Article Damping Characteristics of Viscoelastic Damping Structure under Coupled Condition

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1 Article Damping Characteristics of Viscoelastic Damping Structure uner Couple Conition Jun Wang, Dagang Sun *, Shizhong Liu an Xin Zhang College of Mechanical Engineering, Taiyuan University of Science an Technology, Taiyuan 34, China; (J.W.); (S.L.); (X.Z.) * Corresponence: sungbox@sina.com; Tel.: Acaemic Eitor: Fazal M. Mahome Receive: 18 November 16; Accepte: 3 March 17; Publishe: 15 March 17 Abstract: Temperature has an influence on amping characteristics of the viscoelastic amping structure. The change of the amping characteristics of the structure uner the cycle loa is a ynamic an couple process. The hyperelastic-viscoelastic moel was use to escribe nonlinear eformation an viscoelasticity simultaneously. The temperature istribution an change of the amping characteristics uner the couple conition was analyze by finite element metho (FEM). The maximum value of the simulation results was in agreement with the one calculate by the formula in the literature. Dynamic stiffness an issipate energy were obtaine base on the hysteresis loop. Dynamic stiffness an issipate energy graually ecrease with the increase of the temperature. Keywors: viscoelastic amping structure; hyper-viscoelasticity; amping characteristics; thermalmechanical coupling 1. Introuction The viscoelastic amping structure is wiely applying for vibration an noise reuction. The viscoelastic amping layer, most of which is rubber or rubberlike polymers, represents hysteresis uner a ynamic loa. Part of the mechanical energy is absorbe an finally issipate as heat ue to the internal friction of the molecular chains. Continuous cyclic loa an poor conuction of heat may cause extensive heat to buil up in the structure. The mechanical response of the viscoelastic materials is often highly sensitive to temperature [1 3]. Heat may change the mechanical characteristics of the structure an the amping capability [4 6]. Excessive heat may lea to early fatigue failure or even explosive rupture. Therefore, it is essential to evaluate the temperature istribution an amping characteristics of the viscoelastic amping structure ue to the mechanical energy issipation uring cyclic loaing. Numerous analyses have been propose to estimate the temperature istribution an amping characteristics of viscoelastic material an its structure. Macro-mechanical or micro-mechanical material moels were propose to escribe thermo-viscoelastic behavior [7 1]. The complexity of the theory restricts their engineering application. Habibi et al. investigate structural bamboo at the microscopic an macroscopic level [11]. Two kins of biological cells separately have the responsibility of viscoelasticity at lower or higher frequencies. Davi I. G. Jones introuce simple an effective approaches for escribing the amping-relate properties of viscoelastic materials, with emphasis on the effects of frequency an temperature, an proceee to illustrate simple techniques for measuring the esire properties an for selecting an applying the materials [1]. Johnson an Chen employe a linear viscoelastic moel (the Maxwell soli moel) to solve the couple thermal an large strain history integral. Due to uner-preiction of the size of the hysteresis loop in the linear viscoelasticity, the error in issipate energy preiction at large strain is unacceptable [13]. Math. Comput. Appl. 17,, 7; oi: 1.339/mca17

