Homogenization of dynamic behaviour of heterogeneous beams with random Young's modulus

The paper deals with the differences between effective and homogeneous solution in case of dynamic of continuous media with random micro-structure. In particular, these differences, called “residuals”, are considered for the dynamic linear problem of the Euler-Bernoulli's beam with random Young's modulus . The differential operator with random coefficient, that describes the eigenvalues problem, is taken into account. The convergence to the effective solution is analysed by introducing two measures: the normalized error between apparent and effective Young's moduli and between the modes shapes. The obtained results permit to highlight the dependence of the residuals from the micro-structure dimensionless length and the effect of the modes order; these aspects should be considered in the homogenization of dynamic behaviour of random heterogeneous composites. The assessment of the Rapresentative Volume Element (RVE) by convergence of the Statistical Volume Element (SVE) is also discussed.