Composite Halpin-Tsai

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 ... 

In practical terms the above analysis is tcx) simplistic, particularly in regard to the assumption that the stresses in the fibre and matrix are equal. Generally the fibres are dispersed at random on any cross-section of the composite (see Fig. 3.8) and so the applied force will be shared by the fibres and matrix but not necessarily equally. Other inaccuracies also arise due to the mis-match of the Poisson s ratios for the fibres and matrix. Several other empirical equations have been suggested to take these factors into account. One of these is the Halpin-Tsai equation which has the following form... 

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. 

Note that the expressions for E., and v.,2 are the generally accepted rule-of-mixtures results. The Halpin-Tsai equations are equally applicable to fiber, ribbon, or particulate composites. For example, Halpin and... 

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. 

The term r Vf in Equation (3.71) can be interpreted as a reduced fiber-volume fraction. The word reduced is used because q 1. Moreover, it is apparent from Equation (3.72) that r is affected by the constituent material properties as well as by the reinforcement geometry factor To further assist in gaining appreciation of the Halpin-Tsai equations, the basic equation. Equation (3.71), is plotted in Figure 3-39 as a function of qV,. Curves with intermediate values of can be quickly generated. Note that all curves approach infinity as qVf approaches one. Obviously, practical values of qV, are less than about. 6, but most curves are shown in Figure 3-39 for values up to about. 9. Such master curves for various vaiues of can be used in design of composite materiais. 

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

A.2.3 Composite Moduli Halpin-Tsai Equations. Derivations of estimates for the effective moduli (tensile E, bulk K, and shear G) of discontinuous-fiber-reinforced composite materials are extremely complex. The basic difficulty lies in the complex, and often undefined, internal geometry of the composite. The problem has been approached in a number of ways, but there are three widely recognized... 

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 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. 

The earliest works of trying to model different length scales of damage in composites were probably those of Halpin [235, 236] and Hahn and Tsai [237]. In these models, they tried to deal with polymer cracking, fiber breakage, and interface debonding between the fiber and polymer matrix, and delamination between ply layers. Each of these different failure modes was represented by a length scale failure criterion formulated within a continuum. As such, this was an early form of a hierarchical multiscale method. Later, Halpin and Kardos [238] described the relations of the Halpin-Tsai equations with that of self-consistent methods and the micromechanics of Hill [29],... 

The Halpin-Tsai model (64) yields, for aligned fibre composites and in conditions where the modulus of the fiber, E(, is much higher than that of the unfilled matrix (as in elastomeric composites) ... 

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. 

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... 

Figure 8. Dependence of yield stress on silicone content of BPF carbonate-silicone block polymers. Line is calculated from Halpin-Tsai equations for moduli of composite of rigid matrix containing soft spherical inclusions.

The Chow equations, which constitute a large set that is too long and complex to reproduce here, are sometimes more accurate. Both of these sets of general-purpose equations (Halpin-Tsai and Chow) are applicable to many types of multiphase systems including composites, blends, immiscible block copolymers, and semicrystalline polymers. Their application to such systems requires the morphology to be described adequately and reasonable values to be available as input parameters for the relevant material properties of the individual phases. 

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... 

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 ... 

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. 

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 ... 

The micromechanical models used for the comparison was Halpin-Tsai (H-T) [89] and Tandon-Weng (T-W) [90] model and the comparison was performed for 5 wt% CNT/PP. It was noted that the H-T model results to lower modulus compared to FEA because H-T equation does not account for maximum packing fraction and the arrangement of the reinforcement in the composite. A modified H-T model that account for this has been proposed in the literature [91], The effect of maximum packing fraction and the arrangement of the reinforcement within the composite become less significant at higher aspect ratios [92],... 

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... 

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