Models Of Rubber Elasticity Research Articles

Hyperelasticity and stress softening of filler reinforced polymer networks

AbstractStarting from an analysis of filler networking in bulk polymers, a constitutive micro‐mechanical model of stress softening and hysteresis of filler reinforced polymer networks is developed. It refers to a non‐affine tube model of rubber elasticity, including hydrodynamic amplification of the rubber matrix by a fraction of hard, rigid filler clusters with filler‐filler bonds in the unbroken, virgin state. The filler‐induced hysteresis is described by an anisotropic free energy density, considering the cyclic breakdown and re‐aggregation of the residual fraction of soft filler clusters with already broken, damaged filler‐filler bonds. Experimental investigations of the quasi‐static stress‐strain behaviour of silica and carbon black filled rubbers up to large strain agree well with adaptations found by the developed model. The microscopic material parameters obtained appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler‐filler bonds and the polymer network chain density.

An Average-Stretch Full-Network Model for Rubber Elasticity

An Average-Stretch Full-Network Model for Rubber Elasticity

Constitutive equations for amended non-Gaussian network models of rubber elasticity

New constitutive equations based on an amended form of the Kuhn–Grün probability distribution function due to Jernigan and Flory are derived from the standard James–Guth (JG) 3-chain and Arruda–Boyce (AB) 8-chain non-Gaussian molecular network models. The kinematics describing the stretch of a 1-chain model in an affine deformation shows that the relative stretch of a single molecular chain initially oriented along the diagonal of a cube is determined by the first principal invariant of the Cauchy–Green deformation tensor. The Kuhn–Grün probability distribution for a randomly oriented chain and its more general amended form due to Wang and Guth, are functions of only the relative chain stretch. Hence, any non-Gaussian network model for which the configurational entropy of all chains may be uniform is characterized by an elastic response function that depends on only the first principal invariant of the Cauchy–Green deformation tensor. Both the regular and amended AB 8-chain models are characterized by specific response functions in this class; the regular and amended JG 3-chain models, however, are not. An amended form of the phenomenological composite 3-chain/8-chain model suggested by Wu and van der Giessen is introduced. Analytical relations for several kinds of homogeneous deformations of the standard and amended models are compared with a variety of experimental data by others. It is found that results for the amended 3-chain and 8-chain models do not vary significantly from results for the corresponding regular models. The composite model, on the other hand, shows excellent overall agreement with the diverse data, including equibiaxial deformations for which other models show greater variance; but it offers no improvement in comparison with data for plane strain compression. Some remarks relating the chain parameters of the 3-chain and 8-chain network models, and the limiting chain and continuum stretches for these models are discussed in an appendix.

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I1 of the Cauchy–Green strain tensor, is a simple logarithmic function of I1 and involves just two material parameters, the shear modulus μ and a parameter Jm which measures a limiting value for I1−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Pade approximant for this function. The constants μ and Jm in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function.

Study of the resist deformation in nanoimprint lithography

Numerical simulations and experimental studies are carried out to understand the deformation process of thin polymer film in nanoimprint lithography. Deformation of a thin polymer above its glass transition temperature is studied for various imprinting conditions such as the aspect ratios of a mold pattern, initial thickness of the polymer, and imprinting pressure. Cross-sectional profiles of the deformed polymers are simulated by the finite element method based on a rubber elastic model. The results are compared with experimental data. The areal penetration ratio of the polymer into the recessed groove of the mold and residual thickness underneath the mold are quantitatively evaluated. The simulations and the experimental results agree well with each other.

Multiaxial Deformations of End-linked Poly(dimethylsiloxane) Networks. 2. Experimental Tests of Molecular Entanglement Models of Rubber Elasticity

Five molecular models of rubber elasticity which employ different treatments of entanglement effects (the Kloczkowski−Mark−Erman diffused-constraint model, the Edwards−Vilgis (E−V) slip−link model, the tube models of Gaylord−Douglas (G−D), Kaliske−Heinrich, Rubinstein−Panyukov versions) are assessed using biaxial deformation data for an entanglement-dominated network of end-linked poly(dimethylsiloxane) (PDMS) in which trapped entanglements are dominant in number relative to chemical cross-links. The theoretical stress−strain relations were calculated from the elastic free energy (W) of each model. Using the reduced stress (the nominal stress divided by equilibrium modulus Go), the strain-dependent predictions of each model were tested from two different viewpoints, i.e., the dependence of the reduced stresses on the principal ratio and the Ii dependence of (∂W/∂Ij)/Go (i,j = 1,2), where I1 and I2 are the first and second invariants of deformation tensor (the Rivlin−Saunders method). The diffused-constrai...

