Takayanagi composite models

Takayanagi model belongs to a micromechanical composite models group, allowing empirical description of composite response upon mechanical influence on the basis of constituent it elements properties. One of the possible expressions within the fi ameworks of this model has the following look [38] ... 

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 ... 

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. 

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. 

Another explanation for an abnormal increase in Tgl in polymer blends has been proposed by Manabe, Murakami, and Takayanagi 125). They used a three-layered shell model, which accounts for interaction between the dispsersed and continuous phases of the blend. Abnormal increases in the glass transition of polystyrene in blends with various rubbers were explained by thermal stresses which arise from the difference in thermal expansion coefficients of the component polymers. However shifts in the glass transition temperatures of the SIN s do not appear to arise from differences in the expansion coefficients of the components because samples with the same overall composition and almost identical microstructures have significantly different glass transition temperatures. 

A mechanical model that has been successfully applied to phase-separated systems and composites is that of Takayanagi et al. (1963) which relates the overall modulus of the composite to those of the individual phases and their respective volume fractions. If the matrix is stronger than the filler, the overall modulus of the composite is given by ... 

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 ... 

Figure 5. Shear modulus as a function of composition for PU/PMMA SINs. Models of (a) Takayanagi(35), (b) Budiansky(37), (c) Davies(36), (d) Hourston and Zia(38).

Theory. Basic theories for the prediction of the modulus of a composite from those of the components were derived by, for example, Hashin in 1955 (I), Kerner in 1956 (2), and van der Poel in 1958 (3). Takayanagi (4, 5), and Fujino et al. (6) developed a very promising and instructive model theory which includes calculation of the loss spectra of composites, and it may easily be extended to anisotropic morphologies. Furthermore, Nielsen and coworkers may be cited for fundamental theoretical and experimental contributions (7, 8, 9,10). 

FIGURE 2.6 Viscosity of the composition at a temperature of 180°C (a) and 170 C (b) (theoretical calculations on model of Kemer-Takayanagi A experimental data). 

While the Takayanagi models have proved useful because of their simplicity, the effects of changes in mechanical behavior with composition and phase structure may also be profitably explored using several analytical relations, which include equations derived by Kerner (1956b), Hashin and Shtrikman (1963), and Halpin and Tsai (Ashton et al, 1969, Chapter 5). The most widely applied of these is the Kerner equation, which presents the... 

The above picture is further supported by studies of dynamic mechanical spectroscopy. Thus Matsuo et al. (1970f>) measured E and E" as a function of temperature for a number of compositions, and compared the results to Takayanagi s mechanical model rubber-plastic phase continuity (Takayanagi et al, 1963) (see Section 2.6.4) results are shown in Figure 8.24. 

Polymeric materials have relatively large thermal expansion. However, by incorporating fillers of low a in typical plastics, it is possible to produce a composite having a value of a only one-fifth of the unfilled plastics. Recently the thermal expansivity of a number of in situ composites of polymer liquid crystals and engineering plastics has been studied [14,16, 98, 99]. Choy et al [99] have attempted to correlate the thermal expansivity of a blend with those of its constituents using the Schapery equation for continuous fiber reinforced composites [100] as the PLC fibrils in blends studied are essentially continuous at the draw ratio of 2 = 15. Other authors [14,99] observed that the Takayanagi model [101] explains the thermal expansion. 

Kemmochi K, Takayanagi H, Nagasawa C, Takahashi J, Hayashi R, Manifestation models for closed-loop material recycling in carbon fiber reinforced thermoplastics (CFRTP), High Technology Composites in Modem Applications, Paipetis SA, Youtsos AG eds., 203-213, Sep 1995. 

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. 

Takayanagi s model is a phenomenological model obtained by combination of serial and parallel models. The composite modulus ( c(T)) is determined by (17.4) with X, an adjustment parameter. 

Figure 17.7 shows that Takayanagi s equation seems to be an excellent model to predict the modulus evolution, on the range 0-30 wt.% of filler. The parameter (2) has been determined by adjustment at 4.5. Reuss and Voigt s models are not well adapted to estimate correctly the composite moduli. This is because the matrix and the fillers mechanical characteristics are too different. But, we can show that the composite moduli are comprised between both boundaries, fcCR) and c(V). 

Values of storage modulus at 23 C are plotted as a function of composition in Figure 6.29. Takayanagi s parallel model for the mechanical behavior of a two-component system corresponds to the case in which the stiffer component is continuous, while his series model corresponds to the case in which the softer component is continuous. Clearly the experimental results agree best with the parallel model over most of the concentration... 

More quantitative information depends on the use of models. The Takayanagi models were already mentioned in connection with Figure 6.29. More analytical models have been evolved by Kerner/ Hashin and Shtrikman/ and Davies.Briefly, the first two theories assume spherical particles dispersed in an isotropic matrix. From the modulus of each phase, the composite modulus is calculated. An upper or lower bound modulus is arrived at by assuming the higher or lower modulus phase to be the matrix, respectively. The theory is reviewed elsewhere. 

Modulus VS Composition. The modulus of IPNs as a function of composition 3rields information about their relative phase continuity. The first such study on polyacrylate/polyurethane latex interpenetrating elastomeric networks (lENs) was carried out by Matsuo and co-workers (Fig. 7) (43). Note that in the midrange of composition, the modulus begins to follow the parallel Takayanagi model (44). This suggests that the stiflfer acrylic phase is becoming a continuous phase, as predicted from percolation models of three-dimensional mosaics of compressed spheroidal structures. 

It was realized by Takayanagi [17] that oriented highly erystalline polymers with a clear lamellar texture might be modelled in terms of a two-eomponent composite... 

Derive equations to express Young s modulus as a function of rubber and plastic composition using the Takayanagi models (c) and (d) in Figure 10.6. 

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. 

Another mode of natural nanocomposites reinforcement degree description is micromechanical models application, developed for pol5mier composites mechanical behavior description [1, 37-39]. So, Takayanagi and Kemer models are often used for the description of reinforcement degree on composition for the indicated materials [38, 39]. The authors of Ref. [40] used the mentioned models for theoretical treatment of natural nanocomposites reinforcement degree temperature dependence on the example of PC. 

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