Abstract
Recent results on mathematical modelling of curved rods are presented. More precisely, elastic behavior of a curved rod-like 3-D body is approximated by elastic behavior of its middle curve. The method of approximation is asymptotic expansion with respect to the small parameter (diameter of the cross section of the rod). Certain convergence results are proved and the obtained 1-D approximation is compared with the Cosserat model and arch model.
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References
Álvarez-Dios, J. A. and ViaÁo, J. M. (1995). Une théorie asymptotique de flexion-extension pour les poutres élastiques faiblement courbées. C. R. Acad. Sci. Paris, série I, 321: 1395–1400.
Álvarez-Dios, J. A. and Viaño, J. M. (1998). Mathematical justification of a one-dimensional model for general shallow arches. Math. Meth. Appl. sci., 21: 281–325.
Antman, S. S. (1995). Nonlinear Problems of Elasticity. Springer-Verlag.
Bermúdez, A. and Viaño, J. M. (1984). Une justification des équations de la thermo-élasticité des poutres à section variable par des méthodes asymptotiques. RAIRO Anal. Numb:, 17: 121–136.
Bernadou, M. and Ducatel, Y. (1982). Approximation of General Arch Problems by Straight Beam Elements. Nume, Math., 40: 1–29.
Brezzi, F. and Fortin, M. (1991). Mixed and Hybrid Finite Element Methods. Springer-Verlag. Chatelin, F. (1983). Spectral pproximation of Linear Operators. Academic Press.
Ciarlet, P. G. (1988). Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North-Holland.
Ciarlet, P. G. and Kesavan, S. (1981). Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory. Comp. Meth. Appl. Mech. Engrg., 26: 145–172.
Ciarlet, P. G. and Destuynder, P. (1979). A justification of the two dimensional linear plate model. J. Mécanique, 18: 315–344.
Ciarlet, P. G., Lods, V. and Miara, B. (1996). Asymptotic analysis of linearly elastic shells. II. Justification of flexular shell equations. Arch. Rational Mech. Anal., 136: 163–190.
Cimetière, A., Geymonat, G., Le Dret, H., Raoult, A. and Tutek, Z. (1986). Une dérivation d’un modéle non linéaire de poutres a partir de l’élasticité tridimensionelle. C. R. Acad. Sci. Paris, série I, 302: 697–700.
Cimetière, A., Geymonat, G., Le Dret, H., Raoult, A. and Tutek, Z. (1988). Asymptotic theory and analysis for displacement and stress distributions in nonlinear elastic straight slender rods. J. Elasticity, 19: 111–161.
Dautray, R. and Lions, J. L. (1992). Mathematical Analysis and Numerical Methods for Science and Tehnology. Volume 5 Evolution Problems I. Springer-Verlag.
Davies, E. B. (1995). Spectral Theory and Differential Operators, University Press, Cambridge. Diestel, J. and Uhl, J. J. (1977). Vector measures. AMS, Mathematical Surveys, No 15.
Figueiredo, I. N. and Trabucho, L. (1993). A Galerkin approximation for curved beams. Comp.Meth. Appl. Mech. Engrg. 102: 235–253.
Germain, R (1962). Méchanique des Milieux Continus. Masson.
Giarult, V. and Raviart, R. (1986). Finite Element Methods for Navier-Stokes Equations. Springer-Verlag.
Hay, G. E. (1942). The finite displacement of thin rods. Trans. Amer. Math. Soc., 51: 65–102.
Irago, H. and Kerdid, N. and Viaíio, J. M. (1998). Analyse asymptotique des modes de hautes fréquences dans les poutres minces, C. R. Acad. Sci. Paris, série I, 326: 1255–1260.
Irago, H. and Viaiio, J. M. (1998). Second-order asymptotic approximation of flexural vibrations in elastic rods Mat. Mod. Meth. Appl. Sci.8: 1343–1362.
Jamal, R. (1998). Modélisation asymptotique des comportements statique et vibratoire des tiges courbes élastiques, Thése de doctorat, De L’Université Pierre et Marie Curie, Paris 6.
Jamal, R. and Sanchez-Palencia, É. (1996). Théorie asymptotique des tiges courbes anisotropes. C. R. Acad. Sci. Paris, série I, 322: 1099–1106.
Jurak, M. and Tarnbaca, J. (1999). Derivation and justification of a curved rod model. Math. Mod. Meth. in Appl. Sci., 9: 991–1014.
