe longlong-- termterm deformation.deformation. Accordingly,Accordingly, thethe materialmaterial propertiesproperties ofof thethe rrubberubber showedshowed hyperhyper--elasticelastic behaviorbehavior forfor capturingcapturing nonlinearitynonlinearity elasticityelasticity inin thethe initialinitial deflectiondeflection rangerange andand aa timetime--relatedrelated damagedamage functionfunction forfor representingrepresenting nonlinearnonlinear displacementdisplacement––timetime relationshipsrelationships inin thethe longlong--termterm creepcreep range.range. Therefore,Therefore, thethe constitutivconstitutivee equationequation ofof rubberrubber materialsmaterials isis denoteddenoted asas [1[177]]

W = Whyper + Wcreep, (1) W = Whyper + Wcreep, (1)

Polymers 2019, 11, 988 6 of 19 deformation. Accordingly, the material properties of the rubber showed hyper-elastic behavior for capturing nonlinearity elasticity in the initial deﬂection range and a time-related damage function for representing nonlinear displacement–time relationships in the long-term creep range. Therefore, the constitutive equation of rubber materials is denoted as [17]

W = Whyper + Wcreep, (1) where Whyper and Wcreep are the hyper-elastic model and the time-decay creep model, respectively. In general form, Whyper is denoted by, Whyper = W I + W(J), (2) where W(I) is the deviatoric part of the strain energy density of the primary material response and W(J) is the volumetric part of the strain energy density. For isotropic rubber, W(I) depends on strain invariants, I1, I2, and I3. Strain invariants can be expressed in terms of three principle stretch ratios, λ1, λ2, and λ3, and it is noted that, for incompressible rubbers, λ3 is 1.

2 2 2 I1 = λ1 + λ2 + λ3 (3)

2 2 2 2 2 2 I2 = λ1λ2 + λ2λ3 + λ3λ1 (4) 2 2 2 I3 = λ1λ2λ3 (5) In W(J), J is Jacobian of the deformation gradient and it is a measure of the volume change caused by a deformation q J = det(B) = det(F), (6) where F is the deformation gradient tensor.    + ∂u1 ∂u1 ∂u1   1 ∂x ∂x ∂x   1 2 3  F =  ∂u2 1+ ∂u2 ∂u2 , (7)  ∂x1 ∂x2 ∂x3   ∂u ∂u ∂u   3 3 1 + 3  ∂x1 ∂x2 ∂x3 where u1, u2, and u3 are three-dimensional deformation and x1, x2, and x3 are three-dimensional coordinate axis. For stress calculation of hyper-elasticity, the strain energy potential in polynomial series is expressed as

N N X i j X 1 2i Whyper = Cij I1 3 I2 3 + (Jel 1) , (8) − − Dij − i+j=1 i=1 where Cij and Di are temperature-dependent material parameters and Jel is elastic volume strain. Using Equation (8), some typical hyper-elastic models are derived. For example, if N = 0, the polynomial formulation represents the Neo–Hookean model, which is written as Whyper NH = C10 I1 3 . (9) − − If N = 1, the Mooney–Rivlin hyper-elastic model is obtained as

1 2 Whyper MN = C10 I1 3 + C01 I2 3 + (Jel 1) . (10) − − − D1 − Polymers 2019, 11, 988 7 of 19

Also, using the modiﬁed Equation (10), the Yeoh hyper-elastic model can be obtained:

2 3 Whyper Yeoh = C10 I1 3 + C20 I1 3 + C30 I2 3 . (11) − − − − The hyper-elastic constitutive equation of rubber can be also expressed in high-order polynomial form. For easy implementation and reasonable accuracy, the strain energy potential in terms of Mooney–Rivlin was adopted in the present work. Hyper-elastic material parameters (C01 and C10) at different temperatures were evaluated by the quasi-static experimental results. The detailed identification is: Using the experimental force-deformation curve as shown in Figure4, nominal strain (change in length per unit of original length) and nominal stress (force per unit of original cross-sectional area) are derived. Given experimental nominal stress–strain results, the parameters of the hyper-elastic model are determined by utilizing the least squares fitting algorithm [23]. The identified objective is to minimize the relative error, Ee.

 2 Xn Tth  i  Ee = 1  , (12)  − Ttest  i=1 i

test th where Ti is the stress from the test results and Ti is the nominal stress. Hyper-elastic models as shown in Equation (2–11) are suitable to depict loading portion in a mechanical process; however, the unloading process cannot be predicted. Using a rebound energy approach [24], a modiﬁed hyper-elastic model for describing the complete loading–unloading process is developed: W = [1 (1 θ )β]W I + W(J), (13) hyper − − 0 where θ0 is rebound resilience parameter of rubbers and β is a state variable. In the loading process, β is 0. In the unloading process, at the beginning of unloading, β is 0 and at the end of unloading, β is 1. In the present work, θ0 is 0.55. As shown in Equation (1), a time-decay function with a damage concept, Wcreep, is incorporated into a constitutive equation of rubbers, which describes the nonlinear creep behaviors considering material constitutive structure change and the elapsed time. The creep damage model should be assumed as the creep eﬀect from the deviatoric strain invariants during loading, I1 and I2, and also the elapsed loading time, t. For characterizing nonlinear creep responses, a phenomenological mathematic model of Wcreep is developed. In this work, three kinds of widely nonlinear decay functions are utilized for developing creep damage constitutive models and compared in terms of accuracy, which is in the form of power-law functions, logarithmic functions, and exponential functions. As shown in experimental results, creep parameters should be varied with the temperature. (1) Power-law creep constitutive model is expressed:

r1(T) Wcreep = k1(T)t (I1 + I2). (14)

(2) Logarithmic-based creep constitutive model is shown: h i W = k (T) log t (I + I ). (15) creep 2 r2(T) 1 2

(3) Exponential-based creep constitutive model is expressed:

r3(T)t Wcreep = k3(T)e (I1 + I2), (16) where ki and ri (i = 1, 2, 3) are creep parameters. A trial and error procedure was arranged so that the best adjustment of creep responses could be achieved between numerical and experimental results, and hence creep parameters were identiﬁed. Polymers 2018, 10, x FOR PEER REVIEW 8 of 20

r3 T t Wcreep  k3 ( T )e ( I12 I ) , (16)

where ki and ri (i = 1, 2, 3) are creep parameters. A trial and error procedure was arranged so that the best adjustment of creep responses could be achieved between numerical and experimental results, and hence creep parameters were identified.

