The payne effect for particle-reinforced elastomers (original) (raw)
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Finite viscoelasticity of filled rubber: experiments and numerical simulation
Archive of Applied Mechanics (Ingenieur Archiv), 2003
Constitutive equations are derived for the viscoelastic behavior of particle-reinforced rubbers at isothermal loading with finite strains. A filled rubber is thought of as a composite medium where inclusions with high and low concentrations of junctions between chains are distributed in a bulk material. The characteristic size of inhomogeneities is assumed to be small compared to the dimensions of a specimen and to substantially exceed the radius of gyration for macromolecules. The inclusions with high concentration of junctions are associated with regions of suppressed mobility of chains that surround isolated clusters of filler and its secondary network. The regions with low concentration of junctions arise during the mixing process due to the inhomogeneity in spatial distribution of a cross-linker. With reference to the theory of transient networks, the viscoelastic response of elastomers is ascribed to the thermally activated processes of breakage and reformation of strands in the domains with low concentration of junctions. Stress-strain relations for particlereinforced rubbers 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 (CB) filled rubber. It is evidenced that even at moderate finite deformations (with axial elongations up to 100 %), the characteristic rate of relaxation is noticeably affected by strains. Unlike glassy polymers, where the growth of longitudinal strain results in an increase in the rate of relaxation, the growth of the elongation ratio for natural rubber (unfilled or CB reinforced) implies a decrease in the relaxation rate, which may be explained by partial crystallization of chains in the regions with low concentration of junctions.
Rubber network in elastomer nanocomposites
European Polymer Journal, 2007
The influence of inorganic nanoparticles on crosslinking mechanism of elastomers has been evaluated by applying the tube model on equilibrium statistical mechanics. The results have shown that a highly ordered structure with a huge amount of entanglements, wherein the polymer is nanoscopically confined, is formed by the addition of nanoparticles. These physical links exhibit freedom of movement under stretching, but in a lower volume because of confinement. That is, network molecular parameters such as lateral tube dimensions or average molecular mass of the chains decreased in presence of nanoparticles.
2000
Reinforcement of elastomers is modeled using Monte Carlo simulations on rotational isomeric state chains, to characterize their spatial configurations in the vicinity of filler particles. The resulting filler-perturbed distributions of the chain end-to-end distances are in agreement with experimental results gotten by neutron scattering. The use of these distributions in a standard molecular theory of rubberlike elasticity produces stress-strain isotherms for elongation that are consistent with available experimental results.
Macromolecules, 2006
We present the results of computer simulations of model polymer networks containing stiff and roughly spherical colloidal particles at 20% volume fraction. We employ the coarse-grained "dissipative particle dynamics" model (DPD). The filler particles may be either well dispersed or aggregated, and the strength of their interaction can be tuned by changing the parameters of the polymer-filler nonbonded potentials. By performing nonequilibrium molecular dynamics simulations, we are able to probe directly the viscoelastic behavior of the composites under oscillatory shear deformations of variable amplitude and frequency. The strength of the particleparticle interactions and sample morphology has a certain effect on the small-strain moduli. In some cases, we also observe a nonlinear viscoelastic behavior as a function of increasing shear amplitude. However, the characteristics of this nonlinearity are different from the experimentally observed "Payne effect". Therefore, the origin of this effect does not seem to be entirely and directly related to particle-particle interactions, as is frequently assumed.
Modeling the response of filled elastomers at finite strains by rigid-rod networks
Archive of Applied Mechanics (Ingenieur Archiv), 2002
A constitutive model is developed for the isothermal response of particle-reinforced elastomers at ®nite 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:¯exed and extended. The mechanical energy of a bond in the¯exed 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 ®nite strains, the governing equations are determined by four adjustable parameters which are found by ®tting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several ®lled and un®lled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, ®ller content, crosslink density and temperature on the adjustable constants.
Micromechanics of Rubber Reinforcement by Nano-Structured Fillers
The well known phenomena stress softening and hysteresis in filled rubbers are discussed in the frame of a micro-mechanical model which is an extension of a recently proposed hyperelastic material law. In this model the energy density is based on a generalized tube model of rubber elasticity and an additional contribution of the filler clusters due to their cyclic breakage and reaggregation under cyclic deformation. It has been found that experimental stress strain data from uniaxial deformation test for different strain maxima can be described with this model by a set of fitted parameters which have reasonable values. The stress softening is found to occur substantially at strain values higher than 2-5 %, which is in good agreement with the phenomenological behavior of the Payne effect threshold . 1.) Introduction A recently proposed micro-mechanical model for the stress-strain behavior of filled elastomers has been formulated on the basis of an extended tube model of rubber elasti...
Modeling of the effect of rigid fillers on the stiffness of rubbers
Journal of Applied Polymer Science, 2008
The theories that predict the increase in the modulus of elastomers resulting from the presence of a rigid filler are typically derived from Einstein's viscosity law. For example, Guth and Gold used this approach to predict how the Young's modulus of an elastomer is related to the filler volume fraction. Hon et al. have shown using finite element microstructural models that stiffness predictions at small strains were also possible. Here, microstructural finite element models have been used to investigate the modulus of filled elastomer over a wider range of strains than has been possible previously. At larger strains, finite extensibility effects are significant and here an appropriate stored energy function proposed by Gent was adopted. In this work, the effect of spherical MT-type carbon-black filler behavior was considered. Different models were made and the results were then compared to experimental measurement of the stiffness taken from the literature. It is shown that the boundary conditions of the microstructural unit cell lie between the two extremes of free surfaces and planar surfaces. Also as the filler volume fraction increases then the number of filler particles required in the representative volume to predict the actual stiffness behavior also increases. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008
The stress–strain response and ultimate strength of filled elastomers
Computational Materials Science, 2001
A constitutive model is derived for the mechanical behavior of reinforced elastomers at ®nite strains. A polymer is treated as a rigid-rod network, whose rupture is tantamount to breakage of chains treated as bond scission. Adjustable parameters in the stress±strain relations are found by ®tting observations in tensile tests for ®lled and un®lled ethylene± octene copolymers. It is demonstrated that the model correctly describes stress±strain curves up to the break points. We analyze the eects of temperature, the degree of crystallinity and the ®ller content on Young's modulus and the ultimate strain per bond. It is shown that the dependences of material constants on the volume fraction of carbon black are substantially altered at the critical ®ller contents which correspond to the percolation thresholds found by dc conductivity measurements.