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A self-adjoint variational principle for anisotropic viscoelastic Biot’s equations
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Marcus Wallenberg Laboratory MWL.
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, MWL Structural and vibroacoustics.ORCID iD: 0000-0002-9031-3662
2013 (English)In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 63, 71-83 p.Article in journal (Refereed) Published
Abstract [en]

A variational principle for anisotropic viscoelastic Biot’s equations of motion is presented. It is based upon an extended Hamilton’s principle, also valid for dissipative systems. Using this principle, a functional analogous to the Lagrangian is defined, starting from Biot’s variational formulation based on frame and fluid displacements. Then, a mixed displacement–pressure formulation is presented, which reduces the number of variables of response from six to four. The corresponding functional analogous to the Lagrangian is derived making full use of variational calculus. The derived functionals are self-adjoint and stationary for true motion.

Place, publisher, year, edition, pages
2013. Vol. 63, 71-83 p.
Keyword [en]
Porous materials, Biot’s equations, Variational principles, Dissipative systems
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-118435DOI: 10.1016/j.ijengsci.2012.06.019ISI: 000315363000008OAI: oai:DiVA.org:kth-118435DiVA: diva2:606069
Funder
Swedish Research Council, 621-2005-5754EU, European Research Council, 218508
Note

QC 20130218

Available from: 2013-02-18 Created: 2013-02-18 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Statistical energy analysis and variational principles for the prediction of sound transmission in multilayered structures
Open this publication in new window or tab >>Statistical energy analysis and variational principles for the prediction of sound transmission in multilayered structures
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Multilayered structures have many application in industry and society: they have peculiar properties and serve a variety of purposes, like structural support, thermal insulation, vibrational and acoustic isolation. This thesis concerns the prediction of sound transmission in multilayered structures. Two problems are herein investigated: the transmission of energy through structures and the transmission of energy along structures. The focus of the analysis is on the mid to high frequency range. To predict sound transmission in these structures, statistical energy analysis (SEA) is used.SEA models are devised for the prediction of the sound reduction index for two kinds of multilayered structures, double-walls used in buildings and trim-panels in vehicles; the double-walls comprise an air cavity in between flat plasterboard or glass plates, whereas the trim-panels a porous layer in between curved aluminium and rubber layers. The SEA models are based upon the wave-types carrying energy. The novelty in these SEAs is an element describing the waves in the air cavity, or in the porous layer, fully coupled to the mass-impeded external layers. Compared to measurements, the proposed SEA performs well: for double-walls, it performs better than previous models; for trim-panels, it is an original result. The parameters of the new SEA element, such as modal density, are derived from the coupling equations describing the fully coupled waves. For double-walls, these equations are derived via Newton's laws. For trim-panels, a variational approach based upon a modified Hamilton's principle valid for non-conservative systems is preferred, because it is a powerful machinery for deriving equations of motion and coupling conditions of a medium as complex as the porous layer. The modified Hamilton's principle for non-conservative systems is based upon a self-adjoint functional analogous to the Lagrangian, inspired by Morse and Feshbach's construction. A self-adjoint variational principle for Biot's equations in the displacement formulation is devised. An equivalent mixed formulation is obtained changing the coordinates of the displacement formulation via Lagrange multipliers. From this mixed formulation, the Lagrangian for a porous material with a limp frame is derived, which yields the continuity of the total displacement of the porous layer. Lagrange multipliers help to obtain the correct coupling functionals between a porous material and a solid. The Lagrange multipliers introducing the continuity of the frame and the solid displacements equal the traction of the in-vacuo frame, thus disappearing if the latter is limp. Measurements to gather material parameters for a Biot model of the porous layer have been conducted.The effects of spatial energy decay in the transmission along structures predicted by SEA is studied: a major effect is the increased relevance of indirect coupling loss factors between SEA elements. This may jeopardize the usefulness of SEA at higher frequencies.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. xi, 41 p.
Series
Trita-AVE, ISSN 1651-7660 ; 2013:04
Keyword
statistical energy analysis, porous materials, biot theory, variational principles, hamilton principle, double walls, multilayered structures
National Category
Fluid Mechanics and Acoustics
Research subject
Järnvägsgruppen - Ljud och vibrationer
Identifiers
urn:nbn:se:kth:diva-118427 (URN)978-91-7501-648-1 (ISBN)
Public defence
2013-03-05, F3, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20130218

Available from: 2013-02-18 Created: 2013-02-18 Last updated: 2013-02-18Bibliographically approved

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Finnveden, Svante

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