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Virtual Power and Pseudobalance Equations for Generalized Continua

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Continuous Media with Microstructure 2

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Abstract

In this paper the balance equations of linear and angular momentum are deduced from some regularity properties of the system of contact actions and from the law of action and reaction. This approach provides a simple and unifying formulation of the theories of non polar and polar continua. It also allows for a direct deduction of the classical plate and beam theories as special Cosserat continua, obtained by dimensional reduction induced by appropriate geometrical constraints.

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Notes

  1. 1.

    This assumption corresponds to Newton’s law of action and reaction.

  2. 2.

    That is, if \(\varPi \) and \(\varPi '\) are disjoint sets with a portion S of boundary in common and if \(Q(S)=-Q(-S)\) is the contact action interchanged across S, the contact action on \( \partial (\varPi \cup \varPi ')\) is the sum of the contact actions on \(\partial \varPi \) and on \(\partial \varPi '\).

  3. 3.

    The minus sign on the right is just matter of convenience.

  4. 4.

    The dependence of s on the normal was conjectured by Cauchy, and was currently called the Cauchy postulate. Only in 1959 Noll proved that this conjecture is true, under the assumption that the internal actions have a local character [11]. Since then, the Cauchy postulate has become the Noll theorem.

  5. 5.

    This is the tetrahedron theorem of Cauchy.

  6. 6.

    Alternatively, one can take as primitives the concept of virtual velocity and the existence of two types of actions, distance and contact.

  7. 7.

    In fact, on this assumption is based of the “method of virtual power” developed by Germain [7, 8] and others.

  8. 8.

    Here and in the following, repeated indices are summed.

  9. 9.

    This case includes the micromorphic continua [6] and, in particular, the micropolar continua, also called Cosserat continua.

  10. 10.

    We emphasize that (25) is a consequence of the pseudobalance equation (13) and not a new balance equation. In the literature, it is named balance of micromomentum, microforce balance, equilibrium equation for the macrostress tensor, and is presented, at least tacitly, as a new axiom of mechanics.

  11. 11.

    Ericksen and Truesdell [5], Mindlin [10] and Eringen [6].

  12. 12.

    For reasons of brevity, from here on most of the statements are given without comments and proofs. More detailed treatments can be found in the paper [2] and in the forthcoming lecture notes [4]. For plate and beam theories, see [3].

  13. 13.

    This constraint corresponds to the Cauchy-Born hypothesis, according to which the directors follow the macroscopic deformation.

  14. 14.

    The presence of a microstructure which does not appear explicitly in the expression of the power characterizes this continuum as a continuum with latent microstructure [1].

References

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Universitá di Ferrara, 44100, Ferrara, Italy

    Gianpietro Del Piero

  2. International Research Center M&MoCS, 04012, Cisterna di Latina, Italy

    Gianpietro Del Piero

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  1. Gianpietro Del Piero
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Correspondence to Gianpietro Del Piero .

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Editors and Affiliations

  1. Faculty of Engineering, University of Duisburg-Essen, Essen, Germany

    Bettina Albers

  2. Poznan University of Technology, Faculty of Civil and Environmental Engin, Poznań, Poland

    Mieczysław Kuczma

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Del Piero, G. (2016). Virtual Power and Pseudobalance Equations for Generalized Continua. In: Albers, B., Kuczma, M. (eds) Continuous Media with Microstructure 2. Springer, Cham. https://doi.org/10.1007/978-3-319-28241-1_2

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  • DOI: https://doi.org/10.1007/978-3-319-28241-1_2

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