3.2. Implementation of Finite Element Method The numerical analysis was conducted by the finite element method using Abaqus software. In material libraries, only standard hyper-elastic models are available, such as Neo–Hooke, Mooney– Rivlin, Ogden, Yeoh, polynomial-term, van der Waals, and Arruda–Boyce. In the present work, the proposed model is not a standard model and hence needs to be incorporated via user subroutine. As the UHYPER user subroutine defines the increments of hyper-elastic strain and time-dependent inelastic strain, which is the function of the solution-dependent variables, e.g., deviatoric stress, loading, time-step increment, and temperature [25]. Abaqus provides both explicit and implicit time integration of creep and the choice of the time integration scheme depends on the procedure type, Polymersthe procedure2019, 11, 988 definition, and a geometric non-linearity [1]. The flow chart of implementation8 of of19 constitutive model by UHYPER is illustrated in Figure 7. The finite element model of rubber with mesh and boundary conditions according to the 3.2. Implementation of Finite Element Method experimental arrangement was established. Due to the symmetry of the rubber sample’s geometry and theThe loading numerical condition, analysis an was axial conducted symmetric by themodel finite was element establis methodhed. CAX4HT using Abaqus, which software. is a 4-node In materialthermally libraries,-time coupled only standard plan element hyper-elastic with models3 degrees are of available, freedom such, was as Neo–Hooke,utilized to mesh Mooney–Rivlin, the rubber Ogden,isolator. Yeoh, In the polynomial-term, finite element model, van der the Waals, totaland number Arruda–Boyce. of nodes and In the elements present were work, 1907 the proposedand 838, modelrespectively. is not a For standard the rubber model vibration and hence system, needs to its be finite incorporated element viamodel user wa subroutine.s developed As theby UHYPERC3D8HT userelement subroutine with the definestotal number the increments of nodes and of hyper-elasticelement being strain 63949 and and time-dependent24153. The symmetric inelastic boundary strain, whichcondition is the was function applied of in the the solution-dependent symmetric plan and variables, the bottom e.g., deviatoricof the rubber stress, was loading, constrained. time-step The increment,loading force and was temperature applied on [25 a ].rigid Abaqus body provides along the both vertical explicit direction and implicit and the time degrees integration of freedom of creep of andthe rigid the choice body ofand the the time rubber integration system scheme were coupled. depends The on thenumerical procedure models type, a there shown procedure in Figure definition, 8. In andaddition, a geometric the rigid non-linearity body and [1]. rubber The flow materials chart of implementation were applied ofusing constitutive surface- modelto-surface by UHYPER contact isconditio illustratedns to in prevent Figure 7interpenetration. and the friction coefficient value was 0.2.

Figure 7. Flow chart of UHYPER implementation in ABAQUS.

The finite element model of rubber with mesh and boundary conditions according to the experimental arrangement was established. Due to the symmetry of the rubber sample’s geometry and the loading condition, an axial symmetric model was established. CAX4HT, which is a 4-node thermally-time coupled plan element with 3 degrees of freedom, was utilized to mesh the rubber isolator. In the finite element model, the total number of nodes and elements were 1907 and 838, respectively. For the rubber vibration system, its finite element model was developed by C3D8HT element with the total number of nodes and element being 63949 and 24153. The symmetric boundary condition was applied in the symmetric plan and the bottom of the rubber was constrained. The loading force was applied on a rigid body along the vertical direction and the degrees of freedom of the rigid body and the rubber system were coupled. The numerical models are shown in Figure8. In addition, the rigid body and rubber materials were applied using surface-to-surface contact conditions to prevent interpenetration and the friction coefficient value was 0.2. Polymers 2019, 11, 988 9 of 19 PolymersPolymers 2018 2018, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 99 of of 20 20

((aa)) ((bb)) Figure 8. Numerical model of rubber samples. (a) Rubber materials, (b) rubber device. FigureFigure 8. NumericalNumerical model model of of rubber rubber samples samples.. ( (aa)) R Rubberubber materials, materials, ( (bb)) rubber rubber device. device.

4.4.4. Results Results Results and and and D D Discussionsiscussionsiscussions

4.1. Quasi-Static Analysis 4.1.4.1. Quasi Quasi--SStatictatic A Analysisnalysis SinceSinceSince the the the proposed proposed constitutive constitutive constitutive model model model in in in this this this study study study is is is based based on on strain strain energy energy potential, potential, the the the validationvalidation ofof thethe hyperhyper hyper-elastic--elasticelastic modelmodel isis fundamentalfundamental forfor creepcreep analysis. analysis. LoadLoad Load-deﬂection--ddeflectioneflection historieshistories histories ofof of thethethe simulation simulation simulation and and and experiment experiment experiment result result resultinginging in in quasi quasi quasi-static--staticstatic compr compr compressionessionession are are compared compared in in Figure FigureFigure 9 9.. .

((aa)) T T = = 23 23 °C °C. . ((bb)) T T = = 55 55 °C °C. .

FigureFigureFigure 9. 9. 9. Comparison ComparisonComparison of of of load load load-deﬂection--deflectiondeflection curves curves curves between between between the the the simulation simulation simulation and and and the the the experiment. experiment.

AsAsAs shownshown shown inin inFigureFigure Figure 99, , the9the, the curvescurves curves ofof thethe of numericalnumerical the numerical andand thethe and experimentalexperimental the experimental resultsresults areare results consistent,consistent, are whichwhichconsistent, implies implies which the the implies model model couldcould the model accuratelyaccurately could predict accurately predict the the predict deformation deformation the deformation process process in in process the the static static in loading theloading static–– unloadingunloadingloading–unloading compression. compression. compression. FigureFigureFigure 10 1010 shows showsshows thethe the stressstress stress profileprofile profile ofof of thethe the rubberrubber rubber inin the inthe the identicalidentical identical deformationdeformation deformation ((ɛɛ == (0.16)ε0.16)= 0.16) atat twotwo at tempetempetwo temperatures.ratures.ratures. In Figure 10, it is shown that the stress distribution presents symmetrically, which is due to the loading and boundary conditions. In the loading process, as shown in Figure 10a,c, rubber showed swelling in the radial boundary under compression and in the unloading process in Figure 10b,d, the rubber was elongated under rebound deformation. The maximum Mises stress points at different temperatures were mainly located on the contact edges. Such stress concentration was induced by the non-flat surface of the rubber in the contact area, which was the result of uncontrolled slippage at the rubber–rigid interface [18]. Additionally, at the strain of 16%, the maximum stress values in the loading process are 1.44 MPa and 1.19 MPa at 23 ◦C and 55 ◦C, respectively, which can be explained by

the stiﬀness analysis as compared in Figure4. Compared with Figure 10a,b at 23 ◦C under the same ((aa)) L Loadingoading process, process, T T = = 23 23 °C °C. . ((bb)) U Unloadingnloading process, process, T T = = 23 23 °C °C. . strain, the maximum stress in loading (1.44 MPa) is larger than that (1.00 MPa) in unloading; similar behaviors are also shown at 55 ◦C, hence the numerical model could evaluate the Mullins eﬀect.

Polymers 2018, 10, x FOR PEER REVIEW 9 of 20

(a) (b)

Figure 8. Numerical model of rubber samples. (a) Rubber materials, (b) rubber device.

