The Maxwell Element

The viscoelastic functions exhibited by the Maxwell element can be easily derived and are summarized as follows  

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Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

The dashpot constant, t i, for the Maxwell element is obtained from the slope of the creep curve in the steady state region (see equation (2.32)). 

For viscoelastic materials combinations of these two models can be used, e.g. a spring and a dashpot in series or parallel. The first combination is called the Maxwell element, its response under constant stress is the sum of that of its two components ... 

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 models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

These two basic elements may be combined in series or parallel, giving the Maxwell-element and the Voigt-Kelvin element. 

The Maxwell element (elastic deformation plus flow), represented by a spring and a dashpot in series. It symbolises a material that can respond elastically to stress, but can also undergo viscous flow. The two contributions to the strain are additive in this model, whereas the stresses are equal ... 

In a stress relaxation experiment the Maxwell-element is subjected to an instantaneous deformation sQ which is held constant. It means that ... 

In a stress relaxation we can put de/dt = 0 and obtain for each of the Maxwell elements the equations shown in Equations 13-94. 

In a limited way, the Maxwell element describes a liquid. Similarly, the Kelvin (Voight) element describes a solid. As the relaxation time, Tj, is defined for the Maxwell element, the retardation time, T2, is defined for the Kelvin element. For the Kelvin element under stress. 

The simplest mechanical model which can describe a viscoelastic solution is called Maxwell element. It consists of a spring and a viscous element (dashpot) connected in series. The spring corresponds to a shear modulus Gq and the dashpot to a viscosity r). The behavior of the Maxwell element under harmonic oscillations can be obtained from the following equations ... 

The mechanical response of viscoelastic bodies such as polymers is poorly represented by either the spring or the dashpot. J. C. Maxwell suggested that a better approximation would result from a series combination of the spring and dashpot elements. Such a model, called a Maxwell element, is shown on the right in Figure 3-1. In describing tensile response with the Maxwell element, E, the instantaneous tensile modulus, characterizes the response of the spring while rjE, the viscosity of the liquid in the dashpot, defines the viscous... 

Its behavior (Figure 3-8) reproduces the two transitions observed in real polymers. It is possible to replace one of the Maxwell elements in the generalized Maxwell model with a spring. The stress would decay to a finite value in such a model rather than zero and would approximate the behavior of crosslinked polymers. 

The frequency shift is positive. The n-scaling depends on the value of cytR. In the limit of cytR 1, scaling is found. In this case, the relaxation time is much longer than the period of oscillation and the Maxwell element behaves elastically. The Maxwell model reduces to the simple-spring model (Sect. 2.2). If, on the other hand, the retardation time is short (cor 1), the frequency shift is still positive, but it scales linearly with n. If a positive frequency shift in conjunction with linear n-scaling is found, this in indicative of fast relaxation processes in the contact zone. If this is the case, the damping must also be large. 

In the Maxwell element, both the spring and the dashpot support the same stress. Therefore,... 

As Equation 14.7 shows, the Maxwell element is merely a linear combination of the behavior of an ideally elastic material and pure viscous flow. Now let us examine the response of the Maxwell element to two typical experiments used to monitor the viscoelastic behavior of polymer. 

Solving the equation and noting that the initial strain is C /E, the equation for the Maxwell element for creep can be written as... 

On removal of the applied stress, the material experiences creep recovery. Figure 14.5 shows the creep and the creep recovery curves of the Maxwell element. It shows that the instantaneous application of a constant stress, Oo, is initially followed by an instantaneous deformation due to the response of the spring by an amount Oq/E. With the sustained application of this stress, the dashpot flows to relieve the stress. The dashpot deforms linearly with time as long as the stress is maintained. On the removal of the applied stress, the spring contracts instantaneously by an amount equal to its extension. However, the deformation due to the viscous flow of the dashpot is retained as permanent set. Thus the Maxwell element predicts that in a creep/creep recovery experiment, the response includes elastic strain and strain recovery, creep and permanent set. While the predicted response is indeed observed in real materials, the demarcations are nevertheless not as sharp. 

Figure 14.6 Creep and creep recovery behavior of the Maxwell element.

The rheological equation for the Maxwell element from Equation 14.7 is... 

Ef is the relaxation modulus. For the Maxwell element in a stress relaxation experiment, all the initial deformation takes place in the spring. The dashpot subsequently starts to relax and allows the spring to contract. For times considerably shorter than the relaxation time, the Maxwell element behaves essentially like a spring while for times much longer than the relaxation time, the element behaves like a dashpot. For times comparable to the relaxation time, the response involves the combined effect of the spring and the dashpot. 

Example 14.1 A polystyrene sample of 0.02/m cross-sectional area is subjected to a creep load of 10 N. The load is removed after 30 s. Assuming that the Maxwell element accurately describes the behavior of polystyrene and that viscosity is 5 x 10 P, while Young s modulus is 5 X 10 psi, calculate ... 

Since, as we saw above, the Maxwell element is not perfect, it seems logical to consider a parallel arrangement of the spring and the dashpot. This is the so-called Voigt or Voigt-Kelvin element... 

We note that the Voigt model predicts that strain is not a continuous function of stress that is, the element does not deform continuously with the sustained application of a constant stress. The strain approaches an asymptomatic value given by (Oq/E). The strain of the element at equilibrium is simply that of an ideal elastic solid. The only difference is that the element does not assume this strain instantaneously, but approaches it gradually. The element is shown to exhibit retarded elasticity. In creep recovery, the Maxwell element retracts instantaneously but not completely, whereas the Voigt element exhibits retarded elastic recovery, but there is no permanent set. 

The instantaneous elastic deformation is due to the Maxwell element spring, E,. The primary valence bonds in polymer chains have equihbrium bond angles and lengths. Deformation from these equilibrium values is resisted, and this resistance is accompanied by an instantaneous elastic deformation. 

Irrecoverable viscous flow is due to the Maxwell element dashpot t j. This is associated with shppage of polymer chains or chain segments past one another. 

The motion of the Maxwell element (with modulus E and relaxation time X=t]/E) is given by (see Equation 3.36)... 

The analysis presented above for the Maxwell element to explain the significance of dynamic testing can be extended to the Voigt element and corresponding expressions for moduli can be derived. However, models comprised of single elements are useful only as pedagogical tools. They can be combined in series... 

The equation for solvent transport consists of a diffusional term and a term due to osmotic pressure. The osmotic pressure term arises by using linear irreversible diermodynamics arguments (20). The osmotic pressure is relat to the viscoelastic properties of the polymer through a constitutive equation. In our analysis, the Maxwell element has been used as the constitutive model. Thus, the governing equations for solvent transport in the concentrated regime are... 

Parameter P governs the eontribution of the Maxwell element to effective viscosity, T, (Newtonian viscosity of the solution). Equation [7.2.26] is similar to the Oldroyd-type equation [7.2.15] with the only difference that in the former the upper convective derivative is used to account for nonlinear effects instead of partial derivative, d/dt. 

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