Skip to main content
SpringerLink
Log in

The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool

In memory of Paul Germain (1920–2009)

  • Review Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

In a period of a few decades, the formulation known as the principle of virtual power (PVP) has gained a prominent place among the most efficient tools in the thermomechanics of continua. Strongly marked by a “continental” (French-Italian) influence, it has successfully incorporated the basic invariances of modern continuum mechanics while capturing the spirit of twentieth-century analysis (generalized functions or distributions) in which it became synonymous of weak formulation. It proved to provide the surest and safest way to formulate complex theories of continua (so-called “generalized continuum mechanics”, theory of coupled fields, etc) and approximate or generalized theories of structural members and the associated natural boundary conditions while preparing the way for the full thermomechanical formulation, providing the best setting for the proof of various mathematical theorems, and paving the way for modern numerical methods. The present contribution, illustrated by many examples of varying complexity, emphasizes the role of Paul Germain (1920–2009) in this formulation. The author, himself an active contributor and a never tired propagandist of the method, has participated in these developments during four decades and presents here his witness but critical viewpoint, highlighting the difficult points and also the esthetically pleasing ones where necessary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mach, E.: The science of mechanics; A critical and historical account of its Development, Open Court, Chicago (1911; Paperback 1988) (Original German edition: F.A.Brockhaus, Leipzig, 1883). pp. 59–88, 420–432 in the English text

  2. Dugas, R.: A history of mechanics, Dover reprint, New York (1988; Original edition: Editions du Griffon, Neuchatel, Switzerland, 1955; French and English versions were published simultaneously)

  3. Timoshenko, S.P.: History of strength of materials, Dover, New York (1983; original edition, McGraw-Hill, New York, 1953)

  4. Budo, A.: Theoretische Mechanik, VEB Deutscher Verlag der Wissenschaften, Berlin (DDR), 4th German edition, (1967; original Hungarian edition, Budapest, 1953)

  5. Szabo I.: Geschichte der mechanischen Prinzipien, 3rd edn. Birkhäuser, Basel (1987)

    Book  MATH  Google Scholar 

  6. Crowe M.J.: Mechanics from Aristotle to Einstein. Green Lyon Press, Santa Fe (2007)

    MATH  Google Scholar 

  7. Maugin G.A.: The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech. 35/1, 1–70 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  8. Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus, Première partie : théorie du second gradient. J. de Mécanique (Paris) 12, 235–274 (1973)

    MathSciNet  MATH  Google Scholar 

  9. Germain P.: The method of virtual power in continuum mechanics-II: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Truesdell, C.A., Toupin, R.A.: The classical theory of fields. In: S. Flügge (ed) Handbuch der Physik, Bd.III/1, Springer, Berlin (1960)

  11. Germain P.: Mécanique des milieux continus. Masson Editeurs, Paris (1962)

    Google Scholar 

  12. Germain, P.: Contribution à l’étude des milieux micropolaires et micromorphiques. In: Omaggio a Carlo Ferrari, Levretto e Bella, Torino, pp. 273–297 (1974)

  13. Germain P.:: Duality and convection in continuum mechanics. In: Fichera, D. (eds) Trends in Applications of Pure Mathematics to Mechanics, pp. 107–128. Pitman, London (1976)

  14. Maugin G.A.: Material Inhomogeneities in Elasticity. Chapman and Hall, London (1993)

    MATH  Google Scholar 

  15. Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  16. Dell’Isola F., Seppecher P.: The relationship between edge contact forces, double forces and intersticial working allowed by the principle of virtual power. C.R. Acad. Sci. Paris II b 321, 303–308 (1995)

    MATH  Google Scholar 

  17. Noll W., Virga E.G.: On edge interactions and surface tension. Arch. Rat. Mech. Anal. 111, 1–31 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gurtin M.E., Mizel V.J., Williams W.O.: A note on Cauchy’s stress theorem. J. Math. Anal. Appl. 22, 398–401 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mindlin R.D., Eshel N.N.: On the first strain gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    Article  MATH  Google Scholar 

  20. Mindlin R.D., Tiersten H.F.: Effects of couple stresses in linear elasticity. Arch. Rat. Mech. Anal. 11, 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maugin G.A.: What do we understand by “generalized continuum mechanics”?. In: Maugin, G.A., Metrikine, A.V., Erofeyev, E.V. (eds) Mechanics of Generalized Continua: A Hundred Years After the Cosserats, pp. 3–13. Springer, New York (2010)

  22. Maugin, G.A.: Nonlocal theories or gradient-type theories. A matter of convenience? Arch. Mech. (Poland) 31, 15–26 (1979) [presented at Euromech Colloquium held in Warsaw, 1977]

  23. Le Roux, J.: Etude géométrique de la torsion et de la flexion, dans les déformations infinitésimales d’un milieu continu. Ann. Ecole Norm. Sup. 28, 523–579 (1911) (also, Recherches sur la géométrie des déformations finies, ibid, 30, 193–245, 1913)

  24. Casal P.: La théorie du second gradient et la capillarité. C.R. Acad. Sci. Paris 274A, 1075–1078 (1972)

    MathSciNet  Google Scholar 

  25. Cosserat, E. and F., Théorie des corps déformables. Hermann Editeurs, Paris (1909; Reprint, Editions Gabay, Paris, 2008)

  26. Eringen A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)

    MathSciNet  Google Scholar 

  27. Eringen A.C.: Microcontinuum Field Theories I: Foundations and Solids. Springer, New York (1999)

    Book  MATH  Google Scholar 

  28. Eringen A.C.: Theory of micropolar elasticity. In: Liebowitz, H. (ed.) Fracture: A Treatise, vol. II, pp. 621–729. Academic Press, New York (1968)