4. Results and Discussions

4.1. Quasi-Static Analysis Since the proposed constitutive model in this study is based on strain energy potential, the validation of the hyper-elastic model is fundamental for creep analysis. Load-deflection histories of the simulation and experiment resulting in quasi-static compression are compared in Figure 9.

(a) T = 23 °C. (b) T = 55 °C.

Figure 9. Comparison of load-deflection curves between the simulation and the experiment.

As shown in Figure 9, the curves of the numerical and the experimental results are consistent, which implies the model could accurately predict the deformation process in the static loading– unloading compression.

PolymersFigure2019, 1110, 988 shows the stress profile of the rubber in the identical deformation (ɛ = 0.16) at10 oftwo 19 temperatures.

Polymers 2018, 10, x FOR PEER REVIEW 10 of 20

Polymers 2018, 10, x FOR(c) PEERLoading REVIEW process, T = 55 °C. (d) Unloading process, T = 55 °C . 10 of 20 (a) Loading process, T = 23 °C. (b) Unloading process, T = 23 °C . Figure 10. Stress distribution profile at strain of 16%.

In Figure 10, it is shown that the stress distribution presents symmetrically, which is due to the loading and boundary conditions. In the loading process, as shown in Figure 10a,c, rubber showed swelling in the radial boundary under compression and in the unloading process in Figure 10b,d, the rubber was elongated under rebound deformation. The maximum Mises stress points at different

temperatures were mainly located on the contact edges. Such stress concentration was induced by the non-flat surface of the rubber in the contact area, which was the result of uncontrolled slippage at the rubber–rigid interface [18]. Additionally, at the strain of 16%, the maximum stress values in the

loading process are 1.44 MPa and 1.19 MPa at 23 °C and 55 °C, respectively, which can be explained by (thec) L stiffneoadingss process,analysis asT =compared 55 °C. in Figure 4. Compared(d with) Unloading Figure 1 0process,a,b at 23 T °C = 55under °C . the same strain, the maximumFigureFigure stress 10. Stress Stressin loading distribution distribution (1.44 MPa) profile proﬁle is larger at at strain strain than ofthat 16%. (1.00 MPa) in unloading; similar behaviors are also shown at 55 °C, hence the numerical model could evaluate the Mullins 4.2. Creepeffect. Analysis In Figure 10, it is shown that the stress distribution presents symmetrically, which is due to the loadingThe and4.2. creep Creep boundary numerical Analysis conditions. simulation In was theperformed loading process, in accordance as shown with in the Figure experimental 10a,c, rubber test, inshowed which swellthreeing levels in the ofThe loadingradial creep boundarynumerical of 1.5 kN, simulation under 2.0 kN, compression wasand performed 2.5 kN and at in two in accordance the temperatures unloading with the (23process experimental◦C and in Figure 55 test,◦C) in1 were0 b,d, held the ruforbber 48 h, wawhich respectively.s elongat three levelsed under of loading rebound of 1.5 kN,deformation. 2.0 kN, and The2.5 kN m ataximum two temperatures Mises stress (23 °C points and 55 at°C) different temperaturesTo validatewere held were the for mainly reliability48 h, respectively. locate of thed on numerical the contact simulation edges. andSuch compare stress concentration the accuracy ofwa thes induced three creep by thedamage non-flat functions, surfaceTo validate the of creepthe reliabilityrubber deformations in of thethe numericalcontact obtained area, simulation from which the and wa simulation compares the result the are accuracy of presented uncontrolled of the and three compared slippage creep damage functions, the creep deformations obtained from the simulation are presented and at the rubber–rigid interface [18]. Additionally, at the strain of 16%, the maximum stress values in the with experimentalcompared with results experimental in Figure results 11. in Figure 11. loading process are 1.44 MPa and 1.19 MPa at 23 °C and 55 °C, respectively, which can be explained by the stiffness analysis as compared in Figure 4. Compared with Figure 10a,b at 23 °C under the same strain, the maximum stress in loading (1.44 MPa) is larger than that (1.00 MPa) in unloading; similar behaviors are also shown at 55 °C, hence the numerical model could evaluate the Mullins effect.

4.2. Creep Analysis The creep numerical simulation was performed in accordance with the experimental test, in which three levels of loading of 1.5 kN, 2.0 kN, and 2.5 kN at two temperatures (23 °C and 55 °C) were held for 48 h, respectively. To validate the reliability of the numerical simulation and compare the accuracy of the three Polymers 2018, 10, x FOR PEER REVIEW 11 of 20 creep damage ( functions,a) Power-law the creep creep constitutive deformatio model.ns obtained(b) Logarithmic from-based the creep simulation constitutive aremodel presented. and compared with experimental results in Figure 11.

Figure 11. Comparison of creep deformation between numerical and experiment results (T = 23 °C). Figure 11. Comparison of creep deformation between numerical and experiment results (T = 23 ◦C).

In Figure 11, it is seen that the proposed creep damage functions generally could predict the time-dependent increasing deformation in creep tests under different levels of loadings and (atemperatures.) Power-law Compared creep constitutive with power model-law and. the( blogarithmic) Logarithmic-based-based creep creep constitutive constitutive model, model the . exponential-based creep constitutive model showed steep deformation in the initial creep deformation, which indicates that this model is not suitable for predicting the primary creep stage. As for the logarithmic-based model, the long-term behavior is reasonably evaluated but it shows a relatively large error in the amplitude of initial creep deformation. For clarification, the error analysis of these creep constitutive models is presented in Table 1. Several error indexes are selected, such as the squared correlation coefficient (SCC), the mean absolute percentage error (MAPE) and the mean square error (MSE). The expressions of error indexes are provided in Table 2.

Table 1. Error analysis of different creep damage functions (T = 23 °C). Error Indexes Creep Models Loading SCC MAPE MSE 1.5 kN 0.9991 2.9821 0.0113 Power-low function 2.0 kN 0.9996 1.7582 0.0093 2.5 kN 0.9998 1.6917 0.0087 1.5 kN 0.9995 4.9595 0.0303 Logarithmic function 2.0 kN 0.9990 3.4315 0.0224 2.5 kN 0.9988 3.7001 0.0337 1.5 kN 0.9959 5.3389 0.0497 Exponential function 2.0 kN 0.9976 4.7459 0.0661 2.5 kN 0.9986 4.3447 0.0633

As shown in Table 1, the SCC of these three models were all larger than 0.99, which verifies the validation of these adopted time-dependent nonlinear creep models. For the other two indexes, compared with logarithmic-based and exponential-based models, the average value of MAPE of the power-law creep model decreased by 46.8% and 55.4%, respectively. In the case of 2.0 kN applied loading, the MSE of the power-law creep model decreased by 58.5% and 85.9% respectively than that of logarithmic-based and exponential-based models. Therefore, the power-law creep constitutive model was chosen for the subsequent creep analysis because of the high accuracy.

