Lamina Halpin-Tsai equations

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

The variational energy principles of classical elasticity theory are used in Section 3.3.2 to determine upper and lower bounds on lamina moduli. However, that approach generally leads to bounds that might not be sufficiently close for practical use. In Section 3.3.3, all the principles of elasticity theory are invoked to determine the lamina moduli. Because of the resulting complexity of the problem, many advanced analytical techniques and numerical solution procedures are necessary to obtain solutions. However, the assumptions made in such analyses regarding the interaction between the fibers and the matrix are not entirely realistic. An interesting approach to more realistic fiber-matrix interaction, the contiguity approach, is examined in Section 3.3.4. The widely used Halpin-Tsai equations are displayed and discussed in Section 3.3.5. 

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 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. 

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See also in sourсe #XX -- [ ]

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