Voigt body

The compliance J t) at any time was described by a six-element model made up of an instantaneous compliance, two Kelvin-Voigt bodies, and Newtonian compliance (Sherman 1970) ... 

The key point in the rheological classification of substances is the question as to whether the substance has a preferred shape or a natural state or not [19]. If the answer is yes, then this substance is said to be solid-shaped otherwise it is referred to as fluid-shaped [508]. The simplest model of a viscoelastic solid-shaped substance is the Kelvin body [396] or the Voigt body [508], which consists of a Hooke and a Newton body connected in parallel. This model describes deformations with time-lag and elastic aftereffects. A classical model of viscoplastic fluid-shaped substance is the Maxwell body [396], which consists of a Hooke and a Newton body connected in series and describes stress relaxation. 

The tissue has been modeled from a continuum mechanics point of view by Jfow and students (11,12,13). The model treats cartilage as a blphaslc material (the solid"organic matrix was assumed, for these purposes, to be one "phase" and the water the other). The "solid" organic matrix was further assumed to behave as a single Kelvln-Voigt body whose viscoelastic properties are attenuated by the frictional resistance to fluid flow from the tissue. Attempts were made to use this model to explain the load-strain and load dissipation properties of the tissue for the experimental configuration described In this paper (15),... 

Voigt function to the 540 cm peak. It is very close to the expected value = 0.34 for a two-body Fe-NO oscillator. The calculated value = 0.39 (B3LYP functional) is in agreement with the experimental one. 

In which element or model for a viscoelastic body will the elastic response be retarded by viscous resistance (a) Maxwell or (b) Voigt-Kelvin ... 

Fig. 6. The optimal DNA mutation rate as determined from a model that incorporates one-body and two-body fitness contributions (similar to a spin glass). The genetic code is included in the model. The data are for a N = 50 protein. The fitness improvement is the maximum change in fitness averaged over 10,000 landscapes. To compare the relative location of the optima, the curves have been scaled such that the optima are at 1.0. (a) The optimum mutation rate for the uncoupled landscape as the number of mutants screened increases M= 1000 (O), 10,000 ( ), and 50,000 (A), (b) The optimal mutation rate as the landscape ruggedness increases. The number of coupling interactions is 75 (O), 25 ( ), and 0 (A). As the landscape ruggedness increases, the optimal mutation rate decreases. Reprinted from Voigt et ol. (2000a), with permission.

It should be noted that for a polycrystal composed of cubic crystalhtes, the Voigt and Reuss approximations for the bulk modulus are equal to each other, as they should be since the bulk modulus represents a volume change but not shape change. Therefore, in a cube the deformation along the principal strain directions are the same. Hence, Eqs. 10.39 and 10.40 are equal and these equations also hold for an isotropic body. The... 

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... 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

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. 

Maxwell bodies are obtained if Hookean and Newtonian bodies are connected in series (Figure 11-11). The Kelvin or Voigt model, on the other hand, contains Hookean and Newtonian bodies in a parallel arrangement (Figure 11-11). The Maxwell body is a model for relaxation phenomena and the Kelvin body is a model for retardation processes. 

Real materials exhibit a much more complex behavior compared to these simplified linear viscoelastic models. One way of simulating increased complexity is by combining several models. If, for instance, one combines in series a Maxwell and a Voigt model, a new body is created, called the Burger model (Figure 4-15). 

---