Stability of inflectional elasticae centered at vertices or inflection points
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 187-203
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Stability conditions for inflectional Euler's elasticae centered at vertices or inflection points are obtained. Theoretical results are compared with experimental data for elastic rods.
@article{TM_2010_271_a13,
     author = {Yu. L. Sachkov and S. V. Levyakov},
     title = {Stability of inflectional elasticae centered at vertices or inflection points},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {187--203},
     year = {2010},
     volume = {271},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_271_a13/}
}
TY  - JOUR
AU  - Yu. L. Sachkov
AU  - S. V. Levyakov
TI  - Stability of inflectional elasticae centered at vertices or inflection points
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2010
SP  - 187
EP  - 203
VL  - 271
UR  - http://geodesic.mathdoc.fr/item/TM_2010_271_a13/
LA  - ru
ID  - TM_2010_271_a13
ER  - 
%0 Journal Article
%A Yu. L. Sachkov
%A S. V. Levyakov
%T Stability of inflectional elasticae centered at vertices or inflection points
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2010
%P 187-203
%V 271
%U http://geodesic.mathdoc.fr/item/TM_2010_271_a13/
%G ru
%F TM_2010_271_a13
Yu. L. Sachkov; S. V. Levyakov. Stability of inflectional elasticae centered at vertices or inflection points. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 187-203. http://geodesic.mathdoc.fr/item/TM_2010_271_a13/

[1] Agrachev A.A., Sachkov Yu.L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2004 | Zbl

[2] Korobeinikov S.N., “Vtorichnaya poterya ustoichivosti szhatogo sharnirno opertogo sterzhnya”, Lavrentevskie chteniya po matematike, mekhanike i fizike, Tez. dokl. IV Mezhdunar. konf., In-t gidrodinamiki SO RAN, Novosibirsk, 1995, 104

[3] Krylov A.N., “O formakh ravnovesiya szhatykh stoek pri prodolnom izgibe”, Izbr. trudy, Izd-vo AN SSSR, L., 1958, 486–538

[4] Kuznetsov V.V., Levyakov S.V., “O vtorichnoi potere ustoichivosti eilerova sterzhnya”, Prikl. mekh. i tekhn. fiz., 40:6 (1999), 184–185

[5] Kuznetsov V.V., Levyakov S.V., “Elastika eilerova sterzhnya s zaschemlennymi kontsami”, Prikl. mekh. i tekhn. fiz., 41:3 (2000), 184–186 | MR | Zbl

[6] Lyav A., Matematicheskaya teoriya uprugosti, ONTI, M.; L., 1935

[7] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961

[8] Popov E.P., Teoriya i raschet gibkikh uprugikh sterzhnei, Nauka, M., 1986

[9] Sausvell R.V., Vvedenie v teoriyu uprugosti dlya inzhenerov i fizikov, Izd-vo inostr. lit., M, 1948

[10] Sachkov Yu.L., “Polnoe opisanie stratov Maksvella v obobschennoi zadache Didony”, Mat. sb., 197:6 (2006), 111–160 | DOI | MR

[11] Uitteker E.T., Vatson Dzh.N., Kurs sovremennogo analiza, URSS, M., 2002

[12] Eiler L., “Prilozhenie I: Ob uprugikh krivykh”, Metod nakhozhdeniya krivykh linii, obladayuschikh svoistvami maksimuma libo minimuma, ili reshenie izoperimetricheskoi zadachi, vzyatoi v samom shirokom smysle, Gostekhteorizdat, M.; L., 1934, 447–572

[13] Agrachev A.A., “Geometry of optimal control problems and Hamiltonian systems”, Nonlinear and optimal control theory, Lect. Notes Math., 1932, Springer, Berlin, 2008, 1–59 | DOI | MR | Zbl

[14] Bisshopp K.E., Drucker D.C., “Large deflection of cantilever beams”, Quart. Appl. Math., 3 (1945), 272–275 | MR | Zbl

[15] Born M., Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, Diss., Dieterich, Göttingen, 1906; “Ausgewählte Abhandlungen”, Verzeichnis der wissenschaftlichen Schriften, Bd. 1, Vandenhoeck und Ruprecht, Göttingen, 1963, 5–101

[16] Chen J.-S., Lin Y.-Z., “Snapping of a planar elastica with fixed end slopes”, J. Appl. Mech., 75:4 (2008), Pap. 041024 | DOI

[17] Domokos G., “Global description of elastic bars”, Ztschr. angew. Math. und Mech., 74:4 (1994), T289–T291 | MR | Zbl

