Numerical coupling between shakedown and periodic homogenization for heterogeneous elastic plastic media
Keywords:
elastic plastic shakedown, static approach, kinematic approach, periodic homogenization, numerical modeling, 3D unit cell, convex optimizationAbstract
This paper presents a direct method to numerically study the strength, in the sense of shakedown, of elastic perfectly plastic media with a periodic microstructure, submitted to variable loads. The macroscopic admissible strength domains are obtained by solving constrained nonlinear optimization problems on a three-dimensional unit cell. These problems represent the shakedown analysis problems. Static and kinematic approaches of shakedown are tested by applying the developed method to a layered material and to a periodically perforated sheet.
Downloads
References
[AUN 90] AUNAY S., Architecture de logiciel de modélisation et traitements distribués, Thèse,
Université Technologique de Compiègne, France, 1990.
[BOR 01] BORNERT M., BRETHEAU T., GILORMINI P., Homogénéisation en mécanique des
matériaux, vol. 1, Hermes Sciences Europe Ltd., Paris, 2001.
[BOU 98] BOURGEOIS S., DEBORDES O., PATOU P., “Homogénéisation et plasticité des
plaques minces”, Revue Européenne des éléments finis, vol. 7, n1-2-3, 1998, p. 39-54.
[CAR 99] CARVELLI V., “Shakedown analysis of periodic heterogeneous materials by a kinematic
approach”, Strojnicky Casopis - Mechanical Engineering materials, vol. 50, n4,
, p. 229-240.
[CON 92] CONN A. R., GOULD N. I. M., TOINT P. L., LANCELOT: A Fortran Package for
Large-Scale Nonlinear Optimization, Springer-Verlag, Berlin, 1992.
[DEB 76] DEBORDES O., “Dualité des théorèmes statique et cinématique dans la théorie de
l’adaptation des milieux continus élastoplastiques”, C.R. Acad. Sci., vol. 282, 1976, p. 535-
[DEB 85] DEBORDES O., LICHT C., MARIGO J. J., MIALON P., MICHEL J. C., “Calcul
des charges limites de structures fortement hétérogènes”, Actes du Troisième Colloque
Tendances Actuelles en Calcul de Structures, 1985, p. 55-71.
[DEB 86] DEBORDES O., “Homogenization Computations in the Elastic or Plastic Collapse
Range. Applications to Unidirectional Composites and Perforated Sheets”, 4th. Int. Symp.
Inn. Meth. in Eng., Atlanta, 1986, Springer-Verlag, p. 453-458.
[HAC 01] HACHEMI A., MAGOARIEC H., DEBORDES O., WEICHERT D., “Application of
shakedown theory to layered composites”, ECCM II - 2nd Eur. Conf. on Comp. Mech.,
Cracow, Poland, June 26-29, 2001, Proceedings on Cd-rom (9 pages).
[KOE 78] KOENIG J. A., KLEIBER M., “On a new method of shakedown analysis”, Bull.
Acad. Polon. Sci. Ser. Sci. Techn., vol. 26, 1978, Page 165.
[KOI 60] KOITER W. T., “Progress in solid mechanics”, chapter General theorems for elasticplastic
solids, p. 165–221, Sneddon, I.N., Hill, R., 1960.
[LEN 84] LENE F., Contribution à l’étude des matériaux composites et de leurs endommagements,
Thèse, Université Paris VI, France, 1984.
[MAG 02] MAGOARIEC H., BOURGEOIS S., DEBORDES O., HACHEMI A., WEICHERT D.,
“A numerical approach of shakedown analysis coupled with periodic homogenization for
D elastic plastic media”, WCCM V - Fifth World Congress on Computational Mechanics,
Vienna, Autria, July 7-12 2002, Proceedings on Cd-rom (10 pages).
[MAG 04] MAGOARIEC H., BOURGEOIS S., DEBORDES O., “Elastic plastic shakedown
of 3D periodic heterogeneous media: a direct numerical approach”, Int. J. Plasticity,
vol. 20/8-9, 2004, p. 1655-1675.
[MEL 36] MELAN E., “Theorie statisch unbestimmter Systeme aus ideal-plastischem
Baustoff”, Sitber. Akad. Wiss. Wien, vol. Abt. IIa 145, 1936, p. 195-218.
[PIE 75] PIERRE D. A., LOWE M. J., Mathematical Programming Via Augmented Lagrangien,
An introduction with Computer Programs, Addison-Wesley Publ. Comp., 1975.
[SUQ 83] SUQUET P., “Analyse limite et homogénéisation”, C.R. Acad. Sci., Série II,
vol. 296, 1983, p. 1355-1358.
