The Voigt Element

The viscoelastic functions exhibited by the Voigt element are as follows  

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The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

The Maxwell body is appropriate for the description of stress relaxation, while the Voigt element is more suitable for creep deformation. In a stress relaxation experiment, a strain yo is imposed atr = Oand held constant thereafter (dy/r// = 0) while r is monitored as a function of t. Under these conditions, Eq. (11-29) for a Maxwell body behavior becomes... 

Stress Relaxation. With constant strain, the equation of motion of the Voigt element reduces to Hooke s law with... 

The application of sinusoidal stress and strain is similar to that for a Maxwell body. The results are summarized in Table 3-1 along with the previously derived results for a Maxwell element. Figure 3-6 displays the frequency dependence of D and D" for the Voigt element in tension. The response in shear would be identical with J replacing D. 

The Voigt-Kelvin model is a generalization of the Voigt element that results from connecting Voigt elements in series (Figure 3-9). Here the compliance functions are easily calculated, while the modulus functions are rather complicated. The results are summarized in Table 3-2 a sample calculation is provided below. 

This equation is entirely analogous to the equation of motion for the Voigt element in the mechanical case [equation (3-26)]. 

The total stress supported by the Voigt element is the sum of the stresses in the spring and the dashpot. 

Thus, the rheological equation for the Voigt element is given by... 

The creep and creep recovery curves for the Voigt elements are shown in Figure 14.8. 

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. 

Under creep, the total strain will be due to the instantaneous elastic deformation of the spring of modulus E, and irrecoverable viscous flow due to the dashpot of viscosity 1)2, and the recoverable retarded elastic deformation due to the Voigt element with a spring of modulus E3 and dashpot of viscosity rij. Thus, the total strain is the sum of these three elements. That is. 

In creep recovery, say, the load is removed at time t, the deformation, Oo i> due to the spring of modulus El is recovered instantaneously. This will be followed by the retarded elastic creep reeovery due to the Voigt element given by 83 or... 

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... 

A summary of the important equations for the Maxwell element and the Voigt element can be found in Table 2. A detailed derivation of the equations can be found in the classic texts [1,2]. 

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