Halpin-Tsai theory

Halpin-Tsai theory is used for prediction elastic modulus of unidirectional composites as function of aspect ratio. The longitudinal stiflfiiess, transverse modulus, are expressed in the following general form ... 

For - 0, the Halpin-Tsai theory converged to inversed rule of mixture for stiffiiess. 

The physical properties of a number of other polymer nanocomposites made with clays have been measured. Table 33.3 contains a selection of reported values for some of the most common polymers. Poly(ethylene terephthalate) (PET) and Poly(butylene terephthalate) (PBT) are the most commOTi commercial engineering polymers. The average increase in tensile modulus for most of the PET nanocomposites [21,22,24] is in the range of 35%. This is well below the prediction of a 95% increase for a 5% by weight nanocomposite utilizing Halpin-Tsai theory. The only exception was PET produced by in situ polymerization and tested as fibers [20]. In each one of these references it was acknowledged that full exfoliation had not been reached in the composite. It is reasonable to expect that substantial improvement in properties could be seen if full exfoliation were achieved. The reported increase in tensile modulus for PBT nanocomposites is only in the 36% range [23,24]. 

Based upon Halpin-Tsai theory, a 5% loading of clay in PBT should increase the tensile modulus above 100%. Chisholm [24] claims that the PBT composite made with sulfated PBT was fully exfoliated. Therefore, the degree of exfoliation cannot be the reason for the low increase in modulus. 

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. 

Conversely, when f approaches infinity (fibers of infinite length), the Halpin-Tsai theory reduces to the upper bound rule of mixtures, as already given in equation (10.6). 

The Halpin-Tsai theory as written requires that the fibers be oriented in the direction of the stress. Fornes and Paul (16) show that in the case of completely random orientation in all three orthogonal directions, fiber modulus reinforcement follows the relation... 

The solutions of equations (13.10) and (13.11) for the assumptions of fiber-like and disk-like platelets are given in Figure 13.9. It must be noted that the Mori-Thnaka theory treats fibers and disks as ellipsoidal particles, whereas the Halpin-Tsai theory treats a fiber directly, but assumes that a disk is a rectangular platelet. 

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. 

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

Not much work is available regarding micromechanical theories of strength. However, considerable work as been done on micromechanical theories of stiffness. We will concentrate on those aspects of stiffness theory that are most prominent in usage (e.g., the Halpin-Tsai equations) in addition to those aspects that clearly illustrate the thrust of micromechanics. Available strength information will be summarized with the same intent as for stiffness theories. 

The variational energy principles of classical elasticity theory are used in Section 3.3.2 to determine upper and lower bounds on lamina moduli. However, that approach generally leads to bounds that might not be sufficiently close for practical use. In Section 3.3.3, all the principles of elasticity theory are invoked to determine the lamina moduli. Because of the resulting complexity of the problem, many advanced analytical techniques and numerical solution procedures are necessary to obtain solutions. However, the assumptions made in such analyses regarding the interaction between the fibers and the matrix are not entirely realistic. An interesting approach to more realistic fiber-matrix interaction, the contiguity approach, is examined in Section 3.3.4. The widely used Halpin-Tsai equations are displayed and discussed in Section 3.3.5. 

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 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. 

Many theories have been advanced for predicting the modulus of filled composites. The Kemer theory is often used for the G modulus in the case of filled systems containing spheres Halpin--Tsai modified the Kemer equation in a more general form Lewis and Nielsen suggested a further modification by taking into consideration the packing factor and obtained, in the case of E modulus, the following equation ... 

Crystallites may also be considered to act as reinforcing fillers. For example, the rubbery modulus of poly(vinyl chloride) was shown by lobst and Manson (1970,1972,1974) to be increased by an increase in crystallinity calculated moduli in the rubbery state agreed well with values predicted by equation (12.9). Halpin and Kardos (1972) have recently applied Tsai-Halpin composite theory to crystalline polymers with considerable success, and Kardos et al (1972) have used in situ crystallization of an organic filler to prepare and characterize a model composite system. More recently, the concept of so-called molecular composites —based on highly crystalline polymeric fibers arranged in a matrix of the same polymer—has stimulated a high level of experimental and theoretical interest (Halpin, 1975 Linden-meyer, 1975). 

Maximum strain theory may be modified to predict the strength of randomly oriented short-fiber composites (22). llie Halpin-Tsai equations (14) have established relations for the stiffness of an oriented short-fiber ply from the matrix and fiber properties. These nations show that the longitudinal stiffness of an oriented short-fiber composite is a s itive function of the aspect ratio. 

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. 

What makes the Halpin-Tsai and Mori-Tanaka theories more complex is primarily that both of them correct for end effects. Thus, when a fiber or other particle is of finite length in the stress direction, the particle s ends present points of lesser reinforcement, and clearly the gaps between such particles are not reinforced. 

Values for E obtained in this way are very similar to these calculated on the basis of shear lag theory. A limitation, as pointed out by Hine, Lusti and Gusev [5], relates to the estimates of E and Vf2 using the Halpin-Tsai equation. 

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... 

The Halpin-Tsai and Mori-Tanaka theories based on first-principle arguments adequately model the mechanical properties provided by the reinforcement of montmorillonite as a dispersed phase in all polymers [5,20]. The significant independent variables that correlate to reinforcement are aspect ratio, modulus, and the alignment of the montmorillonite in the direction of the applied stress. It is enlightening to examine some of the general predictions of these theories as they relate to the effect of clay loading, aspect ratio, and the modulus of the pristine... 

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