Halpin-Tsai model

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... 

Elanthikkal et al. investigated the effect of CNCs (from banana waste fibers) (0-10% w/w) on the morphological, thermal, and mechanical properties of poly(ethylene-co-vinyl acetate) (PEVA)/cellulose composites. The produced composites showed superior thermal and mechanical properties as compared to that of the EVA copolymer alone. In this study, three different theoretical models (Halpin-Tsai model, the Kemer model and the Nicolais-Narkis model) have been employed to compare the results with observation of tensile data from mechanical tests. Here, experimental results showed better agreement with the prediction given by the use of the Halpin-Tsai model, assuming that there is perfect wetting of filler by the polymer matrix [217]. 

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. 

Recently, some models (e.g., Halpin-Tsai, Mori- Tanaka, lattice spring model, and FEM) have been applied to estimate the thermo-mechanical properties [247, 248], Young s modulus[249], and reinforcement efficiency [247] of PNCs and the dependence of the materials modulus on the individual filler parameters (e.g., aspect ratio, shape, orientation, clustering) and on the modulus ratio of filler to polymer matrix. 

Compared to other models (e.g., Voigt-Reuss, Halpin-Tsai, modified mixture law, and Cox), the dilute suspension of clusters model promulgated by Villoria and Miravete [255] could estimate the influence of the dispersion of nanofillers in nanocomposite Young s modulus with much improved theoretical-experimental correlation. 

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) ... 

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... 

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 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... 

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. 

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. 

Since the assumption of uniformity in continuum mechanics may not hold at the microscale level, micromechanics methods are used to express the continuum quantities associated with an infinitesimal material element in terms of structure and properties of the micro constituents. Thus, a central theme of micromechanics models is the development of a representative volume element (RVE) to statistically represent the local continuum properties. The RVE is constracted to ensure that the length scale is consistent with the smallest constituent that has a first-order effect on the macroscopic behavior. The RVE is then used in a repeating or periodic nature in the full-scale model. The micromechanics method can account for interfaces between constituents, discontinuities, and coupled mechanical and non-mechanical properties. Their purpose is to review the micromechanics methods used for polymer nanocomposites. Thus, we only discuss here some important concepts of micromechanics as well as the Halpin-Tsai model and Mori-Tanaka model. 

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. 

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... 

An additional advantage of using a fiber-like reinforcement over a platelet-like reinforcement is its higher reinforcing efficiency in case of unidirectionally aligned systems [8], as can be demonstrated by micromechanical models like Halpin-Tsai [9]. A particle is said to reinforce efficiently a polymeric matrix if the increase in Young s modulus is close to the theoretical limit given by the rule of mixtures [10]. 

More refined models of macroscale composite stiffness have been developed (such as the Halpin-Tsai equations [91]) that take into account some of the assumptions made above, e.g., fiber length, orientation, and inefficiencies in... 

The most used materials property model, particularly in engineering design applications, is the Halpin-Tsai model [28] or Halpin-Tsai equations as they are often termed. Although the model has Umitations with respect to its rigor and accuracy, its main advantage is the simple universal form of expressions in its formulation, and its applicabihty to a number of different material forms and their moduli. The general form of Halpin-Tsai equations is given by ... 

Concerning the modulus evaluatirai of the fillers is always problematic. The modulus has been evaluated. Different composites had been processed with increasing LCFo i content. The fillers modulus has been estimated by fitting a semiempiri-cal Halpin-Tsai model on the evolution of the composites Young s modulus as a function of fillers volume fraction. By extrapolation at 100% of fillers, we obtain the filler modulus which is estimated at 6.7 GPa. This value is coherent with wheat straw data given in the literature (Hornsby et al. 1997 KrtMibergs 2000). 

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