Mechanical models for linear viscoelastic response

As an aid in visualizing viscoelastic response, we introduce two linear mechanical models to represent the extremes of the mechanical response spectrum. The spring in Fig, 18.1a represents a linear elastic or Hookean solid whose constitutive equation (relation of stress to strain and time) is simply x = Gy, where G is 

Although the models developed here are to be visualized in tension, the notation used is for pure shear (viscometric) deformation. The equations are equally applicable to tensile deformation by replacing the shear stress t with the tensile stress cr, the shear strain y with the tensile strain e, Hooke s modulus G with Young s (tensile) modulus E, and the Newtonian (shear) viscosity rj with the elongational (Trouton) viscosity tje. 

Some authorities object strongly to the use of mechanical models to represent materials. They point out that real materials are not made of springs and dashpots. True, but they re not made of equations either, and it s a lot easier for most people to visualize the deformation of springs and dashpots than the solutions to equations. 

A word is needed about the meaning of the term linear. For the present, a linear response will be defined as one in which the ratio of overall stress to overall strain, the overall modulus G t), is a fimction of time only, not of the 

If a constant stress tq is suddenly appUed to the dashpot, the strain increases with time according to y = (to/ ) (considering the strain to be zero when the stress is applied) (Fig. 18.3). Doubling the stress doubles the slope of the strain-time line, and at any time, the modulus G(r) = x/y — ij/t = G(t only). So the dashpot is also linear. 

Fundamental Principles of Polymeric Materials, IWid Edition. Christopher S. Brazel and Stephen L. Rosen. 2012 John Wiley Sons, Inc. Published 2012 by John Wil Sons, Inc. 

FIGURE 15.1 Linear viscoelastic models (a) linear elastic (b) linear viscous (c) Maxwell element (d) Voigt-Kelvin element (e) three-parameter (f) four-parameter. 

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IV. MECHANICAL MODELS FOR LINEAR VISCOELASTIC RESPONSE A. MAXWELL MODEL... 

The outline of the review is as follows. First the microscopic starting points, the formally exact manipulations, and the central approximations of MCT-ITT are described in detail. Section 3 summarizes the predictions for the viscoelasticity in the linear response regime and their recent experimental tests. These tests are the quantitatively most stringent ones, because the theory can be evaluated without technical approximations in the linear limit important parameters are also introduced here. Section 4 is central to the review, as it discusses the universal scenario of a glass transition under shear. The shear melting of colloidal glasses and the key physical mechanisms behind the structural relaxation in flow are described. Section 5 builds on the insights in the universal aspects and formulates successively simpler models which are amenable to complete quantitative analysis. In the next section. 

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

The mechanical properties of the nucleus, the stiffest component of the cell, are important for the overall cellular response. It is, probably, even more significant that forces transmitted from the cell surface and acting on the nucleus can alter gene expression and protein synthesis. Kan et al. (1999a) have modeled the nucleus as a viscous fluid and analyzed the effect of the nucleus on the leukocyte recovery. Guilak et al. [2000] have estimated the linear viscoelastic properties of nuclei of chondrocytes. Caille et al. [2002] used a model of nonlinear elastic material to estimate Young s modulus of endothelial cell nuclei. Recently, Dahl et al. [2004], by using the micropipette technique, have estimated the mechanical properties of the cell s nuclear envelope. 

Non-linear viscoelastic mechanical behaviour of a crosslinked sealant was interpreted as due to a Mullins effect. The Mullins effect was observed for a series of sealants under tensile and compression tests. The Mullins effect was partially removed after a mechanical test, when a long relaxation time was allowed, that is the modulus increased over time. Non-linear stress relaxation was observed for pre-strained filler sealants. Time-strain superposition was used to derive a model for the filled sealants. Relaxation over long periods demonstrates that the Mullins effect is caused by non-equilibrium with experimental conditions being faster than return to the initial state. If experiments were conducted over times of the order of a day there may be no Mullins effect. If a filled elastomer were only required to perform its function once per day then each response might be linear viscoelastic. 

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 time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

The relaxation processes described above apply to linear viscoelastic behavior. If the deformation is not small or slow, the orientation of the chain segments may be sufficiently large to cause a nonlinear response. We will see that this effect alone can be accounted for in rheological models by simply replacing the infinitesimal strain tensor by one able to describe large deformations no new relaxation mechanism needs to be invoked. Nonlinear effects related to orientation, such as normal stress differences, can be described in this manner. 

Here Git) is the linear viscoelastic modulus and h(y), the damping function in shear is introduced. For small y, the linear response is recovered. As y increases, we see another relaxation in G y, t) at short times that corresponds to the relaxation of the contour length. Figures 45 and 46 show the expected relaxation behavior and that obtained experimentally for a high molecular weight polymer in solution. The theoretical curves show more nearly exponential decays for both mechanisms than is seen in the actual data, which has a broader relaxational behavior. This is a well-known weakness of the DE model that is related to the relaxation function being nearly exponential in nature (the longest relaxation time is widely separated from the next time which also has a lower intensity). 

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