Tailoring the moduli of composites using hollow reinforcement

Effects of hollow reinforcement on homogenized moduli and stress fields in unidirectional composites are investigated using an elasticity-based homogenization theory for periodic materials with hexagonal symmetries. The theory employs Fourier series representations for hollow fiber and matrix displacement fields in the cylindrical coordinate system which satisfy exactly the equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle. This principle ensures rapid solution convergence in the presence of thick or very thin-walled hollow fibers with relatively few harmonic terms. The solution's analytical framework and stability facilitate parametric and optimization studies aimed at rapid identification of fiber wall-thickness impact on homogenized moduli and local fields in wide range of fiber volume fractions. This is illustrated for two material systems containing hollow glass fibers and alumina nanotubes as reinforcement, revealing new results of interest in the design of multifunctional porous materials. The results of parametric studies are made more precise upon incorporation of the locally-exact homogenization theory into the Particle Swarm Optimization algorithm, with the new computational capability employed to identify tube thickness as a function of the porosity volume fraction which minimizes the differences between target homogenized moduli and the corresponding matrix moduli. Analysis of a generalized Kirsch problem with an underlying microstructure based on targeted homogenized moduli around the cylindrical cavity demonstrates the homogenization theory's utility in tailoring the local stress field with the aim of enhancing material toughness.