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

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A Hyper-Elastic Creep Approach and Characterization Analysis for Rubber Vibration Systems polymers 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 Polymers 2019, 11, 988; doi:10.3390/polym11060988 www.mdpi.com/journal/polymers Polymers 2019, 11, 988 2 of 19 Polymers 2018, 10, x FOR PEER REVIEW 2 of 20 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 toto thethe 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 forresponses depicting are linearlynot 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] [16developed] established a modified a modified Burger viscoelastic model with model tunable in pronydamping series and to stiffness study creep characteristicscoefficients offor compressed calculating rubbercreep productsdeformation 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 complexbe 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
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