A multilevel computational strategy for handling microscopic and macroscopic instabilities

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 27-29, 2009, Pages 2099-2110

  Nezamabadi, S ^a   Yvonnet, J ^b   Zahrouni, H ^a   Potier Ferry, M ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

^b Université Paris Est Créteil   (France)

Author keywords

Asymptotic numerical method; Buckling; Instabilities; Multilevel finite element method; Nonlinear homogenization

Indexed keywords

ASYMPTOTIC DEVELOPMENT; ASYMPTOTIC NUMERICAL METHOD; COMPUTATIONAL STRATEGIES; CONSTITUTIVE RELATIONSHIPS; ELASTIC MATERIALS; FINITE ELEMENT PROBLEMS; FINITE-ELEMENT ANALYSIS; GEOMETRICAL NON LINEARITIES; HETEROGENEOUS MATERIALS; INSTABILITIES; LIMIT POINTS; LINEAR PROBLEMS; LOCAL INSTABILITIES; LOCALIZATION TENSORS; MACRO-SCALE; MICRO-SCALE; MICROSCOPIC SCALE; MULTILEVEL FINITE ELEMENT METHOD; NONLINEAR HOMOGENIZATION; NUMERICAL - TECHNIQUES; NUMERICAL EXAMPLES; SALIENT FEATURES;

ASYMPTOTIC ANALYSIS; BUCKLING; HOMOGENIZATION METHOD; NUMBER THEORY; NUMERICAL METHODS; REAL TIME SYSTEMS;

FINITE ELEMENT METHOD;

EID: 64749087909     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.02.026     Document Type: Article

References (59)

