The Halpin-Tsai Equations

All of the preceding micromechanics results are represented by complicated equations and/or curves. The equations are usually somewhat a ykward to use. The curves are generally restricted to relatively srnall portion of the potential des regime. Thus, a need clearly exists for simple results to be used In the design of composite materials. 

Halpin and Tsai [3-17] developed an interpolation procedure that is an approximate representation of more complicated micromechanics results. The beauty of the procedure is twofoldr-First. it is simple, so it can readily be used in the design process. Second, it enables the generalization of usually limited, although more exact, micromechanics results. Moreover, the procedure is apparently gnltp accurate if the fiber-volume fraction (Vf) does not approach one.  

Theessence of the procedure is that Halpin and Tsai [3-17] showed that Hermans solution [3-14] generalizing Hill s self-consistent model [3-13] can be reduced to the approximate form 

M = composite material modulus E2, G.,2, orv23 M, = corresponding fiber modulus Ej, Gj, or Vj M = corresponding matrix modulus E, G, or v  

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 

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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... 

As in the case of the transverse tensile modulus, 2. the above analysis tends to underestimate the in-plane shear modulus. Therefore, once again it is common to resort to empirical relationships and the most popular is the Halpin-Tsai equation... 

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. 

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. 

Some physical insight into the Halpin-Tsai equations can be gained by examining their behavior for the ranges of values of and t. First, although it is not obvious, can range from 0 to . When = 0,... 

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. 

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 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... 

Observations By virtue of the Halpin-Tsai equation, there is an inherent strength... 

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 ... 

The Halpin-Tsai equation can thus be tailored into a much simpler form for PNCs comprising matrix-filler combinations having inordinately disparate sets of Young s moduli by addressing the shape-, size-, and aggregate-related factors a priori in order to adequately supplant those in (22). 

As seen, the Halpin-Tsai equation has a term a, raised to the power of one, to accommodate the filler aspect ratio. Since IAF intends to supplant the same, the new equation is expected to have a reduced dependence on the aspect ratio. Thus, the presence of aspect ratio in the equation needs to be diluted. Two constitutive equations are suggested the first one contains a correction term in the form of a shape reduction factor (a0 5) (24), while the second (25), is devoid of any extrashape related corrections Modified Halpin-Tsai I ... 

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],... 

FIG. 25.11 Graphical representation of the Halpin-Tsai equation for the representative case F/ M 100. 

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. 

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... 

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. 

These equations are suitable for single calculation and were employed previously for the single ply and angle ply properties. The short fiber composite properties are also given by the Halpin-Tsai equations where the moduli in the fiber orientation direction is a sensitive function of aspect ratio (1/d) at small aspect ratios and has the same properties of a continuous fiber composite at large but finite aspect ratios. 

Figures 4.4 to 4.10 give design charts, derived from the Halpin Tsai equations, for typical E-glass/polyester resin composite laminae, using the material properties given in Table 4.6. For the purposes of generating the graphs it has been assumed that the fibres and matrix are isotropic.

The constitutive equations for the former two techniques are given below, whilst the Halpin-Tsai equations are included in the EUROCOMP Design Code. 

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... 

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