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

In memory of Paul Germain (1920–2009)

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.

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Correspondence to Gérard A. Maugin.

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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].

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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

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Keywords

  • Variational formulation
  • Virtual power
  • Mechanics of deformable solids
  • Thermomechanics
  • Microstructure
  • Discontinuities
  • Mixtures
  • Structures