Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves

The effective response of a periodic fibre reinforced material to SH wave propagation is studied using the method of asymptotic homogenization, complex variable theory and multipole expansions. The quasi-static limit of the effective properties is calculated when the wavelength is much longer than the defining lengthscale of the microstructure. The method developed allows the determination of the elastic properties in the most general (monoclinic) fibre reinforced media and the resulting expressions for the effective moduli are concise. The method is therefore both more general and provides neater closed form solutions than extant methods. Results are shown to be excellent even for very high volume fractions of fibres.