Influence of Preparation Conditions on Network Parameters of Sulfur-Cured Natural Rubber

The effect of initial chain length before cross-linking and sulfur/accelerator amount during curing of natural rubber samples on network parameters (effective chain density, gel fraction, amount of chain ends, entanglement density, trapping factor, and relaxation and correlation times) is investigated by means of proton NMR relaxation, equilibrium swelling, and stress−strain analysis. Remarkable differences are observed in the two sample series depending on the variable initial molar mass. The stress−strain data are evaluated with respect to a non-Gaussian tube model of rubber elasticity that considers the finite extensibility of network chains by referring to the path integral approach of Edwards and Vilgis (Rep. Prog. Phys. 1988, 51, 243; Polymer 1986, 27, 483). According to several experimental indications, we assume a nonaffine tube deformation law as first derived by Heinrich et al. (Adv. Polym. Sci. 1988, 85, 33). The NMR relaxation data are analyzed by considering three types of chains (gel, sol and chain ends). The best fit is obtained by assuming an anisotropic motion of the inter-cross-link chains and chain ends. The swelling data are analyzed by assuming phantom like chains. Within the framework of experimental errors, the network parameters evaluated from the three experimental techniques show fair agreement for both sample series.

On the Length Scale Dependence of Microscopic Strain by SANS

We present a SANS study on the length scale dependence of chain deformation patterns in dense cross-linked elastomers. Three different polyisoprene networks of long primary block-copolymer chains of the HDH-type were analyzed. The total length of the primary chains is identical, while the length of the deuterated middle block was varied in order to cover several length scale regimes of interest. The scattering data are analyzed in the frame of the tube model of rubber elasticity in combination with the random phase approximation (RPA), which is used in order to account for the interchain correlations. We show experimentally that for the longest labeled path the rubber elastic response is dominated by both cross-link and entanglement contributions, whereas it is of clear phantomlike chain behavior if the labeled middle block of the chain is shorter than the distance between two successive entanglements, i.e., the tube diameter. We further propose schemes to quantify nonaffine deformations on various length scales and chain scissioning processes during the radical cross-linking process.

Drawability and attainable mechanical properties of polyamide yarn using true stress – true strain curves

The present paper applies the concept of a molecular network to the analysis of melt spun Polyamide 4.6 fibres (Stanyl® from DSM, The Netherlands), obtained at different wind up speeds, and the yarns, subsequently drawn from these. The mechanical properties of the yarns produced are discussed in terms of the network draw ratio, determined by matching true stress–true strain curves. It is shown that analysis of true stress–true strain curves of as-spun yarns of different orientation, can be used to predict their drawability (final tenacity versus draw ratio). Moreover, the maximum attainable tenacity of drawn yarns under given drawing conditions can be forecasted. It is shown that the Edwards–Vilgis rubber elastic model is able to describe the network deformational behaviour of the as-spun yarns over a wide range of draw ratios. The apparent network draw ratio, obtained by curve matching the drawn yarn true stress–true strain curves with those of its precursor yarn, shows discrepancies with the actually applied draw ratio in the drawing process. These discrepancies are discussed in terms of network structural parameters. It is made plausible that growth of existing crystals during hot-drawing is responsible for this, rather than formation of new crystals.

Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity

In this work a specialized molecular simulation code has been used to provide details of the underlying micromechanisms governing the observed macroscopic behavior of elastomeric materials. In the simulations the polymer microstructure was modeled as a collection of unified atoms interacting by two-body potentials of bonded and non-bonded type. Representative Volume Elements (RVEs) containing a network of 200 molecular chains of 100 bond lengths are constructed. The evolution of the RVEs with uniaxial deformation was studied with molecular dynamics techniques. The simulations enable observation of structural features with deformation including bond lengths and angles as well as chain lengths and angles. The simulations also enable calculation of the macroscopic stress-strain behavior and its decomposition into bonded and non-bonded contributions. The distribution in initial end-to-end chain lengths is consistent with Gaussian statistics treatments of rubber elasticity. It is shown that application of an axial strain of +/ 0.7 (a logarithmic strain measure is used) only causes a change in the average bond angle of /+ 5 degrees indicating the freedom of bonds to sample space at these low-moderate deformations; the same strain causes the average chain angle to change by /+ 20 degrees. Randomly selected individual chains are monitored during deformation; their individual chain lengths and angles are found to evolve in an essentially ane manner consistent with Gaussian statistics treatments of rubber elasticity. The average chain length and angle are found to evolve in a manner consistent with the eight-chain network model of elasticity. Energy quantities are found to remain constant during deformation consistent with the nature of rubber elasticity begin entropic in origin. The stress-strain response is found to have important bonded and non-bonded contributions. The bonded contributions arise from the rotations of the bonds toward the maximum principal stretch axis(es) in tensile(compressive) loading.

A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems

An advanced micro-mechanical model of hyperelasticity and stress softening of reinforced rubbers is presented that combines a non-Gaussian tube model of rubber elasticity with a damage model of stress-induced filler cluster breakdown. The path integral formulation of rubber elasticity is briefly reviewed. Within this framework the consideration of tube-like, topological constraints (packing effects) as well as finite chain extensibility of rubber networks is described. The results are compared to the classical Mooney-Rivlin and inverse Langevin approaches of rubber elasticity. The effect of the filler is taken into account via hydrodynamic reinforcement of the rubber matrix by rigid, self-similar filler clusters, which leads to a quantitative description of stress softening by means of a strain or pre-strain dependent hydrodynamic amplification factor, respectively. Thereby, the pronounced stress softening or high hysteresis of reinforced rubber is referred to an irreversible breakdown of filler clusters during the first deformation cycle. It is shown that the developed concept is in fair agreement with experimental data of unfilled NR-samples in uni-, equibiaxial and pure shear stretching mode. The pronounced stress softening of carbon black filled E-SBR- and EPDM-samples is well described on a quantitative level by an exponential filler cluster decay law.

A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems

An advanced micro-mechanical model of hyperelasticity and stress softening of reinforced rubbers is presented that combines a non-Gaussian tube model of rubber elasticity with a damage model of stress-induced filler cluster breakdown. The path integral formulation of rubber elasticity is briefly reviewed. Within this framework the consideration of tube-like, topological constraints (packing effects) as well as finite chain extensibility of rubber networks is described. The results are compared to the classical Mooney-Rivlin and inverse Langevin approaches of rubber elasticity. The effect of the filler is taken into account via hydrodynamic reinforcement of the rubber matrix by rigid, self-similar filler clusters, which leads to a quantitative description of stress softening by means of a strain or pre-strain dependent hydrodynamic amplification factor, respectively. Thereby, the pronounced stress softening or high hysteresis of reinforced rubber is referred to an irreversible breakdown of filler clusters during the first deformation cycle. It is shown that the developed concept is in fair agreement with experimental data of unfilled NR-samples in uni-, equibiaxial and pure shear stretching mode. The pronounced stress softening of carbon black filled E-SBR- and EPDM-samples is well described on a quantitative level by an exponential filler cluster decay law.

Soft and nonsoft structural transitions in disordered nematic networks

Properties of disordered nematic elastomers and gels are theoretically investigated with emphasis on the roles of nonlocal elastic interactions and crosslinking conditions. Networks originally crosslinked in the isotropic phase lose their long-range orientational order by the action of quenched random stresses, which we incorporate into the affine-deformation model of nematic rubber elasticity. We present a detailed picture of mechanical quasi-Goldstone modes, which accounts for an almost completely soft polydomain-monodomain (PM) transition under strain as well as a "four-leaf clover" pattern in depolarized light scattering intensity. Dynamical relaxation of the domain structure is numerically studied using a simple model. The peak wave number of the structure factor obeys a power-law-type slow kinetics and goes to zero in true mechanical equilibrium. The effect of quenched disorder on director fluctuation in the monodomain state is analyzed. The random frozen contribution to the fluctuation amplitude dominates the thermal one, at long wavelengths and near the PM transition threshold. We also study networks obtained by crosslinking polydomain nematic polymer melts. The memory of the initial director configuration acts as correlated and strong quenched disorder, which renders the PM transition nonsoft. The spatial distribution of the elastic free energy is strongly dehomogenized by external strain, in contrast to the case of isotropically crosslinked networks.

