Energy flow relations from quadratic quantities in three-dimensional isotropic medium and exact formulation for one-dimensional waves

• Published in: Journal of sound and vibration 2006-12, Vol.298 (4), p.934-957
• Main Authors: Joly, N., Pascal, J.C.
• Format: Article
• Language: English
• Subjects: Acoustics; Conversion; Density; Displacement; Equivalence; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematical models; Physics; Solid mechanics; Static elasticity (thermoelasticity...); Structural acoustics and vibration; Structural and continuum mechanics; Three dimensional; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2006.06.043

From the basic equations of continuum mechanics in a three-dimensional (3D) isotropic and damped medium excited by harmonic body forces, exact expressions are obtained from quadratic variables for time-averaged energy quantities: kinetic- and strain-energy densities, structural intensity, structural intensity divergence and curl. These energy quantities are split into four components: longitudinal, shear and two mixed ones. Each component is governed by similar relations of different quadratic variables. For 1D wave fields, an exact formulation based on quadratic variables is derived. The fundamental solutions of this formulation are analyzed for unloaded, and for concentrated loaded systems. The energy-models reported in the literature consider only some components of these solutions. The energy density and structural intensity components obtained from the quadratic formulation and from the usual displacement formulation are equivalent; this is illustrated for the energy transfers modeled by the quadratic formulation, in comparison with the displacement formulation, for a one-dimensional, longitudinal- and shear-wave field with wave conversion at one end.