Elastic relaxation modulus

These points can be illustrated by comparing the elastic relaxation modulus ,(0 for crystalline (isotactic), amorphous, and chemically cross-hnked (atactic) polystyrene samples, as shown in Figure 15.9. Crystalhnity has little effect below Tg, but as the molecular motion increases above T, the modulus of the amorphous polymCT drops more sharply. The value of E t) remains high for the crystalline polymer throughout this range until the rapid decrease at the melting temperature is recorded. The cross-hnked sample maintains its modulus level at this temperature as the crosslinks are not thermally labile and do not melt. 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

The stress relaxation properties of a high molecular weight polybutadiene with a narrow molecular weight distribution are shown in Figure 1. The behavior is shown in terms of the apparent rubber elasticity stress relaxation modulus for three differrent extension ratios and the experiment is carried on until rupture in all three cases. A very wide rubber plateau extending over nearly 6 decades in time is observed for the smallest extension ratio. However, the plateau is observed to become narrower with increasing extension... 

The model of a viscoelastic body with one relaxation time used above has one principal disadvantage it does not describe the viscous flow of the reactants before the gel-point at t < t. Thus it is important to use a more general model of a viscoelastic medium to interpret the results obtained. The model must allow for flow and may be constructed by combining viscous and viscoelastic elements the former has viscosity rp and the latter has a relaxation modulus of elasticity Gp and viscosity rp,... 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

The rubber elasticity theory predicts that the relaxation modulus is given by... 

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]...

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

The value of the stress relaxation modulus at the relaxation time G(x) is of the order of kT per chain in either the Rouse or Zimm models, just as the strands of a network in Chapter 7 stored of order kT of elastic energy ... 

Figure 2. Mechanical loss tangent (tan 5) and elastic storage modulus at 102 c.p.s. and spin-lattice relaxation time (20) vs. temperature for poly (4-methyl-l-pentene) crystallized from dilute solution...

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