Polymers 2019, 11, 988 11 of 19

In Figure 11, it is seen that the proposed creep damage functions generally could predict the time-dependent increasing deformation in creep tests under different levels of loadings and temperatures. Compared with power-law and the logarithmic-based creep constitutive model, the exponential-based creep constitutive model showed steep deformation in the initial creep deformation, which indicates that this model is not suitable for predicting the primary creep stage. As for the logarithmic-based model, the long-term behavior is reasonably evaluated but it shows a relatively large error in the amplitude of initial creep deformation. For clarification, the error analysis of these creep constitutive models is presented in Table1. Several error indexes are selected, such as the squared correlation coefficient (SCC), the mean absolute percentage error (MAPE) and the mean square error (MSE). The expressions of error indexes are provided in Table2.

Table 1. Error analysis of diﬀerent creep damage functions (T = 23 ◦C).

Error Indexes Creep Models Loading SCC MAPE MSE 1.5 kN 0.9991 2.9821 0.0113 Power-low function 2.0 kN 0.9996 1.7582 0.0093 2.5 kN 0.9998 1.6917 0.0087 1.5 kN 0.9995 4.9595 0.0303 Logarithmic function 2.0 kN 0.9990 3.4315 0.0224 2.5 kN 0.9988 3.7001 0.0337 1.5 kN 0.9959 5.3389 0.0497 Exponential function 2.0 kN 0.9976 4.7459 0.0661 2.5 kN 0.9986 4.3447 0.0633

Table 2. Expression of error evaluation indexes.

Error Index Formula Analysis

 n 2  P ( ) ( )   [Fsim i Fexp i ]  SCC  i=1 ·  Larger is better  r n r n   P 2 P 2   Fsim(i) Fexp(i)   i=1 · i=1  n P Fsim(i) Fexp(i) 100 − MAPE n F (i) Smaller is better · i=1 exp n 2 1 P MSE n Fsim(i) Fexp(i) Smaller is better · i=1 −

As shown in Table1, the SCC of these three models were all larger than 0.99, which veriﬁes the validation of these adopted time-dependent nonlinear creep models. For the other two indexes, compared with logarithmic-based and exponential-based models, the average value of MAPE of the power-law creep model decreased by 46.8% and 55.4%, respectively. In the case of 2.0 kN applied loading, the MSE of the power-law creep model decreased by 58.5% and 85.9% respectively than that of logarithmic-based and exponential-based models. Therefore, the power-law creep constitutive model was chosen for the subsequent creep analysis because of the high accuracy. To further study the eﬀect of creep parameters on responses in the power-law creep model, sensitivity analysis was conducted. The detailed process of parameter sensitivity analysis is as follows: (1) A set of creep parameters as the reference values was selected. In the present work, the reference parameters were adopted in testing conditions of 2.0 kN loading at 23 ◦C, denoted as k0 and r0. (2) Polymers 2018, 10, x FOR PEER REVIEW 12 of 20

Table 2. Expression of error evaluation indexes.

Error Index Formula Analysis 2 n F i F i  sim exp SCC i1 Larger is better nn F i22 F i sim   exp   ii11 n 100 Fsim  i  Fexp  i MAPE  Smaller is better ni1 Fexp  i 1 n 2 MSE  Fsim  i Fexp  i Smaller is better n i1

To further study the effect of creep parameters on responses in the power-law creep model, sensitivity analysis was conducted. The detailed process of parameter sensitivity analysis is as follows: (1) A set of creep parameters as the reference values was selected. In the present work, the Polymers 2019, 11, 988 12 of 19 reference parameters were adopted in testing conditions of 2.0 kN loading at 23 °C, denoted as k0 and r0. (2) One parameter’s value was varied and another was unchanged; the changed proportion range ofOne each parameter’s parameter value was approximately was varied and – another20% to +20%. was unchanged; The effects the of varying changed parameters proportion rangeare shown of each in Figureparameter 12. was approximately –20% to +20%. The eﬀects of varying parameters are shown in Figure 12.

(a) Change in k. (b) Change in r.

Figure 12. EffectsEﬀects of of varying parameters in creep deformation responses.

As shown shown in in Figure Figure 1 212, when, when k wak wass equal equal to 0.8 to 0.8k0 andk0 and 1.2 k 1.20, thek0 ,maximum the maximum deformations deformations in creep in werecreep 0.93 were δ 0.930 andδ 01.07and δ 1.070, respectively,δ0, respectively, in which in which δ0 waδs0 thewas maximum the maximum deformation deformation of k0 ofandk0 andr0. Forr0. varyingFor varying r, ther, thedifference diﬀerence in maximum in maximum deformations deformations wa wass 0.96 0.96δ0 andδ and 1.04 1.04δ0 forδ for +20%+20% r0 andr and −20%20% r0, 0 0 0 − respectively.r0, respectively. In Inthe the phenomenal phenomenal aspect, aspect, k kcontrolcontrolledled the the amplitude amplitude of of creep creep deformation, which describes the the strength strength of the of creepthe creep damage, damage, and r determines and r determine the inclinations the inclination degree ofcreep degree deformation. of creep deformation.Creep compliance, J(t), is a representative index for evaluating creep performances. Using the power-lawPolymersCreep 2018 , creep compliance,10, x FOR model, PEER J (REVIEWt)), under is a representative diﬀerent levels index of loading for evaluating and temperature creep performances. is presented in Us Figureing13 ofthe 13 20. power-law creep model, J(t) under different levels of loading and temperature is presented in Figure 13.

(a) T = 23 °C . (b) T = 55 °C.

Figure 13. Comparison of numerical and experimental creep compliance.

As shown in Figure 13 13,, it it waswas concluded concluded that whenwhen thethe loading loading levels levels increased,increased, the creep compliance reduced.reduced. T Thishis phenomen phenomenonon is is presented presented at at different different temperatures; additionally, at the same loading level, the increment ofof temperaturetemperature leadsleads toto thethe enhancementenhancement ofof creepcreep compliance.compliance. During creep, the stress distribution of rubber changes with time. However, these results cannot be observed by experimental testing. In the present work, the maximum principle stress profilesprofiles of rubbersrubbers are studied byby finitefinite elementelement simulation,simulation, asas shownshown inin FigureFigure 1414..

(a) In the initial creep time. (b) In the final creep time.

Figure 14. Maximum principle stress profiles (T = 23 °C; F = 2.5 kN).

In Figure 14, it is shown that the tensile stress distributes around the free surfaces and the maximum tensile principle stress is observed at the center of the free surface, while the maximum compressive stress occurs at the contact edge. Compared with Figure 14a,b, the maximum tensile stress was 0.99 MPa in the initial creep and 1.48 MPa in the final creep, which indicates that the maximum principle stress was enhanced during creep process. For detailed comparison, the time- dependent maximum principle stress of a reference point where the maximum tensile principle stress was located is plotted in Figure 15.

