Maxwell element

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. 

A dynamic modulus can also be evaluated by following a procedure similar to that used for the dynamic compUance above. The modulus is most easily approached by considering a Maxwell element in which case the differential equation analog to Eq. (3.77) is... 

We observed above that the Rouse expression for the shear modulus is the same function as that written for a set of Maxwell elements, except that the summations are over all modes of vibration and the parameters are characteristic of the polymers and not springs and dashpots. Table 3.5 shows that this parallel extends throughout the moduli and compliances that we have discussed in this chapter. In Table 3.5 we observe the following ... 

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 dashpot constant, t i, for the Maxwell element is obtained from the slope of the creep curve in the steady state region (see equation (2.32)). 

Solution As shown in the previous Example, the modulus for a Maxwell element may be expressed as... 

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. 

Note 5 The relaxation spectrum (spectrum of relaxation times) describing stress relaxation in polymers may be considered as arising from a group of Maxwell elements in parallel. 

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

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. 

A Maxwell element shows an instantaneous elastic deformation and thereafter unlimited flow. For polymers in the solid condition the latter is not realistic. For a fluid polymer it is more relevant moreover, the instantaneous elastic deformation is in accordance with the real behaviour when the stress is released a polymer fluid shows an instantaneous recoil. 

Apparently the stress relaxation proceeds in two phases, each with a stress decrease of 1 MPa at a strain e = 1. It seems logical to think of a parallel arrangement of two Maxwell elements, both with a spring constant E = I MPa, but with relaxation times which differ by a factor 10,000. Inspection of the stress values (with a = (To exp(-i/T)) easily results in T = sec, = 10,000 sec. The viscosities are then r = vE) 10 and 10 Pa s. 

From the G-M equation, while still in the co-rotational frame, we can choose a specific form of the relaxation modulus. Thus, for a single Maxwell element we can obtain... 

The Single Maxwell Element LVE Constitutive Equation Consider the single Maxwell mechanical element shown in the following figure. The element was at rest for t < 0. A shear strain y12(t) is applied at t = 0. By stating that the stress is the same in the dashpot and spring, while the total strain is the sum of those... 

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 ... 

The pattern in Figure 6.15, read from the right to the left on the log t scale, resembles the relaxation behaviour, the decrease of with increasing time under a constant strain, for two Maxwell elements in parallel (Figure 6.16), though over a much broader interval of log t. It could, therefore, be considered as the behaviour of two broad clusters of Maxwell elements. The log -log a> curve thus indicates the existence of two broad relaxation mechanisms, each with a spectrum of relaxation... 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

These two basic elements may be combined in series or parallel, giving the Maxwell-element and the Voigt-Kelvin element. 

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 a stress relaxation experiment the Maxwell-element is subjected to an instantaneous deformation sQ which is held constant. It means that ... 

FIG. 13.17 Left time dependent reduced stress, constant strain vs. log reduced time, log (t/t). Right time dependent reduced strain, of a Voigt-Kelvin... 

The time dependent modulus of a Maxwell element is given by... 

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