2 Math. Comput. Appl. 17,, 7 of 7 Shah et al. stuie the couple problem of eformation of a linear-viscoelastic composite cyliner by applying the corresponence principle [14]. Pesek et al. propose a mathematical moel base on a weak formulation of the partial ifferential equation by FEMLAB (original release of COMSOL Multiphysics) to investigate the thermo-mechanical interaction in a pre-stresse rubber block use for resilient elements of compose tram wheels. A proportional amping moel an the equality of the heat energy ensity an the issipation energy ensity were compute to perform coupling between the mechanical an thermal equations uner selecte simple stress states [15]. Banic et al. investigate the temperature of a rubber amper uner cyclic loaing ue to hysteresis losses by finite element analysis (FEA). The visco-plastic constitutive moel establishe by Bergstrom- Boyce was use to preict the heat generation an hysteresis in the rubber-metal spring of railway raw gear [16]. Luo et al. preicte the heat generation of a rubber spring instrument uring the spring-accelerate fatigue test. A static hysteresis loop was obtaine via the FEA approach experimentally [17]. Kamran an Anastasia use a non-linear visco-elastic constitutive moel (propose by Schapery) to analyze the effect of coupling between the thermal an mechanical response, which was attribute to the issipation of energy, heat conuction an temperatureepenent material parameters on the overall response of visco-elastic solis [18]. Hwang an Yeong analyze the temperature istribution of a couple 3D ynamic rolling simulation of a tire by finite element metho (FEM) [19]. The heat generation rate was assume to be equal to the strain energy ensity function multiplie by the hysteresis coefficient. Fenza et al. investigate the amping characteristics of a viscoelastic embee composite fuselage structure by experiments at ifferent temperatures []. Kerchman an Cheng [1] evaluate the heat generation an transient temperature using linearize constitutive moel an FEM. Frequency-epenence on the Viscoelastic Damping VED structure is also a matter of major concern. The sanwich viscoelastic amping structure is analyze by the metho of moel reuction to reuce the high-orer finite element moels to a smaller size in irect ynamic analysis []. The frequency response analysis of viscoelastic beams an plane frames with an arbitrary number of Kelvin-Voigt viscoelastic ampers was concerne. The exact frequency response in all frame members was also obtaine in close analytical form [3]. Since the amping characteristics of viscoelastic material are sensitive to temperature, also ue to the low conuctivity of the material, the amping ability is a ynamic couple process. Some works treate the couple problem with a one-way coupling approach, especially common in the rolling resistance fiel of rubber tires. Some couple thermo-mechanical constitutive moels nee to user-efine material (UMAT) subroutine by support. Meanwhile, ue to the couple effect, the temperature istribution of the viscoelastic amping structure may not be constant or homogeneous. In this paper, a fully couple analysis of the viscoelastic amping structure will be investigate to escribe the change of the amping characteristics more accurately. The investigation by commercial finite coe ABAQUS (Dassault Systèmes, Vélizy-Villacoublay, France) will be valiate by comparing it with the available fiel ata in the present literature.. Simulation Metho.1. Hyper-Viscoelastic Moel Viscoelastic materials usually represent large eformation an ynamic characteristics simultaneously. The linear viscoelastic moels o a poor job of replicating the hysteresis loop in a loa/unloa cycle of eformation because the hysteresis loops generate are narrow. Therefore, the combine usage of the hyperelasticity an linear viscoelasticity (usually escribe as a Prony series) is use for the hysteresis analysis. The stress response of the hyper-viscoelastic moel consists of a nonlinear elastic part an a viscous part. The elastic response is instantaneous while the viscous part is prolonge over time. More etails of the constitutive moel as formulations an implementation are available in the literature [4,5]. The hyperelastic response (time-inepenent) is generally erive from stress-strain ata by tensile, compression, shear an/or biaxial tests. There stress-strain ata may be fitte as a particular strain energy function by the linear or nonlinear least squares

3 Math. Comput. Appl. 17,, 7 3 of 7 metho. The ynamic mechanical characteristics may be obtaine by the ynamic mechanical analyzer (DMA). The Prony series parameters were fitte by the optimization metho. For simplicity, the constitutive parameters of the material were from the example in the ABAQUS Example Problem Manual article [6,7]. All units of British thermal units (BTU) parameters were translate into the international system of units (SI). The hyperelastic response is escribe as the Neo Hooke moel: 1 U C ( I 3) ( J el 1) 1 1 (1) D1 3 where C1 is the positive material parameter, C , psi = Pa, an D 1 controls 6 compressibility, D The N-term Prony series: where g an i N gt ( ) g gi exp( t/ i) () i1 g are normalize imensionless constants an g N gi 1. The Prony parameters were as follows: g1.396, t g.118, t g3.858, t Finite Element Metho (FEM) Implementation For temperature epenence, most viscoelastic materials are usually assume as thermal rheological simple (TRS) material near or above the glass transition temperature. The TRS translation function is expresse by the Williams Lanel Ferry (WLF) equation [8]: i1 C1( ) log A C ( ) (3) where the values are efine as = 1.7 C, C 1 = 56.1 C, an C = 1 C. By means of the backwar ifference metho, the temperature is integrate into the Newton metho for solving nonlinear equations. Its exact solution algorithm [8]: Kuu Ku u Ru Ku K (4) R where u an are the respective corrections to the incremental isplacement an temperature, Kij are sub-matrices of the fully couple Jacobian matrix, Ru an R are the mechanical an thermal resiual vectors, respectively. A viscoelastic amping structure consisting of two elastic constraine layers an one viscoelastic amping layer was accomplishe by the solution of temperature isplacement in ABAQUS. The length was.9 m with 45 elements. The thickness of the elastic constraine layer an the viscoelastic amping layer was.1 m with eight elements. The elastic constraine layer was iscretize into 4- noe plane strain thermally couple quarilateral (CPE4T) elements an 4-noe plane strain thermally couple quarilateral (CPE4HT) elements for the viscoelastic amping layer. A sinusoial isplacement U.64sint ( f, f = 1 Hz) in the x irection was applie on the top left-han corner. The bottom sie was completely fixe (Figure 1). Heat generation by hysteresis is associate with intrinsic issipation ue to plasticity, which is usually escribe by inelastic heat friction (IHF). IHF is efine by the ratio of the issipate energy to the plastic work [9]. For viscoelastic materials, the IHF is assume to be in the ABAQUS help ocumentation. The properties of materials refer to the ABAQUS Example Problem Manual article (Table 1). The initial temperature was 1.7 C. An aaptive time stepping was etermine with a creep strain error tolerance of

