[Instabilités en coin dans un cylindre mince non linéairement élastique : solutions analytiques mécanisme déformation]
Dans cette Note, on étudie les instabilités « en coin » dans un cylindre mince formé d'un matériau non linéairement élastique. Partant des équations nonlinéaires tri-dimensionnelle, nous obtenons par une méthode nouvelle un système dynamique singulier. On montre que ce système retient les instabilités en coin. Nous obtenons également les expressions analytiques des solutions. On met aussi en évidence le fait que l'effet de couplage entre la nonlinéarité du matériau et la longueur caractéristique est le mécanisme qui provoque l'apparition de coins.
In this Note, we study the corner instabilities in a slender cylinder constituted by a nonlinearly elastic material. Starting from the three-dimensional nonlinear field equations, we derive, through a novel method, a singular dynamical system as the normal form equation. It is shown that this system can capture the corner instabilities. We are also able to obtain analytical expressions of the solutions. The mechanism that causes corner formations is also found.
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@article{CRMATH_2007__345_1_55_0,
author = {Dai, Hui-Hui and Wang, Fan-Fan},
title = {Corner instabilities in a slender nonlinearly elastic cylinder: analytical solutions and formation mechanism},
journal = {Comptes Rendus. Math\'ematique},
pages = {55--58},
publisher = {Elsevier},
volume = {345},
number = {1},
year = {2007},
doi = {10.1016/j.crma.2007.05.020},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.05.020/}
}
TY - JOUR AU - Dai, Hui-Hui AU - Wang, Fan-Fan TI - Corner instabilities in a slender nonlinearly elastic cylinder: analytical solutions and formation mechanism JO - Comptes Rendus. Mathématique PY - 2007 SP - 55 EP - 58 VL - 345 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.05.020/ DO - 10.1016/j.crma.2007.05.020 LA - en ID - CRMATH_2007__345_1_55_0 ER -
%0 Journal Article %A Dai, Hui-Hui %A Wang, Fan-Fan %T Corner instabilities in a slender nonlinearly elastic cylinder: analytical solutions and formation mechanism %J Comptes Rendus. Mathématique %D 2007 %P 55-58 %V 345 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.05.020/ %R 10.1016/j.crma.2007.05.020 %G en %F CRMATH_2007__345_1_55_0
Dai, Hui-Hui; Wang, Fan-Fan. Corner instabilities in a slender nonlinearly elastic cylinder: analytical solutions and formation mechanism. Comptes Rendus. Mathématique, Tome 345 (2007) no. 1, pp. 55-58. doi : 10.1016/j.crma.2007.05.020. http://www.numdam.org/articles/10.1016/j.crma.2007.05.020/
[1] Nonlinear Problems of Elasticity, Springer, New York, 1995
[2] Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Appl. Mech. Rev., Volume 40 (1987), pp. 1699-1734
[3] Mathematical Elasticity I: Three-Dimensional Elasticity, Elsevier, North-Holland, 1988
[4] Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods, Wave Motion, Volume 28 (1998), pp. 367-381
[5] Phase transitions in a slender cylinder composed of an incompressible elastic material. I. Asymptotic model equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 462 (2006), pp. 75-95
[6] Asymptotically approximate model equations for weakly nonlinear long waves in compressible elastic rods and their comparisons with other simplified model equations, Math. Mech. Solids, Volume 9 (2004), pp. 61-79
[7] Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, Proc. R. Soc. Lond. A, Volume 456 (2000), pp. 331-363
[8] Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods, Acta. Mech., Volume 157 (2002), pp. 97-112
[9] Global bifurcation in nonlinear elasticity with an application to barrelling states of cylindrical columns, J. Elasticity, Volume 71 (2003), pp. 33-58
[10] Global continuation in nonlinear elasticity, Arch. Rational Mech. Anal., Volume 143 (1998), pp. 1-28
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