MIXTURE THEORY FOR LONGITUDINAL WAVE-PROPAGATION IN UNIDIRECTIONAL COMPOSITES WITH CYLINDRICAL FIBERS OF ARBITRARY CROSS-SECTION .1. FORMULATION

A binary mixture theory is developed for propagation of longitudinal waves in a unidirectional, fibrous composite containing a two-dimensional periodic array of cylindrical fibers of arbitrary cross section. The case considered concerns a class of problems for which the waves propagate in the direction of the fiber axis. Model construction is based upon an asymptotic scheme wherein the signal wavelength is assumed to be large compared to composite micro-dimension; this premise is appropriate for many applications. The resulting theory, which retains information on the displacement profile in the microstructure, is closed in the sense that all the mixture properties that it contains can be determined once the geometry of the unit cell, shape of the fiber cross section and the properties of the constituents are known. In the theory formulated herein, calculations of the mixture properties require the solution of time-independent microstructure boundary value problems (MBVPs) defined over a unit cell. In this part of the paper, the MBVPs are derived, and the relationship between their solution and mixture properties is established. The description of a finite-element procedure to solve the MBVPs is deferred to the second part of the paper, which also contains an application of mixture theory to a study of dispersion of time-harmonic waves in composites with several microstructural geometries and combination of material properties. © 1979.