Multi-scale stochastic method to perform a composite longitudinal bar

In this paper the solution of a composite periodic longitudinal bar is investigated in the framework of the homogenization and stochastic techniques. A two scale asymptotic homogenization technique is used to provide the equivalent mechanical characterization of the bar. Such approach isolates a macro and micro-scale problem. The macro-scale problem describes the dynamics of the bar, while the micro-scale problem supplies the equivalent Young modulus and density. The micro-scale is made of two different materials: the portion of each material in the micro-scale is known in a statistical sense and a probability density function (pdf) is assumed to take into account such uncertainty. Consequently, the macro-scale material properties are random and their probability density functions are calculated in closed form. The random properties of the material represent the coefficients of the governing equation of the bar. The pdf of the solution (eigensolution) is obtained analytically. A Monte-Carlo simulation provides an alternative solution compared to the provided one.