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Variational Principles in Numerical Practice

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Encyclopedia of Continuum Mechanics

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Synonyms

Bone-resorbable graft interaction; Dissipation in concrete; PEM vibration damping

Definitions

Variational principles represent a general framework for determining the mechanical state of a system, by identifying its motion as a minimum of a pertinent functional. Moreover, finite element methods are naturally based on variational principles and provide a very powerful tool for numerically solving many mechanical as well as other multi-physics problems. The purpose of the present note is to illustrate some recent applications with special reference to biomechanics and dissipation in quasi-brittle materials and piezo-electromechanical structures, in order to confirm the validation and to highlight the bright prospects of this method.

Introduction

Theoretical elegance of the variational approach applied to mechanical problems has long been extensively and thoroughly formalized in fundamental scientific works (Landau and Lifshitz, 1976; Marsden et al., 2001; Berdichevsky, 2009;...

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References

  • Alessandroni S, dell’Isola F, Frezza F (2001) Optimal piezo-electro-mechanical coupling to control plate vibrations. Int J Appl Electromagn Mech 13(1–4):113–120

    Google Scholar 

  • Alessandroni S, dell’Isola F, Porfiri M (2002) A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators. Int J Solids Struct 39(20):5295–5324

    Google Scholar 

  • Alessandroni S, Andreaus U, dell’Isola F, Porfiri M (2004) Piezo-electromechanical (PEM) Kirchhoff–Love plates. Eur J Mech A Solids 23(4):689–702

    Google Scholar 

  • Alessandroni S, Andreaus U, dell’Isola F, Porfiri M (2005) A passive electric controller for multimodal vibrations of thin plates. Comput Struct 83(15):1236–1250

    Google Scholar 

  • Andreaus U, Giorgio I, Madeo A (2015) Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. Z Angew Math Phys 66(1):209–237. https://doi.org/10.1007/s00033-014-0403-z

    Article  MathSciNet  MATH  Google Scholar 

  • Batra RC, dell’Isola F, Vidoli S, Vigilante D (2005) Multimode vibration suppression with passive two-terminal distributed network incorporating piezoceramic transducers. Int J Solids Struct 42(11):3115–3132

    Google Scholar 

  • Berdichevsky V (2009) Variational principles of continuum mechanics: I. Fundamentals. Springer Science & Business Media, Berlin/Heidelberg

    MATH  Google Scholar 

  • Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33(4):1482–1498. https://doi.org/10.1063/1.1728759

    Article  MathSciNet  MATH  Google Scholar 

  • Bowland AG, Weyers RE, Charney FA, Dowling NE, Murray TM, Ramniceanu A (2012) Effect of vibration amplitude on concrete with damping additives. ACI Mater J 109(3):371–378

    Google Scholar 

  • Cazzani A, Malagù M, Turco E (2016a) Isogeometric analysis of plane-curved beams. Math Mech Solids 21(5):562–577. https://doi.org/10.1177/1081286514531265

    Article  MathSciNet  MATH  Google Scholar 

  • Cazzani A, Malagù M, Turco E, Stochino F (2016b) Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math Mech Solids 21:183–209. https://doi.org/10.1177/1081286515577043

    MathSciNet  MATH  Google Scholar 

  • Cowin SC (1999) Bone poroelasticity. J Biomech 32(3):217–238. https://doi.org/10.1016/S0021-9290(98)00161-4

  • dell’Isola F, Porfiri M, Vidoli S (2003a) Piezo-electromechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers. Comptes Rendus Mecanique 331(1):69–76

    Google Scholar 

  • dell’Isola F, Santini E, Vigilante D (2003b) Purely electrical damping of vibrations in arbitrary PEM plates: a mixed non-conforming FEM-Runge-Kutta time evolution analysis. Arch Appl Mech 73(1–2):26–48

    Google Scholar 

  • dell’Isola F, Andreaus U, Placidi L (2015) At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math Mech Solids 20(8):887–928. https://doi.org/10.1177/1081286513509811

  • Den Hartog JP (1947) Mechanical vibrations. McGraw-Hall Book Company, New York

    MATH  Google Scholar 

  • Eugster SR, dell’Isola F (2017) Exegesis of the introduction and sect. I from “Fundamentals of the Mechanics of Continua”** by E. Hellinger. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 97(4):477–506

    Google Scholar 

  • Giorgio I, Scerrato D (2017) Multi-scale concrete model with rate-dependent internal friction. Eur J Environ Civil Eng 21(7–8):821–839

    Article  Google Scholar 

  • Giorgio I, Culla A, Del Vescovo D (2009) Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch Appl Mech 79(9):859–879

    Article  MATH  Google Scholar 

  • Giorgio I, Galantucci L, Della Corte A, Del Vescovo D (2015) Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int J Appl Electromagn Mech 47(4):1051–1084

    Article  Google Scholar 

  • Giorgio I, Andreaus U, Scerrato D, dell’Isola F (2016) A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomech Model Mechanobiol 15(5):1325–1343. https://doi.org/10.1007/s10237-016-0765-6

  • Greco L, Cuomo M (2014) An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput Methods Appl Mech Eng 269:173–197. https://doi.org/10.1016/j.cma.2013.09.018

    Article  MATH  Google Scholar 

  • Greco L, Cuomo M (2016) An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput Methods Appl Mech Eng 298:325–349. https://doi.org/10.1016/j.cma.2015.06.014

    Article  Google Scholar 

  • Landau L, Lifshitz E (1976) Mechanics: volume 1 (course of theoretical physics), 3rd edn. Pergamon Press, Oxford

    Google Scholar 

  • Lekszycki T, dell’Isola F (2012) A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. Z Angew Math Mech 92(6):426–444. https://doi.org/10.1002/zamm.201100082

  • Marsden JE, Pekarsky S, Shkoller S, West M (2001) Variational methods, multisymplectic geometry and continuum mechanics. J Geom Phys 38(3):253–284

    Article  MathSciNet  MATH  Google Scholar 

  • Misra A (1998) Stabilization characteristics of clays using class C fly ash. Transp Res Rec: J Transp Res Board 1611:46–54

    Article  Google Scholar 

  • Oliveto G, Cuomo M (1988) Incremental analysis of plane frames with geometric and material nonlinearities. Eng Struct 10(1):2–12

    Article  Google Scholar 

  • Scerrato D, Giorgio I, Madeo A, Limam A, Darve F (2014) A simple non-linear model for internal friction in modified concrete. Int J Eng Sci 80:136–152

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. DISG, University of Rome La Sapienza, Rome, Italy

    Ugo Andreaus

  2. International Research Center M&MoCS, L’Aquila, Italy

    Ivan Giorgio

Authors
  1. Ugo Andreaus
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  2. Ivan Giorgio
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Corresponding author

Correspondence to Ugo Andreaus .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Dipartimento di Ingegneria, Strutturale e Geotecnica, Università di Roma "La Sapienza", Via Eudossiana, 18, 00184, Roma, Italia

    Francesco dell'Isola

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Andreaus, U., Giorgio, I. (2018). Variational Principles in Numerical Practice. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_175-1

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  • DOI: https://doi.org/10.1007/978-3-662-53605-6_175-1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

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