Polymers 2018, 10, x FOR PEER REVIEW 13 of 20 Polymers 2018, 10, x FOR PEER REVIEW 13 of 20

(a) T = 23 °C . (b) T = 55 °C. (a) T = 23 °C . (b) T = 55 °C. Figure 13. Comparison of numerical and experimental creep compliance. Figure 13. Comparison of numerical and experimental creep compliance. As shown in Figure 13, it was concluded that when the loading levels increased, the creep As shown in Figure 13, it was concluded that when the loading levels increased, the creep compliance reduced. This phenomenon is presented at different temperatures; additionally, at the compliance reduced. This phenomenon is presented at different temperatures; additionally, at the same loading level, the increment of temperature leads to the enhancement of creep compliance. same loading level, the increment of temperature leads to the enhancement of creep compliance. During creep, the stress distribution of rubber changes with time. However, these results cannot During creep, the stress distribution of rubber changes with time. However, these results cannot be observed by experimental testing. In the present work, the maximum principle stress profiles of be observed by experimental testing. In the present work, the maximum principle stress profiles of ruPolymersbbers2019 are, 11studied, 988 by finite element simulation, as shown in Figure 14. 13 of 19 rubbers are studied by finite element simulation, as shown in Figure 14.

(a) In the initial creep time. (b) In the final creep time. (a) In the initial creep time. (b) In the final creep time. Figure 14. Maximum principle stress profiles (T = 23 °C; F = 2.5 kN). Figure 14.14. Maximum principle stress proﬁlesprofiles ((TT == 23 °C;◦C; F = 2.52.5 kN). kN). In Figure 14, it is shown that the tensile stress distributes around the free surfaces and the In Figure 1144, it it is shown that the tensile stress distributes around the free surfaces and the maximum tensile principle stress is observed at the center of the free surface, while the maximum maximum tensile principle stress stress is is observed at at the center of the free surface,surface, while the maximum compressive stress occurs at the contact edge. Compared with Figure 14a,b, the maximum tensile compressive stressstress occursoccurs at at the the contact contact edge. edge. Compared Compared with with Figure Figure 14a,b, 14 thea,b, maximum the maximum tensile tensile stress stress was 0.99 MPa in the initial creep and 1.48 MPa in the final creep, which indicates that the stresswas 0.99 wa MPas 0.99 in M thePa initialin the creep initial and creep 1.48 and MPa 1.48 in theMPa ﬁnal in creep,the final which creep, indicates which thatindicates the maximum that the maximum principle stress was enhanced during creep process. For detailed comparison, the time- maximumprinciple stress principle was enhancedstress was duringenhanced creep during process. creep For process. detailed For comparison, detailed comparison, the time-dependent the time- dependent maximum principle stress of a reference point where the maximum tensile principle stress dependentmaximum principlemaximum stress principle of a reference stress of pointa reference where point the maximum where the tensile maximum principle tensil stresse principle was located stress was located is plotted in Figure 15. wasis plotted locate ind is Figure plotted 15 .in Figure 15.

Figure 15. Time variation of maximum principle stress of reference point (T = 23 ◦C).

In Figure 15, it is clearly seen that, at a fixed loading level, the maximum principle stress of the reference point increased over time. This phenomenon is called “stress hardening”, which can be explained by the engineering principle, in that the extra geometric deformation during creep adds to the mechanical deformation [17]. It is also shown that, with the increase in loading levels, the maximum principle stress and its hardening degree (the slope of the time varied maximum principle stress) increased. To study the effect of temperature and loading levels on the stress hardening effect, the maximum tensile principle stresses under different conditions are compared in Table3. The hardening degree, λ, is calculated as ! σmax f σmax i λ(%) = − − − 100%, (17) σmax i × − where σmax f and σmax i are the maximum tensile principle stresses in the final and initial creep times, − − respectively. As shown in Table3, at room and high temperatures, σmax i and σmax f increased with increasing − − loading level. It was also seen that λ at room temperature was in the range of 40–50%, while at high temperatures, λ was less than 10%. Hence, a slower development of maximum principle stress increase occurs at high temperature. Polymers 2018, 10, x FOR PEER REVIEW 14 of 20

Figure 15. Time variation of maximum principle stress of reference point (T = 23 °C).

In Figure 15, it is clearly seen that, at a fixed loading level, the maximum principle stress of the reference point increased over time. This phenomenon is called “stress hardening”, which can be explained by the engineering principle, in that the extra geometric deformation during creep adds to the mechanical deformation [17]. It is also shown that, with the increase in loading levels, the maximum principle stress and its hardening degree (the slope of the time varied maximum principle stress) increased. To study the effect of temperature and loading levels on the stress hardening effect, the maximum tensile principle stresses under different conditions are compared in Table 3. The hardening degree, λ, is calculated as

maxfi max (%) 100% , (17)  maxi where  max f and  maxi are the maximum tensile principle stresses in the final and initial creep times, respectively.

Table 3. Maximum tensile principle stress under different conditions. Polymers 2019, 11, 988 14 of 19 Temperature Loading  maxi (MPa) (MPa) λ (%) 1.5 kN 0.0413 0.0598 44.95% Table 3. Maximum tensile principle stress under diﬀerent conditions. 23°C 2.0 kN 0.0669 1.0025 53.21%

Temperature2.5 kNLoading 0.0997σmax i (MPa) 1.σ0max485 f (MPa) 48.95%λ (%) − − 1.5 kN 1.5 kN0.0403 0.04130.0439 0.0598 44.95%8.93% 55°C 2.0 kN 0.0645 0.0691 7.13% 23 ◦C 2.0 kN 0.0669 1.0025 53.21% 2.5 kN 0.0913 0.0923 1.10% 2.5 kN 0.0997 1.0485 48.95%

As shown in Table 4, at room1.5 kN and high temperatures, 0.0403 0.0439 and 8.93% increased with 55 ◦C 2.0 kN 0.0645 0.0691 7.13% increasing loading level. It was also seen that  at room temperature was in the range of 40%–50%, 2.5 kN 0.0913 0.0923 1.10% while at high temperatures, was less than 10%. Hence, a slower development of maximum principle stress increase occurs at high temperature. Additionally,Additionally, the the axial axial creep creep deformation deformation profiles proﬁles at atdifferent diﬀerent times times are are shown shown in in Figure Figure 1 616. .

(a) Time = 1 min. (b) Time = 48 h.

FigureFigure 16. 16. AxialAxial creep creep deformation deformation profiles proﬁles at atdifferent diﬀerent times times (T (=T 23= 23°C;◦ C;F =F 1.5= 1.5kN). kN).

