Relaxation time generalized Maxwell model

This equation, based on the generalized Maxwell model (e.g. jL, p. 68), indicates that G (o) can be determined from the difference between the measured modulus and its relaxational part. A prerequisite, however, is that the relaxation spectrum H(t) should be known over the entire relaxation time range from zero to infinity, which is impossible in practice. Nevertheless, the equation can still be used, because this time interval can generally be taken less wide, as will be demonstrated below. 

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 pertinent parameters used in a generalized Maxwell model can be easily presented in graphical form as shown in Figure 3-10. Here three spring constants (1010, 108, and 106) are associated with relaxation times (102, 104, and 105 respectively). Clearly the stress relaxation modulus for this particular generalized Maxwell model could be easily calculated using equation (3-34). 

Now one may associate these relaxation times with the relaxation times of a generalized Maxwell model. Thus the stress relaxation behavior for the bead-and-spring model is given as... 

Note that v is the number density of Rouse segments in chains of N segments in length and that are monodisperse. The indexp is the eigenmode from the solution to the equation of motion. Furthermore, equation 65 is the equation for a special form of the generalized Maxwell model having constant coefficients Gj = vkBT (see eq. 18). The relaxation times are given by Xr g/ p. ... 

The relaxation function R X,t) can be described by a generalized Maxwell model, as illustrated in Fig. 5.43 with three elasticity moduli fJ of different relaxation times Xj... 

A single Maxwell model or Voigt model has a single characteristic time, T. To express the viscoelastic properties of real polymer materials, the generalized Maxwell model with multiple relaxation times t s, as shown in Fig. 4, is much more convenient than the single Maxwell model. In this case, and " are the sums of each element and are given as... 

Fig. 4 The generalized Maxwell model, containing multiple relaxation times.

Comparison of the forms of equations 58 to 61 with equations 21 to 23 of Chapter 9 and equations 23 and 24 of Chapter 3 shows that the time and frequency dependence correspond to a generalized Maxwell model as in the Rouse theory and its various modifications, but here the spring constants (or discrete contributions to the relaxation spectrum) are not necessarily all equal they are proportional to the concentrations of the various types of strands, v e. The molecular weight does not enter explicitly, but it may be expected that the higher the molecular weight the greater the concentrations of strands which find it difficult to leave the network and hence have large values of the time parameter... 

If several Maxwell models are used in series to represent polymer response, as in the generalized Maxwell model (see Chapter 6), and if the spring moduli and relaxation times are judiciously chosen, the transition region broadens as shown in Fig. 7.17. The parameters used for the curves... 

Table 7.1 Parameters used in generalized Maxwell models for Fig. 7.17. Spring constants in Pa, relaxation times in s.

To illustrate the abihty of a generalized Maxwell Model (Prony Series) to fit long term data, consider the master curve data from Fig. 7.5 for polyisobutylene. A complete data set at 25°C was constructed as shown in Fig. 7.18. Thirty relaxation times evenly spaced in log time between 10 " and 10 were chosen and the sign control method used to calculate the Prony series representation seen in Fig. 7.19. The modulus E(t) calculated from... 

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 small shear strain i.e. 1 % was used for relaxation experiment of the LCPs (Rahman 2013). In all relaxation tests, two distinct steps were observed for all LCPs initial sharp decay followed by plateau values for long time. The Generalized Maxwell model ((4.1)) has been used to fit the experimental data to obtain the relaxation spectrum of the LCPs (Majumder et al. 2007) (Fig. 4.4). 

The notional transition from a finite number of elements to an infinite number (i.e., a spectrum) of elements leads to models where (1) expresses the probability density of relaxation times in the generalized Maxwell model. The relaxation function of the generalized Maxwell spectrum then is ... 

If the number of elements in the generalized Maxwell model is increased toward infinity, one arrives at the continuous spectrum function, F(t), where F r) dr is the contribution to G(t) due to Maxwell elements having relaxation times between rand t+ dr. The relaxation modulus is related to the spectrum function as shown by Eq. 4.18. 

Maxwell element A in creep at a constant stress of 10,000 Pa exhibits a total elongation of 0.65 strain nnits after 100 h and a permanent set of 1.00 unit on removal of the load after a total of 250 h. Now a generalized Maxwell model made up of elements A, B, and C is constructed. B has twice the spring modulus and one-half the relaxation time of A, whereas C has one-half the spring modnlns and twice the relaxation time of A. If a sudden elongation of 1.80 nnits is applied and held constant on the generalized Maxwell model, what stress will be exerted by it after 100 h ... 

Two Maxwell elements are tested separately in creep experiments in which the stress is equal to Oq. The time to reach an elongation 8 of 150% is measured. At that exact time, the stress is removed and the permanent set is measured. Next, the two elements are combined in parallel (a generalized Maxwell model) and subjected to a stress relaxation experiment with a total elongation of 8q. What is the ratio of the stress at 250.0 s compared to that at 50.0 s ... 

Figure 3 displays the viscoelastic properties of PP-2 as a function of angular frequency (a) measured using both the SPR and parallel plate rheometer at 180°C and 200°C. The generalized Maxwell model [1] (Equations (7) and (8)) is the most commonly used model to describe hnear viscoelasticity, where, a is the angular frequency, Gt is the relaxation strength and A, is the relaxation time of Maxwell elements i to N, and as a result, this model was chosen to assess the quality of the SPR measurements. 

The response to an applied strain of 200% (e = 2) was then studied. As shown, the logarithmic term of Eq. (6) may be neglected for large N, and the response of the model then becomes equivalent to that of a generalized Maxwell body (Weichert body) with 48 discrete relaxation times, in parallel with a spring. 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

The only parameter in (11) having dimensions of time is C. Although this is not a relaxation time per se, it can be associated with a "Maxwell-type relaxation time, as follows. Although the linear Maxwell model predicts a constant (Newtonian) viscosity, it may be generalized by utilizing a co-rotational reference frame which follows the local rotation and translation of each fluid element [9]. When a term is added to account for the high shear limiting behavior, the result is the co-rotational form of the Jeffreys model ... 

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