Micromechanics Halpin-Tsai equations

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. 

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 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 equations were based upon purely mechanics arguments from micromechanics. It was not until Talreja [14—16], Chang and Allen [239],... 

The micromechanical equations of Halpin and Kardos [8,9] (historically referred to as the Halpin-Tsai equations) and of Chow [5] are particularly useful and versatile. The Halpin-Tsai equations have the advantage of simplicity. See Equation 20.1 for the lamellar fiber-reinforced matrix model , one of several useful forms in which these equations have been expressed. Af... 

Tucker et al. [50] prepared an application review of different classes of micromechanical models. The authors remarked that Halpin-Tsai equations [46] are the most widely used, but the Mori-Tanaka type models [45] give the best results for large aspect ratio fillers. 

Expectedly quantitative results vary considerably from one fiber-rubber system to another, and other compounding ingredients may induce additional effects (positive or negative). The qualitative effects are completely in line with the expected role of short fibers, in agreement with micromechanic considerations (see Section 7.2). Certain authors have used well established micromechanic approaches, e.g., Voigt and Reuss averages (Equations 7.1 and 7.2), and Halpin-Tsai equations (Equation 7.5) to consider the effects on short natural fibers in rubber compounds. ... 

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 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],... 

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