The nonlinear viscoelastic response of carbon black-filled natural rubbers

Three series of tensile relaxation tests are performed on natural rubber filled with various amounts of carbon black. The elongation ratio varies in the range from λ=2.0 to 3.5. Constitutive equations are derived for the nonlinear viscoelastic behavior of filled elastomers. Applying a homogenization method, we model a particle-reinforced rubber as a transient network of macromolecules bridged by junctions (physical and chemical cross-links, entanglements and filler clusters). The network is assumed to be strongly heterogeneous at the meso-level: it consists of passive regions, where rearrangement of chains is prevented by surrounding macromolecules and filler particles, and active domains, where active chains separate from temporary nodes and dangling chains merge with the network as they are thermally agitated. The rate of rearrangement obeys the Eyring equation, where different active meso-domains are characterized by different activation energies. Stress–strain relations for a particle-reinforced elastomer are derived by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data. It is demonstrated that the filler content strongly affects the rearrangement process: the attempt rate for separation of strands from temporary nodes increases with elongation ratio at low fractions of carbon black (below the percolation threshold) and decreases with λ at high concentrations of filler.     

Finite viscoelasticity of filled rubber: experiments and numerical simulation

Summary  Constitutive equations are derived for the viscoelastic behavior of particle-re-inforced elastomers at isothermal deformation with finite strain. A filled rubber is thought of as a composite medium where inclusions with high and low concentrations of junctions between chains are randomly distributed in the bulk material. The characteristic length of the inhomogeneities is assumed to be small compared to the size of the specimen and substantially exceed the radius of gyration for macromolecules. Inclusions with high concentration of junctions are associated with regions of suppressed mobility of chains that surround isolated clusters and/or the secondary network of filler. Regions with low concentration of junctions arise during the preparation process due to a heterogeneity in the spatial distribution of the cross-linker and the filler. With reference to the concept of transient networks, the time-dependent response of an elastomer is attribute d to thermally activated rearrangement of strands in the domains with low concentration of junctions. Stress–strain relations for particle-reinforced rubber are developed by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data in tensile relaxation tests for several grades of unfilled and carbon black-filled rubber. It is demonstrated that even at moderate finite deformations (with axial elongations up to 100%), the characteristic rate of relaxation is noticeably affected by strain. Unlike glassy polymers, where the rate of relaxation increases with longitudinal strain, the growth of the elongation ratio results in a decrease in the relaxation rate for natural rubber (unfilled or particle-reinforced). The latter may be explained by (partial) crystallization of chains in the regions with low concentration of junctions. Received 16 October 2001; accepted for publication 25 June 2002 Present address: A. D. Drozdov Department of Production, Aalborg University, Fibigerstraede 16, DK-9220 Aalborg, Denmark We would like to express our gratitude to Dr. K. Fuller (TARRC, UK) for providing us with rubber specimens and to Prof. P. Haupt and Dr. S. Hartmann (University of Kassel, Germany) for sending their experimental data. We are indebted to Mr. G. Seifritz for his assistance in performing mechanical tests. ADD acknowledges stimulating discussions with Prof. N. Aksel (University of Bayreuth, Germany).     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.     

Constitutive equations from Gaussian slip-link network theories in polymer melt rheology

Allowing for flow-dependent slip in the junctions of a temporary junction network, we derive the constitutive equations of temporary slip-link networks. The stress tensor is determined by three material functions, namely, the time-dependent linear-viscoelastic memory function, and two strain-dependent functions describing slip and disentanglement of network strands. Slip and disentanglement are related via a mass balance for network strands.By specifying slip and disentanglement, the constitutive equations of Lodge, Wagner, Doi-Edwards, and Marrucci are shown to be special temporary slip-link constitutive equations. To demonstrate the predictive power of temporary slip-link network theories, we compare predictions and extensional flow data with step change in flow direction.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Finite viscoplasticity of non-affine networks: stress overshoot under shear

A constitutive model is derived for the viscoplastic behavior of polymers at finite strains. A polymer is treated as an equivalent network of chains bridged by permanent junctions. The elastic response of the network is attributed to elongation of strands, whereas its plastic behavior is associated with sliding of junctions with respect to their reference positions. A new kinetic equation is proposed that expresses the rate of sliding of junctions in terms of the Cauchy stress tensor. Constitutive equations for an equivalent non-affine network are developed by using the laws of thermodynamics, where internal dissipation of energy reflects two processes at the micro-level: sliding of chains along entanglements and friction of strands between junctions. A similarity is revealed between these relations and the pom-pom model. The governing equations are applied to study stress overshoot at simple shear of an incompressible medium. Adjustable parameters in the stress-strain relations are found by fitting experimental data on polycarbonate melt reinforced with short glass fibers and polystyrene solution. Fair agreement is demonstrated between the observations and the results of numerical simulation.Received: 6 January 2003, Accepted: 28 April 2003     

