Linear viscoelastic behaviour of complex polymeric materials: a fractional mode representation (original) (raw)
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International Journal of Polymeric Materials, 2004
Polymeric materials are known to be more or less dispersive and absorptive. Dispersion has a consequence that the dynamic modulus is frequency dependent, and absorption is exhibited by the fact that these materials have the ability to absorb energy under vibratory motion. The phenomenon of dispersion in conjunction with the powerful notion of complex Modulus of Elasticity (MOE), permits to establish the relation between the real and the imaginary components of the MOE, that is, respectively the Storage and loss moduli. The loss factor is simply determined through taking the Ratio of these two MOE components. The theoretical background for the interrelations between the Storage modulus and the loss modulus is found in the Kramers-Kronig relations. However, due to the mathematical difficulties encountered in using the exact expressions of these relations, approximations are necessary for applications in practical situations. On the other hand, several simple models have been proposed to explain the viscoelastic behavior of materials, but all fail in giving a full account of the phenomenon. Among these models, the standard viscoelastic model, better known as the Zener model, is perhaps the most attractive. To improve the performance of this model, the concept of fractional derivates has been incorporated into it, which results in a four-parameter model. Applications have also shown the superiority of this model when theoretical predictions are compared to experimental data of different polymeric materials. The aim of this article is to present the results of applying this model to rubber, both natural and filled, and to some other selected more general polymer.
A variable order fractional constitutive model of the viscoelastic behavior of polymers
International Journal of Non-linear Mechanics, 2019
The multiple timescale evolution of polymers' microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. In the presented work, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.
International Journal of Plasticity, 2003
Following the modelling of Zener, we establish a connection between the fractional Fokker-Planck equation and the anomalous relaxation dynamics of a class of viscoelastic materials which exhibit scale-free memory. On the basis of fractional relaxation, generalisations of the classical rheological model analogues are introduced, and applications to stress-strain relaxation in filled and unfilled polymeric materials are discussed. A possible generalisation of Reiner's Deborah number is proposed for systems which exhibit a diverging characteristic relaxation time. #
Memory of rheological stress in polymers using Fractional Calculus
arXiv: Soft Condensed Matter, 2020
The rheological properties of viscoelastic materials like polymer melts are greatly affected by factors like salinity, temperature, concentration and pH of the solution. In this study, the memory of the stress affected by each of these factors is shown to be trapped in the order of the fractional derivative of the dynamical equation describing stress and strain in the material. To demonstrate this, the rheological properties of the polymer melt hydrolyzed polyacrylamide HPAM have been modeled using a two element Maxwell model. The model has successfully reproduce existing experimental data on elastic modulus and complex viscosity for these stress factors, besides predicting the development of creep compliance with shear rate. The work also establishes that it is possible to tailor a particular rheological property by suitably tuning a pair of properties, complementary conjugates, that offset each others effects on the rheology. The study shows that HPAM has at least two pairs of com...
Bulletin of the Institute for Chemical Research, Kyoto University, 1989
The relaxation modulus, G(t, y), at finite magnitude of shear, y, was measured for a solution of a star-branched polystyrene, composed of four chains with molecular weight Mb each connected at one end at a point. The result was compared with that for a linear polymer with molecular weight 2Mb, i.e., with the same end-to-end chain length as the branched polymer. The stress relaxation of the branched polymer was much slower than that of the linear polymer. On the other hand, the function h(t, y)=G(t, y)/G(t, 0) was approximately the same for two polymers. The observation may be in accord with the tube model theory of polymer entanglement; the branch point should hinder the sliding motion of the chain as a whole along the tube-like cage formed by surrounding chain molecules while it should not disturb the shrink of chain along the tube, which gives rize to the nonlinear characteristics of relaxation modulus.
Framework for analyzing hyper-viscoelastic polymers
2018
Hyper-viscoelastic polymers have multiple areas of application including aerospace, biomedicine, and automotive. Understanding their mechanical responses is therefore extremely important, particularly because they often exhibit strong rate and temperature dependence, including a low temperature brittle transition. Relationships between the response at various strain rates and temperatures are investigated and a framework developed to predict response at rates where experiments are unfeasible. A master curve showing the rate dependences of the storage and loss moduli at a reference temperature is constructed using the results of a dynamic mechanical analysis (DMA) experiment. A frequency sweep spanning two decades and a temperature range from pre-glass transition to pre-melt is used. A fractional derivative model is fitted to the experimental data, and this models parameters are used to show how the stress-strain relationships at a desired strain rate could be derived.
Viscoeleastic properties of solutions of star-branched polystyrene
Macromolecules, 1990
The relaxation modulus, G(t, y), at finite magnitude of shear, y, was measured for a solution of a star-branched polystyrene, composed of four chains with molecular weight Mb each connected at one end at a point. The result was compared with that for a linear polymer with molecular weight 2Mb, i.e., with the same end-to-end chain length as the branched polymer. The stress relaxation of the branched polymer was much slower than that of the linear polymer. On the other hand, the function h(t, y)=G(t, y)/G(t, 0) was approximately the same for two polymers. The observation may be in accord with the tube model theory of polymer entanglement; the branch point should hinder the sliding motion of the chain as a whole along the tube-like cage formed by surrounding chain molecules while it should not disturb the shrink of chain along the tube, which gives rize to the nonlinear characteristics of relaxation modulus.
2019
Leibniz fractional (L-Fractional) derivative is used to mo del viscoelastic mechanical systems. Since this derivativ e has important physical and mathematical meaning, it would be in t resting to compare the theoretical with experimental dat a. Specifically the relaxation behaviour of the Zener fractional viscoelas tic model is verified by experiment. The experimental result s of the viscoelastic relaxation behaviour in a polymer mesh used for the surgical tre tment of female urinary incontinence are used in order t o check the applicability of fractional modelling in these systems. Da ta from relaxation experiments are used in combination with theoretical analysis to prove the Zener-model fractional analysis conc ept.
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.