Free vibrations of randomly inhomogeneous plates

Consider a large collection of elastic rectangular plates with random inhomogeneities, but otherwise indistinguishable in any overall sense. An expression is obtained for the natural frequency, μ, of such plates, vibrating freely under simply supported boundary conditions, in the form μ = μ⁽⁰⁾ + ?μ⁽¹⁾ + ?²μ⁽²⁾ + … where μ⁽⁰⁾ is the natural frequency of a homogeneous comparison plate, ? is a small real parameter measuring the degree of inhomogeneity, and the coefficients μ⁽¹⁾, μ⁽²⁾, …, are given explicitly. By constructing geometrically a correlation function for a special type of composite plate, μ⁽¹⁾ is computed and hence, to first order in ?, the variance of μ. The paper concludes with a theorem linking the mean and variance of μ to the volume concentration and geometry of the inclusions in the plate.