Nonlinear dependence of viscosity in modeling the rate-dependent response of natural and high damping rubbers in compression and shear: Experimental identification and numerical verification

The rate-dependent behavior of filled natural rubber (NR) and high damping rubber (HDR) is investigated in compression and shear regimes. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity founded on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts is proposed. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the concept of the Zener model. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme that ascertains the fundamental relation between the inelastic strain rate and the overstress tensor of the Mandel type by evaluating simple relaxation test results is proposed. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this experimentally based motivation, an evolution equation using power laws is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated in the finite strain viscoelasticity model in a thermodynamically consistent way. Simulation results for simple relaxation tests, multi-step relaxation tests and monotonic tests at different strain rates using the developed model show an encouraging correlation with the experiments conducted on HDR and NR in both compression and shear regimes. Finally, an approach to extend the proposed evolution equation for rate-dependent cyclic processes is proposed. The simulation results are critically compared with the experiments.

[1]  O. H. Yeoh,et al.  Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates , 1990 .

[2]  S. Reese,et al.  A theory of finite viscoelasticity and numerical aspects , 1998 .

[3]  Stefan L. Burtscher,et al.  Aspects of Cavitation Damage in Seismic Bearings , 2000 .

[4]  Christian Miehe,et al.  Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation , 2000 .

[5]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .

[6]  E. Krempl,et al.  Models of viscoplasticity some comments on equilibrium (back) stress and drag stress , 1987 .

[7]  Ch. Tsakmakis,et al.  Finite deformation viscoelasticity laws , 2000 .

[8]  L. Mullins.  Softening of Rubber by Deformation , 1969 .

[9]  Large Strain Viscoelastic Constitutive Models for Rubber, Part II: Determination of Material Constants , 1995 .

[10]  Jacob Lubliner,et al.  On fading memory in materials of evolutionary type , 1969 .

[11]  Sanjay Govindjee,et al.  A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins' effect , 1991 .

[12]  Sanjay Govindjee,et al.  Mullins' effect and the strain amplitude dependence of the storage modulus , 1992 .

[13]  A. I. Leonov.  Nonequilibrium thermodynamics and rheology of viscoelastic polymer media , 1976 .

[14]  Mary C. Boyce,et al.  Large strain time-dependent behavior of filled elastomers , 2000 .

[15]  J. L. Sullivan,et al.  A Non-Linear Viscoelastic Characterization of a Natural Rubber Gum Vulcanizate , 1980 .

[16]  R. Germundsson,et al.  Mathematica Version 4 , 2000 .

[17]  R. Ogden,et al.  A pseudo–elastic model for the Mullins effect in filled rubber , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  Ozgen U. Colak,et al.  Modeling deformation behavior of polymers with viscoplasticity theory based on overstress , 2005 .

[19]  Alan N. Gent.  Relaxation Processes in Vulcanized Rubber. II. Secondary Relaxation Due to Network Breakdown , 1962 .

[20]  Javier Bonet,et al.  Large strain viscoelastic constitutive models , 2001 .

[21]  A. Lion,et al.  A physically based method to represent the thermo-mechanical behaviour of elastomers , 1997 .

[22]  Yoshihiro Yamashita,et al.  Approximated form of the strain energy-density function of carbon-black filled rubbers for industrial applications. , 1992 .

[23]  Satoshi Fujita,et al.  Research, Development and Implementation of Rubber Bearings for Seismic Isolation , 1990 .

[24]  Oscar Lopez-Pamies,et al.  Time and temperature dependent response and relaxation of a soft polymer , 2002 .

[25]  P. R. Pinnock,et al.  The mechanical properties of solid polymers , 1966 .

[26]  B. D. Coleman,et al.  An approximation theorem for functionals, with applications in continuum mechanics , 1960 .

[27]  M. Boyce,et al.  Constitutive models of rubber elasticity: A review , 2000 .

[28]  D. Besdo,et al.  Properties of rubberlike materials under large deformations explained by self-organizing linkage patterns , 2003 .

[29]  James M. Kelly,et al.  Earthquake-Resistant Design with Rubber , 1993 .

[30]  Gerhard A. Holzapfel,et al.  A new viscoelastic constitutive model for continuous media at finite thermomechanical changes , 1996 .

[31]  Yoshiaki Okui,et al.  An improved hyperelasticity relation in modeling viscoelasticity response of natural and high damping rubbers in compression: experiments, parameter identification and numerical verification , 2002 .

[32]  Jacob Lubliner,et al.  Normality rules in large-deformation plasticity , 1986 .

[33]  Mary C. Boyce,et al.  Constitutive modeling of the large strain time-dependent behavior of elastomers , 1998 .

[34]  Yoshiaki Okui,et al.  Hyperelasticity Model for Finite Element Analysis of Natural and High Damping Rubbers in Compression and Shear , 2006 .

[35]  R. Christensen.  A Nonlinear Theory of Viscoelasticity for Application to Elastomers , 1980 .