1. Abeyaratne R., and Triantafyllidis N. An investigation of localization in a porous elastic material using homogenization theory. J. Appl. Mech. 51 (1984) 481-486
2. Aggoune W., Zahrouni H., and Potier-Ferry M. Asymptotic numerical method for unilateral contact. Int. J. Numer. Methods Engrg. 68 (2006) 605-631
3. Boutyour E.H., Zahrouni H., Potier-Ferry M., and Boudi M. Bifurcation points and bifurcated branches by an asymptotic numerical method and padé approximants. Int. J. Numer. Methods Engrg. 60 (2004) 1987-2012
4. Cadou J.M., Cochelin B., and Potier-Ferry M. A numerical method for the computations of bifurcation points in fluid mechanics. Eur. J. Mech. B 25 (2006) 234-254
5. Cochelin B. A path following technique via an asymptotic numerical method. Comput. Struct. 53 (1994) 1181-1192
6. Cochelin B., Damil N., and Potier-Ferry M. Méthode asymptotique numTrique (2007), Hermès Science Publications
7. Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures vol. 1 (1991), Wiley, Chichester
8. deBotton G., Hariton I., and Socolsky E.A. Neo-Hookean fiber reinforced composites in finite elasticity. J. Mech. Phys. Solids 54 (2006) 533-559
9. J. Descamps, H.L. Cao, M. Potier-Ferry, An asymptotic numerical method to solve large strain viscoplastic problems, in: D.R.J. Owen, E. Oñate, E. Hinton (Eds.) Computational Plasticity, Fundamentals and Applications, C.I.M.N.E., Barcelona, vol. 1, 1997, pp. 393-400.
10. Drapier S., Grandidier J.C., and Potier-Ferry M. A structural approach of plastic microbuckling in long fibre composites: comparison with theoretical and experimental results. Int. J. Solids Struct. 38 (2001) 3877-3904
11. Erami K., Ohno N., and Okumura D. Long-wave in-plane buckling of elastoplastic square honeycombs. Int. J. Plast. 22 (2006) 1569-1585
12. 2 multiscale approach for modeling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials. Comput. Methods Appl. Mech. Engrg. 183 (2000) 309-330
13. 2) to describe the response of highly non-linear structures using generalized continua. Comput. Methods Appl. Mech. Engrg. 192 (2003) 3233-3244
14. Geymonat G., Muller S., and Triantafyllidis N. Homogenization of nonlinearly elastic materials: microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rational Mech. Anal. 122 (1993) 231-290
15. Ghosh S., Lee K., and Raghavan P. A multilevel computational model for multi-scale damage analysis in composite and porous media. Int. J. Solids Struct. 38 (2001) 2335-2385
16. Gong L., Kyriakides S., and Triantafyllidis N. On the stability of Kelvin cell foams under compressive loads. J. Mech. Phys. Solids 53 (2005) 771-794
17. Grandidier J.C., Ferron G., and Potier-Ferry M. Microbuckling and strength in long-fiber composites - theory and experiments. Int. J. Solids Struct. 29 (1992) 1753-1761
18. Hashin Z. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29 (1962) 143-150
19. R. Kouhia, Techniques for the analysis of non-linear systems with applications to solids and structural mechanics, Thesis, In the series 'Acta Polytechnica Scandinavia', Espoo, Finland, 1999.
20. Kouznetsova V.G., Geers M.G.D., and Brekelmans W.A.M. Multi-scale second order computational homogenization pf multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Engrg. 193 (2004) 5525-5550
21. Kyriakides S., Arseculeratne R., Perry E.J., and Liechti K.M. On the compressive failure of fiber-reinforced composites. Int. J. Solids Struct. 32 (1995) 689-738
22. Ladevèze P. Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation (1999), Springer-Verlag, Berlin
23. Ladevèze P., Loiseau O., and Dureisseix D. A micro-macro and parallel computational strategy for highly heterogeneous structures. Int. J. Numer. Methods Engrg. 52 (2001) 121-138
24. Lahmam H., Cadou J.M., Zahrouni H., Damil N., and Potier-Ferry M. High-order predictor-corrector algorithms. Int. J. Numer. Methods Engrg. 55 (2002) 685-704
25. Lee S.H., and Waas A.M. Compressive response and failure of fiber reinforced unidirectional composites. Int. J. Fract. 100 (1999) 275-306
26. Lopez-Pamies O., and Ponte Castañeda P. Second-order estimated for the macroscopic response and loss of ellipticity in porous rubbers at large deformations. J. Elast. 76 (2005) 247-287
27. Lopez-Pamies O., and Ponte Castañeda P. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I - theory. J. Mech. Phys. Solids 54 (2006) 807-830
28. Lopez-Pamies O., and Ponte Castañeda P. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II - application to cylindrical fibers. J. Mech. Phys. Solids 54 (2006) 831-863
29. Lopez-Pamies O., and Ponte Castañeda P. Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I - analysis. J. Mech. Phys. Solids 55 (2007) 1677-1701
