non-Gaussian Theory Research Articles

On cavitation, post-cavitation and yield in amorphous polymer–rubber blends

The deformation behaviour of amorphous polymer–rubber blends is investigated in terms of an axisymmetric unit cell model containing an initially spherical rubber particle. The behaviour of the rubber is described by an incompressible non-Gaussian network theory, while for the matrix we adopt a recent large strain elastic–viscoplasticity model that incorporates the intrinsic softening upon yield and the subsequent progressive orientation hardening typical for amorphous glassy polymers. Guided by simple analytical estimates, cavitation of the rubber particle is interpreted in terms of the unstable growth of a pre-existing small void. It is shown that cavitation and yield are essentially coupled processes. On the macroscopic scale, both are softening mechanisms : If macroscopic yield takes place before the limit stress for cavitation is reached, cavitation is prohibited. Furthermore, and contrary to common belief, it is found from the interfacial stress history that, using realistic material parameters, the rubber particle continues to significantly affect plasticity in the matrix in the post-cavitation regime, i.e. after it has cavitated, so that cavitated particles cannot always be considered to be equivalent to particle-sized voids.

Application of the network alteration theory for modeling the time-dependent constitutive behaviour of rubbers. Part I. General theory

The long-term time dependent irreversible deformation behaviour of rubber-like materials is characterized by large (irreversible) deformation, stress-relaxation, creep, permanent set and aging induced flexibility changes. These types of material behaviour are being described by combining the general (rubber) hyperelastic theory and Tobolsky's (network) alteration theory. To generalize Tobolsky's idea, the long-term time-dependent irreversible deformation behaviour is postulated as being caused by alterations of the original microstructure into new microstructures. This concept yields a general time-dependent constitutive model for rubberlike materials. Besides, the short-term rubber hysteresis behaviour (Mullins' effect) is adopted into the present model by implying the rubber hysteresis model of Simo. The experimental characterization of the model parameters – using the eight non-Gaussian chain theory for rubber elasticity as point of departure – as well as the experimental model verification are performed using modified Natural Indonesian Rubber samples. The experimental evidences will be presented in Part II (Septanika, E.G., Ernst, L.J., 1998. Application of the network alteration theory for modeling the time-dependent constitutive behaviour of rubbers. Part II. Further evaluation of the general theory and experimental verification. Mech. Mater. 30, 265–273). The present paper is concerned with the model development.

General formula for conditional meanusing higher order statistics

A general formula is given for conditional mean, which is the well-known minimum variance estimator, in terms of cross-cumulants and locally optimum nonlinearities. This formula will greatly facilitate the solution of various problems emerging in nonlinear system identification, non-Gaussian estimation theory, and nonlinear signal processing.

Gauge-ball spectrum of the four-dimensional pure U(1) gauge theory

We ibvestigate the continuum limit of the gauge-ball spectrum in the four-dimensional pure U(1) lattice gauge theory. In the confinement phase we identify various states scaling with the correlation length exponent ν ⋍ 0.35. The square root of the string tension also scales with this exponent, which agrees with the non-Gaussian fixed point exponent recently found in the finite-size studies of this theory. Possible scenarios for constructing a non-Gaussian continuum theory with the observed gauge-ball spectrum are discussed. The 0 ++ state, however, scales with a Gaussian value ν ⋍ 0.5. This suggests the existence of a second, Gaussian continuum limit in the confinement phase and also the presence of a light or possibly massless scalar in the non-Gaussian continuum theory. In the Coulomb phase we find evidence for a few gauge balls, being resonances in multi-photon channels; they seem to approach the continuum limit with as yet unknown critical exponents. The maximal value of the renormalized coupling in this phase is determined and its universality confirmed.

Prediction of deformation behavior of glassy polymers based on molecular chain network model

In order to develop a constitutive equation for polymeric materials covering a wide class of deformation processes a three-dimensional nonaffine model has been developed by adopting the non-Gaussian network theory with the generalized Argon double-kink model of intermolecular resistance. A phenomenological relation between the non-Gaussian molecular chain network structure and entanglement of chains is formulated and a simple evolutional equation of the number of entanglements of chains during the deformation process is developed. The validity of the proposed constitutive equation has been proven through comparison between the predicted results and experimental results. Furthermore, its application to the deformation analysis of circular polymeric bars under tension has been discussed

Molecular behaviour of elastomeric materials under large deformation: 2. Rheological model of polymer networks

The molecular mechanism of real rubber vulcanizates, both unfilled and filled, is discussed in relation to the classical non-Gaussian theory, in which the deviation from theory at large deformation seems to be attributable to the concept of fixed crosslinks in the theory. A new slider model that possesses the characteristic viscous property is introduced to represent physical crosslinks. The slider model is effectively available for interpreting the typical phenomena observed in real rubber vulcanizates, such as the crossing of the stress-strain curves measured at different temperatures and the Mullins softening under cyclic deformation.