Jurak, M. and Tambaca, J. (2001). Linear curved rod model. General curve. Math. Mod. Meth. Appl. Sci., 11: 1237–1253.
Jurak, M., Tambaca, J. and Tutek, Z. (1999). Derivation of a curved rod model by Kirchhoff assumptions. ZAMM, 79: 455–463.
Jurak, M. and Tutek, Z. (1999). Wrinkled rod. Math. Mod. Meth. Appl. Sci., 9: 665–692.
Kerdid, N. (1993). Comportement asymptotique quand l’épaisseur tend vers zéro du problème de valeurs propres pour une poutre mince encastré linéare, C. R. Acad. Sci. Paris, série I, 316: 755–758.
Kikuchi, F. (1982). Accuracy of some finite element models for arch problems. Comp. Meth. Appl. Mech. Engrg., 35: 315–345.
Landau, L. D. and Lifshitz, E. M. (1970). Theory of Elasticity. Pergamon Press.
Le Dret, H. (1989). Modeling of the junction between two rods. J. Math. Pures. Appl.68: 365–397.
Le Dret, H. (1991). Problémes variationneles dans les multi-domains. Masson.
Li-ming, X. (1998). Asymptotic analysis of dynamic problems for linearly elastic shells–Jus-tification of equations for dynamic membrane shells. Asymptotic Analysis, 17: 121–134.
Nečas, J. and Hlaváček, I. (1981) Mathematical Theory of Elastic and Elasto-Plastic Bodies:An Introduction. Elsevier.
Raoult, A. (1985). Construction d’un modèle d’évolution de plaques avec terme d’inerte de rotation. Annali di Matematica Pura et Applicata, Serie Quarta, 139: 361–400.
Saleeb, A. R. and Chang, T. Y. (1987). On the hybrid-mixed formulation of C° curved beam elements. Comp. Meth. Appl. Mech. Engrg., 60: 95–121.
Sanchez-Hubert, J. and Sanchez-Palencia, É. (1999). Statics of curved rods on account of torsion and flexion. Eur. J. Mech. A/Solids, 18: 365–390.
Sanchez-Palencia, É. (1992). Asymptotic and spectral properties of a class of singular-stiff problems. J. Math. Pures. Appl., 71: 379–406.
Tambaca, J. (1999). One-dimensional models in theory of elasticity,master’s thesis, Department of Mathematics, University of Zagreb. (In Croatian).
Tambaca, J. (2000). Evolution model of curved rods. PhD thesis, Department of Mathematics, University of Zagreb. (in Croatian).
Tambaca, J. and Tutek, Z. (2000). Dynamic Curved Rod Model. Proceedings of the Conference on Applied Mathematics and Computation, Dubrovnik 1999, eds. V. Hari et al.
Tambaèa, J. and Tutek, Z. (2000). Evolution model of curved rods. Proceedings of the Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000), SIAM: 197–201.
Tambaca, J. (2001). One-dimensional approximations of the eigenvalue problem of curved rods. Math. Meth. Appl. Sci., 24: 927–948.
Tambaca, J. (2002). Justification of the dynamic model of curved rods. Submitted to Asymptotic Analysis.
Trabucho, L. and Viaiio, J. M. (1996). Mathematical Modelling of Rods in Handbook of Numerical Analysis, Vol. IV, Eds. P. G. Ciarlet, J. L. Lions. North-Holland.
Tretter, C. (2000). Linear operator pencils A - AB with discrete spectrum. Integral Eq. Oper. Th., 37: 357–373.
Tutek, Z. and Aganovié, I. (1986). A justification of the one-dimensional linear model of elastic beam, Math. Meth. in the Appl. Sci., 8: 1–14.
Weinberger, H. F. (1974). Variational Methods for Eigenvalue Approximation,SIAM.
Zemer, M. (1994). An asymptotically optimal finite element scheme for the arch problem. Numer. Math., 69: 117–123.
Zhang, Z. (1992). Arch beam models: Finite element analysis and superconvergence. Numer. Math., 61: 117–143.
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Jurak, M., Tambača, J., Tutek, Z. (2002). Modelling of Curved Rods. In: Drmač, Z., Hari, V., Sopta, L., Tutek, Z., Veselić, K. (eds) Applied Mathematics and Scientific Computing. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4532-0_4
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DOI: https://doi.org/10.1007/978-1-4757-4532-0_4
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