AsAs shown shown in inFigure Figure 16 16, the, the axial axial deformation deformation profiles profiles present presenteded a layer a layer phenomenon: phenomenon: At Atthe the bottom,bottom, it wa its was the theminimum minimum (approximately (approximately zero) zero) due to due the to boundary the boundary condition condition,, at the attop, the it wa top,s it thewas maximum the maximum because because of the loading of the loading condition, condition, and in other and in areas other it areasgradually it gradually varied. varied.This layer This characteristiclayer characteristic stayed stayedunchanged unchanged over different over diff erent creep creep times. times. Additionally, Additionally, the the maximum maximum creep creep deformationdeformation increase increasedd over over time time and and bulging bulging of ofthe the free free surfaces surfaces enhance enhancedd with with increasing increasing time. time. ToTo evaluat evaluatee the the creep creep degree, thethe creep creep deformation deformation percentage, percentage,Creep Creep(%), (%), under under different different loading loadinglevels levels and temperatures, and temperatures is compared, is compared in Table in4 Table. 5. ! Dt D0 Creep(%) = − 100%, (18) De D × − 0

where Dt is the creep deformation after t, De is the ﬁnal creep deformation, and D0 is the creep deformation at the end of applied loading.

Table 4. Creep (%) of rubber materials during the creep test at diﬀerent times.

Loading 1.5 kN 2.0 kN 2.5 kN

Temperature 23 ◦C 55 ◦C 23 ◦C 55 ◦C 23 ◦C 55 ◦C 1 min 27.17% 26.78% 31.90% 29.50% 30.71% 25.81% 30 min 72.50% 58.88% 74.31% 60.50% 74.28% 57.00% Time 1 h 77.50% 66.70% 78.62% 68.14% 78.50% 61.55% 6 h 89.33% 83.27% 88.97% 80.65% 88.48% 81.86% 12 h 92.67% 89.07% 93.97% 89.23% 92.32% 85.94% 24 h 96.00% 95.42% 97.24% 94.69% 96.16% 91.37% 48 h 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% Polymers 2018, 10, x FOR PEER REVIEW 15 of 20

DDt  0 Creep(%) 100% , (18) DDe  0 where Dt is the creep deformation after t, De is the final creep deformation, and D0 is the creep deformation at the end of applied loading.

Table 5. Creep (%) of rubber materials during the creep test at different times.

Loading 1.5 kN 2.0 kN 2.5 kN Temperature 23 °C 55 °C 23 °C 55 °C 23 °C 55 °C 1 min 27.17% 26.78% 31.90% 29.50% 30.71% 25.81% 30 min 72.50% 58.88% 74.31% 60.50% 74.28% 57.00% 1 h 77.50% 66.70% 78.62% 68.14% 78.50% 61.55% Time 6 h 89.33% 83.27% 88.97% 80.65% 88.48% 81.86% 12 h 92.67% 89.07% 93.97% 89.23% 92.32% 85.94% 24 h 96.00% 95.42% 97.24% 94.69% 96.16% 91.37% 48 h 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% Polymers 2019, 11, 988 15 of 19 As shown in Table 5, at a fixed loading level, the creep deformation percentage in room temperature was higher than that in high temperature; this temperature-related phenomenon is As shown in Table4, at a fixed loading level, the creep deformation percentage in room temperature increasingly obvious with increasing loading. For example, at the time interval of 1 min, the was higher than that in high temperature; this temperature-related phenomenon is increasingly obvious difference in creep deformation percentage between 23 °C and 55 °C, δ, is 0.39% at a loading of 1.5 with increasing loading. For example, at the time interval of 1 min, the difference in creep deformation kN and 4.9 % at a loading of 2.5 kN, a 12.6 times increase. In addition, δ changes in different creep percentage between 23 ◦C and 55 ◦C, δ, is 0.39% at a loading of 1.5 kN and 4.9 % at a loading of 2.5 kN, stages. For instance, when the applied loading was 2.5 kN, δ was 17.28% at 30 min and 4.79% at 24 h. a 12.6 times increase. In addition, δ changes in different creep stages. For instance, when the applied Hence, at a fixed applied loading, δ in initial creep was greater than that in stable creep, which loading was 2.5 kN, δ was 17.28% at 30 min and 4.79% at 24 h. Hence, at a fixed applied loading, δ in indicates that high temperature leads to a faster development of creep deformation in stable creep. initial creep was greater than that in stable creep, which indicates that high temperature leads to a Relative creep rate (RC) is another significant index for depicting creep behaviors, which is faster development of creep deformation in stable creep. defined as [16] Relative creep rate (RC) is another significant index for depicting creep behaviors, which is defined as [16] DD  t 0 RC  Dt D0 , (19) RC = −H , (19) H where Dt isis the the creep creep deformation after t and H is the original thickness. FFigureigure 1177 shows the relativerelative creep rate versusversus timetime plot.plot.

(a) F = 1.5 kN. (b) F = 2.0 kN. (c) F = 2.5 kN.

FigureFigure 17. CreepCreep rate rate for for rubbers rubbers under under different diﬀerent conditions.

As shown in Figure 1717,, underunder didifferentfferent loading loading levels levels and and temperatures temperatures the the relative relative creep creep rate rate is isapproximately approximately proportional proportional to to the the logarithm logarithm of of time time in in the the stable stable creep creep stage, stage, which which characteristic characteristic of ofphysical physical creep creep [26 [2].6 Physical]. Physical creep creep is dueis due to to the the viscoelasticity viscoelasticity of of rubber rubber materials materials and and the the slippages slippages in cross links of rubbers molecules under loading. In general, physical creep is primarily dominated in short-time creep tests (less than 103 min). It was also seen that, at room temperature, RC at loading levels of 1.5 kN, 2.0 kN, and 2.5 kN were 4.37%, 3.95%, and 3.61%, respectively, while in high temperatures, RC at the final creep time at loading levels of 1.5 kN, 2.0 kN, and 2.5 kN were 2.90%, 2.71%, and 2.69%, respectively. These results demonstrate that the high temperature mode shows more creep resistance than room temperature at a fixed loading. This can be explained by a slower increase of maximum principle stress due to “stress hardening” and a more uniform stress distribution in the case of high temperature, as shown in Table3. To further evaluate the engineering potential of the proposed constitutive model, the creep behavior of the rubber vibration system was predicted and compared with experimental results, as shown in Figure 18. It is seen that the proposed model in which the materials’ parameters are identified by rubber-material testing could depict the different creep stages of the rubber system and the numerical solutions match the experimental results. Polymers 2018, 10, x FOR PEER REVIEW 16 of 20 Polymers 2018, 10, x FOR PEER REVIEW 16 of 20 in cross links of rubbers molecules under loading. In general, physical creep is primarily dominated inin short cross-time links creep of rubbers tests (less molecules than 10 3under min). loading.It was also In seengeneral, that, physical at room creep temperature, is primarily RC at dominated loading 3 levelsin short of - 1.5time kN, creep 2.0 tests kN, (andless than 2.5 kN10 weminre). It4.37% was, also3.95% seen, and that, 3.61% at room, respectively temperature,, while RC at in loading high temperaturelevels of 1.5s, RC kN, at 2.0 the kN final, and creep 2.5 time kN atwe loadingre 4.37% levels, 3.95% of ,1.5 and kN, 3.61% 2.0 kN, respectively, and 2.5 kN, whilewere 2.90% in high, 2.71%temperature, and 2.69%s, RC, respectively. at the final creep These time results at loading demonstrate levels ofthat 1.5 the kN, high 2.0 kNtemperature, and 2.5 kN mode were shows 2.90% , more2.71% creep, and resistan 2.69%,ce respectively. than room temperatureThese results at demonstrate a fixed loading. that Thisthe highcan be temperature explained bymode a slower shows increasemore creep of maximumresistance than principle room stresstemperature due to at “ astress fixed hardeningloading. This” and can abe more explained uniform by a stressslower distributionincrease of in maximumthe case of principlehigh temperature, stress due as shown to “stress in Table hardening 3. ” and a more uniform stress distributionTo further in evaluatthe casee ofthe high engineering temperature, potential as shown of the in Tableproposed 3. constitutive model, the creep behaviorTo offurther the rubber evaluat vibratione the engineering system wa s potential predicted of and the comparedproposed with constitutive experimental model, results, the creep as shownbehavior in Figureof the rubber18. It ivibrations seen that system the proposewas predictedd model and in compared which the with materials experimental’ parameters results, are as identifiedshown in by Figure rubber 18-material. It is seen testing that could the depict propose thed differentmodel in creep which stages the of materials the rubber’ parameters system and are theidentified numerical by solutionsrubber-material match ttestinghe experimental could depict results. the different creep stages of the rubber system and Polymersthe numerical2019, 11, 988 solutions match the experimental results. 16 of 19

