Viscoelastic behaviour, linear

In this chapter, we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the timescale of the response to an applied deformation. Finally, we show that in alternating sttain experiments the presence of the viscous component leads to a phase difference between suess and strain. 

Newton s law of viscosity defines viscosity rj by stating that stress o is proportional to the velocity gradient in the liquid  

It can be seen that the shear stress is directly proportional to the rate of change of shear strain with time. This formulation brings out the analogy between Hooke s law for elastic solids and Newton s law for viscous liquids. In the former 

Hooke s law describes the behaviour of a linear elastic solid and Newton s law that of a linear viscous liquid. A simple constitutive relation for the behaviour of a linear viscoelastic solid is obtained by combining these two laws  

A simple possible formulation of linear viscoelastic behaviour combines these equations, making the assumption that the shear stresses related to strain and strain rate are additive  

The equation represents one of the simple models for linear viscoelastic behaviour, the Kelvin or Voigt model, and is discussed in detail in Section 4.2.3 below. 

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N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behaviour, Springer-Verlag, Berlin, 1989. 

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

Note 1 In linear viscoelastic behaviour, stress and strain are assumed to be small so that the squares and higher powers of crand f may be negleeted. 

Model of the linear viscoelastic behaviour of a liquid in which (q ) + P)cT= Dy... 

Note 3 Comparison with the general definition of linear viscoelastic behaviour shows that the polynomials P(D) and 0(D) are of order one, qo =Q,pq = pia and a= a. Hence, a... 

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

Note 4 For linear viscoelastic behaviour, a sinusoidal stress (o) results from the sinusoidal strain with... 

Note 5 For linear viscoelastic behaviour interpreted in terms of complex stress and strain (see notes 2 and 3)... 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

The dynamics of block copolymers melts are as intriguing as their thermodynamics leading to complex linear viscoelastic behaviour and anisotropic diffusion processes. The non-linear viscoelastic behaviour is even richer, and the study of the effect of external fields (shear, electric. ..) on the alignment and orientation of ordered structures in block copolymer melts is still in its infancy. Furthermore, these fields can influence the thermodynamics of block copolymer melts, as recent work has shown that phase transition lines shift depending on the applied shear. The theoretical understanding of dynamic processes in block copolymer melts is much less advanced than that for thermodynamics, and promises to be a particularly active area of research in the coming years. 

For a quenched lamellar phase it has been observed that G = G"scales as a>m for tv < tvQ. where tvc is defined operationally as being approximately equal to 0.1t and r is a single-chain relaxation time defined as the frequency where G and G" cross (Bates et al. 1990 Rosedale and Bates 1990). This behaviour has been accounted for theoretically by Kawasaki and Onuki (1990). For a PEP-PEE diblock that was presheared to create two distinct orientations (see Fig. 2.7(c)), Koppi et al. (1992) observed a substantial difference in G for quenched samples and parallel and perpendicular lamellae. In particular, G[ and the viscosity rjj for a perpendicular lamellar phase sheared in the plane of the lamellae were observed to exhibit near-terminal behaviour (G tv2, tj a/), which is consistent with the behaviour of an oriented lamellar phase which flows in two dimensions. These results highlight the fact that the linear viscoelastic behaviour of the lamellar phase is sensitive to the state of sample orientation. 

With all these models, the simple ones as well as the spectra, it has to be supposed that stress and strain are, at any time, proportional, so that the relaxation function E(t) and the creep function D(t) are independent of the levels of deformation and stress, respectively. When this is the case, we have linear viscoelastic behaviour. Then the so-called superposition principle holds, as formulated by Boltzmann. This describes the effect of changes in external conditions of a viscoelastic system at different points in time. Such a change may be the application of a stress or also an imposed deformation. 

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. 

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

Schwarzl (1970) studied the errors to be expected in the application of this type of equations, starting from the theory of linear viscoelasticity. Flis results are given schematically in Fig. 13.59. For non-linear viscoelastic behaviour, the exactitude of the approximate equations cannot be predicted. 

Dynamic mechanical analysis involves the determination of the dynamic properties of polymers and their mixtures, usually by applying a mechanical sinusoidal stress For linear viscoelastic behaviour the strain will alternate sinusoidally but will be out of phase with the stress. The phase lag results from the time necessary for molecular rearrangements and this is associated with the relaxation phenomena. The energy loss per cycle, or damping in the system, can be measured from the loss tangent defined as ... 

The purpose of this study was to give an insight into molecular properties which imderlie the linear viscoelastic behaviour of molten polymers. Properties were probed from proton magnetic dipoles attached to polymeric chains or to smadl molecules in concentrated polymeric solutions. 

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). 

Three complex functions may be used to characterize the linear viscoelastic behaviour in the frequency domain ... 

Although the (Simplex shear modulus is not the most appropriate function to use in all c ses, we wUl describe the linear viscoelastic behaviour in terms of this last function, which is tiie most referred to experimentally furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus. 

Figure 4 Master curve for the linear viscoelastic behaviour of entangd polymers in the terminal region of relaxation V Polystyrene, bulk (M=860000, T=190°C) Polyethylene, bulk (M=340000, T=130°C) A Polybutadiene solution (M=350000, <)) polymer=43%, T=20°C) [ om ref.4].

Constitutive equations of the Maxwell-Wiechert tjq)e have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of pol3maer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple spring-dashpot mechanical analogy leading to the linear equation ... 

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