Halpin-Tsai method

In the absence of experimental testing and manufacturers data, lamina stiffness may be calculated using constituent properties and the Halpin-Tsai method. 

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 Halpin-Tsai method relies on the selection of appropriate 4 values. Table 4.5 of the EUROCOMP Design Code gives suitable values for the various directional properties. The Halpin-Tsai method is an empirical one and the C have been obtained by comparing the results with experimental data. 

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],... 

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. 

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. 

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]. 

They Compare their results with Odegard, et al., [48] the micromechanic method was BEM Halpin-Tsai Eq. [68] with aligned fiber by perfect bonding. 

Clay is very anisotropic and small with a large surface area and modulus. To account for these characteristics of clay in a polymer, Fornes and Paul evaluated two analytical methods, those of Halpin-Tsai [6,7,8] and Mori-Tanaka [9], that were developed to calculate the Young s moduli for the types of morphology that can be associated with clay particles. Assuming that the polymer and the reinforcing dispersed phase are the only components in the polymer composite, the volume... 

There have been a number of attempts to produce more appropriate predictions of the Young s modulus of particulate reinforced composites, without having such widely-separated bounds on the predictions. They have also been reviewed recently by Young et a/. A number of years ago, Halpin and Tsai developed an approach based upon the self-consistent micromechanics method of Hill that enabled prediction the elastic behaviour of a composite for a variety of both fibre and particulate geome-... 

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