The simple Takayanagi model

Even for block copolymers, in which the phase separation can be distinguished in electron micrographs, there are problems in matching parameters such as Poisson s ratios of the two components nevertheless the simple Takayanagi models, particularly when extended by a treatment to account for the finite length of the reinforcing component, can describe numerous features of static and d3mamic elastic behaviour. 

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 ... 

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

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. 

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. 

Takayanagi and co-workers transformed the spring and dashpot relaxation models (Section 1.5.6) to plastic and rubber elements in an effort to better explain the mechanical behavior of poly blends (Takayanagi et ai, 1963). Some simple combinations of the Takayanagi models are shown in Figure 2.11. The plastic phase is denoted by P and the rubber phase by R, while the quantities X and (p are functions of the volume fractions of parallel and series elements, respectively. 

The Takayanagi models were remarkably successful in providing a simple interpretation of the d3mamic mechanical behaviour of crystalline polymers and polymer blends. The theoretical basis is contained in Equations (8.1) to (8.6) of Section 8.2, and is deficient in two respects. First, only tensile deformations are considered and shear deformations are ignored. Secondly, as emphasized in Chapter 7, Voigt and Reuss schemes (i.e. parallel and series) only provide bounds to the true behaviour. 

In multicomponent polymeric systems such as polymer blends or blocks, each phase stress relaxes independently (39-41). Thus each phase will show a glass-rubber transition relaxation. While each phase follows the simple superposition rules illustrated above, combining them in a single equation must take into account the continuity of each phase in space. Attempts to do so have been made using the Takayanagi models (41), but the results are not simple. 

The first attempts in the 1960s and 70s proposed to predict elastic properties by using simple analytical models derived from the analysis of fiber-filled media. Takayanagi [210] described the elastic behavior of miidirectional oriented processed fibers. Halpin and Kardos [211] proposed to use the Halpin-Tsai model [212]. Phillips and Patel [213] also applied this model to PE with an adjustable paiameter linked to the crystallite shape ratio. However, these models require the assmnptiou that lamellae be regarded as fibers. Moreover, they are known to fit only the experimental data at low volume fr actions of filler and this is not the case for semicrystalline materials, for which the crystallinity can often reach 60 to 70%. 

In comparison with the simple case of para-aramid fibres, formulation of the structure-modulus relationships for semicrystalline fibres is much more complex. This is due to the segregation of crystalline and noncrystalline regions and to chain folding. The usual approach is based on recasting the structural models into mechanical models proposed by Takayanagi et as shown in... 

To fit and to estimate the modulus evolution, different simple models have been tested such as the models of Voigt (17.2), Reuss (17.3), and Takayanagi (17.4). The composite modulus ( c) is determined from f and which are the filler and the matrix moduli, respectively. 

International Conference on the Physics of Electronic and Atomic Collisions, Kyoto , K. Takayanagi and N. Oda, eds., The Society for Atomic Collision Research, Japan (1979), pp. 888-889 J. P. Dwyer and A. Kuppermann, Resonances in collinear reactive scattering A simple hyperspherical coordinate model, manuscript in preparation. 

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