Viscoelasticity Takayanagi models

Lipatov et al. [116,124-127] who simulated the polymeric composite behavior with a view to estimate the effect of the interphase characteristics on composite properties preferred to break the problem up into two parts. First they considered a polymer-polymer composition. The viscoelastic properties of different polymers are different. One of the polymers was represented by a cube with side a, the second polymer (the binder) coated the cube as a homogeneous film of thickness d. The concentration of d-thick layers is proportional to the specific surface area of cubes with side a, that is, the thickness d remains constant while the length of the side may vary. The calculation is based on the Takayanagi model [128]. From geometric considerations the parameters of the Takayanagi model are related with the cube side and film thickness by the formulas ... 

The Takayanagi model was developed to account for the viscoelastic relaxation behaviour of two phase polymers, as recorded by dynamic mechanical testing. " It was then extended to treat both isotropic and oriented semi-crystalline polymers. The model does not deal with the development of mechanical anisotropy on drawing, but attempts to account for the viscoelastic behaviour of either an isotropic or a highly oriented polymer in terms of the response of components representing the crystalline and amorphous phases. Hopefully, comparisons between the predictions of the model and experimental results may throw light on the molecular processes occurring. 

In a manner similar to the application of springs and dashpots to the theory of linear viscoelasticity, we note that for units in parallel the total stress is (T = (Ti -f (T2 + (T3 -f , and that for units in series the total strain is 6 = -f 82 -f 3 -f . Finally, application of Hooke s law, cr = sE, allows the complex modulus E of the Takayanagi models in Figures 2.11a-d to be represented by the following equations, respectively ... 

Qualitatively, the modulus of polymer blends and composites is expected to be intermediate between the modulus of the materials involved. Quantitatively, the picture is more complex, depending on phase morphology and continuity. The Takayanagi models (Section 10.1.2.3) explore the basic methods of calculating not only the modulus, but many other viscoelastic quantities. While the original models were developed with glassy and rubbery polymers in mind, they are quite general and useful for composite systems as well. 

Figure 9.12 Temperature dependence of storage and loss moduli for a polyvinyl chloride-nitrile bidadiere rubber film bonded in parallel to a polyvinyl chloride film. Takayanagi model type (a) gives better fit to experiment. (Reproduced from Takayanagi, M. (1963) Viscoelastic properties of crystalline polymers. Memoirs of the Faculty of Engineering Kyushu Univ., 23,41. Copyright (1963) Kyushu University.)...

We now consider some models of polymer structure and ascertain their usefulness as representative volume elements. The Takayanagi48) series and parallel models are widely used as descriptive devices for viscoelastic behaviour but it is not correct to use them as RVE s for the following reasons. First, they assume homogeneous stress and displacement throughout each phase. Second, they are one-dimensional only, which means that the modulus derived from them depends upon the directions of the surface tractions. If we want to make up models such as the Takayanagi ones in three dimensions then we shall have a composite brick wall with two or more elements in each of which the stress is non-uniform. 

Takayanagi, M., Harima, H., and Iwata, Y. (1963). Viscoelastic behavior of polymer blends and its eomparison with model experiments. Mem. Faculty Eng. Kyushu Univ. 23, 1-13. 

Viscoelastic stress analysis of two component systems shows that a broadening of the dispersion zone is to be expected 166,167), even if the disperse phase (filler) is purely elastic 166) and it is not necessary to ascribe different molecular properties to the continuous phase. The simplest way to visualize this mechanical interaction is by the use of phenomenological mechanically equivalent models. The model of Takayanagi (/68) is illustrated in Fig. 16. The elastic solution for this model is easily derived from elementary considerations. By the correspondence principle of viscoelastic stress analysis 169), the viscoelastic solution is obtained simply by substituting complex moduli in place of purely elastic moduli... 

Takayanagi [17] devised series-parallel and parallel-series models as an aid to understanding the viscoelastic behaviour of a blend of two isotropic amorphous polymers in terms of the properties of the individual components. For an A phase dispersed in a B phase there are two extreme possibilities for the stress transfer. For efficient stress transfer perpendicular to the direction of tensile stress we have the series-parallel model (Figure 8.9(a)) in which the overall modulus is given by the contribution for the two lower components in parallel (as in Equation (8.3)) in series with the contribution for the upper component (as in Equation (8.5)) ... 

PIG. 14-18. Schematic representation of a partialiy crystalline polymer system (crystalline white, amorphous black) by a mechanical model with blocks whose viscoelastic properties correspond to those of the individual phases, arranged partly in series and partly in parallel in accordance with two empirical ratios Ip and X. (Takayanagi. )... 

Takayanagi [50] devised series-parallel and parallel-series models as an aid to understanding the viscoelastic behaviour of a blend of two isotropic amorphous polymers in terms of the... 

Using the theories considered above, one shordd have in mind that they do not take into account the distribution of particles by their size and shape. In many cases, there is a need to modify these equations. The mechanical models are also very useful for calculation of viscoelastic moduli and mechanical loss in particulate-filled pol3miers. The most well-known model was proposed by Takayanagi and is described in many books. The application of this model to... 

A composite is described with the aid of the same Takayanagi s models which were discussed above. The equivalence of both the models enables them to be described with the aid of one equation. If viscoelastic Poisson s ratio (i = i i- ii ) is equal to v = v =p, then the dynamic shear modulus... 

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