The flip side of buckling

Continuum Mechanics and Thermodynamics

Volumn 19, Issue 3-4, 2007, Pages 211-243

  Grabovsky, Yury ^a   Truskinovsky, Lev ^b  

^a TEMPLE UNIVERSITY   (United States)

^b ECOLE POLYTECHNIQUE   (France)

Author keywords

Buckling; Flip instability; Infinitesimal stability

Indexed keywords

COMPRESSIVE LOADING; ELASTIC MATERIALS; SLENDER STRUCTURES; SLENDERNESS PARAMETERS;

BUCKLING; COMPRESSIVE STRENGTH; FAILURE ANALYSIS; FEATURE EXTRACTION; PARAMETER ESTIMATION;

STRUCTURAL INTEGRITY;

EID: 34547757049     PISSN: 09351175     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00161-007-0044-y     Document Type: Article

References (45)

1. Beatty M.F. (1971). Estimation of ultimate safe loads in elastic stability theory. J. Elast. 1(2): 95-120
2. Berdichevsky V.L. (1983). Variational Principles of Continuum Mechanics. (Russian) Nauka, Moscow
3. Biot M.A. (1965). Mechanics of Incremental Deformations. Wiley, New York
4. Budiansky B. (1974). Theory of buckling and post-buckling behavior of elastic structures. Adv. Appl. Mech. 14: 1-65
5. Capriz G. and Podio-Guidugli P. (1979). The role of Fredholm conditions in Signorini's perturbation method. Arch. Ration. Mech. Anal. 70(3): 261-288
6. Chillingworth D.R.J., Marsden J.E. and Wan Y.H. (1982). Symmetry and bifurcation in three-dimensional elasticity. I. Arch. Ration. Mech. Anal. 80(4): 295-331
7. Ciarlet, P.G.: Mathematical elasticity. Vol. II, volume 27 of Studies in Mathematics and its Applications. North-Holland, Amsterdam (1997). Theory of plates
8. Ciarlet P.G. and Destuynder P. (1979). A justification of nonlinear model in plate theory. Comput. Methods Appl. Mech. Eng. 17(18): 227-258
9. Davies P.J. (1991). Buckling and barrelling instabilities of non-linearly elastic columns. Q. Appl. Math. 49(3): 407-426
10. Del Piero G. (1980). Lower bounds for the critical loads of elastic bodies. J. Elast. 10(2): 135-143
11. Del Piero, G., Rizzoni, R.: Two sided estimates for local minimizers in compressible elasticity. Preprint
12. Euler, L.: Additamentum I. De curvis elasticis. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti. Series prima Opera mathematica, 24, 231-297, Lausanne, 1744. Edited by Constantin Carathéodory, Birkhaüser, Bäsel, Switzerland, 1967.
13. English translation by Oldfather, W. A., Ellis, C. A. and Brown, D. M. in Isis, 20(1), 72-160 (1933)
14. Föppl, A.: In Vorlesungen über technische Mechanik, vol. 5, pp. 132-139. Leipzig (1907)
15. Fosdick R.L. and Shield R.T. (1963). Small bending of a circular bar superposed on finite extension or compression. Arch. Ration. Mech. Anal. 12: 223-248
16. Friesecke G., James R.D. and Müller S. (2002). A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11): 1461-1506
17. Friesecke G., James R.D. and Müller S. (2006). A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180(2): 183-236
18. Fu Y.B. and Ogden R.W. (1999). Nonlinear stability analysis of pre-stressed elastic bodies. Contin. Mech. Thermodyn. 11(3): 141-172
19. Healey T.J. and Montes-Pizarro E.L. (2003). Global bifurcation in nonlinear elasticity with an application to barrelling states of cylindrical columns. J. Elast. 71(1-3): 33-58
20. Hill R. (1957). On uniqueness and stability in the theory of finite elastic strain. J. Mech. Phys. Solids 5: 229-241
21. Hill R. (1975). On the elasticity and stability of perfect crystals at finite strain. Math. Proc. Camb. Philos. Soc. 77: 225-240
22. Holden J.T. (1964). Estimation of critical loads in elastic stability theory. Arch. Ration. Mech. Anal. 17: 171-183
23. Horgan C.O. (1995). Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37(4): 491-511
24. Kirchhoff G. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40: 51-88
25. Koiter, W.T.: On the stability of elastic equilibrium. Ph.D. Thesis, Technische Hogeschool (Technological University of Delft), Delft, Holland (1945)
26. Lecumberry, M., Müller, S: Stability of slender bodies under compression and validity of the von Kármán theory. Preprint
27. Levinson M. (1968). Stability of a compressed neo-Hookean rectangular parallelepiped. J. Mech. Phys. Solids 16: 403-415
28. Love A.E.H. (1927). A treatise on the mathematical theory of elasticity. Dover, New York
29. Marsden J.E. and Hughes T.J.R. (1994). Mathematical foundations of elasticity. Dover, New York
30. Monneau R. (2003). Justification of the nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method. Arch. Ration. Mech. Anal. 169(1): 1-34
31. Mora M.G. and Müller S. (2004). A nonlinear model for inextensible rods as a low energy Γ-limit of three-dimensional nonlinear elasticity. Ann. Inst. H. Poincaré Anal. NonLinéaire 21(3): 271-293
32. Pantz O. (2001). Une justification partielle du modéle de plaque en flexion par Γ-convergence. C.R. Acad. Sci. Paris Ser. I, 332: 587-592
33. Pantz O. (2003). On the justification of the nonlinear inextensional plate model. Arch. Rational Mech. Anal. 167: 179-209
34. Pearson C.E. (1956). General theory of elastic stability. Q. Appl. Math. 14: 133-144
35. Podio-Guidugli P. (2003). A new quasilinear model for plate buckling. J. Elast. 71(1-3): 157-182
36. Ryzhak E.I. (1999). Korn's constant for a parallelepiped with a free face or pair of faces. Math. Mech. Solids 4(1): 35-55
37. Scherzinger W. and Triantafyllidis N. (1998). Asymptotic analysis of stability for prismatic solids under axial loads. J. Mech. Phys. Solids 46(6): 959-1007
38. Signorini, A.: Sulle deformazioni termoelastiche finite. In: Proceedings of the 3rd International Cong. Appl. Mech., vol. 2, pp. 80-89 (1930)
39. Simpson, H.C., Spector, S.J.: On bifurcation in finite elasticity: buckling of a rectangular rod. Preprint
40. Simpson H.C. and Spector S.J. (1987). On the positivity of the second variation in finite elasticity. Arch. Ration. Mech. Anal. 98(1): 1-30
41. Stoker J.J. (1968). Nonlinear Elasticity. Gordon and Breach Science Publishers, New York
42. Truesdell C. and Noll W. (2004). The non-linear field theories of mechanics. Springer, Berlin
43. Van Hove, L.: Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Nederl. Akad. Wetensch., Proc., 50, 18-23=Indagationes Math. 9, 3-8 (1947)
44. von Kármán, T.: Festigkeitsprobleme im Maschinenbau. In: Encyclopdie der Mathematischen Wissenschaften, vol. IV/4, pp. 311-385. Leipzig (1910)
45. Zubov L.M. and Rudev A.N. (1993). On the peculiarities of the loss of stability of a non-linearly elastic rectangular beam. Prikl. Mat. Mekh. 57(3): 65-83