Halpin-Tsai predictions

The experimental values are compared with the Guth and Halpin-Tsai predictions using the respective aspect ratios of 70 and 90 to fit the data (Figure 12.11). These values are lower than expected from the average dimensions of the MWNTs but much higher than those previously published for MWNTs for hydrocarbon rubber / MWNTs composites (22,31) which is a result from a better filler dispersion. 

With these values, a comparison was made between the predicted values from the Halpin-Tsai and Mori-Tanaka models and the measured Young s modulus as a function of montmorillonite concentration in the polymer. When the experimentally determined number-average aspect ratio, 57, was employed, Halpin-Tsai predicted higher values and Mori-Tanaka predicted lower values than the experimental results. When the aspect ratio for the theoretical perfect exfoliation of montmorillonite was utilized in the Mori-Tanaka model, 97, the values were virtually identical to the experimental values. When the Halpin-Tsai model was altered to accommodate dispersed phases that have more than... 

The Halpin-Tsai prediction for the modulus at 2 wt.% loading of fully exfoliated montmorillonite with an aspect ratio of 100 in this polymer is duplicated by the ammonium ion-exchanged montmorillonite that has an experimentally determined aspect ratio of approximately 53.3. The enhanced reinforcing efficiency of this organomontmorillonite in SAN as a function of aspect ratio was not anticipated. 

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid orthotropic material are introduced. In Section 3.3, the elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. 

The mere existence of different predicted stiffnesses for different arrays leads to an important physical observation Variations in composite material manufacturing will always yield variations in array geometry and hence in composite moduli. Thus, we cannot hope to predict composite moduli precisely, nor is there any need to Approximations such as the Halpin-Tsai equations should satisfy all practical requirements. 

There is much controversy associated with micromechanical analyses and predictions. Much of the controversy has to do with which approximations should be used. The Halpin-Tsai equations seem to be a commonly accepted approach. 

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 close fit of the experimental data and the values predicted by the constitutive modified Halpin-Tsai equations I and II (24) and (25), as seen in Fig. 43 (for NR) illustrates the appropriate definition of the IAF. Table 10 also confirms that newly devised equations (24) and (25) provide astounding results because their predictions conform to the experimental data. The introduction of IAF imparts a definitive change to the predicting ability of the constitutive equations for polymer/filler nanocomposites (Fig. 43 Table 10). 

It is concluded from the above that the mechanical characteristics of CNT composites are not yet well established. In order to have a better insight into the expected performance, idealized upper bounds for various mechanical properties would be useful to have. Although many sophisticated models for predicting the mechanical properties of fiber-reinforced polymers exist, the two most common and simplest ones are the rule of mixtures and the Halpin-Tsai... 

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 ... 

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. 

Various composite models such as parallel model, series model, Halpin-Tsai equation, and Kerner s model can be used to predict and compare the mechanical properties of polymer blends [43-45]. For the theoretical prediction of the tensile behavior of PMMA/EMA blends, some of these models... 

The Halpin-Tsai model is a well-known composite theory to predict the stiffness of unidirectional composites as a functional of aspect ratio. In this model, the lorrgitudi-nal Ejj and transverse engineering modtrli are expressed in the following general form ... 

Luo, et al. [80] have used multi-scale homogenization (MH) and FEM for wavy and straight SWCNTs, they have compare their results with Mori-Tanaka, Cox, and Halpin-Tsai, Fu, et al. [81] and Lauke [82], Trespass, et al. [83] used 3D elastic beam for C-C bond, 3D space frame for CNT, and progressive fracture model for prediction of elastic modulus, they used rule of mixture for compression of their results. Their assumption was embedded a single SWCNT in polymer with perfect bonding. The multi-scale modeling, MC, FEM, and using equivalent continuirm method was used by Spanos and Kontsos [84] and compared with Zhu, et al. [85] and Paiva, et al. [86] results. 

A number of micro-mechanical models have been developed over the years to predict the mechanical behavior of particulate composites [23-2. Halpin-Tsai model has received special attention owing to better prediction of the properties for a variety of reinforcement geometries. The relative tensile modulus is expressed as... 

Figure 12.3a shows the Halpin-Tsai micromechanical predictions for unidirectional PP composites filled with 5vol% of aligned ID needle-like and 2D cylindrical platelet-like nanoclays of different aspect ratios, assuming a Young s modulus of 200 GPa for both... 

There are several methods for predicting the elastic properties of unidirectional laminae. Three of the most suitable methods are presented. These are (a) the rule of mixtures method, (b) the self-consistent doubly embedded method and (c) Halpin-Tsai method. 

The difficulties associated with predicting accurately the elastic properties of a unidirectional lamina using mathematical closed form solutions, prompted development of a number of semi-empirical relationships. The Halpin-Tsai method (the equations for which are included in the design document) provides the most popular and widely used relationships. 

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]. 

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. 

Peponi et al. [19] further extended the above-mentioned PDF estimation approach to the Halpin-Tsai equation in the case of PP-composites to predict their properties of random distribution of these fibers in PP matrix. 

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