. . FigureFigure 18. 18. ComparisonComparison of of numerical numerical and and experimental experimental creep creep deformation deformation of of the the rubber rubber system system at at roomroomFigure temperature. temperature. 18. Comparison of numerical and experimental creep deformation of the rubber system at room temperature. TThehe axial axial creep creep deformation deformation profiles proﬁles at at different diﬀerent times times are are shown in Figure 19. The axial creep deformation profiles at different times are shown in Figure 19.

(a) Time = 20 s. (b) Time = 48 h. (a) Time = 20 s. (b) Time = 48 h. FigureFigure 19. 19. CreepCreep profiles profiles of of the the rubber rubber system system at at different different time timess ( T = 2323 °C;◦C; FF == 2.2.00 kN). kN). Figure 19. Creep profiles of the rubber system at different times (T = 23 °C; F = 2.0 kN). InIn Figure Figure 1919, ,it it is is seen seen that that the the deformation deformation distribution distribution at at different different creep creep time timess wa wass similar; similar; however,however,In Figure the the maximum maximum 19, it is creep creepseen deformationthat deformation the deformation increase increasedd distribution with with time time elapsing. elapsing. at different It It is is noticedcreep noticed time that thats at atwa the thes similar;initial initial creepcreephowever, time time (20the (20 s s),maximum), self-contactself-contact creep hadhad deformation occurredoccurred inside inside increase the thed top topwith of of thetime the rubber rubberelapsing. system, system, It is whichnoticed which lasted that lasted at during the during initial the thecreepcreep creep deformation. time deformation. (20 s), self Diff- contacterentDifferent from had from rubber occurred rubber materials, inside materials, the axialtop the of creep axialthe rubber deformationcreep system,deform presented ationwhich present lasted an irregular edduring an irregularlayerthe creep phenomenon layer deformation. phenomenon due to Different the due structural to fromthe structural geometryrubber materials, geometry effect. theeffect. axial creep deformation presented an irregularDuringDuring layer creep, creep, phenomenon the the Mises Mises stress stressdue todistribution distribution the structural of of the the geometry rubber rubber system systemeffect. at at different different creep creep time timess is is shown shown Polymersinin Figure FigureDuring 2018 2 20,0 10. . , creep,x FOR PEER the Mises REVIEW stress distribution of the rubber system at different creep times is17 shown of 20 in Figure 20.

(a) Time = 20 s (b) Time = 48 h

FigureFigure 20. 20. MisesMises stress stress distribution ofof thethe rubberrubber system system at at di differentfferent creep creep times time (Ts =(T23 = ◦23C; °C;F = F2.0 = kN).2.0 kN). As seen in Figure 20, the patterns of the two stress profiles look similar; the maximum Mises stress wasAs located seen inin theFigure region 20,where the patterns the stiff ofness the is two weakest stress (e.g., profiles the central look similar region).; the Using maximum the analysis Mises of stressthe rubber was locate systemd in mentioned the region above, where the the proposed stiffness model is weakest and its ( numericale.g., the central approach region). could Using provide the a analysisgood prediction of the rubber for creep system evaluation mentioned of rubber-based above, the proposed engineering model cases. and its numerical approach could provide a good prediction for creep evaluation of rubber-based engineering cases.

T , is expressed as

CTT1  r  Log()T  , (20) CTT2 ()r where C1 and C2 are two constants which are related to the reference temperature Tr and the type of rubber materials. According to ISO4664-1, when the glass transition temperature (Tg) is regarded as the reference temperature (Tr), then C1 and C2 are 17.44 K and 51.6 K, respectively [31]. Then, the time– temperature equivalent shift factor calculated in the term of glass transformation temperature is

17.44TTg  Log()T  . (21) 51.6 (TTg )

One of the key issues of applying the WLF equation is the determination of the glass transition temperature of rubber materials. In the present work, the glass transformation temperature was tested utilizing the dynamic differential scanning calorimetry (DSC) method [31]. Measurements were conducted on a Mettler–Toledo DSC instrument (Type: QJ-X03) in a fluid nitrogen atmosphere. Rubber samples were prepared weighing 5.80 mg of the compound in aluminum crucibles. The weight was measured by the Mettler–Toledo scale (type: XP105) with a resolution of 0.01mg. Before testing, the rubbers were heated from −100 °C to 100 °C for erasing in-balance thermal effects, then this was repeated using constant heating and cooling rates of 10 °C/min. During DSC measurement, liquid nitrogen was released at 10 mL/min. The thermal flow curve of rubber samples is shown in Figure 21. The inflection point of this figure represents the glass transition temperature of rubber samples, hence the glass transition temperature was –25.70 °C as shown by data processing. Additionally, according to GB/T 29611-2013, titled as “Determination of the rubber’s glass transition temperature by differential scanning calorimetry (DSC) method”, the labels of exo and endo are added in Figure 21.

