Halpin-Tsai equations modulus

As in the case of the transverse tensile modulus, 2. the above analysis tends to underestimate the in-plane shear modulus. Therefore, once again it is common to resort to empirical relationships and the most popular is the Halpin-Tsai equation... 

The modulus of the composites can be theoretically calculated using the well-known Halpin-Tsai equation [181], given by ... 

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

The effect of polymer-filler interaction on solvent swelling and dynamic mechanical properties of the sol-gel-derived acrylic rubber (ACM)/silica, epoxi-dized natural rubber (ENR)/silica, and polyvinyl alcohol (PVA)/silica hybrid nanocomposites was described by Bandyopadhyay et al. [27]. Theoretical delineation of the reinforcing mechanism of polymer-layered silicate nanocomposites has been attempted by some authors while studying the micromechanics of the intercalated or exfoliated PNCs [28-31]. Wu et al. [32] verified the modulus reinforcement of rubber/clay nanocomposites using composite theories based on Guth, Halpin-Tsai, and the modified Halpin-Tsai equations. On introduction of a modulus reduction factor (MRF) for the platelet-like fillers, the predicted moduli were found to be closer to the experimental measurements. 

The elastic modulus of composite materials reinforced by discontinuous cylindrical fibers or lamellar shapes is expressed by the Halpin-Tsai equations [106, 230], as shown in the following equation ... 

Since the polymer-filler interaction has direct consequence on the modulus, the derived function is subjected to validation by introducing the function in established models for determination of composite modulus. The IAF is introduced in the Guth-Gold, modified Guth-Gold, Halpin-Tsai and some variants of modified Halpin-Tsai equations to account for the contribution of the platelet-like filler to Young s modulus in PNCs. These equations have been plotted after the introduction of IAF into them. 

The modulus and yield kinetic parameters of the block polymer B can be related to those of the homopolymer in terms of a microcomposite model in which the silicone domains are assumed capable of bearing no shear load. Following Nielsen (10) we successfully applied the Halpin-Tsai equations to calculate the ratio of moduli for the two materials. This ratio of 2 is the same as the ratio of the apparent activation volumes. Our interpretation is that the silicone microdomains introduce shear stress concentrations on the micro scale that cause the polycarbonate block continuum to yield at a macroscopic stress that is half as large as that for the homopolymer. The fact that the activation energies are the same however indicates that aside from this geometric effect the rubber domains have little influence on the yield mechanism. 

The Chow equations [5] and the Halpin-Tsai equations [8,9] are also useful in modeling the effects of the crystalline fraction and of the lamellar shape (see Bicerano [23] for an example) on the moduli of semicrystalline polymers. Grubb [24] has provided a broad overview of the elastic properties of semicrystalline polymers, including both their experimental determination and their modeling. Janzen s work in modeling the Young s modulus [25-27] and yielding [27] of polyethylene is also quite instructive. 

Although the reinforcement by aromatic polyamides was remarkable, the composites prepared by these researchers were not molecular. As mentioned above, the molecules were coagulated into microfibrils of 15 un. 30 nm in thickness. Using the Halpin-Tsai equation modified by Nielsen (1975) and the modulus values of Ei = 0.91 GPa and E2 = 182 GPa for nylon 6 and PPTA as well as the observed modulus of the composites, the calculated L/D ratio was 15 for low molecular weight PPTA and 25 for high molecular... 

Use the Halpin-Tsai equations to determine the five elastic constants of a unidirectional fiber composite in which alumina fibers are dispersed in a glass matrix. The Young s modulus and Poisson s ratio of polycrystalline AljOj are 400 GPa and 0.23 and for the glass, 70 GPa and 0.20. 

In another work, SWNT-epoxy composites gave dT/dFf of 107.3 GPa. However, PAMAM-O-functionalised SWNT-epoxy composites had a higher dr/dFf of 153.6 GPa. In this paper, the authors used the Halpin-Tsai equation to predict the modulus of fibre reinforced composites.The experimental values were only half of their model prediction. The reason for this was that most of the SWNTs in epoxy showed significant curvature. If the experimental values of their work were scaled up, their theoretical maximum values would be dI7dFf 300 GPa, which is in excellent agreement with previous theoretical predictions. 

