A non-linear viscoelastic model with structure-dependent relaxation times (original) (raw)
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A Nonlinear Network Viscoelastic Model
Journal of Rheology, 1978
A nonlinear constitutive equation for polymer melts and concentrated solutions is derived from a Lodge-Yamamoto type of network theory. The network junctions are postulated to move nonaffinely in a well-defined manner. The functional form of the creation and destruction rates of junctions is assumed to depend on the average extension of the network strand and the absolute temperature in such a way to allow for the time-temperature superposition principle. The theory shows good agreement with all data examined. The paper concludes with a strong flow problem (melt spinning). The results indicate the validity of the model in this flow regime.
Le Journal de Physique IV, 1996
We survey the important problems in polymer viscoelasticity that impede progress in this field. These problems manifest themselves as spectacular anomalies in the viscoetastic data that have been reproduced in different polymers and in different laboratories, but have largely been put aside for lack of explanation. Without solving these difficult problems, there can be no satisfactory understanding of the viscoelastic properties of polymers spanning across the local segmental motion, the glass-rubber softening dispersion, the rubbery plateau and the terminal dispersion. The coupling model, which has general applicablity to various relaxation mechanisms of polymers and beyond polymers, is able to resolve all these problems.
Journal of Applied Mechanics and Technical Physics, 1996
In order to model polymer fluid flows within the framework of continuum mechanics, it is necessary to write a theological state equation that establishes a relationship between the stress tensor for a polymer system and the velocity-gradient tensor. This can be done either by a phenomenological approach [1], generalizing the available experimental data, or by using some model concepts of the structure of polymer materials [2][3][4][5][6][7][8][9][10][11][12][13]. However, both approaches will probably not provide us .with a simple enough rheological constitutive relation suitable for a description of various flows of linear polymer solutions and melts. Therefore, the problem of construction of a succession of rheologieal constitutive relations taking new and more subtle effects into account at each step is of great importance. The success of such a procedure is determined by the selection of an initial approximation and by the rules of transition to subsequent approximations.
A nonlinear viscoelastic model for polymer solutions and melts—II
Chemical Engineering Science, 1968
appraisal of the rheological model proposed in Part I is presented. The experimental data on six fluids (four polymer solutions. one soap solution and one polymer melt) are used to test the model in various flow situations: steady simple shearing; oscillatory, small-amplitude simple shearing; stress relaxation; and stress growth. The model parameters were determined by a nonlinear least squares method in fitting four material functions.
Constitutive equations for a polymer fluid based on the concept of non-affine networks
Constitutive equations are developed for a polymer fluid, which is treated as a permanent network of strands bridged by junctions. The junctions are assumed to slide with respect to their reference positions under loading. Governing equations are derived by using the laws of thermodynamics under the assumption that the vorticity tensor for the flow of junctions is proportional to that for macro-deformation. Explicit expressions are developed for the steady elongational viscosity, as well as for the steady shear viscosity and normal stress functions. To verify the constitutive relations, three sets of experimental data are approximated on polystyrene solutions with various molecular weights. It is demonstrated that the model can correctly describe stress overshoot for the shear stress and first normal stress difference in start-up tests with various strain rates. Adjustable parameters in the governing equations change consistently with the strain rate, molecular weight and concentrat...
Journal of Non-Newtonian Fluid Mechanics, 2009
The constitutive models for the viscoelasticity of polymers are presented for determining molecular weight distributions (MWDs) from viscosity measurements. The inversion of this model derived from control theory and melt calibration procedure connects the relaxation modulus, viscosity, and other flow properties of a polymer. The linear principle enables simultaneous and accurate modelling of the relaxation modulus and of viscosity flow curves over a wide range. Starting from viscosity measurements, the new model is used to determine the MWD, linear viscoelastic relaxation moduli, and the relaxation spectra of polyethylene of different grades. In addition, two benchmark analyses of bimodal polystyrene are reported, and the capability of the model is proven by the two-box test of Malkin. The error of the modelled viscosity is smaller than that for previously reported models. One of the main features of this work is that no relaxation time or spectrum procedures were used to generate and model linear viscoelasticity.
The main concern of this paper is the development of a three dimensional viscoelastic model at finite strain to describe nonfactorizable behavior of rubber-like materials. The model is developed within the framework of rational thermodynamics and internal state variable approach such that the second law of thermodynamics in the form of Clausius-Duhem inequality is satisfied. The nonfactorizable aspect of the behavior is introduced via a strain dependent relaxation times. The model is applied to describe the response of the isotropic Pipkin multi-integral viscoelastic model and the Bromobutyl (BIIR) material, several parameters involved are then identified using quasi-static and dynamic experiments thanks to a least-square minimization procedure. The proposed model is able to reproduce quasi-static response and show a good ability to predict the dynamic response of nonfactorizable rubber-like materials (BIIR) and the multi-integral model of Pipkin in a wide range of strain. (A. Zine). cle by Wineman . A significant class of models have been developed following the internal variable approach which consists on a generalization to a three dimensional model of the one dimensional Maxwell model which was firstly suggested by Schapery [4] and followed by the authors in [5-7] and [8] among others. The advantage of these models is their simplicity to be implemented into Finite element industrial software and applied to engineering application such as the work by Ansari and Hassanzadeh-Aghdam . Other contributions to this approach used the fractional derivatives from the Maxwell model to obtain a fractional representation of the constitutive equations, see [10] and [11] among others.
Polymer Engineering and Science, 1978
Shear stress and first normal stress difference data are presented for materials which exhibit a constant viscosity and yet at the same time exhibit elasticity levels of the same order as polymer melts. Flow pattern observations in circular die entry flows in conjunction with independent shear and normal stress measurement strongly suggest that these fluids would make excellent model fluids for melt studies. Studies in which the influence of elasticity in the absence of shear thinning and fluid inertia can easily be made. Furthermore it is clearly shown that a realistic solution to the die entry flow problem is not obtained using second order flow theory. In the second order region the secondary cell is observed to be almost identical in size to the cell observed for an inelastic Newtonian fluid in creeping flow. Marked growth in the secondary cell as a function of elasticity is not observed until the shear rates exceed the region of second order behavior. This growth in cell size as a result of elasticity is followed at higher shear rates by a spiraling flow instability like that observed for some polymer melts.
A Rouse-tube model of dynamic rubber viscoelasticity
Journal of Physics A: Mathematical and Theoretical, 2007
The dynamic-mechanical response of a polymer network has been calculated using a stress-based Rouse model formalism. In contrast to the previous work, this improved formulation incorporates appropriate boundary conditions and provides a smooth crossover from the classical equilibrium result of rubber elasticity to the short timescale relaxation. We develop a consistent implementation of the classical tube model, which is merged with the Rouse dynamics to take into account the entanglement effects. In a polymer network, crosslinks prevent the global reptation and constraint release. Entanglements thus acquire a different topological meaning and have a much stronger effect on the resulting mechanical response. We construct a dynamic stress tensor for a polymer network, which naturally covers the whole frequency/time range. Using this stress tensor, we first examine the equilibrium response to small shear and uniaxial deformations, and then investigate the linear dynamic response of a network for all the cases where the stress-tensor computations are analytically tractable.
Constitutive equations for non-affine polymer networks with slippage of chains
Continuum Mechanics and Thermodynamics, 2005
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). Macrodeformation 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.