The Zener model

This model can be posed in two ways ((a) or (b) of Fig. 4.14.) which are—as will become apparent— identical. The horizontal parallel lines in Fig. 4.14 indicate elements which have constant strain e.g. and // of (a) have the same strain. We now derive the differential equation describing the relationship between 7, y, a, and y for (a). Let the stress in the dashpot be 0. The stress in is then cr — Ti, so that 

Now the quantity T (y,j / Jr) is another time constant, which it is useful to write as T... The ratio of the two time constants is thus 

It is left as a problem for the reader to show that the alternative variant of the Zener model (Fig. 4.14(b)) also leads to eqn (4.47). A solid which conforms to the predictions of this model is termed a Zener solid. The model shows all the significant characteristics of polymer relaxations it has to be generalized slightly (see 4.3.2) to be a precise fit. The solutions which we now state for creep, stress relaxation, and dynamic response do, however, have considerable illustrative significance. 

A constant stress o is applied at r = 0. da/dt thereafter equals zero, so that from eqn (4.47) 

The strain y i) then relaxes to its equilibrium value of Jr Oq with time constant x . The solution to eqn (4.48) is 

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Gai. jMn.jAs. Furthermore, the scaling theory of electronic states near the MIT, discussed in the previous sections, makes it possible to explain the presence of the ferromagnetism on the both sides of the MIT, and a non-critical evolution of 7fc across the critical point (Matsukura et al. 1998b). A comparison between theoretical and experimental data in a wider range of Mn and hole concentrations requires reliable information on the hole density in particular samples, which is not presently available. In appears, however, that in the case of both Gai-jjMnjAs and Im jMnjAs on the insulator side of the MIT, the experimental values of Tc are systematically higher than those expected from the Zener model. 

Figure 10.7 Response of the Zener model (Fig. 10.6a) to a shear step stress input.

The main disadvantage of the Maxwell model is that the static shear modulus p0 vanishes in this model, while the drawback of the Kelvin-Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin-Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131]... 

For a standard linear medium (the Zener model) taking into account Eq. (401) we have [131]... 

Young s modulus relaxation to occur between a value Eq at zero frequency, and a value E o at infinite frequency. The difference E o — Eq is the magnitude of the relaxation process. An empirical modification of the Zener model, known as the Cole-Cole equation, has the complex modulus equation... 

The simplest way to obtain the behavior discussed in connection with Fig. 5.9 is to place a second spring in series with a Voigt model. This is shown in Fig. 5.14(b) and is known as the Zener model or standard linear solid. The constitutive equation is found by simply adding the strains from the spring and Voigt model (Eqs. (5.41) and (5.48))... 

The Zener model (or standard linear solid). The model may be represented as a spring in series with a Kelvin model, as in (a), or as a spring in parallel with a Maxwell model, as in (b). The significant properties inherent in the Zener model include (i) two time constants, one for constant stress and one for constant strain r (ii) an instantaneous strain at t>0 when subject to a step-function stress and (iii) full recovery following removal of the stress. For the Kelvin and Maxwell models, see Problem 4.8. 

Frequency response The frequency dependence of the dynamic mechanical properties of the Zener model can be obtained 1 substituting eqns 4.20 and 4.21 into eqn 4.47. It then follows from eqn 4.25 after reduction that... 

The reader will note that the definition of a phenomenological theory given at the start of this section has been satisfied by the Zener model. Measurements in creep, stress relaxation, or dynamic response are interrelated by three of the model parameters, say /r, and r. ... 

In order to fit isothermal data (for example the linear polyethylene data for 15°C, see Figure 4.4) it is necessary to fit to the data the three adjustable parameters of the Zener model, which are J, /r, and t. Since the Zener model is supposed equalfy valid for creep, stress relaxation, and dynamic response, these same parameters should then fit (for linear polyethylene at IS C) G(t), G (tu) and J ([Pg.146]

The Zener model almost succeeds, but not quite it fails only in that the relaxations observed are broader than the predictions. For example, note that in Figure 4.15 the predicted rate effects effectively run their complete course in 3 decades of time for J(t) the active relaxation region extends from r s 1 s to r s 10 s and for G[Pg.146]

The Zener model gives a lucid description of the origin of the large temperature dependence of the viscoelastic properties of polymers. Of the three parameters and Tg. the dominant parameter is r,. Both... 

Smce the Zener model is a linear viscoelastic model, it obeys the Boltz-mann superposition principle. In this problem we are concerned with a strain history which is a smooth varying function of time, with y undergoing sinusoidal oscillations. Therefore the integral form of the BSP is the most straightforward one to apply ... 

Figure 9.29 Data for grain size of AI2O3 (after annealing for 100 hour at 1700 C) as a function of SiC inclusion volume fraction, compared with the predictions of the Zener model and computer simulations. (From Ref. 60.)...

The Zener model leads lo a linear equation in stress and strain and their first time derivatives... 

Fig. 4.16. Solution to the Zener model for a dynamic experiment. Model with Jy=1GPa" . and r,= 100s. Note that maxima occur in J" at...

The failure of the Zener model is easily rectified by assuming that the mechanism is a set of relaxation processes with a band of relaxation times which are closely spaced. The a-process shown in Fig. 4.4 is the interlamellar (intercrystal) slip process, which is analogous to the grain boundary creep process in metals. The heterogeneity of the polymeric solid is the origin of the fact that the relaxation times occur in a distribution all relaxations in polymers are found to be described by distributed relaxation times. 

The Zener model gives a lucid description of the origin of the large temperature dependence of the viscoelastic properties of polymers. Of the three parameters Jy, Jr, and the dominant parameter is t . Both J and Jr do vary with temperature, but the effect is small. For convenience, let the relaxation time at temperatures T and To be written r and tr,- Then the temperature dependence can be described by a parameter Oy... 

Show, without recourse to the model itself, that the Zener model whose stress relaxation modulus is given by eqn (4.53)... 

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