Homogenization of the heterogeneous beam dynamics: The influence of the random Young's modulus mixing law

This paper concerns the homogenization of the dynamic response of Euler Bernoulli's beam with random Young's modulus. Considering the eigenvalue problem, special attention is dedicated to the homogenization residuals (correctors) analysis, i.e. the difference between the random heterogeneous solution and the homogenized solution. Several correlation (mixing) laws of the Young's modulus are considered and a dimensionless characteristic scale length, based on the correlation length, is introduced. The effects of the mixing law on the residuals are analyzed using numerical approaches both for sampling the random Young’ modulus and for examining the beam eigenvalue problem. Two measurements are introduced to estimate the residuals between apparent and effective solution: the normalized difference of the Young's modulus and the normalized difference of the modes' shape. The effect of the mode's order is also highlighted with reference to forced vibrations.