Short fibers Halpin-Tsai equations

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. 

Maximum strain theory may be modified to predict the strength of randomly oriented short-fiber composites (22). llie Halpin-Tsai equations (14) have established relations for the stiffness of an oriented short-fiber ply from the matrix and fiber properties. These nations show that the longitudinal stiffness of an oriented short-fiber composite is a s itive function of the aspect ratio. 

It is well known that the elastic modulus of short fiber reinforced composite can be evaluated by means of the Halpin-Tsai equation ... 

The Halpin-Tsai equations (Halpin 1969 Halpin and Kaidos 1976) have long been popular for predicting the properties of short fiber composites. Tucker and Liang (1999) reviewed the application of several composite rtKxlels for fiber-reinforced corrqxfsites. They reported that the Halpin-Tsai theory offered reasonable predictions for composite modulus. 

Typical curves as calculated with Halpin-Tsai equations and parameters for short glass fiber-polypropylene composites parameters used in calculation short glass fibers = 77.0 GPa,... 

Calculating the Average Orientation from Measured Modulus and Halpin-Tsai Equations for PBT and PA/PAT Composites with Short Glass Fibers... 

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. ... 

A7.2 Halpin-Tsai Equations for Short Fibers Filled Systems Numerical Illustration... 

An interpolation procedure applied by Halpin and Tsai [17,18] has led to general expressions for the moduli of composites, as given by Eqs. (2.18) and (2.19). Note that for = 0, Eq. (2.18) reduces to that for the lower hmit, Eq. (2.8), and for = infinity, it becomes equal to the upper limit for continuous composites, Eq. (2.7). By empirical curve fitting, the value of = 2(l/d) has been shown to predict the tensile modulus of aligned short-fiber composites in the direction of the fibers, and the value of = 0.5 can be used for the transverse modulus. Other mathematical relationships for modulus calculations of composites with discontinuous fillers include the Takaya-nagi and the Mori-Tanaka equations [20]. 

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