    Google Scholar 

  29. Maugin G.A.: Sur la dynamique des milieux déformables avec spin magnétique- Théorie classique. J. Mécanique (Paris) 13, 75–96 (1974)

    MathSciNet  MATH  Google Scholar 

  30. Maugin G.A.: A continuum theory of deformable ferrimagnetic bodies-I-field equations. J. Math. Phys. (USA) 17, 1727–1738 (1976)

    Article  ADS  MATH  Google Scholar 

  31. Collet B., Maugin G.A.: Couplage magnétoélastique de surface dans les milieux ferromagnétiques. C.R. Acad. Sci. Paris 280A, 1641–1644 (1975)

    MathSciNet  Google Scholar 

  32. Daher N., Maugin G.A.: The method of virtual power in continuum mechanics : application to media presenting singular surfaces and interfaces. Acta Mech. 60, 217–240 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Daher N., Maugin G.A.: Virtual power and thermodynamics for electromagnetic continua with interfaces. J. Math. Phys. (USA) 27, 3022–3035 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Daher N., Maugin G.A.: Deformable semiconductors with interfaces: basic equations. Int. J. Eng. Sci. 25, 1093–1129 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  35. Truesdell C.A.: Rational Thermodynamics. Springer, New York (1984)

    Book  MATH  Google Scholar 

  36. Quiligotti S., Maugin G.A., dell’Isola F.: An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mech. 160, 45–60 (2003)

    Article  MATH  Google Scholar 

  37. Kestin J.: Local equilibrium formalism applied to mechanics of solids. Int. J. Solids Struct. 29, 1827–1836 (1992)

    Article  MATH  Google Scholar 

  38. Maugin G.A.: The thermomechanics of nonlinear irreversible behaviours. World Scientific, New Jersey (1999)

    Google Scholar 

  39. Frémond M., Nedjar B.: Endommagement et principe des puissances virtuelles. C.R. Acad. Sci. Paris II-317, 857–864 (1993)

    Google Scholar 

  40. Maugin G.A.: On the thermomechanics of continuous media with diffusion and /or weak nonlocality. Arch Appl. Mech. 75, 723–738 (2006)

    Article  ADS  MATH  Google Scholar 

  41. Gold’enweizer A.L.: Methods for justifying and refining the theory of shells. Appl. Math. Mech. (P.M.M.) 32, 704–718 (1968)

    Article  Google Scholar 

  42. Ciarlet P.G., Destuynder P.: A justification of a nonlinear model in plate theory. Comput. Methods Appl. Mech. Eng. 17/18, 227–258 (1979)

    Article  ADS  Google Scholar 

  43. Destuynder, P.: On a justification of models of plates and shells by asymptotic methods (in French). State Doctoral Thesis, University of Paris 6, Paris (1980)

  44. Germain, P.: Four lectures on the Foundations of shell theory, 75 p., Lectures at the Laboratorio de Computaçäo Cientifica, LCC/CNP q , Rio de Janeiro, Brazil (July 1982) (unfortunately not published in any other form)

  45. Touratier M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991)

    Article  MATH  Google Scholar 

  46. Touratier M., Maugin G.A.: Heterogeneous elastic wave guides of rectangular cross section. J. Acoust. Soc. Am. 78, 1806–1825 (1985)

    Article  ADS  Google Scholar 

  47. Salençon, J.: Mécanique des milieux continus, Tome I: Concepts généraux, Editions de l’Ecole Polytechnique, Palaiseau (2001)

  48. Nayroles B.: Opérations algébriques en mécanique des structures. C.R. Acad. Sci. Paris 273A, 1075–1078 (1971)

    MathSciNet  Google Scholar 

  49. d’Alembert, J. Le Rond: Traité de dynamique, 1st edn, Paris (1743) [Reprinted by Gauthier-Villars Publishers, in two volumes, Paris, 1925; also Fac-simile Reprint by Editions Gabay, Paris, 1998]

  50. Maugin, G.A.: A relativistic version of the principle of virtual power [K.Kondo Anniversary volume]. Int. J. Eng. Sci. 19, 1719–1730 (1981)

    Google Scholar 

  51. Maugin, G.A., Paul Germain: (1920–2009), L’Archicube (Bulletin de l’Association des Anciens élèves de l’Ecole Normale Supérieure de Paris), 7bis, Special issue, pp. 126–129, Février 2010

  52. Maugin, G.A., Drouot, R., Sidoroff, F. (eds.): Continuum Thermomechanics : The Art and Science of Modelling Material Behaviour (Paul Germain’s Anniversary Volume), Kluwer Academic Publishers, Dordrecht, The Netherlands (2000; Contribution by Paul Germain: “My discovery of Mechanics”, pp. 1–24, English text by Eleni Maugin)

Download references

Author information

Authors and Affiliations

  1. Université Pierre et Marie Curie, Institut Jean le Rondd’Alembert, UMR CNRS 7190, Case 152, 4 place Jussieu, 75252, Paris Cedex 05, France

    Gérard A. Maugin

Authors
  1. Gérard A. Maugin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard A. Maugin.

Additional information

Communicated by Andreas Öchsner.

This contribution is not intended for providing a history of the principle of virtual power in pre-d’Alembertian times or of its development by applied mathematicians, mathematical physicists and mechanical engineers in the nineteenth century. For these, we refer the reader to historical and critical reviews such as those given in more general works by Mach [1], Dugas [2], Timoshenko [3], Budo [4], Szabo [5] and Crowe [6].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maugin, G.A. The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Continuum Mech. Thermodyn. 25, 127–146 (2013). https://doi.org/10.1007/s00161-011-0196-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-011-0196-7

Keywords

Search

Navigation