Structural theory of elastomeric composites based on fiber systems — Invariant description

Abstract

A three-dimensional theory of elastomeric composites with elastomeric matrices reinforced by systems of fibers is presented. The theory is based on a structural approach in which the matrix and the reinforcement of the composite are considered separately without reduction to a medium having continuously changing characteristics. The approach is based on the idea of a vector field of macroscopic displacements given by the positions of the axial lines of the fibers in the curret (deformed) configuration of the composite. The vector field determines the current macroscopic configuration, the tensor fields of the measures of macroscopic strain, and the field of the macroscopic stress tensor in the composite. The displacement, strain, and stress fields in the elastomeric matrix and the fibers of the reinforcing systems are regarded as derivatives of the field of macroscopic displacements of the medium. Relations are presented to describe the kinematics of the fibers in the current configuration of the composite, including the evolution of their orientation and the frequency of their planar and spatial distribution. Equations are obtained for the macroscopic motion of the fiber-reinforced matrix, and the dynamic variational principle that governs this motion is established. The elastic macroscopic potential of the matrix is found and related to the components of the macroscopic stress tensor. The procedure to be followed in constructing the constitutive equations of the composite is described. The proposed system of equations, relations, and algorithms is closed and can be used to solve problems involving the deformation of products made of fiber-reinforced elastomers and the creation of elastomeric composite products, based on fiber systems, that possess the requisite properties.