Exact micromechanical stress analysis of long fiber composites under uniform tensile loading

In this paper, based on a systematic procedure in the field of theory of elasticity, it is tried to establish an exact closed form solution for the problem of a long-fiber composite representative volume element (RVE) under uniform tension loading condition. To this end, the RVE of long fiber composites is considered. The RVE is composed of two concentric cylinders where the inner and outer ones are of fiber and matrix materials, respectively. Consequently, this study deals with an axially symmetric problem of the theory of elasticity. Using the Love function for axisymmetric problems, the Navier equilibrium equations for displacement components are converted to a single biharmonic equation in terms of the Love function. Then, a general expression for the separable solutions of biharmonic equation in cylindrical coordinates is derived. Next, according to the boundary conditions and physical interpretations, appropriate terms from separable solutions are chosen and a solution is suggested. In the final step, the unknown constants in the suggested solution are computed by using the boundary conditions. The obtained exact solution satisfies the entire field equations of theory of elasticity. Using this solution, the exact distribution of stress and displacement field components of the RVE is determined and related numerical results are presented. Based on the exact solution, an expression for the equivalent longitudinal modulus of composite is obtained and compared with the well-known rule of mixture formula. This comparison reveals that the rule of mixture can be a good engineering approximation for the equivalent longitudinal modulus.