Voight model

The rule of mixtures is useful in roughly estimating upper and lower bounds of mechanical properties of an oriented fibrous composite, where the matrix is istropic and the fiber orthotropic, with coordinate 1 the principal fiber dfrection and coordinate 2 transverse to it. For the upper bound, the Voight model is used (Fig. 12.3), where it is assumed that the strain is the same in the fiber and matrix. For the lower bound, the Reuss model is used, where the stress is assumed to be the same. This gives the following equations for composite moduli ... 

The rule of mixtures equations have several drawbacks. The isostrain assumption in the Voight model implies strain compatibility between the phases, which is very unlikely because of different Poisson s contractions of the phases. The isostress assumption in the Reuss model is also unrealistic since the libers cannot be treated as a sheet. Despite this, these equations are often adequate to predict experimental results in unidirectional composites. A basic limitation of the rule of mixtures occurs when the matrix material yields, and the stress becomes constant in the matrix while continuing to increase in the fiber. 

Voigt element n. This is a Voight model which is a component, together with other Voight or Maxwell components, of a more complex viscoelastic model system, such as the standard linear solid. 

However, various combinations of eiastic and viscous elements have been used to approximate the material behavior of polymer melts. Some models are combinations of springs and dashpots to represent the elastic and viscous responses, respectively. The most common ones being the Maxwell model for a polymer melt and the Kelvin or Voight model for a solid. One model that represents shear thinning behavior, normal stresses in shear flow and elastic behavior of certain polymer melts is the K-BKZ model [28-29]. 

The retardation spectrum corresponding to the Rouse theory is also discrete, equivalent to a generalized Voight model, but not with equal magnitudes of compliance contributions the pth contribution Jp to the compliance falls ofif rapidly with increasing p, being approximately proportional to the retardation time itself as well as to l/nkT. They have been calculated by Betry the retardation times are spaced between the relaxation times, as shown in the lower panel of Fig. 9-7. From the discrete retardation times and Jp values, the creep compliance can be calculated in principle by equation 18 of Qrapto 3. 

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

Figure H3.3.5 The creep response of a food (circles) was fitted to a Burger model with one Kelvin-Voight unit. The goodness of fit is shown as the continuous curve and the standard error. The values of compliance and viscosity of the respective springs and dashpots were outcomes of the fitting process.

The contact force exerted on the tup by the specimen dynamically balances this force. The tup/specimen contact is modelled using a Kelvin-Voight element having stiffness kc and damping coefficient r. Thus the contact force is given by ... 

B. Voight-Kelvin (Creep) Model Superposition Principles... 

The gel layer is treated as a Kelvin-Voight viscoelastic material where the gel shear is acting in parallel. This viscoelastic material stress has both an elastic and viscous component acting in parallel. This viscoelastic material model was substituted into the momentum equation and the resulting gel layer equation of motion is... 

Calculate using the Voight and Reuss models the bounds for Young s modulus of Mg0-Al203 composites as a function of volume fraction. 

In the last two centuries, a lot of attempts and discussion have been made on the elucidation and development of the various constitutive models of liquids. Some of the theoretical models that can be mentioned here are Boltzmann, Maxwell (UCM, LCM, COM, 1PM), Voight or Kelvin, Jeffrey, Reiner-Rivelin, Newton, Oldroyd, Giesekus, graded fluids, composite fluids, retarded fluids with a strong backbone and fading memory, and so on. Further and deeper knowledge related to the physical and mathematical consequences of the structural models of liquids and of the elasticity of liquids can be found in Ref. [6]. 

FIGURE 11.11 Creep response for (a) Maxwell model, (b) Voight-Kelvin model, and (c) four-parameter model for constant stress applied at r, and removed at if... 

FIGURE 123 Composite stress models (a) Voight or isostiain (h) Reuss or isostiess. Arrows indicate tension force direction. 

To use the model at higher temperatures, derivatives of these coefficients with respect to T and P are determined by fitting other kinds of data, such as density, heat of dilution, or heat capacity. A discussion of recent developments is Voight (2001). 

Voight, W., 2001, Solubility equUibra in multicomponent oceanic salt systems from t = 200 to 200°C. Model parameterization and databases. Pure Appl. Chem., v. 73, pp. 831-844. 

BONANOS In most experimental situations the system of electrodes, interfaces, bulk, grain boundary properties etc. are effectively connected in series. What is the relevance of the Voight and naxwell models in such situations ... 

The elastic behaviour of the partially polymerized lattice is model dependent. The two simplest cases are vhen (a) the two con nents are subjected to equal strain and (b) the two con nents are subjected to equal stress (22). Case (a), the Voight limit, corresponds to con nents coupled in pairallel, i.e. long polymer chains, and (b), the Reuss limit, corresponds to con nents coupled in series, i.e. short polymer chains. Baughmann (20) used case (a) to calculate the lattice strain. 

The early stage of development, following the works of Maxwell, Kelvin and Voight, has constituted the starting point for many other researchers. In particular in the mid 40s different modeling strategies have actually been used to describe nonlinear viscoelastic solids. 

Strobl, G.R., Schneider, M.J., and Voight-Martin, I.G. (1980) Model of partial crystallization and melting derived from small-angle X-ray scattering and electron microscopic studies on low-density polyethylene. J. Polym. Sci. Polym. Phys. Ed., 18, 1361-1381. 

An important aspect of the development of forecasts on the basis of physical models is how to optimize models given observed data. Most physics-based forecasting models (including those underpinned by Voight s relation) take the... 

Hence, we suggest a simple, serial equivalent circuit analog that describes the impedance behavior of carbon electrodes as seen in Figure 12. It contains a Voight-type analog in series with R-C, which reflects the charge transfer, a potential-dependent Warburg -type element (solid state diffusion of Li-ions), and, finally, a capacitive potential-dependent element that reflects the accumulation of lithium. This relatively simple model has already been discussed in depth [105-107]. 

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