Maxwell element/model

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Table 3.2 Calculated Values of G(t) at Various Times Based on a Model Consisting of Two Maxwell Elements in Parallel...

In addition to the set of Voigt elements, a Maxwell element could also be included in the model. The effect is to include a contribution given by Eq. (3.69) to the calculated compliance. This long time flow contribution to the compliance is exactly what we observe for non-cross-linked polymers in Fig. 3.12. 

The purpose of this problem is to consider numerically the effect of including more than two Maxwell elements in the model for a relaxation experiment. Prepare a table analogous to Table 3.2 for a set of four Maxwell elements having the following properties ... 

The creep curve for polypropylene at 4.2 MN/m (Fig. 2.5) is to be represented for times up to 2 X 10 s by a 4-element model consisting of a Maxwell unit and a Kelvin-Voigt unit in series. Determine the constants for each of the elements and use the model to predict the strain in this material after a stress of 5.6 MN/m has been applied for 3 x 10 seconds. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Maxwell element or model Model in which an ideal spring and dashpot are connected in series used to study the stress relaxation of polymers, modulus Stress per unit strain measure of the stiffness of a polymer, newtonian fluid Fluid whose viscosity is proportional to the applied viscosity gradient. 

For viscoelastic materials combinations of these two models can be used, e.g. a spring and a dashpot in series or parallel. The first combination is called the Maxwell element, its response under constant stress is the sum of that of its two components ... 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

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. 

The reality is better approximated by a generalized Maxwell model (Figure 6.8), consisting of a large number of Maxwell elements in parallel, each with its own relaxation time, T , and its own contribution, ), to the total stiffness. This system can be described by ... 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

The Maxwell element (elastic deformation plus flow), represented by a spring and a dashpot in series. It symbolises a material that can respond elastically to stress, but can also undergo viscous flow. The two contributions to the strain are additive in this model, whereas the stresses are equal ... 

In practice the stress relaxation behaviour has to be described expressed with N Maxwell elements connected in parallel, each with its own spring constant E and relaxation time t (the so-called Maxwell-Wiechert model) ... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

One obvious way of introducing a range of relaxation and retardation times into the problem is to construct mathematical models thai are equivalent to a number of Maxwell and/or Voigt models connected in parallel (and/or series). The Maxwell-Wiechert model (Figure 13-96), for example, consists of an arbitrary number of Maxwell elements connected in parallel. For simplicity let s see what you get with, say, three Maxwell elements and then extrapolate later to an arbitrary number, n. 

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

What is commonly called the three-element standard, or simply the standard solid (or Zener s solid), is a combination of either a Kelvin-Voigt element in series with a spring or, alternatively, a Maxwell element in parallel with a spring (see Fig. 10.6). The strain response of the first model to the stress input CT = cjoH(t) can be written as... 

The simplest mechanical model which can describe a viscoelastic solution is called Maxwell element. It consists of a spring and a viscous element (dashpot) connected in series. The spring corresponds to a shear modulus Gq and the dashpot to a viscosity r). The behavior of the Maxwell element under harmonic oscillations can be obtained from the following equations ... 

The rheological consequences of the Maxwell model are apparent in stress relaxation phenomena. In an ideal solid, the stress required to maintain a constant deformation is constant and does not alter as a function of time. However, in a Maxwellian body, the stress required to maintain a constant deformation decreases (relaxes) as a function of time. The relaxation process is due to the mobility of the dashpot, which in turn releases the stress on the spring. Using dynamic oscillatory methods, the rheological behavior of many pharmaceutical and biological systems may be conveniently described by the Maxwell model (for example, Reference 7, Reference 17, References 20 to 22). In practice, the rheological behavior of materials of pharmaceutical and biomedical significance is more appropriately described by not one, but a finite or infinite number of Maxwell elements. Therefore, associated with these are either discrete or continuous spectra of relaxation times, respectively (15,18). 

The mechanical response of viscoelastic bodies such as polymers is poorly represented by either the spring or the dashpot. J. C. Maxwell suggested that a better approximation would result from a series combination of the spring and dashpot elements. Such a model, called a Maxwell element, is shown on the right in Figure 3-1. In describing tensile response with the Maxwell element, E, the instantaneous tensile modulus, characterizes the response of the spring while rjE, the viscosity of the liquid in the dashpot, defines the viscous... 

This widely used model consists of an arbitrary number of Maxwell elements connected in parallel, as shown in Figure 3-7. 

Its behavior (Figure 3-8) reproduces the two transitions observed in real polymers. It is possible to replace one of the Maxwell elements in the generalized Maxwell model with a spring. The stress would decay to a finite value in such a model rather than zero and would approximate the behavior of crosslinked polymers. 

In considering systems where there are very many Maxwell elements employed in the model, that is, z in equation (3-34) is large, it is often convenient to replace the summation in the equation by an integration. Thus ... 

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