Voigt solid

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Figure 10.5 Response of a Kelvin-Voigt solid to a shear step stress input.

Fig. 13 Isotopic line splitting of the V3 stretching vibration in single crystalline (see also Fig. 12(a)), after [108, 109], The origin of each absorption band is indicated by an isotopomer present in crystals of natural composition. While the absorption could be fitted by a Lorentzian band profile, the remaining peaks were dominated by the Gaussian contribution in the Voigt band shapes (solid lines below the spectrum). The sum result of fitting the isotopic absorption bands is inserted in the measured spectrum as a solid line...

Eckert H,Elbers S,Epping JD, Janssen M, Kalwei M, StrojekW, Voigt U (2005) Dipolar Solid State NMR Approaches Towards Medium-Range Structure in Oxide Glasses. 246 195-233... 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

Figure 3.11 Qualitative creep function of a Voigt polymer solid...

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) ... 

Note 4 Comparison with the general definition of linear viscoelastic behaviour shows that the polynomial /"(D) is of order zero, 0(D) is of order one, ago = a and a = p. Hence, a material described by the Voigt-Kelvin model is a solid (go > 0) without instantaneous elasticity (/"(D) is a polynomial of order one less than 0(D)). 

Analyses of the results obtained depend on the shape of the specimen, whether or not the distribution of mass in the specimen is accounted for and the assumed model used to represent the linear viscoelastic properties of the material. The following terms relate to analyses which generally assume small deformations, specimens of uniform cross-section, non-distributed mass and a Voigt-Kelvin solid. These are the conventional assumptions. 

Note 4 Damping curves are conventionally analysed in terms of the Voigt-Kelvin solid giving a decaying amplitude and a single frequency. 

Note 5 Given the properties of a Voigt-Kelvin solid, a damping curve is described by the equation... 

Note 2 The logarithmic decrement can be used to evaluate the decay constant, p. From the equation for the damping curve of a Voigt-Kelvin solid. 

Note 2 For a Voigt-Kelvin solid, with P(D)=1 and Q y)=a+pD, where a is the spring constant and P the dashpot constant, the equation describing the deformation becomes... 

Note 7 Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency. 

Note 1 For a Voigt-Kelvin solid of negligible mass, the absolute modulus can be evaluated from the ratio of the flexural force (/o) and the amplitude of the flexural deflection (y) with... 

Note 3 A material specimen which behaves as a Voigt-Kelvin solid under forced oscillation , with a mass added at the point of application of the applied oscillatory force... 

Evidently a fluid polymer cannot be considered in the model the deformation approaches to a limit. For a solid polymer the model seems more appropriate, though is represents neither a spontaneous elastic deformation nor permanent flow. Therefore a combination of a Kelvin-Voigt element with a spring and with a dashpot in series is, in principle, more appropriate. 

There are several models to describe the viscoelastic behavior of different materials. Maxwell model, Kelvin-Voigt model, Standard Linear Solid model and Generalized Maxwell models are the most frequently applied. 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

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. 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

Fig. 1. The Voigt lineshape is plotted for three different values of the Voigt parameter, a - WL/WG, namely a = 0 (Gaussian, dotted line), a = 1 (solid line, with the component Lorentzian and Gaussian lineshapes having an equal width given by tl/(, W 0.6107 H-) and a x (Lorentzian, dashed line). The frequency scale is given in units of the FWHM of the Voigt lineshape, W.

Fig. 11. (Upper) Splitting of pHe+ states due to magnetic interactions, and observable laser transitions between the F+ and F states according to Bakalov and Korobov [33]. (Lower) Observed hyperfine splitting of the unfavoured laser transition (n, L) = (38,34) —> (37, 35) [16]. The laser bandwidth is 1.2 GHz. The solid line is the result of a fit of two Voigt functions (a Gaussian fixed to the laser bandwidth convoluted with a Lorentzian to describe the intrinsic line width) to the spectrum. The intrinsic width of each lines was found to 0.4 0.1 GHz. From Widmann et al. [16]...

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