Theory of composites modeled by interpenetrating solid continua

Abstract

The differential equations and boundary conditions describing the behavior of composite materials are derived by means of a model of interpenetrating solid continua, which are permitted to displace with respect to each other to produce interaction between them. The equations for two and three constituent compos~tes are obtained, and it is shown that the equations obtained for three constituent composites can easily be generalized for n-constituent composites. A variational principle is also presented, which is shown to yield the aforementioned system of equations and boundary conditions in the absence of dissipation and heat flow. The equations and boundary conditions for non-continuous fiber reinforced composites are obtained from the general equations of a two constituent composite. The linear differential equations and boundary conditions in small field variables, for small fields superposed on large static biasing states, are obtained from these generally interactable nonlinear equations.

A suitable invariant form of the energy function is obtained for the isotropic and transversely isotropic composite materials, and the linear constitutive equations and the equations of motion are found using these expressions. Few static problems are considered for transversely isotropic composites, and for isotropic composites the static and dynamic potentials are obtained and the concentrated force problem is considered. The propagation of plane waves, surface waves and thickness vibration are considered, in both isotropic and transversely isotropic composites, and the existence of higher order modes and dispersive properties of surface waves are established. By means of a simplified model the numerical solutions to surface waves and plane waves are obtained and the graphs of frequency as a function of wavenumber are obtained for various percentages of reinforcement and different propagation directions relative to the fiber direction.