30. Lopez-Pamies O., and Ponte Castañeda P. Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II - results. J. Mech. Phys. Solids 55 (2007) 1702-1728
31. Michel J.C., Lopez-Pamies O., Ponte Castanñeda P., and Triantafyllidis N. Microscopic and macroscopic instabilities in finitely strained porous elastomers. J. Mech. Phys. Solids 55 (2007) 900-938
32. Miehe C., Schroder J., and Becker M. Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interactions. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4971-5005
33. Miehe C. Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput. Methods Appl. Mech. Engrg. 192 (2003) 559-591
34. Monteiro E., Yvonnet J., and He Q.-C. Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Comput. Mater. Sci. 42 (2008) 704-712
35. Muller S. Homogenization of non-convex integral functional and cellular elastic materials. Arch. Ration. Mech. Anal. 99 (1987) 189-212
36. Nemat-Nasser S., and Hori M. Micromechanics: overall properties of heterogeneous materials. North-Holland Series, Applied Mathematics and Mechanics vol. 37 (1993), North-Holland, Amsterdam
37. Nestorović M., and Triantafyllidis N. Onset of failure in finitely strained layered composites subjected to combined normal and shear loading. J. Mech. Phys. Solids 52 (2004) 941-974
38. Ohno N., Okumura D., and Noguchi H. Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation. J. Mech. Phys. Solids 50 (2002) 1125-1153
39. Ohno N., Okumura D., and Niikawa T. Long-wave buckling of elastic square honeycombs subject to in-plane biaxial compression. Int. J. Mech. Sci. 46 (2004) 1697-1713
40. Okumura D., Ohno N., and Noguchi H. Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression. Int. J. Solids Struct. 39 (2002) 3487-3503
41. Okumura D., Ohno N., and Noguchi H. Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids. J. Mech. Phys. Solids 52 (2004) 641-666
42. Okumura D., Okada A., and Ohno N. Buckling behavior of Kelvin open-cell foams under [0 0 1], [0 1 1] and [1 1 1] compressive loads. Int. J. Solids Struct. 45 (2008) 3807-3820
43. Ponte Castañeda P., and Suquet P. Nonlinear composites. Advances in Applied Mechanics vol. 34 (1998), Academic Press, New York
44. Riks E. Some computational aspects of the stability analysis of nonlinear structures. Comput. Methods Appl. Mech. Engrg. 47 (1984) 219-259
45. Schraad M.W., and Triantafyllidis N. Scale effects in media with periodic and nearly periodic microstructures. II - failure mechanisms. J. Appl. Mech. 64 (1997) 763-771
46. Smit R., Brekelmans W., and Meijer H. Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput. Methods Appl. Mech. Engrg. 155 (1998) 181-192
47. Suquet P. Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palenzia E., and Zaoui A. (Eds). Homogenization Techniques for Composites Materials. Lecture Notes in Physics vol. 272 (1987), Springer, Berlin 193-278
48. Terada K., and Kikuchi N. A class of general algorithms for multi-scale analysis of heterogeneous media. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5427-5464
49. Triantafyllidis N., and Bardenhagen S.G. The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. J. Mech. Phys. Solids 44 (1996) 1891-1928
50. Triantafyllidis N., and Maker B.N. On the comparison between microscopic and macroscopic instability mechanisms in a class of fiber-reinforced composites. J. Appl. Mech. 52 (1985) 794-800
51. Triantafyllidis N., and Schraad M.W. Onset of failure in aluminium honeycombs under general in-plane loading. J. Mech. Phys. Solids 46 (1998) 1089-1124
52. Triantafyllidis N., Nestorović M.D., and Schraad M.W. Failure surfaces for finitely strained two-phase periodic solids under general in-plane loading. J. Appl. Mech. 73 (2006) 505-515
53. Vannucci P., Cochelin B., Damil N., and Potier-Ferry M. An asymptotic-numerical method to compute bifurcated branches. Int. J. Numer. Methods Engrg. 41 (1998) 1365-1389
54. Willis J.R. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics vol. 21 (1981), Academic Press, New York 1-78
55. Yvonnet J., and He Q.C. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J. Comput. Phys. 223 (2007) 341-368
56. Yvonnet J., Zahrouni H., and Potier-Ferry M. A model reduction method for the post-buckling analysis of cellular microstructures. Comput. Methods Appl. Mech. Engrg. 197 (2007) 265-280
57. Zahrouni H., Aggoune W., Brunelot J., and Potier-Ferry M. Asymptotic numerical method for strong nonlinearities. Rev. Eur. Elem. Finis 118 (2004) 13-97
58. Zahrouni H., Braikat B., Damil N., and Potier-Ferry M. Solving plasticity problems by a perturbation technique. Proc. Appl. Math. Mech. 5 (2005) 455-456
59. Zahrouni H., Cochelin B., and Potier-Ferry M. Computing finite rotations of shells by an asymptotic-numerical method. Comput. Methods Appl. Mech. Engrg. 175 (1999) 71-85