Polymers 2019, 11, 988 17 of 19

5. Time–Temperature Equivalent Analysis It is evident that creep responses measured by long-term loading creep tests are expensive and time-consuming. To reduce the experimental cost, time–temperature equivalent analysis should be conducted. Some accelerated methods have been developed to predict long-term creep performances of rubber materials based on short-time experiments [27,28]. The principle of these accelerated methods lies in the fact that, in the creep test, the effect of longer time is similar to the effect of higher temperature [29]. Among them, the well-known equation for describing the temperature–time equivalent principle is Williams–Landel–Ferry (WLF) equation [30]; its time-dependent shift factor, αT, is expressed as C1(T Tr) Log(αT) = − − , (20) C + (T Tr) 2 − where C1 and C2 are two constants which are related to the reference temperature Tr and the type of rubber materials. According to ISO4664-1, when the glass transition temperature (Tg) is regarded as the reference temperature (Tr), then C1 and C2 are 17.44 K and 51.6 K, respectively [31]. Then, the time–temperature equivalent shift factor calculated in the term of glass transformation temperature is 17.44 T Tg Log(αT) = − − . (21) 51.6 + (T Tg) − One of the key issues of applying the WLF equation is the determination of the glass transition temperature of rubber materials. In the present work, the glass transformation temperature was tested utilizing the dynamic differential scanning calorimetry (DSC) method [31]. Measurements were conducted on a Mettler–Toledo DSC instrument (Type: QJ-X03) in a fluid nitrogen atmosphere. Rubber samples were prepared weighing 5.80 mg of the compound in aluminum crucibles. The weight was measured by the Mettler–Toledo scale (type: XP105) with a resolution of 0.01 mg. Before testing, the rubbers were heated from 100 C to 100 C for erasing in-balance thermal effects, then this was − ◦ ◦ repeated using constant heating and cooling rates of 10 ◦C/min. During DSC measurement, liquid nitrogen was released at 10 mL/min. The thermal flow curve of rubber samples is shown in Figure 21. The inflection point of this figure represents the glass transition temperature of rubber samples, hence the glass transition temperature was –25.70 ◦C as shown by data processing. Additionally, according to GB/T 29611-2013, titled as “Determination of the rubber’s glass transition temperature by differential scanning calorimetry (DSC)

Polymersmethod”, 2018 the, 10,labels x FOR PEER of exo REVIEW and endo are added in Figure 21. 18 of 20 Exo

Endo Figure 21. Determination of T by diﬀerential scanning calorimetry (DCS) method. Figure 21. Determination of Tgg by differential scanning calorimetry (DCS) method.

By integrating glass transition temperature into Equation (21), the master curve of the equivalent By integrating glass transition temperature into Equation (21), the master curve of the equivalent shift factor, αT, is illustrated in Figure 22. shift factor, T , is illustrated in Figure 22.

Figure 22. Temperature dependence of instantaneous modulus of rubber materials.

Based on the equivalent shift factor master curve, the creep compliance, J(T1) at temperature T1, can be converted to the creep compliance, J(T2) at temperature T2 [31], which is expressed as  T JTJT  log1 . (22) 12 T2

6. Conclusions This paper provides a hyper-elastic creep constitutive model to evaluate creep characteristics of rubber materials under different conditions. The numerical implementation of the proposed phenomenological model is presented and validated. The time-dependent, loading-related, and temperature-induced creep behaviors of rubber materials are studied. The proposed method is further utilized to predict creep performances of a rubber vibration system for validating its engineering potential. A time-temperature equivalent analysis by WLF equation in glass transition temperature is also introduced. By comparing numerical and experimental results, the proposed creep models could depict nonlinear creep behaviors of rubber materials and rubber vibration systems, which provides an option for rubber system design and its creep prediction.

Author Contributions: conceptualization, D. L. and G.L.; methodology, D.L. and K.X.; software, K.X.; validation, Y.M. and L.Q.; formal analysis, Y.M.; investigation, G.L.; resources, L.Q.; data curation, L.Q.; writing—original draft preparation, D.L.; writing—review and editing, G.L.; visualization, L.Q.; supervision, G.L.; project administration, D.L.; funding acquisition, D. L. and Y.M.

Polymers 2018, 10, x FOR PEER REVIEW 18 of 20 Exo

Endo

Figure 21. Determination of Tg by differential scanning calorimetry (DCS) method.

By integrating glass transition temperature into Equation (21), the master curve of the equivalent Polymers 2019, 11, 988 18 of 19 shift factor, T , is illustrated in Figure 22.

Figure 22. Temperature dependence of instantaneous modulus of rubber materials. Figure 22. Temperature dependence of instantaneous modulus of rubber materials.

Based on the equivalent shift factor master curve, the creep compliance, J(T1) at temperature T1, Based on the equivalent shift factor master curve, the creep compliance, J(T1) at temperature T1, can be converted to the creep compliance, J(T2) at temperature T2 [31], which is expressed as can be converted to the creep compliance, J(T2) at temperature T2 [31], which is expressed as " ! # αT1 J(T1) = Jlog  T2 . (22) α T1 · JTJT  logT2 . (22) 12 T2 6. Conclusions  This paper provides a hyper-elastic creep constitutive model to evaluate creep characteristics 6. Conclusions of rubber materials under different conditions. The numerical implementation of the proposed phenomenologicalThis paper provides model a hyper is presented-elastic andcreep validated. constitutive The model time-dependent, to evaluate creep loading-related, characteristics and of rubbertemperature-induced materials under creep different behaviors conditions. of rubber materials The numerical are studied. implementation The proposed of method the proposed is further phenomenologicalutilized to predict creepmodel performances is presented of and a rubber validated. vibration The time system-dependent, for validating loading its-related engineering, and temperaturepotential. A- time-temperatureinduced creep behaviors equivalent of analysis rubber by materials WLF equation are studied. in glass The transition proposed temperature method is is furtheralso introduced. utilized Byto comparingpredict creep numerical performances and experimental of a rubber results, vibration the proposed system creep for validating models could its engineeringdepict nonlinear potential. creep behaviorsA time-temperature of rubber materials equivalent and analysis rubber by vibration WLF equation systems, in which glass provides transition an temperatureoption for rubber is also system introduced. design and By comparing its creep prediction. numerical and experimental results, the proposed creep models could depict nonlinear creep behaviors of rubber materials and rubber vibration Author Contributions: conceptualization, D.L. and G.L.; methodology, D.L. and K.X.; software, K.X.; validation, systemY.M. ands,L.Q.; which formal provides analysis, an Y.M.; option investigation, for rubber G.L.; system resources, design L.Q.; and data its curation, creep prediction. L.Q.; writing—original draft Authorpreparation, Contributions: D.L.; writing—review conceptualization, and editing, D. L. and G.L.; G.L. visualization,; methodology, L.Q.; D.L. supervision, and K.X. G.L.;; software, project K. administration,X.; validation, D.L.; funding acquisition, D.L. and Y.M. Y.M. and L.Q.; formal analysis, Y.M.; investigation, G.L.; resources, L.Q.; data curation, L.Q.; writing—original draftFunding: preparation,This work D.L was.; writing funded—review by National and Naturalediting, ScienceG.L.; visualization, Foundation ofL.Q China.; supervision, (No. 51709248), G.L.; project Open administration,Foundation of HenanD.L.; funding Key Laboratory acquisition, ofUnderwater D. L. and Y.M. Intelligent Equipment (KL03B1803) and National Natural Science Foundation of Shandong Province (No. 201503014). Conflicts of Interest: The authors declare no conflict of interest.

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