Kelvin model generalized

Generalized Kelvin model Generalized Maxwell model... 

Boltzmann s constant, and T is tempeiatuie in kelvin. In general, the creep resistance of metal is improved by the incorporation of ceramic reinforcements. The steady-state creep rate as a function of appHed stress for silver matrix and tungsten fiber—silver matrix composites at 600°C is an example (Fig. 18) (52). The modeling of creep behavior of MMCs is compHcated because in the temperature regime where the metal matrix may be creeping, the ceramic reinforcement is likely to be deforming elastically. 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

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)). 

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. 

Table 3-2. Behavior of Generalized Maxwell and Voigt-Kelvin Models in Various Experiments...

The creep of dam concrete includes two parts instantaneous elastic deformation and viscous deformation. A generalized Kelvin model consisting of two standard Kelvin model in series is used to describe the time-dependent deformation of the dam concrete as shown in Figure 1. 

If the viscous strain of Kelvin model is f at to t = to + Ar, and the stress remains constant during the Al time increment, the viscous strain increment of the generalized Kelvin model during At can be derived from equation (4) as... 

The model s strain is the sum of the initial elastic strain and the strains of the two Kelvin models, so that the partial strain e, of the generalized Kelvin model is expressed as... 

The generalized Voigt element or the Voigt-Kelvin model is a series arrangement of an arbitrary number of Voigt elements (Figure 14.12). Under creep, the creep response of each individual element is given by... 

The theory of non-isothermal viscoelastic behavior as developed by Hopkins [2] and Haugh [3] may be based on the representation of linear viscoelastic behavior by mechanical models. The linear viscoelastic behavior of polymers in simple shear at constant temperature and prescribed stress history may be expressed in terms of the deformation of a generalized Kelvin model. Spring constants and dashpot viscosity constants of the model have to be appropriately chosen the choice depends on temperature. For the non-isothermal treatment, the elasticity of the springs and the viscosities of the dashpots have to be inserted as functions of temperature. Due to the prescribed temperature history, they become functions of time. 

We have seen that the Maxwell model describes the stress relaxation of a viscoelastic solid to a first approximation, and the Kelvin model the creep behaviour, but that neither model is adequate for the general behaviour of a viscoelastic solid where it is necessary to describe both stress relaxation and creep. 

The behaviour of the generalized Kelvin model of damper can be described by means of the following equations ... 

The equation of the generalized Kelvin model written in the local coordinate system can be finally presented in the form ... 

The amplitude of displacements is equal to 0.01 m iu all of the cousidered cases. A comparison of dissipated energy calculated for the considered models of a VE damper is shown in Figure 7. From this calculation, it may be concluded that the dissipation energies of the fractional-derivative Kelvin model and both generalized models are approximately equal in the range 0 - 15.0 rad/sec of excitation frequency. This range of frequency covers the range of the first three natural frequencies of vibration of the con-... 

The results of optimization of problems in which VE dampers are modelled using the generalized Kelvin model and the generalized Maxwell model are almost identical... 

The explicit form of the matrices used to describe the generalized Kelvin model is given by ... 

Figure 2.13 Three-element model for viscoelastic behavior of liquids (general relaxation model) as a combination of Maxwell and Voigt-Kelvin models.

More complex arrangements of elements are often used, especially if multiple relaxations are involved or if accurate representations of engineering data are required. The Maxwell-Weichert model consists of a very large (or infinite) number of Maxwell elements in parallel (2). The generalized Voigt-Kelvin model places a number of Kelvin elements in series. In each of these models, a spring or a dashpot may be placed alone, indicating elastic or viscous contributions. 

However, this model still has no instantaneous elasticity and a free spring is normally included in series with the generalized Kelvin solid with the result (sometimes referred to as the Voigt-Kelvin model),... 

These equations can be used to represent a cross-Unked material. Although the Generalized Kelvin Model can be solved for the case of relaxation, due to the forms of the differential equations and ease of solution. Maxwell elements in parallel are typically used for relaxation while Kelvin elements... 

It is possible using transform methods to convert viscoelastic problems into elastic problems in the transformed domain, allowing the wealth of elasticity solutions to be utilized to solve viscoelastic boundary value problems. Although there are restrictions on the applicability of this technique for certain types of boundary conditions (discussed further in Chapter 9), the method is quite powerful and can be introduced here by building on the framework provided by mechanical models. Recall the differential equation for a generalized Maxwell or Kelvin model,... 

Give sketches for generalized Maxwell and Kelvin models. Label all elements. 

A distribution of retardation times based on a generalized Kelvin model leads to a retardation spectrum, L(x), defined by,... 

Naturally, corresponding forms can be found for the complex compliance function for a generalized Kelvin model. Verification of these expressions is left as an exercise for the reader. 

While the above description suggests that various master curve shapes can be represented by generalized Maxwell or Kelvin models, it does not mean that the determination of the proper values of the spring moduli or relaxation times to obtain a precise fit is trivial. 

If only the spring moduli are nonlinear, a nonlinear generalized Kelvin model can be represented by. 

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