[Représentation irréductible des distributions surfaciques et transformation de Piola des charges externes soutenables par un continuum de troisième gradient]
Dans cet article, on trouve les transformations de Piola qui relient les charges externes Eulériennes et Lagrangiennes que les milieux continuous de troisième gradient peuvent soutenir. Comme l’ont montré Gabrio Piola et Paul Germain, le schéma de postulation le plus efficace en mécanique est basé sur le principe des travaux virtuels et, par conséquent, la mécanique des continuums doit être mathématiquement fondée sur la théorie des distributions. En utilisant le principe des travaux virtuels, l’ensemble des charges externes admissibles soutenables par les continuums de troisième gradient comprend : i) la densité de force volumique, ii) la densité de surface de la force de contact, iii) la densité de surface de la double force de contact, iv) la densité de surface de la triple force de contact, v) la densité linéaire des forces de contact de bord, vi) la densité linéaire des doubles forces de contact de bord et vii) les forces de contact concentrées sur les points de coin. Suivant la nomenclature introduite par Paul Germain, les forces sont duales en travail virtuel des déplacements virtuels, les forces doubles de surface et de ligne sont duales des dérivées des déplacements virtuels dans la ou les directions normales des surfaces et des bords constituant la frontière du continuum et les forces triples de surface sont duales des dérivées secondes normales des déplacements virtuels. Les forces de volume et de surface se transforment comme dans les milieux continus de Cauchy à premier gradient. En plus, nous trouvons que a) le travail virtuel dépensé par la force triple de surface Eulérienne, lorsqu’elle est transformée en description Lagrangienne, doit être représentée comme le travail dépensé par tous les types de charges Lagrangiennes externes énumérées aux points i)-vii) ; b) la force double de surface Eulérienne se transforme en force double de surface, en force de contact de surface et en force de ligne de contact Lagrangiennes, c) la force double de ligne de contact Eulérienne se transforme en doubles forces de ligne de contact Lagrangiennes, en forces de ligne et en forces de coin concentrées, d) les forces de ligne de contact de bord et de coin Eulériennes se transforment uniquement en leur contrepartie Lagrangienne. Les formules de transformation de Piola déduites dans cet article dépendent des premier, deuxième et troisième gradients du placement. Les résultats présentés permettent la formulation de problèmes de conditions aux limites bien posés pour les milieux continus de troisième gradient dans la description Lagrangienne et sont pertinents en mécanique computationnelle. Compte tenu des formules de transformation de Piola obtenues, le concept de charges mortes doit être aussi modifié. Nous pensons avoir donné un exemple de la façon dont la « Mécanique à la française » , telle qu’elle a été développée à partir des idées de D’Alembert et de Lagrange, est toujours un outil de découverte fertile.
In this paper Piola transformations are found that relate the Eulerian and Lagrangian external loads which third gradient continua can sustain. As shown by Gabrio Piola and Paul Germain, the most effective postulation scheme in mechanics is based on the principle of virtual work and therefore continuum mechanics must be mathematically founded based on the theory of distributions and on differential geometry. Using the principle of virtual work, the set of admissible external loads sustainable by third gradient continua is seen to include: i) volume force density, ii) surface density of contact force, iii) surface density of contact double force, iv) surface density of contact triple force, v) line density of edge contact forces, vi) line density of contact edge double forces and vii) contact forces concentrated on wedge points. Following the nomenclature introduced by Paul Germain, forces are dual in virtual work of virtual displacements, surface and line double forces are dual of the derivatives of virtual displacements in the normal direction(s) of the surfaces and edges constituting the boundary of the continuum, and surface triple forces are dual of the second normal derivatives of virtual displacements. Volume and surface forces transform as in first gradient Cauchy continua. Moreover we find that: a) the virtual work expended by Eulerian surface triple force, when transformed into the Lagrangian description, must be represented as the work expended by all the kinds of external Lagrangian loads listed in i)-vii); b) Eulerian surface double force transforms into Lagrangian surface double force, surface contact force and edge contact line force; c) Eulerian edge contact line double force transforms into Lagrangian edge contact line double forces, edge line forces and point concentrated wedge forces; d) Eulerian edge and wedge contact line forces transforms into their Lagrangian counterpart only. The Piola transformation formulas deduced in this paper depend on the first, second and third gradients of placement. The presented results allow for the formulation of well-posed boundary condition problems for third gradient continua in the Lagrangian description, and are relevant in computational mechanics. In view of the obtained Piola transformation formulas, the concept of dead loads needs to be modified. We believe to have given an example of how the Mechanics in the French Style, as developed on the ideas by D’Alembert and Lagrange, is still a fertile tool of invention.
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Mot clés : Continuum de troisième gradient, Principe des Travaux Virtuels, Transformations de Piola, Description lagrangienne, Description Eulerienne, Théorie des distributions, Géométrie différentielle
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@article{CRMECA_2023__351_S3_A4_0,
author = {Francesco dell{\textquoteright}Isola and Roberto Fedele},
title = {Irreducible representation of surface distributions and {Piola} transformation of external loads sustainable by third gradient continua},
journal = {Comptes Rendus. M\'ecanique},
publisher = {Acad\'emie des sciences, Paris},
year = {2023},
doi = {10.5802/crmeca.157},
language = {en},
note = {Online first},
}
TY - JOUR AU - Francesco dell’Isola AU - Roberto Fedele TI - Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua JO - Comptes Rendus. Mécanique PY - 2023 PB - Académie des sciences, Paris N1 - Online first DO - 10.5802/crmeca.157 LA - en ID - CRMECA_2023__351_S3_A4_0 ER -
%0 Journal Article %A Francesco dell’Isola %A Roberto Fedele %T Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua %J Comptes Rendus. Mécanique %D 2023 %I Académie des sciences, Paris %Z Online first %R 10.5802/crmeca.157 %G en %F CRMECA_2023__351_S3_A4_0
Francesco dell’Isola; Roberto Fedele. Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua. Comptes Rendus. Mécanique, Online first (2023), pp. 1-30. doi : 10.5802/crmeca.157.
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