The effect of temperature on the viscoelastic behavior of linear low-density polyethylene

Summary Observations are reported on linear low-density polyethylene in isothermal torsional oscillation and relaxation tests at various temperatures ranging from room temperature to 110 C. Constitutive equations are derived for the viscoelastic response of a semicrystalline polymer at small strains. The polymer is treated as an equivalent network of strands bridged by junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is thought of as an ensemble of meso-regions with various potential energies for rearrangement of strands. Two types of meso-domains are introduced: active, where strands separate from temporary junctions as they are excited by thermal fluctuations, and passive, where detachment of strands is prevented by the surrounding macromolecules. The time-dependent behavior of the ensemble reflects separation of active strands from their junctions and merging of dangling strands with the network. Stress–strain relations are developed by using the laws of thermodynamics. The governing equations involve six material constants that are found by fitting the experimental data. The study focuses on the effects of (i) temperature, (ii) the deformation mode (torsion versus bending), and (iii) the loading program (oscillations versus relaxation) on the adjustable parameters.This work was partially supported by the West Virginia Research Challenge Grant Program     

Influences of reinforcing particle and interface bonding strength on material properties of Mg/nano-particle composites

In the present study, an effective model is proposed to predict the effective elastic behavior of the three-phase composite containing spherical inclusions, each of which is surrounded by an interphase layer. The constitutive equations are derived for the stress and strain of each phase of the composite subjected to a far-field tension. Based on these constitutive laws, the effective bulk, shear and Young’s modulus are obtained. A statistical debonding criterion is adopted to characterize the varying probability of the evolution of interphase debonding. Influences of debonding damage, particle volume fraction, interphase properties and bonding strength on overall mechanical behavior of composites are also discussed. Numerical analyses are carried out on particle-reinforced composites and the predictions have a good agreement with the experimental results.     

炭黑填充橡胶材料的修正三链模型和Mullins模型

本文探讨了炭黑颗粒填充橡胶材料的本构模型。考虑到橡胶单个分子链与周围分子网络的约束作用和炭黑颗粒对橡胶的补强作用,提出了一种修正三链模型,用Edwards管模型描述分子链之间的相互作用和约束,采用应变放大因子来考虑炭黑含量的影响。并在修正三链模型的基础上,利用橡胶分子网络重构理论,提出了一种适合表征橡胶Mullins现象的本构模型。通过与实验数据比较分析,修正三链模型可较准确地表征未填充橡胶材料不同变形模式的力学性能和炭黑颗粒填充橡胶材料的单向拉伸力学行为,Mullins模型也可较好地描述橡胶材料的Mullins现象。     

A micro-mechanical model for the response of filled elastomers at finite strains

Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.     

Polymer Networks with Slip-links: 2. Constitutive Equations for a Cross-linked Network

Stress–strain relations are derived for the mechanical response of elastomers at arbitrary three-dimensional deformations with finite strains. An elastomer is treated as an incompressible network of chains bridged by permanent (chemical cross-links and physical cross-links whose lifetime exceeds the characteristic time of deformation) and temporary (entanglements modeled as slip-links) junctions. Two types of chains are introduced in the network to distinguish between permanent and temporary nodes. Type-I chains have free ends, and their motion at the micro-level is constrained by a random number of slip-links. Type-II chains are Gaussian chains permanently connected to the network. Concentration of type-I chains is fixed, while the number of type-II chains per unit volume can change under deformation. The governing equations involve two (networks with constant concentrations of type-II chains) or three (networks where the content of type-II chains is affected by mechanical factors) material parameters. These parameters are found by fitting observations on rubbers, thermoplastic–elastomers, and thermoplastic-elastomer composites. Good agreement is demonstrated between the experimental data in uniaxial tensile tests and the results of numerical simulation at elongations up to 1,000%. It is shown that the adjustable parameters are affected by chemical composition and molecular architecture of polymers in a physically plausible way.     

The influence of pre-loading and thermal recovery on the viscoelastic response of filled elastomers

Experimental data are reported in tensile relaxation tests on carbon black-filled natural rubber at strains up to 200%. Constitutive equations are derived for the time-dependent response of a particle-reinforced elastomer at finite strains. Adjustable parameters in the model are found by fitting observations. The effects on mechanical pre-loading and thermal recovery are analyzed on the material constants.     

Deformation of Particulate Composite with Physically Nonlinear Inclusions and Microdamageable Matrix

The structural theory of short-term damage is generalized to the case where the matrix of a particulate composite has microdamages and the inclusions deform nonlinearly. The basis for this generalization is the stochastic elasticity equations of a porous-matrix particle-reinforced composite. Microvolumes of the matrix meet the Huber-Mises failure criterion. A balance equation for damaged microvolumes is derived. The balance equation and the equations relating macrostresses and macrostrains of a particulate composite with porous matrix and physically nonlinear inclusions constitute a closed-form system. The system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the microdamage-macrostrain relationship and plotting deformation diagrams are proposed. Uniaxial tension curves are plotted for the case where the material of inclusions is linearly hardening__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 3–11, February 2005.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.      

