Models for viscoelastic behaviour

The following type of differential equation is encountered in the text, for example, in the analysis of the models for viscoelastic behaviour ... 

Figure 6.1 Complex shear modulus (a) and complex shear compliance (b) for standard polyisobutylene reduced to 25 °C. Points from averaged experimental measurements curves from a theoretical model for viscoelastic behaviour. (Reproduced with permission from Marvin and Oser, J. Res. Natl. Bur. Stand. B, 66,171 (1962))...

Fig. 14. Spring and dash-pot model for viscoelastic behaviour. Elastic and viscous element in scries.

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

We now consider some models of polymer structure and ascertain their usefulness as representative volume elements. The Takayanagi48) series and parallel models are widely used as descriptive devices for viscoelastic behaviour but it is not correct to use them as RVE s for the following reasons. First, they assume homogeneous stress and displacement throughout each phase. Second, they are one-dimensional only, which means that the modulus derived from them depends upon the directions of the surface tractions. If we want to make up models such as the Takayanagi ones in three dimensions then we shall have a composite brick wall with two or more elements in each of which the stress is non-uniform. 

In this paper, optical techniques (Fabry Perot and self beating correlation) are first described and used to measure the viscoelastic parameters of 5P4E as a function of the pressure and temperature. These results represent an extension of the data available in the literature. Then, various rheological models for fluids behaviour in an elastohydrodynamic contact (E.H.D.) are described. 

N.W Johnson, J. Segalmann A model for viscoelastic fluid behaviour which allows Non- Affine deformation Journal of Non-Newtonian Fluid Mechanics, 1977, Vol. 2, p. 255-270. 

Within this section of the chapter we concentrate on the success of the tube model of viscoelastic behaviour in uncrosslinked material. In uncrosslinked material entanglements are supposed to form a tube around every polymer it they can reptate out of the tube creating a new tube. There is considerable experimental support for this picture (see ref. 14 for a general reference) which should still be valid when reptation is stopped by permanent crosslinks. Once the tube concept is accepted a rich theory of rubbery, perhaps even glassy, material can be developed. 

Example 2.17 Establish and plot the variation with frequency of the storage and loss moduli for materials which can have their viscoelastic behaviour described by the following models... 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

The grade of polypropylene whose creep curves are given in Fig. 2.5 is to have its viscoelastic behaviour fltted to a Maxwell model for stresses up to 6 MN/m and times up to ICKX) seconds. Determine the two constants for the model and use these to determine the stress in the material after 900 seconds if the material is subjected to a constant strain of 0.4% throughout the 900 seconds. 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

The reptation-tube model, being used for interpretation of viscoelastic behaviour of the system, has allowed to obtain (Doi and Edwards 1986) the relation for terminal characteristics... 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

For a more complete description of the time and the temperature dependence of the fibre strength a theoretical description of the viscoelastic and plastic tensile behaviour of polymer fibres has been developed. Baltussen (1996) has shown that the yielding phenomenon, the viscoelastic and plastic extension of a polymer fibre can be described by the Eyring reduced time model. This model uses an activated site model for the plastic and viscoelastic shear deformation of adjacent chains in the domain, in which the straining of the intermolecular bonding is now modelled as an activated shear transition between two states, separated by an energy barrier. It provides a relation between the lifetime, the creep load and the temperature of the fibre, which for PpPTA fibres has been confirmed for a range of temperatures (Northolt et al., 2005). 

Fig. 16.9 shows the low frequency slopes of 2 and 1, respectively, as expected for viscoelastic liquids and the high frequency slopes Vi and 2/3 for Rouse s and Zimm s models, respectively. Experimentally it appears that in general Zimm s model is in agreement with very dilute polymer solutions, and Rouse s model at moderately concentrated polymer solutions to polymer melts. An example is presented in Fig. 16.10. The solution of the high molecular weight polystyrene (III) behaves Rouse-like (free-draining), whereas the low molecular weight polystyrene with approximately the same concentration behaves Zimm-like (non-draining). The higher concentrated solution of this polymer illustrates a transition from Zimm-like to Rouse-like behaviour (non-draining nor free-draining, hence with intermediate hydrodynamic interaction).