Relative tensile modulus of OMMT-polypropylene nanocomposites as a function of inorganic volume fraction. The solid line represents the fitting by using unmodified Halpin-Tsai equation. (Reproduced from Mittal, V., /. Thermoplast. Compos. Mater., 22, 453, 2009. With permission from Sage Publishers.)... 

It is well known that the elastic modulus of short fiber reinforced composite can be evaluated by means of the Halpin-Tsai equation ... 

On the other hand, the tensile properties of PP composites of these fibers analyzed through Halpin-Tsai equation revealed that the theoretical values of both Young s modulus and tensile strength were higher than those of experimentally obtained values. The latter could be understood as due to the poor matrix-fiber interface as is generally explained in the case of most of polymer-based composites. 

The Halpin-Tsai equation can explain the experimental results fairly well at low volume fraction of fibre. However, the modulus of a composite, calculated using the Halpin-Tsai eqnation often shows lower value in comparison with experimental data. At a low fibre volume fraction, the change in modulus as a function of fibre concentration can be expressed by the following equation ... 

More specifically, transverse modulus is given by the modified Halpin-Tsai equation (Equation 6.12) ... 

Again, the Halpin Tsai equations give a set of elastic constants to a good approximation, provided that the axial Young s modulus 3 is modified to... 

There are two simplifications of the Halpin-Tsai equations, en f approaches zero, approximating spheroidal structures, the Halpin-Tsai theory converges to the inverse rule of mixtures, and provides a lower bound modulus. Equation (10.8) expresses this relation, with an appropriate change in subscripts. 

Figure 13.9 (16,17) summarizes the results of the Halpin-Tsai equations, and also those of the Mori-Tanaka theory, to be discussed below. Note that for fibers, the Halpin-Tsai equations predict equal moduli for the 2 and 3 directions, but for platelets, the moduli are equal for the 1 and 2 directions. Also, note the symbolism where n and x represent the composite modulus parallel and perpendicular to the major axis of the filler. 

According to the Halpin-Tsai equations, what is the Young s modulus of a... 

Based on the Halpin-Tsai equations, what value of Young s modulus do you predict for a 5 wt.-% of well-bonded, single-waUed carbon nanotubes randomly dispersed in poly(vinyl alcohol) The SWNTs are 100 nm long and 1 nm in diameter. How does your answer compare with that shown in Table 13.6 ... 

The Halpin-Tsai equations (Halpin 1969 Halpin and Kaidos 1976) have long been popular for predicting the properties of short fiber composites. Tucker and Liang (1999) reviewed the application of several composite rtKxlels for fiber-reinforced corrqxfsites. They reported that the Halpin-Tsai theory offered reasonable predictions for composite modulus. 

Deng et al. [31] obtained PLA/cdHAp nanocomposites by solvent casting. SEM observations confirmed close contact between the polymer matrix and the filled nanocrystals, and the homogeneous dispersion of nanocrystals in the polymer matrix at a microscopic level. The tensile modulus for the nanocomposites increased with cdHAp loading. Theoretical predictions of the modulus (by assuming that the nanocomposites behave as a short-fibre-filled system) based on the Halpin-Tsai equation show excellent agreement with the experimental results. 

Many equations have been proposed for the transport properties of two-phase systems and in-depth details of the existing models are discussed elsewhere [4]. Noticing that virtually all the early theories neglected the effects of the particle shape, their packing density, and the possible formation of anisotropic clusters, Lewis and Nielsen modified the Halpin-Tsai equation for the elastic modulus of composite materials by incorporating the maximum volume fraction of filler cpm while still maintaining a continuous matrix phase [33,34]. Transposed to thermal conductivity Lewis and Nielsen s equation becomes... 

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