Collisions and fractures: a model in S ⁢ B ⁢ D

Elena Bonetti; Michel Frémond

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 1, page 47-57
  • ISSN: 1120-6330

Abstract

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We investigate collisions (assumed to be instantaneous) and fractures of three-dimensional solids. Equations of motion and constitutive laws provide a set of partial differential equations, whose corresponding variational problem may be solved in the space of special functions with bounded deformations ( S ⁢ B ⁢ D ), exploiting the direct method of calculus of variations.

How to cite

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Bonetti, Elena, and Frémond, Michel. "Collisions and fractures: a model in $SBD$." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.1 (2004): 47-57. <http://eudml.org/doc/252334>.

@article{Bonetti2004,
abstract = {We investigate collisions (assumed to be instantaneous) and fractures of three-dimensional solids. Equations of motion and constitutive laws provide a set of partial differential equations, whose corresponding variational problem may be solved in the space of special functions with bounded deformations ($SBD$), exploiting the direct method of calculus of variations.},
author = {Bonetti, Elena, Frémond, Michel},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Collisions; Fractures; Velocities of bounded deformations; velocities of bounded deformations; variational formulation},
language = {eng},
month = {3},
number = {1},
pages = {47-57},
publisher = {Accademia Nazionale dei Lincei},
title = {Collisions and fractures: a model in $SBD$},
url = {http://eudml.org/doc/252334},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Bonetti, Elena
AU - Frémond, Michel
TI - Collisions and fractures: a model in $SBD$
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/3//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 1
SP - 47
EP - 57
AB - We investigate collisions (assumed to be instantaneous) and fractures of three-dimensional solids. Equations of motion and constitutive laws provide a set of partial differential equations, whose corresponding variational problem may be solved in the space of special functions with bounded deformations ($SBD$), exploiting the direct method of calculus of variations.
LA - eng
KW - Collisions; Fractures; Velocities of bounded deformations; velocities of bounded deformations; variational formulation
UR - http://eudml.org/doc/252334
ER -

References

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  1. ALBERTI, G. - MANTEGAZZA, C., A note on the theory of S ⁢ B ⁢ D functions. Boll. Un. Mat. Ital., v. 11, 1997, 375-382. Zbl0877.49001MR1459286
  2. AMBROSIO, L. - COSCIA, A. - DAL MASO, G., Fine properties of functions with bounded deformations. Arch. Rat. Mech. Anal., v. 139, 1997, 201-238. Zbl0890.49019MR1480240DOI10.1007/s002050050051
  3. AMBROSIO, L. - FUSCO, N. - PALLARA, D., Special functions of bounded variations and free discontinuity problems. Oxford University Press, Oxford2000. Zbl0957.49001MR1857292
  4. ATTOUCH, H. - BUTTAZZO, G. - MICHAILLE, G., Variational analysis in Sobolev and B ⁢ V spaces. Application to P ⁢ D ⁢ E and optimization. MPS/SIAM series in optimization, 2004. Zbl1095.49001
  5. BELLETTINI, G. - COSCIA, A., Una caratterizzazione dello spazio B ⁢ D ⁢ Ω per sezioni unidimensionali. Seminario di Analisi Matematica, Dip. Matematica Univ. Bologna, 1993. 
  6. BELLETTINI, G. - COSCIA, A. - DAL MASO, G., Compactness and lower semicontinuity in S ⁢ B ⁢ D ⁢ Ω . Math. Z., v. 228, 1998, 337-351. Zbl0914.46007MR1630504DOI10.1007/PL00004617
  7. BONETTI, E. - FRÉMOND, M., Collisions and fracture: a 1 - D theory. How to tear off a chandelier from the ceiling. 2003, submitted. Zbl1072.74004
  8. BONETTI, E. - FRÉMOND, M., Collisions and fractures. Proceedings of Nonlinear analysis and mechanics of continuous media, ICOMA-MECOM (Ho-Chi-Minh ville, 2003), to appear. Zblpre05237385
  9. BRAIDES, A., Approximation of free-discontinuity problems. Springer-Verlag, Berlin1998. Zbl0909.49001MR1651773
  10. FRÉMOND, M., Non-smooth thermomechanics. Springer-Verlag, Berlin2001. Zbl0990.80001
  11. FRÉMOND, M., Collisions and damage. Tendências em Matemática Aplicada e Computacional, v. 3, 2002, n. 1, SBMAC, Sao Carlos-SP, ISBN 85-86883-06-9, Brazil. 
  12. MATTHIES, H. - STRANG, G. - CHRISTIANSEN, E., The saddle point of a differential program. In R. GLOWINSKI - E. RODIN - O.C. ZIENKIEWICZ (eds.), Energy Methods in Finite Elements Analysis. Wiley and Sons, New York1979, 309-318. MR537013
  13. MOREAU, J.J., Sur les lois de frottement, de viscosité et de plasticité. C.R. Acad. Sci., Paris, v. 271, 1970, 608-611. 
  14. SUQUET, P.M., Existence et régularité des solutions des équations de la plasticité parfaite. C.R. Acad. Sc., Paris, série A, v. 286, 1978, 1201-1204. Zbl0378.35057MR501114
  15. TEMAM, R., Problèmes mathématiques en plasticité. Gauthier-Villars, Paris1983. Zbl0547.73026MR711964

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