Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: a generalization of the Cauchy tetrahedron argument

  • Francesco dell’Isola
  • Pierre Seppecher
  • Angela Madeo
Part of the CISM Courses and Lectures book series (CISM, volume 535)


The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D’Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salençon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an effective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defined by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)–(1965), Toupin (1962), Mindlin (1964)–(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell’Isola and Seppecher (1995)–(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.


Contact Force Regularity Assumption Gradient Theory Gradient Continuum Virtual Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Authors and Affiliations

  • Francesco dell’Isola
    • 1
    • 4
  • Pierre Seppecher
    • 2
    • 4
  • Angela Madeo
    • 3
    • 4
  1. 1.Laboratorio Strutture e Materiali IntelligentiUniversità di Roma “La Sapienza” Dipartimento di Ingegneria Strutturale e GeotecnicaItaly
  2. 2.Institut de Mathématiques de Toulon U.F.R. des Sciences et TechniquesUniversité de Toulon et du VarToulonFrance
  3. 3.Institut National des Sciences Appliquées (INSA) GCU et LGCIEUniversité de LyonLyonFrance
  4. 4.International Research Centre on “Mathematics & Mechanics of Complex Systems” M& MOCSCisterna di LatinaItaly

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