Energy flow in plates: Analysis of bending waves and exact quadratic formulation for one-dimensional fields Cédric Devaux * , Nicolas Joly, Jean-Claude Pascal Laboratoire d’Acoustique de l’Université du Maine, CNRS, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France article info Article history: Received 20 July 2007 Received in revised form 12 March 2008 Accepted 27 March 2008 Available online 7 April 2008 Keywords: Energy methods Intensity Energy densities Quadratic variables Boundary conditions abstract 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 den- sities, 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 sim- plifications then occurring. The fundamental solutions for quadratic variables in an unloaded portion of plate are analyzed; only some of their components, those with the lar- ger 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 for- mulation that needs more intensive computations than the displacement formulation for numerical applications. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Estimating the amplitude of the vibration fields in complex structures when there are too many modes to compute, or when the global response of the Statistical Energy Analysis removes resonances, has caused much attention and research during the last two decades. Using energy quantities to solve these problems was often proposed as it is generally consid- ered, notably for simple cases like rods and beams, that the relationship between input power and energy is simple and ro- bust [1]. Using the conservation of energy, which is based on the definition of the structural intensity expressed in terms of wave components and consists of basic simplifications, results in an equation similar to that governing the heat transfer [2,3]. This approach comes to energetic equations that give correct results in the least complicated configuration of one- dimensional systems [3–10]. However using those methods for solving industrial problems implies they can give results on usual elements like plates [11] and shells [12]. Unfortunately, as often shown [5,13,14], the thermal analogy approach never gave accurate results in the case of plates. This failure can be attributed to excessive simplifications [15]:  in two-dimensional models, wave interactions are not taken into account, and the vibration field is described as a plane wave superposition [14–20],  evanescent waves are neglected, though they can significantly contribute to vibration and energy transfer as far as bound- aries or singularities are considered [21], 0165-2125/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2008.03.005 * Corresponding author. Tel.: +33 0679339043. E-mail address: cedric.devaux@univ-lemans.fr (C. Devaux). Wave Motion 45 (2008) 895–907 Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wavemoti