Fiber-reinforced composite with cubic symmetry constituents Oscar C. Valdiviezo-Mijangos a , Federico J. Sabina b, * , Julia ´n Bravo-Castillero c,1 , Reinaldo Rodrı ´guez-Ramos c,d,1 , Rau ´l Guinovart-Dı ´az c,1 a Posgrado en Ciencias de la Tierra, Instituto de Geofı ´sica, UNAM, Ciudad Universitaria, 04510 Me ´xico, D.F. Mexico b Instituto de Investigaciones en Matema ´ticas Aplicadas y en Sistemas, Universidad Nacional Auto ´noma de Me ´xico, Apartado Postal 20-726 c/o Delegacione de Alvaro Obregon, 01000 Me ´xico, D.F. Mexico c Universidad de la Habana, Facultad de Matema ´tica y Computacio ´n, San La ´zaro y L, Vedado, Havana 4, CP-10400, Cuba d Instituto de Ingenierı ´a, Universidad Nacional Auto ´noma de Me ´xico, Apartado Postal 70-472, Delegacio ´n Coyoaca ´n, 04510 Me ´xico, D.F., Mexico Received 12 June 2001; received in revised form 14 December 2001; accepted 18 December 2001 Abstract An elastic material with unidirectional cylindrical fibers periodically distributed is analyzed. Each periodic cell is a binary homogeneous elastic medium with cubic symmetry constituents. Perfect bonding conditions at the interface are considered. Simple closed-form formulae are obtained for the overall properties using the asymptotic homogenization method. The analytical solution of the required resulting plane- and antiplane-strain local problems, which turns out to be only three, makes use of potential methods of a complex variable and properties of Weierstrass elliptic and related functions of periods (1,0) and (0,1). Two new exact relations are derived without solving any local problem and are valid for any shape of the interface compatible with the square symmetry. Some numerical examples are shown. D 2002 Published by Elsevier Science B.V. Keywords: Fiber-reinforced composite; Exact relations; Asymptotic homogenization method; Cubic symmetry; Tetragonal symmetry; Elliptic functions; Effective properties 1. Introduction An important technological question is the pre- diction of the material properties of binary phases based upon the knowledge of the properties of the constituents and their volumetric fractions. Quite a large number of important examples of reinforced materials and other characteristics are presented in Ref. [1]. Recently, the problem of a fiber-reinforced composite under a square periodic arrangement for transversely isotropic phases is studied in Ref. [2]. It produced closed-form formulae for the effective properties. The same technique, viz., the asymptotic homogenization method, used by them can be uti- lized to produce similar closed-form formulae when the constituents have cubic symmetry. More referen- ces to relevant fiber-reinforced papers and the math- ematical technique can be found in Ref. [2]. General formulae presented in Ref. [2] were made use; references to equations and figures in Ref. [2] are given the prefix I. 0167-577X/02/$ - see front matter D 2002 Published by Elsevier Science B.V. PII:S0167-577X(02)00479-2 * Corresponding author. Tel.: +52-56-22-35-63; fax: +52-56- 22-35-64. E-mail addresses: fjs@uxmym1.iimas.unam.mx (F.J. Sabina), jbravo@matcom.uh.cu (J. Bravo-Castillero), reinaldo@matcom.uh.cu (R. Rodrı ´guez-Ramos), guino@matcom.uh.cu (R. Guinovart-Dı ´az). 1 Tel.: +537-795-771; fax: +537-333-373. www.elsevier.com/locate/matlet October 2002 Materials Letters 56 (2002) 339 – 343