A two-scale asymptotic expansion method for periodic composite Euler beams

This paper develops a two-scale asymptotic expansion solution method for periodic composite Euler beams for the first time. In this method, a two-scale solution in the asymptotic expansion form is achieved for the fourth-order uniformly elliptic differential equation with periodic oscillating coefficients, and the first-order perturbed displacement is zero. The analytical solutions of unit cell problems are found with the help of four normalization conditions proposed in this work, and it follows that the homogenized elastic modulus of periodic composite beam is exactly the harmonic mean of material moduli. Another contribution of this work is to mathematically verify the convergence of the two-scale asymptotic expansion solution through the two-scale convergence method. In addition, the paper reveals the effects of higher-order expansion terms on the two-scale displacements and concludes that the second order or even higher-order perturbed displacements are necessary for obtaining accurate curvatures relevant to micro stresses. Finally, numerical experiments validate that the present method is rigorous in mathematics and physically acceptable.