Energy flow in plates: Analysis of bending waves and exact quadratic formulation for one-dimensional fields

Thickness-averaged energy densities and energy flow are analyzed in plates, accounting for the hysteretic damping of the material and the external exciting load distribution, without any more approximation than those of the kinematic models. Bending waves, which are governed by bi-harmonic equations, are studied. Complex kinetic- and strain-energy densities, structural intensity and, its divergence and curl are written using some quadratic variables. An exact quadratic formulation, which is based on energy densities and another quadratic variable, is obtained for one-dimensional bending fields, on account of the simplifications then occurring. The fundamental solutions for quadratic variables in an unloaded portion of plate are analyzed; only some of their components, those with the larger space scale, are often considered in energy models of the literature. Then the quadratic formulation for one-dimensional bending waves is applied to a semi-infinite plate excited by a concentrated force. This proves the equivalence of the quadratic and the displacement formulations for modeling energy fields in plates, and the limitation of the quadratic formulation that needs more intensive computations than the displacement formulation for numerical applications.