Unrelaxed modulus

The creep modulus will vary with time, i.e. decrease as time increases, in a manner similar to that shown for the relaxation modulus. The classical variation of these moduli is illustrated in Fig. 2.9. On log-log scales it is observed that there is a high value of creep or relaxation modulus at short times. This is referred to as the Unrelaxed Modulus and is independent of time. Similarly at long times there is a low value Relaxed Modulus which is also independent of time. 

It was shown earlier that the variation of creep or relaxation moduli with time are as illustrated in Fig. 2.9. If we now introduce temperature as a variable then a series of such curves will be obtained as shown in Fig. 2.58. In general the relaxed and unrelaxed modulus terms are independent of temperature. The remainder of the moduli curves are essentially parallel and so this led to the thought that a shift factor, aj, could be applied to move from one curve to another. 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

It is apparent that there are a considerable number of parameters to be determined. According to equation (8) and equations (2-7) there are 6N+2 parameters where N is the number of relaxations present (it is not 8N because the relaxed modulus of one process is equal to the unrelaxed modulus of the next process in a sequence). In practice, it is found that with the large number of experimental points available in a scan (typically 50-100) the determinaton usually proceeds satisfactorily. However, in coitimon with many statistical fitting situations, it can happen that parameter determination is not unique. Our experience has shown that problems can arise when the relaxation strength is small or when only part of a peak is recorded. The problem with small relaxaton strength is associated with equation (1) where it is seen that the activation energy is related to the ratio of peak area and relaxation strength E(j- Ep. When the process is quite... 

Now we will discuss a procedure of reconstruction the temperature dependence of the relaxed and unrelaxed elastic moduli. We proposed before that the unrelaxed modulus, which describes the Jahn-Teller contribution, vanishes. Actually, the dynamic modulus measured in an experiment is the total one containing the contribution of the Jahn-Teller system as a summand. So, even the dynamic modulus which contains the unrelaxed Jahn-Teller contribution should be non-zero and can have a certain temperature dependence that is not associated with the Janh-Teller impurities. As well, the relaxed modulus for this reason can differ from one described with the expression (45). To deal with the impurity s contribution only, we can measure the temperature dependence of the dynamic modulus for an un-doped crystal and subtract it from one obtained for the the doped crystal. But it requires two specimens (doped and un-doped) and two experiments. More easy is to reconstruct the relaxed and unrelaxed moduli with the help of the data relating to the doped crystal. To derive the necessary expressions we will use the (20) and (21) and... 

Fig. 8 Elastic moduli ci = (cim + C1122 + 2ci3i3)/2 vs. inverse temperature obtained for 54.4 MHz in ZnSe Cr with concentration of the dopand ncr = 10 ° cm. Filled circles represent the real part of the dynamic modulus (q — co)/cq, open circles represent the relaxed modulus (c —co)/cq, and open triangles represent the unrelaxed modulus (c —cq )/cq. The initial reference modulus Co was taken as an extrapolation of ci(T) to F = 0 K. After Fig. 6 in [17]...

In the Hnear viscoelastic regime, the presence of diblock introduces an additional relaxation time which is apparent at low frequency and is due to the relaxation of the free end of the diblock in the elastomeric matrix. The more diblock in the adhesive, the more pronounced is the jump in modulus between the relaxed and the unrelaxed modulus. 

Equations (7.26) are called dispersion relations and analogous equations can be derived for /, and J2 (see problem 7.7) and for more general models. They can also be derived for the real and imaginary parts of the dieleetrie constant (see section 9.2.4). The limiting values of G, and /, at low frequencies are called the relaxed modulus and compliance, G, and J, and the limiting values at high frequencies are called the unrelaxed modulus and compliance, G and / ... 

It is useful to consider the behavior of the SLS as a function of dimensionless frequency (o)T, as shown in Fig. 5.16. At low frequencies, the dashpot has sufficient time to open and close and elastic behavior is obtained (relaxed modulus). The stress and strain are in phase. At very high frequencies, there is no time for the anelastic strain to develop. The stress and strain are again in phase but the modulus is higher, with only the spring of modulus , opening and closing (unrelaxed modulus). For intermediate frequencies, a lag develops and... 

It is usual to separate out the instantaneous elastic response in terms of the unrelaxed modulus G, giving... 

A concept that is of value in considering the relationship of viscoelastic behaviour to physical and chemical structure is that of relaxation strength . In a stress relaxation experiment, the modulus relaxes from a value G at very short times to Gr, at very long times (Figure 5.20(b)). Similarly in a dynamic mechanical experiment, the modulus changes from Gr, at low frequencies to G at very high frequencies. G is the unrelaxed modulus and Gr is the relaxed modulus (Figure 5.20(c)). 

The Rouse-model has also intrinsic limitations at short times. According to Eq. (6.76), the unrelaxed modulus is determined by the number density of Rouse-sequences, cr, since we find... 

For very high velocities the unrelaxed modulus must be used to describe the stiffness of the material in the vicinity of the crack tip, giving the following for the... 

In this equation, Gq is the high frequency limiting vzdue of the modulus (the unrelaxed modulus), %ww is the relaxation time, and P a shape parameter. The KWW function has been found to describe various processes. Most importantly for polymers, the local segmental relaxation dynamics conform closely to form of equation 1. 

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