4 Math. Comput. Appl. 17,, 7 4 of 7 U Viscoelastic amping layer Elastic constraine layer Fixe Figure 1. Finite Element (FE) moel of viscoelastic structure. Table1. Properties of materials. Property Elastic Constraine Layer Viscoelastic Damping Layer Density (kg/m 3 ) Moulus of elasticity (Pa) Poisson ratio.3 - Thermal expansion coefficient (1 6 (m/m K)) 1.8 Specific heat capacity (J/kg C) 5 19 Heat conuction(w/(m C) 7.14 Non-elastic thermal friction coefficient Results an Discussion 3.1. Temperature Distribution The temperature istribution in the en was contoure on a postprocessor (Figure ). The temperature ajacent to the elastic constraine layer nearly maintaine the ambient temperature (1.7 C) ue to pure elasticity an excellent thermal conuctivity. The temperature was increase from the outsie to the insie as concentric circle in the viscoelastic amping layer. The maximal value max appeare in the center, an reache 8.1 C. The same result coul be foun in the ABAQUS Example Problem Manual article. The value for Figure. Temperature ( C) contour in the en. max can also be calculate by the next formula [3]: max fu H (5) 8K where is the initial temperature, f is the frequency of the loaing cycles, H is the thickness of the viscoelastic component, K is the heat conuction of the viscoelastic material, an U is the amount of energy generate per cycle: U E (6) where E is the loss moulus of the viscoelastic material an is the amplitue of the strain. Further, max calculate at 8.9 C by Equation (5) matche well with the simulation (8.1 C). The simulation error was.7%, so it is clear that the propose metho has the avantages of high accuracy to evaluate the maximal value of the temperature istribution.

5 Math. Comput. Appl. 17,, 7 5 of Dynamic Damping Characteristics In this part, the change of the issipate energy an ynamic stiffness were iscusse. The hysteresis loop is generate by loa an isplacement ue to hysteresis behavior. The integration area of the hysteresis loop is the issipate energy E, which represents the ability of the viscoelastic amping structure. The reaction force of every noe RFn on the bottom an isplacement x of left corner on the top were extracte separately on the postprocessor. The reaction force of the structure RF was summe by RF n. The combine RF with x the hysteresis curve was rawn (Figure 3a). The hysteresis loop i not form completely in the first perio, so E was calculate from the secon perio to the en. E was reuce with the increase of the temperature. With the temperature increase, the relaxation time of the intermolecular motion was shortene. The issipate energy was ecreasing because the motion of the molecule graually kept up with the external force [13]. Dynamic stiffness is efine as GB/T [31] F F y X x (7) where F is ouble amplitue in maximal isplacement, X is ouble amplitue corresponing to the transmitting force, x is the isplacement per unit length on the x-axis, an y is the force per unit length on the y-axis (Figure 3b). Thirty percent of the ynamic stiffness was ecrease in the perio of loaing. A similar result was foun by Liu [3]. Both the stiffness an amping coefficient ecrease with the temperature increasing. Creep was aggravate by the temperature rising while the carrying capacity of the viscoelastic amping structure ecrease. With the increase of the total number of cycles, the elastic moulus or stiffness of the material ecreases. It is a common phenomenon that the structural stiffness cannot bear external loa before the material has been estroye. The ability of the issipate energy will strongly ecline. F F a b o X X (a) (b) Figure 3. Hysteresis loops (a) Couple hysteresis loop; (b) Schematic iagram. 4 E 4 E Dissipate energy E(J) Dissipate energy E(J) Time( t / h) (a) Time( t / h) (b) Figure 4. Change of ynamic amping characteristics. (a) Dissipate energy (E); (b) Dynamic stiffness (F).