A simplified non-Gaussian approach to digital optical receiver design with avalanche photodiodes: experimental

Detailed experimental measurements of various digital receiver parameters taken on a receiver using a low noise silicon avalanche photodiode are compared with simple theories that incorporate Gaussian and non-Gaussian models for the probability distribution functions for the avalanche gain. The authors analyze the salient aspects of the theories, particularly those parameters that give results different from the Gaussian approximation, and those parameters that are readily extracted from the theory (e.g., the influence of background current due either to dark current or to imperfectly extinguished light sources). Experiments were performed in which a receiver was characterized for bit error rate (BER) and optimum gain as the optical power at a modulation rate of 10 Mb/s (NRZ) was varied and as additional background light was allowed to fall on the detector. Excellent agreement with the author's newly developed simplified approximation to the non-Gaussian theory for receiver sensitivity (ibid., vol.9, p.1019-26, Aug. 1991) is obtained. >

The Net Charge Process for Interacting, Signed Diffusions

We consider a system of $N (\rightarrow \infty)$ interacting one-dimensional diffusions, in which each diffusion is assigned a random charge $(\pm 1)$, and study the behavior of the net charge distribution through space and time. The diffusion equations are a slight variation of those considered in the initial studies of "the propagation of chaos," but the interaction involves the signs of the diffusions and triplet rather than pairwise interactions. This has the effect of leading to a non-Gaussian fluctuation theory, which turns out to be close to the $P(:\Phi^4:)$ models of Euclidean quantum field theory. The main tools of the proofs involve the Stroock-Varadhan martingale theory and a general theory of $U$-statistics.

Fluctuation effects in high Tc superconductors in strong fields

A quasi-one dimensional critical regime induced by a strong magnetic field is important for HTSC due to its short coherence length. A brief summary is given of our effort to understand the observed broad resistive transitions of HTSC in the field on the basis of a non-Gaussian fluctuation theory. We then try to delineate problems we encounter in extending it to lower temperature. Preliminary results for specific heat and |ψ|2 correlation function are described.

Correctly averaged non-Gaussian theory of rubberlike elasticity. Application to the description of the behavior of poly(dimethylsiloxane) bimodal networks

A new theoretical relationship between the rubberlike elasticity of a polymer network and the distribution of the vector connecting junction points (or the end-to-end vector distribution function of the chain joining two of these points) is presented. In this approach the free energy of the chains is correctly averaged over the network structure, a difference with respect to older treatments in which a preaveraged deformation dependence is considered. The new theory and a slight modification in the previous procedure to analyze bimodal network data lead to a more satisfactory explanation of these data (presenting a pronounced upturn for high extensibilities due to non-Gaussian effects) in terms of molecular magnitudes. © 1986, American Chemical Society. All rights reserved.

Quantitative molecular characterization of cold-drawn semi-crystalline polymers. Case of polyethylene terephthalate

The influence of pre-orientation on the natural draw ratio achieved in necking has been studied for poly(ethylene terephthalate). The results are analysed with the aid of the non-Gaussian theory of rubber elasticity assuming that the amorphous phase behaves as a network formed by entanglements of chains. A model for separating the influence of amorphous and strain-induced crystalline regions on deformation is proposed and compared with the pseudo-affine model.

A non-Gaussian theory of rubberlike elasticity based on rotational isomeric state simulations of network chain configurations. III. Networks prepared from the extraordinarily flexible chains of polymeric sulfur and polymeric selenium

Distribution functions for the end-to-end separation of polymeric sulfur [-S-] and selenium [-Se-] are obtained from Monte Carlo simulations which take into account the chains’ geometric characteristics and conformational preferences. Comparisons with the corresponding information on polyethylene [-CH2-] demonstrate the remarkable equilibrium flexibility or compactness of these two molecules. Use of the S and Se distribution functions in the three-chain model for rubberlike elasticity in the affine limit gives elastomeric properties very close to those of simple Gaussian networks, even though their distribution functions appear to be significantly non-Gaussian. Specifically, unlike the case of polyethylene, even very short S and Se chains show only slight upturns in modulus at high elongations; this is consistent with their unusually high flexibility.

Rubber elasticity in relation to polyester networks

The equation of rubber elasticity has been applied to the dynamic shear modulus of styreneunsaturated polyester networks. Due to the non-ideal nature of the networks, the apparent front factor approached the theoretical value of 0.5 only when the crosslink density was low. At higher crosslink densities, it has been found that either the non-Gaussian theory of Smith or the steric interaction approach of Blanchard and Wooton satisfactorily describes the results.