Stress-Strain Relations for Viscoelastic Materials. The viscoelastic behaviour of an elastomer varies with temperature, pressure, and rate of strain. This elastic behaviour varies when stresses are repeatedly reversed. Hence any single mathematical model can only be expected to approximate the elastic behaviour of actual substances under limited conditions 2J. ... 

We will begin with a brief survey of linear viscoelasticity (section 2.1) we will define the various material functions and the mathematical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts and concentrated solutions in a purely rational and phenomenological way (section 2.2) the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6). 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

The purpose of this paper is to explore various aspects of the rheological behaviour of lyotropic liquid crystalline systems. Lyotropics are often used as model systems for thermotropics because their viscoelastic behaviour seems to be quite similar (1) and solutions are much more easier to handle and can be studied more accurately than melts. The emphasis is on transient data as these are essential for verifying viscoelastic models but are hardly available in the literature. Transient experiments can also provide insight in the development of flow—induced orientation and structure. The reported experiments include relaxation of the shear stress and evolution of... 

The adsorption kinetics of a surfactant to a freshly formed surface as well as the viscoelastic behaviour of surface layers have strong impact on foam formation, emulsification, detergency, painting, and other practical applications. The key factor that controls the adsorption kinetics is the diffusion transport of surfactant molecules from the bulk to the surface [184] whereas relaxation or repulsive interactions contribute particularly in the case of adsorption of proteins, ionic surfactants and surfactant mixtures [185-188], At liquid/liquid interface the adsorption kinetics is affected by surfactant transfer across the interface if the surfactant, such as dodecyl dimethyl phosphine oxide [189], is comparably soluble in both liquids. In addition, two-dimensional aggregation in an adsorption layer can happen when the molecular interaction between the adsorbed molecules is sufficiently large. This particular behaviour is intrinsic for synergistic mixtures, such as SDS and dodecanol (cf the theoretical treatment of this system in Chapters 2 and 3). The huge variety of models developed to describe the adsorption kinetics of surfactants and their mixtures, of relaxation processes induced by various types of perturbations, and a number of representative experimental examples is the subject of Chapter 4. 

The above analysis of the viscoelastic behaviour for adsorption layers of a reorientable surfactant leads to important conclusions. It is seen that the most important prerequisite for a realistic prediction of the elastic properties is the adequacy of the theoretical model used to describe the equilibrium adsorption of the surfactant. For example, when we use the von Szyszkowski-Langmuir equation instead of the reorientation model to describe the interfacial tension isotherm, this rather minor difference drastically affects the elasticity modulus of the surface layer. The elasticity modulus, therefore, can be regarded to as a much more sensitive parameter to find the correct equation of state and adsorption isotherm, rather than the surface or interfacial tension. Therefore the study of viscoelastic properties can give much more insight into the nature of subtle phenomena, like reorientation, aggregation etc. 

When plastics are unloaded, the creep strain is recoverable. This contrasts with metals, where creep strains are permanent. The Voigt linear viscoelastic model predicts that creep strains are 100% recoverable. The fractional recovered strain is defined as 1 — e/cmax, where e is the strain during recovery and Cmax is the strain at the end of the creep period. It exceeds 0.8 when the recovery time is equal to the creep time. Figure 7.9 shows that recovery is quicker for low Cmax and short creep times, i.e. when the creep approaches linear viscoelastic behaviour. 

Before the models are described, the two simple aspects of viscoelastic behaviour already referred to - creep and stress-relaxation - are considered. For the full characterisation of the viscoelastic behaviour of an isotropic solid, measurements of at least two moduli are required, e.g. Young s modulus and the rigidity modulus. A one-dimensional treatment of creep and stress-relaxation that will model the behaviour of measurements of either of these (or of other measurements that might involve combinations of them) is given here. Frequently compliances, rather than moduli, are measured. This means that a stress is applied and the strain produced per unit stress is measured, whereas for the determination of a modulus the stress required to produce unit strain is measured. When moduli and compliances are time-dependent they are not simply reciprocals of each other. 

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