[18] Domokos G., Fraser W.B., Szeberényi I., “Symmetry-breaking bifurcations of the uplifted elastic strip”, Physica D, 185:2 (2003), 67–77 | DOI | MR | Zbl

[19] Fried I., “Stability and equilibrium of the straight and curved elastica—finite element computation”, Comput. Methods Appl. Mech. and Eng., 28 (1981), 49–61 | DOI | Zbl

[20] Frisch-Fay R., Flexible bars, Butterworths, London, 1962 | Zbl

[21] Glassmaker N.J., Hui C.Y., “Elastica solution for a nanotube formed by self-adhesion of a folded thin film”, J. Appl. Phys., 96:6 (2004), 3429–3434 | DOI

[22] Greenhill A.G., The applications of elliptic functions, Macmillan, New York, 1892 | Zbl

[23] van der Heijden G.H.M., Neukirch S., Goss V.G.A., Thompson J.M.T., “Instability and self-contact phenomena in the writhing of clamped rods”, Intern. J. Mech. Sci., 45 (2003), 161–196 | DOI | Zbl

[24] Jairazbhoy V.A., Petukhov P., Qu J., “Large deflection of thin plates in cylindrical bending—non-unique solutions”, Intern. J. Solids and Struct., 45 (2008), 3203–3218 | DOI | Zbl

[25] Jin M., Bao Z.B., “Sufficient conditions for stability of Euler elasticas”, Mech. Res. Commun., 35 (2008), 193–200 | DOI | MR | Zbl

[26] Jurdjevic V., Geometric control theory, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[27] Kuznetsov V.V., Levyakov S.V., “Complete solution of the stability problem for elastica of Euler's column”, Intern. J. Non-Lin. Mech., 37 (2002), 1003–1009 | DOI | Zbl

[28] Lardner T.J., “A note on the elastica with large loads”, Intern. J. Solids and Struct., 21 (1985), 21–26 | DOI | MR

[29] Lawden D.F., Elliptic functions and applications, Springer, New York, 1989 | MR | Zbl

[30] Levyakov S.V., “Stability analysis of curvilinear configurations of an inextensible elastic rod with clamped ends”, Mech. Res. Commun., 36 (2009), 612–617 | DOI | Zbl

[31] Levyakov S.V., Kuznetsov V.V., “Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads”, Acta mech., 211 (2010), 73–87 | DOI | Zbl

[32] Maddocks J.H., “Stability of nonlinearly elastic rods”, Arch. Ration. Mech. Anal., 85 (1984), 311–354 | DOI | MR | Zbl

[33] Mikata Y., “Complete solution of elastica for a clamped–hinged beam, and its applications to a carbon nanotube”, Acta mech., 190 (2007), 133–150 | DOI | Zbl

[34] El Naschie M.S., “Thermal initial post buckling of the extensional elastica”, Intern. J. Mech. Sci., 18 (1976), 321–324 | DOI

[35] Panayotounakos D.E., Theocaris P.S., “Analytic solutions for nonlinear differential equations describing the elastica of straight bars: Theory”, J. Franklin Inst., 325:5 (1988), 621–633 | DOI | MR | Zbl

[36] Raboud D.W., Lipsett A.W., Faulkner M.G., Diep J., “Stability evaluation of very flexible cantilever beams”, Intern. J. Non-Lin. Mech., 36 (2001), 1109–1122 | DOI | Zbl

[37] Sachkov Yu.L., “Maxwell strata in the Euler elastic problem”, J. Dyn. and Control Syst., 14:2 (2008), 169–234 | DOI | MR | Zbl

[38] Sachkov Yu.L., “Conjugate points in the Euler elastic problem”, J. Dyn. and Control Syst., 14:3 (2008), 409–439 | DOI | MR | Zbl

[39] Seide P., “Large deflections of a simply supported beam subjected to moment at one end”, J. Appl. Mech., 51 (1984), 519–525 | DOI

[40] Stampouloglou I.H., Theotokoglou E.E., Andriotaki P.N., “Asymptotic solutions to the non-linear cantilever elastica”, Intern. J. Non-Lin. Mech., 40 (2005), 1252–1262 | DOI | MR | Zbl

[41] Tang T., Glassmaker N.J., “On the inextensible elastica model for the collapse of nanotubes”, Math. and Mech. Solids., 15:5 (2010), 591–606 | DOI | Zbl

[42] Wang C.Y., “Post-buckling of a clamped-simply supported elastica”, Intern. J. Non-Lin. Mech., 32 (1997), 1115–1122 | DOI | Zbl

[43] Wolfram S., Mathematica: A system for doing mathematics by computer, Addison-Wesley, Reading (MA), 1991