6 Math. Comput. Appl. 17,, 7 6 of 7 4. Conclusions The aim of this research was to investigate the temperature istribution an change of the amping characteristics of the viscoelastic amping structure uner a couple conition. A hyperviscoelastic constitutive moel was employe to escribe the hyperelastic response an viscoelastic response. A two-imensional planar FEM moel was analyze in the couple temperatureisplacement solver in ABAQUS. The temperature istribution at the en of the loa illustrate that built-up heat appeare in the center of the structure. The maximal temperature is in agreement with the one calculate by the empirical formula. With the increase of the temperature cause by inelastic issipation comes a reuction in the issipate energy an ynamic stiffness. Since viscoelastic materials exhibit better issipation behavior when subjecte to shear loa, in the future research the analysis will be implemente in more styles of loa such as ben, compression an hybri conition. Besies, the IHF coefficient is a key factor for etermining the issipate energy ue to the inelastic eform. The value of the IHF will be evaluate precisely by an experiment in the next step. Acknowlegments: The work was fune by the National Natural Science Founation for Young Scientists of China (Grant No ), the National Natural Science Founation for Young Scientists of China (Grant No ). Author Contributions: Jun Wang put forwar to the metho an contribute the manuscript. Dagang Sun gave some writing an theory suggestion. Shizhong Liu helpe simulate couple solution. Xin Zhang provie some program for post process. Conflicts of Interest: The authors eclare no conflict of interest. References 1. War, I.M.; Sweeney, J. Chapter The mechanical properties of polymers: General consierations. In Mechanical Properties of Soli Polymers, n e.; Wiley: Hoboken, NJ, USA, 13.. Christensen, R. Chapter 3 Thermoviscoelasticity. In Theory of Viscoelasticity, n e.; Acaemic Press: Waltham, MA, USA, Ferry, J.D. Chapter 11 Depenence of viscoelastic behavior on temperature an pressure. In Viscoelastic Properties of Polymers; Wiley: Hoboken, NJ, USA, Chang, K.C.; Soong, T.T.; Oh, S.; Lai, M.L. Effect of Ambient Temperature on Viscoelastically Dampe Structure. J. Struct. Eng. 199, 118, , oi:1.161/(asce) (199)118:7(1955) 5. Fang, Q.H.; Zhang, F.P.; Huang, B.Z. Stuy on Extension Characteristics of Vulcanize Rubber at Different Temperatures. J. Buil. Mater. 5, 8, /j.issn Liu, D.H.; Fan, D.; Ouyang, Y.F.; Wu, C.P. Temperature Effect on Mechanical Properties of Rubber Isolators. Noise Vib. Control 14, 3, /j.issn Boukamel, A.; Méo, S.; Débores, O.; Jaeger, M. A Thermo-viscoelastic Moel for Elastomeric Behaviour an Its Numerical Application. Arch. Appl. Mech. 1, 71, , oi:1.17/s Roas, C.O.; Zaïri, F.; Naït-abelaziz, M. A Finite Strain Thermo-viscoelastic Constitutive Moel to Describe the Self-heating in Elastomeric Materials uring Low-cycle Fatigue. J. Mech. Phys. Solis 13, 64, DOI:1.116/j.jmps Stefanie, R. A Micromechanically Motivate Material Moel for the Thermo-viscoelastic Material Behaviour of Rubber-like Polymers. Int. J. Plast. 3, 19, DOI:1.116/S () Holzapfel, G.A.; Simo, J.C. A New Viscoelastic Constitutive Moel for Continuous Meia at Finite Thermomechanical Changes. Int. J. Solis Struct. 1996, 33, DOI: 1.116/-7683(95) Habibi, M.K.; Tam, L.H.; Lau, D.; Yang, L. Viscoelastic Damping Behavior of Structural Bamboo Material an Its Microstructural Origins. Mech. Mater. 16, 97, , oi:1.116/j.mechmat Jones, D.I.G. Hanbook of Viscoelastic Vibration Damping; 1st E.; Wiley: Hoboken, NJ, USA, Johnson, A.R.; Chen, T.K. Approximating Thermo-viscoelastic Heating of Largely Straine Soli Rubber Components. Comput. Methos Appl. Mech. Engrg. 5, 194, , oi:1.116/j.cma Shah, S.; Muliana, A.; Rajagopal, K.R. Couple Heat Conuction an Deformation in a viscoelastic Composite Cyliner. Mech. Time-Depen. Mater. 9, 13, DOI: 1.17/s

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