Wave propagation in semi-infinite bar with random imperfections of density and elasticity module

Mathematical modeling and properties of a linear longitudinal wave propagating in a slender bar with random imperfections of material density and Young modulus of elasticity is discussed. Fluctuation components of material properties are considered as continuous stochastic functions of the length coordinate. Two types of fluctuation and their influence on response properties have been investigated, in particular the delta correlated and a diffusion-type processes. Investigation itself is based on Markov processes and corresponding Fokker–Planck–Kolmogorov equation. The stochastic moments closure as a solution method has been used. Many effects due to the stochastic nature of the problem have been detected. Along the bar a drop of the mean value of the response with the simultaneous increase of the response variance have been observed. This effect does not represent any conventional damping, but a gradual drop of the deterministic and an increase of the stochastic components of the overall response. The rate of the response indeterminacy increases with the increase of the length coordinate. Increasing values of material imperfection variances and the rising excitation frequency can lead to a critical state when the length of the propagating wave is comparable with the correlation length of imperfections. This state will manifest itself as a radical change of the response character. The problem will pass beyond the boundaries of stochastic mechanics and lose its physical meaning. Similar effects can be observed in the FEM analysis, where there is also a certain permissible upper boundary of the excitation frequency corresponding with the size and type of the element used.