Kelvin and Maxwell Models

In Chapter 2 when the Maxwell and Kelvin models were analysed, it was found that the time constant for the deformations was given by the ratio of viscosity to modulus. This ratio is sometimes referred to as the Relaxation or Natural time and is used to give an indication of whether the elastic or the viscous response dominates the flow of the melt. 

For a more complete discussion of Maxwell and Kelvin models and viscoelasticity (96-98,101). 

The viscoelasticity properties are also important, because they can supply information directly related to the form of the macromolecules. The models of the linear viscoelasticity are developed from two elements a spring and a dashpot. Two of those elements in line constitute the Maxwell model and in parallel the Kelvin model (or Vogt).20 Normally, those models don t represent the behavior of complex materials satisfactorily. Other models such as the Burgers model, where the Maxwell and Kelvin models are connected in line, are used to determine the modulus of elasticity (Yj and Y2) and the coefficients of viscosity ( and t]2).21... 

A combination of the Maxwell and Kelvin models comprising four elements is shown in Figure 3.14a. 

The response of this model to creep, relaxation, and recovery situations is thus the sum of the effects described previously for the Maxwell and Kelvin models and is illustrated in Figure 3.14b. 

A series crmnection of the Maxwell and Kelvin models makes the four-element model, known as the Burger s model (Burgers 1935), which can describe the viscoelastic creep behaviors of polymers, as given by... 

In a mechanical model, each spring or dashpot represents a mechanical analogue to the response of the material. However, the most complex mechanical model may not be able to describe polymer concrete. In the case of rPET polymer concrete, the Maxwell and Kelvin models have elements which allow representation of the viscoelastic response. A combination of these two models in series satisfactorily describes the creep response of rPET polymer concrete. This multiparameter model is shown in Figure 4.12. 

When the load is removed at time t the response is also the composite of the response corresponding to the Kelvin and Maxwell models. The strain equation for this mechanical model is represented as a combination of the response of the Maxwell and Kelvin models as follows ... 

The combination of Equations 4.30-4.32 and the elimination of the subscripts for the Maxwell and Kelvin models give the third-order linear differential constitutive equation ... 

There is no need here to solve this differential equation formally, since the same result can be achieved by superimposing the polymer concrete creep test results from the Maxwell and Kelvin models. This is possible since the eight-parameter model is simply the Kelvin and Maxwell models in series. Thus, the desired solution of Equation 4.42 is obtained ... 

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

In cases where the rate of change of strain or stress is very small, the creep compliance and relaxation modulus may be approximately the inverse of each other. Consideration of simple Maxwell and Kelvin models confirm the condition given by Eq. 6.42 as in Homework problem 6.7. 

It is next required to obtain a quantitative description of stress relaxation and creep that will help to form a link with the original mathematical description in terms of the Boltzmann integrals. It is simple and instructive to do this by development of the Maxwell and Kelvin models. 

Generalized Maxwell and Kelvin models are combinations of several Maxwell elements in parallel or Kelvin elements in series respectively. They were introduced to describe discrete relaxation times. The generalized Maxwell model is written as... 

A variety of models have been employed to explain the viscoelastic behaviour of polymeric materials. The Maxwell unit, consisting of a spring and dashpot in series, and the Kelvin (or Voigt) unit, consisting of a spring and dashpot in parallel, are the simplest of these models (see Figures 7 and 8). The Maxwell and Kelvin models lead to analogous equations and have similar limitations. Here we consider only the Maxwell model. 

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