The Application of Statistical Theories of Rubber Elasticity to Polymer Melts

This paper discusses the limitations of statistical theories of elasticity when applied to polymer melts. Several workers have used such theories to predict the magnitude of die swell, but have largely failed to recognize the fact that the classical relationships for elasticity do not hold at large deformations. The relationships between stress and strain are derived for tensile and shear deformations, based on the nonGaussian statistical theory, which is valid for the large deformations encountered in polymer melt flow. Results of die swell experiments are presented, which suggest that the theory can successfully be applied to polymer melts despite the major assumptions which its use entails.

Non-Gaussian theory of rubber elasticity

Non-Gaussian theory for rubber elasticity has been developed using the distribution function which was proposed for the excluded volume problem. An equation of state which is based on a very simple model yields remarkably good results when the non-Gaussian parameters are adjusted. A distribution function for network study has been suggested. The computed simple shear-strain curve also closely agrees with that of the experimental curve predicting the relationship to be nonlinear. The equations for swelling phenomenon have been derived using the non-Gaussian network concept.

Network characterization from stress–strain behavior at large extensions

AbstractThe stress–strain properties of natural rubber and synthetic cis‐polyisoprene vulcanizates have been investigated over a large range of extensions. The results are in qualitative agreement with those of Mullins, in that the upturn of a Mooney‐Rivlin plot is ascribed to finite extensibility effects. It is further found that the stress–strain curves can be adequately described by a modification of existing non‐Gaussian theory. Analysis of the curves shows significant finite extensibility effects even at extensions well below the upturn. Thus, division of stress–strain curves into Gaussian and non‐Gaussian regions is to be avoided. Values for n, the number of statistical segments between crosslinks, were calculated using the proposed treatment. The values of n found were relatively unaffected by temperature. Further, the molecular weight of a statistical segment was found to be about the same for synthetic cis‐polyisoprene as for natural rubber despite differences in the crystallization behavior. Crystallization effects are therefore thought to be of little importance in determining the position of the upturn. The equivalent statistical segment was found to be 4.3 isoprene units for natural rubber and cis‐polyisoprene.

Determination of Degree of Crosslinking in Natural Rubber Vulcanizates. Part IV. Stress-Strain Behavior at Large Extensions

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter LinkedIn Tools Icon Tools Get Permissions Cite Icon Cite Search Site Citation L. Mullins; Determination of Degree of Crosslinking in Natural Rubber Vulcanizates. Part IV. Stress-Strain Behavior at Large Extensions. Rubber Chemistry and Technology 1 March 1961; 34 (1): 290–300. doi: https://doi.org/10.5254/1.3540199 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu nav search search input Search input auto suggest Search

Determination of degree of crosslinking in natural rubber vulcanizates. Part IV. Stress‐strain behavior at large extensions

AbstractDepartures from behavior predicted by the statistical theory of rubberlike elasticity, which occur at high extensions, are ascribed to the finite extensibility of network chains. By use of a non‐Gaussian statistical theory, a relation is obtained between the extension ratio λ*, the extension at which departures are small but significant and the degree of crosslinking. Experimental results on the dependence of λ* on the degree of crosslinking and on the degree of swelling are in good agreement with theoretical predictions. It is shown that the additional determination of λ* permits the calculation of both the degree of crosslinking and the extent of network flaws due to chain ends, from a single simple extension stress‐strain curve. The calculation of these two basic parameters has hitherto been unattainable from measurements on vulcanized rubbers alone. This development provides a new technique of particular value in studies of the degradation of vulcanized networks.

Statistical Theory of Networks of Non-Gaussian Flexible Chains

The aim of this paper is to develop a selfconsistent theory of rubber-like materials consisting of networks of non-Gaussian chain molecules. Three kinds of series developments are derived for the distribution function of perfectly flexible single chains from the Fourier integral solution of Rayleigh; namely, (1) long chains with actual extension much less than the maximum extension, (2) long chains with actual extension comparable to the maximum extension, and (3) short chains. In the non-Gaussian network theory, the leading term of the series (2) is used as the starting point for the individual chains of the network. Calculations are made for the case where the free junctions are moving with no restriction, and for the case where the free junctions are assumed to be at their most probable positions. The final expressions of the elastic energy for the two cases are compared, and it is shown that the percentage difference of the two expressions is of the order 1/n (n being the average number of links per chain), which is negligible for sufficiently large n. Finally an expression of the elastic energy is obtained with the assumption that all junctions are fixed and is shown to be, in general, a function of three strain invariants. The interdependence of the coefficients of the invariants is shown. Comparison of theory and experiment is given. Because of the interdependence of the coefficients only part of the observed deviations from Gaussian theory can be explained by our molecular theory. The remaining discrepancies must be ascribed to van der Waals forces. This should show up in the (not yet investigated) temperature dependence of these discrepancies.