Nonlinear Weakly Curved Rod by Γ-Convergence
Source abstract
Abstract
We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use the approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity and do not presuppose any constitutional behavior. To derive the model, by means of $Γ$-convergence, we need to propose how is the order of strain energy related to the thickness of the body $h$. We analyze the situation when the strain energy (divided by the order of volume) is of the order $h^4$. That is the same approach as the one when Föppl-von Kármán model for plates and the analogous model for rods are obtained. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Elasticity and Material Modeling · Advanced Mathematical Modeling in Engineering · Advanced Numerical Methods in Computational Mathematics
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (29)
Show source-provided references
1. Acerbi, E., Butazzo, G., Percivale, D.: A variational definition of the strain energy for an elastic string. J. Elast. 25, 137–148 (1991)10.1007/BF00042462 ↗
2. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
3. Alvarez-Dios, J.A., Viano, J.M.: A bending and stretching asymptotic theory for general elastic shallow arches. In: Proc. ESAIM, vol. 2, pp. 145–152 (1997)
4. Alvarez-Dios, J.A., Viano, J.M.: Mathematical justification of a one-dimensional model for general elastic shallow arches. Math. Methods Appl. Sci. 21(4), 281–325 (1998)10.1002/(SICI)1099-1476(19980310)21:4<281::AID-MMA951>3.0.CO;2-O ↗
5. Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Applied Mathematical Sciences, vol. 107. Springer, New York (2005)
6. Braides, A.: Γ-Convergence for Beginners. Oxford University Press, Oxford (2002)10.1093/acprof:oso/9780198507840.001.0001 ↗
7. Ciarlet, P.G.: Mathematical Elasticity. Vol. I, Three-Dimensional Elasticity. North-Holland, Amsterdam (1988)
8. Ciarlet, P.G.: Mathematical Elasticity. Vol. II. Theory of Plates. Studies in Mathematics and Its Applications, vol. 27. North-Holland, Amsterdam (1997)
9. Ciarlet, P.G.: Mathematical Elasticity. Vol. III. Theory of Shells. Studies in Mathematics and Its Applications, vol. 29. North-Holland, Amsterdam (2000)
10. Dal Maso, G.: An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications. Birkäuser, Basel (1993)10.1007/978-1-4612-0327-8 ↗
11. Fox, D.D., Raoult, A., Simo, J.C.: A justification of nonlinear properly invariant plate theories. Arch. Ration. Mech. Anal. 124, 157–199 (1993)10.1007/BF00375134 ↗
12. Friesecke, G., James, R.D., Müler, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)10.1002/cpa.10048 ↗
13. Friesecke, G., James, R.D., Müler, S.: A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)10.1007/s00205-005-0400-7 ↗
14. Friesecke, G., James, R.D., Müler, S.: The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C. R. Math. 335(2), 201–206 (2002)10.1016/S1631-073X(02)02388-9 ↗
15. Friesecke, G., James, R., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Γ-convergence. C. R. Math. Acad. Sci. Paris 336(8), 697–702 (2003)10.1016/S1631-073X(03)00028-1 ↗
16. Le Dret, H., Raoult, A.: The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74, 549–578 (1995)
17. Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6(1), 59–84 (1996)10.1007/BF02433810 ↗
18. Lecumberry, M., Müller, S.: Stability of slender bodies under compression and validity of the von Kármán theory. Arch. Ration. Mech. Anal. 193(2), 255–310 (2009)10.1007/s00205-009-0232-y ↗
19. Lewicka, M., Mora, M.G., Pakzad, M.: Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 5, IX, 1–43 (2010)
20. Marigo, J.J., Meunier, N.: Hierarchy of one-dimensional models in nonlinear elasticity. J. Elast. 83, 1–28 (2006)10.1007/s10659-005-9036-y ↗
21. Mora, M.G., Scardia, L.: Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. J. Differ. Equ., in press
22. Mora, M.G., Müller, S.: Derivation of the nonlinear bending-torsion theory for inextensible rods by Gamma-convergence. Calc. Var. 18, 287–305 (2003)10.1007/s00526-003-0204-2 ↗
23. Mora, M.G., Müller, S.: A nonlinear model for inextensible rods as a low energy Gamma-limit of three-dimensional nonlinear elasticity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 271–293 (2004)10.1016/j.anihpc.2003.08.001 ↗
24. Mora, M.G., Müller, S.: Convergence of equilibria of three-dimensional thin elastic beams. Proc. R. Soc. Edinb., Sect. A, Math. 138, 873–896 (2008)10.1017/S0308210506001120 ↗
25. Müller, S., Packzad, M.R.: Convergence of equilibria of thin elastic plates: the von Kármán case. Commun. Partial Differ. Equ. 33(6) 1018–1032 (2008)10.1080/03605300701629443 ↗
26. Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992)
27. Scardia, L.: The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity. Asymptot. Anal. 47, 317–343 (2006)
28. Scardia, L.: Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence. Proc. R. Soc. Edinb., Sect. A, Math. 139, 1037–1070 (2009)10.1017/S0308210507000194 ↗
29. Velcic, I.: Shallow shell models by Γ-convergence, submitted paper. Preprint, http://web.math.hr/ivelcic