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 parameters are related with filler concentration and interphase thickness by the following simple relationships ... 

Surface structure determination 325—331 Suspension springs 244—248 Takayanagi model See DAS model Tamm states... 

Where a melt-crystallized polymer has been processed by drawing, rolling or other means to produce an aligned structure in which lamellae as well as polymer chains have discernible order, a pseudocrystalline unit cell is present. Provided that this unit cell contains elements of the crystals as well as the boundaries between crystals and that it is entirely typical of the material as a whole then it could be considered as a RVE within the meaning defined above. The lamella crystal itself sometimes considered as embedded in an amorphous matrix would not seem to be an acceptable RVE for reasons similar to those advanced against the Takayanagi model, namely that its modulus is dependent upon the surface tractions. The boundaries between lamella crystals in the matrix must be included in an acceptable RVE. 

The foregoing summary of applications of composites theory to polymers does not claim to be complete. There are many instances in the literature of the use of bounds, either the Voigt and Reuss or the Hashin-Shtrikman, of simplified schemes such as the Halpin-Tsai formulation84, of simple models such as the shear lag or the two phase block and of the well-known Takayanagi models. The points we wish to emphasize are as follows. 

In most simple applications of models all that is really being achieved is curve fitting. This applies to the Takayanagi models (which are one dimensional and assume uniformity of stress and strain within each element) as well as to simple fibre models such as the shear lag. 

Analysis of Stress—Optical Data. The slight, if indeed real, improvement of the isotropic model over the Takayanagi model would be of little consequence were it not for a more pronounced difference between the two models in their ability to describe the stress-optical data. When the parameters obtained from the dynamic data (Table IV) are substituted into Equations 8 and 9, Equation 8 produces results which are uniformly too low. Equation 9 also underestimates the magnitude of Ka but only by an average 7% (Figure 14). For most blends the discrepancy is less than 5%, and all calculated values show the characteristic elevation of the birefringence attributed to the multiphase structure. 

In this work the mechanical model proposed by Takayanagi (9) will be used. Two variants were proposed, both assuming that the two phases are connected partly in parallel and partly in series (Figures 7a and 7b). Kaplan and Tschoegl (10) have shown that the two variants of the Takayanagi model are equivalent. The series model (Figure 7b) will be used for our calculations. The modulus is given by ... 

Table II. Takayanagi Model Parameters of Volume Fraction of PG-PST Blends...

For correlating the consistency coefficient (A ) of the power law model and the Casson yield stress (aoc), again the lower bound form of Takayanagi model (Ross-Murphy, 1984) was satisfactory ... 

In Chapter 2, simple Takayanagi models for the modulus of two-component blends were discussed that are applicable to the shear modulus Gc of a composite formed from components (polymers) x and y with shear moduli Gx and Gy, respectively. If volume fractions of the two components, the eqmtions for the upper and lower bound values are ... 

It is reasonable to expect the major component to form the continuous phase, that is, at 4>x > 0-5. More complex Takayanagi models have been reviewed by Manson and Sperling (1976). 

Fig. 11.10 The series-parallel and parallel-series versions, (b) and (c), respectively, for the Takayanagi model (a), which allows for noncrystalline (shaded) material being partly in series and partly in parallel with crystalline material.

Deduce the equations corresponding to equations (11.16) and (11.17) for the Takayanagj models shown in figs 11.10(b) and (c). A polymer that is 75% crystalline by volume has amorphous and crystalline moduli E3 and equal to 1.0 x 10 and 1.0 x lO Pa, respectively, parallel to the draw direction. Calculate the modulus parallel to the draw direction for the simple series and simple parallel Takayanagi models and for the series-parallel and parallel-series models, assuming that in the last two models a = ft = 0.5. 

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. 

Fig. 5. Parallel and series Takayanagi models. (Adapted from Ref 64.)...

Fig, 6, Takayanagi models for A-phuse dispersed in B-phase. i Adapted from Ref 64. 

Fig. 7. Temperature dependence ofstorage and loss modulifor P VC-NBR film bonded in parallel to a PVC film. Takayanagi model type / gives better fit to experiment.

The performance of polymer blends could be well represented by a Takayanagi model in which the relative values of /. and were related to the shape of the dispersed phase, e.g. /. = for homogeneous dispersions, and / > for dispersions in the form of elongated molecular aggregates. But for semi-crystalline polymers, with A and B representing amorphous and crystalline components, the dispersions were generally broader than predicted, suggesting that the unordered material was not identical with that in a completely amorphous state. 

Fig. 8. High density polyethylene. Schematic representation ofrariation ofe.xtensional (E ) and transverse (E> o > moduli of oriented specimens with temperature, in terms of a Takayanagi model (Adapted from Ref. 67.)...

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