A Theory of Elasticity with Microstructure for Fiber-Reinforced Composites

Abstract

In this Chapter the basic ideas of Chapter 1 are applied to derive a system of displacement equations of motion for a unidirectionally fiber-reinforced composite. The system of equations is derived in three stages. The first stage of the derivation involves certain assumptions and calculations within a representative cell of the actual fiber-reinforced composite. In particular it is assumed that the motion of a fiber and the neighboring matrix material can be described by linear expansions in a system of local coordinates. The kinematic variables that are introduced in the expansions are defined at the centerlines of the fibers only. On the basis of these expansions the strain energy and the kinetic energy in a representative cell are subsequently computed. The averages over the cell next yield energy densities which are defined at the centerline of the fiber. In the next stage of the derivation a transition is achieved from the system of discrete cells to a homogeneous continuum model in the manner discussed in Chapter 1. In the final stage Hamilton’s principle in conjunction with certain continuity relations and the use of Lagrangian multipliers yields a set of displacement equations of motion.