Halpin-Tsai rule of mixtures

Figure 4.4 Design of unidirectional composite laminae Halpin-Tsai rule of mixtures. Relationship of weight to volume fraction.

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 Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

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... 

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. 

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

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]. 

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 mechanical properties of molecular composites have also been artalyzed theoretically. A theory by Halpin and Tsai relates the composite modulus to the individual moduli of the compmrents (Halpin and Tsai 1973 Halpin and Kardos 1976). hr the limiting case where the reinforcing fiber aspect ratio approaches infinity, the composite modulus and tensile strength are predicted to follow a finear rule of mixtures. That is, the composite properties are a finear functimt of the fiber and the matrix properties and volume fraction. This ultimate rule of mixture reinforcement behavior was achieved for composites of PBT in arcHnatic, heterocycUc matrix polymers (Krause et al. 1986,1988 Hwang et al. 1983) and work was also dmte to achieve that same effect with thermoplastic matrix materials (WickcUffe 1986 Tsai and Arnold 1982). 

Preliminary results on the elastic and fracture properties of several types of microoonposites, including hybrids, have recently been obtained. In Figure 1 the modulus of Kevlar 149/epoxy microoomposites made of 1 single fibre, and 8 single fibres, is seen to follow quite well the Halpin-Tsai (or rule-of-mixtures, RoM) equation, even at such low volume fraction. In Figure 2 the strength is shown to increase linearly according to the RoM. 

Generally, the two simplest and most common models for the mechanical properties of fiber reinforced composites are the rule of mixtures and the Halpin-Tsai equations [71]. The computational methods for the investigation of CNTs and CNT-filled composites can be categorized into two classes continuum methods and atomistic methods [31]. 

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). 

Models of increasing sophistication have been developed to predict the elastic properties of composite materials from the properties of their constituent parts. These range from the simple rule-of-mixtures approach to the Halpin-Tsai and Mori-Tanaka analyses, where the geometry - essentially, the aspect ratio - of the reinforcing particles can be taken into account. This has the potential to model the effects of extreme aspect ratios that are seen in nanocomposites. Direct finite element simulation of the microstructure is an option that is becoming increasingly feasible at both the micro and nano levels. 

When >oo, the Halpin-Tsai model equations reach the upper bound, which is normally called Voigt rule of mixtures (ROM) [54] where fiber and matrix have the same uniform strain (i.e., isostrain approach) ... 

Conversely, when 0, the Halpin-Tsai model equations reach the lower bound under equal stress (ie., isostress approach), the so-called Reuss inverse rule of mixtures (IROM) [55] ... 

Micromechanical models have been widely used to estimate the mechanical and transport properties of composite materials. For nanocomposites, such analytical models are still preferred due to their predictive power, low computational cost, and reasonable accuracy for some simplified stmctures. Recenfly, these analytical models have been extended to estimate the mechanical and physical properties of nanocomposites. Among them, the rule of mixtures is the simplest and most intuitive approach to estimate approximately the properties of composite materials. The Halpin-Tsai model is a well-known analytical model for predicting the stiffness of unidirectional composites as a function of filler aspect ratio. The Mori-Tanaka model is based on the principles of the Eshelby s inclusion model for predicting the elastic stress field in and around the eflipsoidal filler in an infinite matrix. 

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