Constitutive Models of Rubber Elasticity: A Review

Abstract A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory—Erman model for low stretch deformation and the Arruda—Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also addressed.

Direct Comparison of Some Recent Rubber Elasticity Models

Abstract The paper compares the Arruda—Boyce model, the van der Waals model and the Reduced Polynomial model—a generic class of polynomial models of which Yeoh's cubic model is a special case—in their ability to predict multiaxial deformation states on the basis of uniaxial measurements. These models are reviewed in the light of novel experimental data, giving ample space to the derivation of the equations needed for optimization of the material parameters. The technological relevance of these findings is exemplified in the finite element analysis (FEA) of a complex membrane.

Mechanical and Optical Behavior of Double Network Rubbers

Stress, strain, and optical birefringence were measured for a series of peroxide-cured natural rubbers, having both isotropic and double network structures. The residual stretch (permanent set) for the latter ranged from 2.0 to 4.5, with elastic moduli that were an increasing function of this residual strain, as found in previous works. A small birefringence, ca. 10-5, was observed for the unstressed double networks, and its magnitude increased with increasing residual strain. The sign of this birefringence corresponded to extension of the (undeformed) double networks. Under stress, the birefringence followed the stress optical law. The double network properties were interpreted using the constrained-chain model of rubber elasticity, with the assumption of independent, additive contributions from the two component networks. The calculated results differed from the experimental findings, in particular underestimating the residual strain. This failure is a consequence of the overprediction of the stresses during compression, a limitation common to molecular theories of rubber elasticity. The modeling of the double networks does account qualitatively for the sign of their unstressed birefringence, which is due to the stress- optical coefficient being larger in tension than in compression. This particular deviation from the stress- optical law is known from both theory and experiment.

Analytic stress-strain relationship for isotropic network model of rubber elasticity

James and Guth [3] have used the inverse Langevin function-based expression of tension in a polymer chain to build the so-called 3-chain model of uncompressible rubber elasticity. It is an analytical model, but it is not isotropic. If one considers an isotropic superposition of an infinite number of chains in all the directions, one obtains an isotropic model, but it is no more tractable analytically. Following Cohen [12], we propose to replace the inverse Langevin function by its Padé approximant, the two functions being very close. The isotropic model is then analytically integrable, and yields large strain strain-stress relationship.

Elasticity of polydiene networks in tension and compression

Previously we reported experimental data, on natural rubber networks in tension and compression, which deviated from the constraint models of rubber elasticity. Equivalent experiments were carried out on crosslinked polybutadiene, and similar discrepancies with theory observed. An alternative approach, based on the tube model of chain entanglements, was found to provide a more accurate description of the experimental data. However, physical interpretation of the values for the model's parameters is problematic.

Analysis of variation of molecular parameters of natural rubber during vulcanization in conformational tube model. II. Influence of sulfur/accelerator ratio

A natural rubber (NR) with a conventional sulfur cure system and a ratio of sulfur/accelerator (V) equal to 3 was investigated. The network structure of the NR during vulcanization was analyzed using a model of rubber elasticity based on the tube concept, which was applied to the treatment of the stress-strain measurements. The influence of cure time and temperature on the chemical crosslinks density was ana- lyzed. The values were compared with those obtained by means of an equilibrium volume swelling measurement. The differences between samples of NR cured with V5 3 and 1.5 were analyzed. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 2747-2755, 1999

An Extended Tube-Model for Rubber Elasticity: Statistical-Mechanical Theory and Finite Element Implementation

Abstract A novel model of rubber elasticity—the extended tube-model—is introduced. The model considers the topological constraints as well as the limited chain extensibility of network chains in filled rubbers. It is supplied by a formulation suitable for an implementation into a finite element code. Homogeneous states of deformation are evaluated analytically to yield expressions required e.g., for parameter identification algorithms. Finally, large scale finite element computations compare the extended tube-model with experimental investigations and with the phenomenological strain energy function of the Yeoh-model. The extended tube-model can be considered as an interesting approach introducing physical considerations on the molecular scale into the formulation of the strain energy function which is on the other hand the starting point for the numerical realization on the structural level. Thus, the gap between physics and numerics is bridged. Nevertheless, this study reveals the importance of a proper parameter identification and adapted experiments.