Note on a mixed variational principle in finite elasticity

Gérard A. Maugin; Carmine Trimarco

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992)

  • Volume: 3, Issue: 1, page 69-74
  • ISSN: 1120-6330

Abstract

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In the present context the variation is performed keeping the deformed configuration fixed while a suitable material stress tensor S and the material coordinates are required to vary independently. The variational principle turns out to be equivalent to an equilibrium problem of placements and tractions prescribed at the boundary of a body of finite extent.

How to cite

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Maugin, Gérard A., and Trimarco, Carmine. "Note on a mixed variational principle in finite elasticity." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 3.1 (1992): 69-74. <http://eudml.org/doc/244257>.

@article{Maugin1992,
abstract = {In the present context the variation is performed keeping the deformed configuration fixed while a suitable material stress tensor \( \mathcal\{S\} \) and the material coordinates are required to vary independently. The variational principle turns out to be equivalent to an equilibrium problem of placements and tractions prescribed at the boundary of a body of finite extent.},
author = {Maugin, Gérard A., Trimarco, Carmine},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elastostatics; Conservation laws; Fracture; conservation laws; variation; deformed configuration; material stress tensor; equilibrium problem},
language = {eng},
month = {3},
number = {1},
pages = {69-74},
publisher = {Accademia Nazionale dei Lincei},
title = {Note on a mixed variational principle in finite elasticity},
url = {http://eudml.org/doc/244257},
volume = {3},
year = {1992},
}

TY - JOUR
AU - Maugin, Gérard A.
AU - Trimarco, Carmine
TI - Note on a mixed variational principle in finite elasticity
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1992/3//
PB - Accademia Nazionale dei Lincei
VL - 3
IS - 1
SP - 69
EP - 74
AB - In the present context the variation is performed keeping the deformed configuration fixed while a suitable material stress tensor \( \mathcal{S} \) and the material coordinates are required to vary independently. The variational principle turns out to be equivalent to an equilibrium problem of placements and tractions prescribed at the boundary of a body of finite extent.
LA - eng
KW - Elastostatics; Conservation laws; Fracture; conservation laws; variation; deformed configuration; material stress tensor; equilibrium problem
UR - http://eudml.org/doc/244257
ER -

References

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  1. OGDEN, R. W., Nonlinear Elastic Deformations. J. Wiley & S., Ellis Horwood Series, Chichester1984. Zbl0541.73044MR770388
  2. ESHELBY, J. D., The Elastic Energy Momentum Tensor. J. of Elasticity, 5, 1975, 321-335. Zbl0323.73011MR489190
  3. BATRA, R. C., The force on a Lattice Defect in an Elastic Body. J. of Elasticity, 17, 1981, 3-8. , Zbl0602.73018
  4. EPSTEIN, M. - MAUGIN, G. A., The Energy Momentum Tensor and Material Uniformity in Finite Elasticity. Acta Mechanica, 83, 1990, 127-133. Zbl0714.73029MR1076041DOI10.1007/BF01172974
  5. HERRMANN, A. G., Material Momentum Tensor and Path independent Integrals of Fracture Mechanics. Int. J. Solids and Structures, 18, 1982, 319-324. Zbl0479.73094
  6. ESHELBY, J. D., The force on a Disclination in a Liquid Crystal. Philosophical Magazine, 42A, 1980, 359-367. 
  7. RICE, J. R., A path independent Integral and the approximate Analysis of strain Concentrations by notches and cracks. J. Appl. Mechs., 35, 1968, 379-386. 
  8. BUI, D., Dualité entre les intégrales indépendantes du contour dans la théorie des solides fissurés. C. R. Acad. Sc. Paris, Série A, t. 276, 1973, 1425-1428. Zbl0292.73060MR317635
  9. REISSNER, E., On a Variational Theorem for Finite Elastic Deformations. J. Math. Physics, 32, 1953, 129-135. Zbl0051.16303MR58411
  10. MANACORDA, T., Sopra un Principio Variazionale di E. Reissner per la Statica dei Mezzi Continui. Boll. U.M.I., 9 (232A), 1954; 154-159. Zbl0057.40006MR65363
  11. TRUESDELL, C. A. - TOUPIN, R. A.. In: Encyclopedia of Physics, vol. III/1, Springer, Berlin1960. 
  12. HELLINGER, E., Die allgemeinen Ansätze der Mechanik der Kontinua. Enz. Math. Wiss., 4, 1914, 602-694. JFM45.1012.01
  13. GREEN, A. E. - RIVLIN, R. S., The Mechanics of Non-Linear Materials with Memory. Arch. Rat. Mech. An., 1, 1957, 1-21; 470. Zbl0079.17602MR95628

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