Constitutive equations in finite elasticity of rubbers

A constitutive model is derived for the elastic behavior of rubbers at arbitrary three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk material. With reference to the concept of constrained junctions, the chain ends are assumed to be located at some distances from appropriate junctions. These distances are not fixed, but are altered under deformation. An explicit expression is developed for the distribution function of vectors between junctions (an analog of the end-to-end distribution function for a flexible chain with fixed ends). An analytical formula is obtained for the strain energy density of a polymer network, when the ratio of the mean-square distance between the ends of a chain and appropriate junctions is small compared with the mean-square end-to-end distance of chains. Stress–strain relations are derived by using the laws of thermodynamics. The governing equations involve three adjustable parameters with transparent physical meaning. These parameters are found by fitting experimental data on plain and particle-reinforced elastomers. The model ensures good agreement between the observations at uniaxial tension and the results of numerical simulation, as well as an acceptable prediction of stresses at uniaxial compression, simple shear and pure shear, when its parameters are found by matching observations at uniaxial tensile tests.     

A constitutive equation for filled rubber under cyclic loading

This paper describes experiments and the development of constitutive equations to predict the steady-state response of filled rubber under cyclic loading. An MTS servo-hydraulic machine was used to obtain the dynamic hysteresis curves for a filled rubber compound in uniaxial tension-compression. The material tests were performed with mean strains from −0.1 to 0.1, strain amplitudes ranging from 0.02 to 0.1, and strain rates between 0.01 and 10 s⁻¹. Temporary material set, the Payne effect and rate-dependence were observed from the experimental results. A hyper-viscoelastic constitutive model was developed to characterize the dynamic response of the rubber. A cornerstone of this constitutive modeling was to devise a scheme to evaluate material set and a finite strain, non-linear viscoelastic law from the test data. Predictions of the dynamic hysteresis curves using the proposed constitutive equation were found to be in good agreement with the uniaxial test results.     

Constitutive equations for non-affine polymer networks with slippage of chains

A model is derived for isothermal three-dimensional deformation of polymers with finite strains. A polymer fluid is treated as a permanent network of chains bridged by junctions (entanglements). Macro-deformation of the medium induces two motions at the micro-level: (i) sliding of junctions with respect to their reference positions that reflects non-affine deformation of the network, and (ii) slippage of chains with respect to entanglements that is associated with unfolding of back-loops. Constitutive equations are developed by using the laws of thermodynamics. Three important features characterize the model: (i) the symmetry of relations between the elongation of strands and an appropriate configurational tensor, (ii) the strong nonlinearity of the governing equations, and (iii) the account for the volumetric deformation of the network induced by stretching of chains. The governing equations are applied to the numerical analysis of extensional and shear flows. It is demonstrated that the model adequately describes the time-dependent response of polymer melts observed in conventional rheological tests.     

Macroscopic constitutive equations of thermo-poroviscoelasticity derived using eigenstrains

Macroscopic constitutive equations for thermo-viscoelastic processes in a fully saturated porous medium are re-derived from basic principles of micromechanics applicable to solid multi-phase materials such as composites. Simple derivations of the constitutive relations and the void occupancy relationship are presented. The derivations use the notion of eigenstrain or, equivalently, eigenstress applied to the separate phases of a porous medium. Governing coupled equations for the displacement components and the fluid pressure are also obtained.     

Design sensitivity analysis for the homogenized elasticity tensor of a polymer filled with rubber particles

The main purpose of this work is the computational simulation of the sensitivity coefficients of the homogenized tensor for a polymer filled with rubber particles with respect to the material parameters of the constituents. The Representative Volume Element (RVE) of this composite contains a single spherical particle, and the composite components are treated as homogeneous isotropic media, resulting in an isotropic effective homogenized material. The sensitivity analysis presented in this paper is performed via the provided semi-analytical technique using the commercial FEM code ABAQUS and the symbolic computation package MAPLE. The analytical method applied for comparison uses the additional algebraic formulas derived for the homogenized tensor for a medium filled with spherical inclusions, while the FEM-based technique employs the polynomial response functions recovered from the Weighted Least-Squares Method. The homogenization technique consists of equating the strain energies for the real composite and the artificial isotropic material characterized by the effective elasticity tensor. The homogenization problem is solved using ABAQUS by the application of uniform deformations on specific outer surfaces of the composite RVE and the use of tetrahedral finite elements C3D4. The energy approach will allow for the future application of more realistic constitutive models of rubber-filled polymers such as that of Mullins and for RVEs of larger size that contain an agglomeration of rubber particles.