[36]  Peter Haupt,et al.  Continuum Mechanics and Theory of Materials , 1999 .

[37]  A. Lion,et al.  On finite linear viscoelasticity of incompressible isotropic materials , 2002 .

[38]  Aleksey D. Drozdov,et al.  A micro-mechanical model for the response of filled elastomers at finite strains , 2003 .

[39]  D. Besdo,et al.  A phenomenological constitutive model for rubberlike materials and its numerical applications , 2003 .

[40]  A. Drozdov.  A constitutive model for nonlinear viscoelastic media , 1997 .

[41]  H. G. Kilian,et al.  Universal properties in filler-loaded rubbers , 1994 .

[42]  J. C. Simo,et al.  On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects , 1987 .

[43]  Yozo Fujino,et al.  CONSTITUTIVE MODEL OF HIGH-DAMPING RUBBER MATERIALS , 2004 .

[44]  Jacob Lubliner,et al.  A model of rubber viscoelasticity , 1985 .

[45]  S. Ahzi,et al.  A two-phase self-consistent model for the deformation and phase transformation behavior of polymers above the glass transition temperature: application to PET , 2005 .

[46]  Shape optimization of a rubber bearing , 1991 .

[47]  Sanjay Govindjee,et al.  Transition from micro-mechanics to computationally efficient phenomenology: Carbon black filled rubbers incorporating mullins' effect , 1992 .

[48]  P. Haupt,et al.  Viscoplasticity of elastomeric materials: experimental facts and constitutive modelling , 2001 .

[49]  A. Castellani,et al.  ELASTOMERIC MATERIALS USED FOR VIBRATION ISOLATION OF RAILWAY LINES , 1998 .

[50]  M. A. Johnson,et al.  A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments , 1993 .

[51]  Lucien Laiarinandrasana,et al.  Visco-hyperelastic model with internal state variable coupled with discontinuous damage concept under total Lagrangian formulation , 2003 .

[52]  Haoyue Zhang,et al.  Finite deformation of a polymer: experiments and modeling , 2001 .

[53]  Jacob Lubliner,et al.  On the structure of the rate equations of materials with internal variables , 1973 .

[54]  A. Lion,et al.  The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales , 2004 .

[55]  M. A. Johnson,et al.  The mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string , 1993 .

[56]  P. Mason.  The strain dependence of rubber viscoelasticity. Part II. The influence of carbon black , 1960 .

[57]  John F. Stanton,et al.  Elastomeric Bearings: State‐of‐the‐Art , 1983 .

[58]  Athol J. Carr,et al.  Compression behaviour of bridge bearings used for seismic isolation , 1996 .

[59]  E. Meinecke,et al.  Effect of Carbon Black on the Mechanical Properties of Elastomers , 1988 .

[60]  A. Tobolsky,et al.  A New Approach to the Theory of Relaxing Polymeric Media , 1946 .

[61]  Alexander Lion,et al.  A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation , 1996 .

[62]  H. Rothert,et al.  Formulation and implementation of three-dimensional viscoelasticity at small and finite strains , 1997 .

[63]  J. Yang,et al.  A Review of Methods to Characterize Rubber Elastic Behavior for Use in Finite Element Analysis , 1994 .

[64]  D. Seibert,et al.  Direct Comparison of Some Recent Rubber Elasticity Models , 2000 .

[65]  F. Bueche,et al.  Mullins Effect and Rubber-Filler Interaction , 1962 .

[66]  George T. Gray,et al.  Three-dimensional, finite deformation, viscoplastic constitutive models for polymeric materials , 1997 .

[67]  R. Rivlin.  Large Elastic Deformations of Isotropic Materials , 1997 .

[68]  S. Edwards,et al.  Elasticity of entangled networks , 1981 .

[69]  Arthur R. Johnson,et al.  Rubber Viscoelasticity Using the Physically Constrained System's Stretches as Internal Variables , 1993 .

[70]  Erhard Krempl,et al.  Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers , 2003 .

[71]  Joey Mead,et al.  Large Strain Viscoelastic Constitutive Models for Rubber, Part I: Formulations , 1994 .

[72]  Alan N. Gent,et al.  Relaxation processes in vulcanized rubber. I. Relation among stress relaxation, creep, recovery, and hysteresis , 1962 .

[73]  Gerhard A. Holzapfel,et al.  ON LARGE STRAIN VISCOELASTICITY: CONTINUUM FORMULATION AND FINITE ELEMENT APPLICATIONS TO ELASTOMERIC STRUCTURES , 1996 .

[74]  A. Amin,et al.  Measurement of lateral deformation in natural and high damping rubbers in large deformation uniaxial tests , 2003 .

[75]  Glaucio H. Paulino,et al.  Computation of Mixed-Mode Stress Intensity Factors for Cracks in Three-Dimensional Functionally Graded Solids , 2004 .

[76]  R. Rivlin.  LARGE ELASTIC DEFORMATIONS OF ISOTROPIC MATERIALS. I. FUNDAMENTAL CONCEPTS , 1997 .

[77]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .