Abstract
We study the influence of the type of loading on the asymptotic behavior of linearly elastic, isotropic and homogeneous slender circular rings. By using formal asymptotic expansions, we obtain three families of models depending on the properties of the loads. If the loads expend work in inextensional displacements, then we find the classical model where the leading term of the energy corresponds to the bending-torsion energy of inextensional displacements. If the loads do no work in inextensional displacements, the model must be refined and we obtain two other types of models. In these other models, which depend on the type of loading, the leading term of the energy contains additional terms such as, for the second class, an extension energy due to the circumferential stretching of the ring, and even, for the third class, specific load-dependent contributions. This classification is illustrated in several examples.
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References
E. Acerbi, G. Buttazzo and D. Percivale, A variational definition for the strain energy of an elastic string. J. Elasticity 25 (1991) 137–148.
S.S. Antman, The Theory of Rods, Handbuch der Physik, Vol. VIa/2. Springer, Berlin (1972) pp. 641–703.
S.S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences, Vol. 107. Springer, Berlin (1995).
A. Bermúdez and J.M. Viaño, Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques. RAIRO Anal. Numér. 18 (1984) 347–376.
P.G. Ciarlet, Mathematical Elasticity, Vol. I. North-Holland, Amsterdam (1988).
A. Cimetière, G. Geymonat, H.L. Dret, A. Raoult and Z. Tutek, Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. J. Elasticity 19 (1988) 111–161.
G. Geymonat, F. Krasucki and J.-J. Marigo, Stress distribution in anisotropic elastic composite beams. In: P. Ciarlet and E. Sanchez-Palencia (eds), Applications of Multiple Scalings in Mechanics. Masson, Paris (1987) pp. 118–133.
G. Geymonat, F. Krasucki and J.-J. Marigo, Sur la commutativité des passages à la limite en théorie asymptotique des poutres composites. C. R. Acad. Sci. Paris Sér. II 305 (1987) 225–228.
M.E. Gurtin, The Linear Theory of Elasticity, Handbuch der Physik, Vol. VIa. Springer, Berlin (1972) pp. 1–296.
R. Jamal and E. Sanchez-Palencia, Théorie asymptotique des tiges courbes anisotropes. C. R. Acad. Sci. Paris Sér. I 322 (1996) 1099–1106.
A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1927) fourth edition.
K. Madani, Etude des structures élastiques élancées à rayon de courbure faiblement variable: Une classification des modèles asymptotiques. C. R. Acad. Sci. Paris Sér. IIb 326 (1998) 605–608.
J. Marigo, H. Ghidouche and Z. Sedkaoui, Des poutres flexibles aux fils extensibles: Une hiérarchie de modèles asymptotiques. C. R. Acad. Sci. Paris Sér. IIb 326 (1998) 79–84.
J.-J. Marigo and K. Madani, Quelques modèles d-anneaux élastiques suivant les types de chargement. C. R. Acad. Sci. Paris Sér. IIb 326 (1998) 805–810.
M.G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Gamma-convergence. Preprint No. 93, Max-Planck Institut für Mathematik, 2002, 1–19.
F. Murat and A. Sili, Comportement asymptotique des solutions du système de l’élasticité linéarisée anisotrope hétérogène dans des cylindres minces. C. R. Acad. Sci. Paris Sér. I 328 (1999) 179–184.
A. Rigolot, Sur une théorie asymptotique des poutres. J. Mécanique 11(2) (1972) 673–703.
J. Sanchez-Hubert and E. Sanchez-Palencia, Couplage flexion torsion traction dans les poutres anisotropes á sections hétérogènes. C. R. Acad. Sci. Paris 312 (1991) 337–344.
J. Sanchez-Hubert and E. Sanchez-Palencia, Statics of curved rods on account of torsion and flexion. European J. Mech. A Solids 18 (1999) 365–390.
S.P. Timoshenko, Strength of Materials, Parts 1 and 2, 3rd edn. Krieger Publishing (1983).
L. Trabucho and J.M. Viaño, Mathematical modelling of rods. In: P.G. Ciarlet and J.-L. Lions (eds), Handbook of Numerical Analysis, Vol. IV. Norh-Holland, Amsterdam (1996).
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Marigo, JJ., Madani, K. The influence of the type of loading on the asymptotic behavior of slender elastic rings. J Elasticity 75, 91–124 (2005). https://doi.org/10.1007/s10659-005-3397-0
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DOI: https://doi.org/10.1007/s10659-005-3397-0