In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a -dimensional energy for elastic strings by using techniques from Γ-convergence.
E.Acerbi, G.Buttazzo and D.Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity.25 (1991), 137–148. doi:10.1007/BF00042462.
2.
J.F.Babadjian and M.Baía, analysis of a thin film with periodic microstructure, Proc. Roy. Soc. Edinburgh, Section A136 (2006), 223–243. doi:10.1017/S0308210500004534.
3.
D.Blanchard and G.Griso, Asymptotic behavior of structures made of straight rods, J. Elasticity108(1) (2012), 85–118. doi:10.1007/s10659-011-9357-y.
4.
A.Braides, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl., Vol. 22, Oxford University Press, Oxford, 2002.
5.
A.Braides, I.Fonseca and G.Francfort, asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J.49 (2000), 1367–1404. doi:10.1512/iumj.2000.49.1822.
6.
L.Carbone, A.Gaudiello and P.Hernández-Llanos, T-junction of ferroelectric wires, ESAIM Math. Model. Numer. Anal.54(5) (2020), 1429–1463. doi:10.1051/m2an/2020001.
7.
P.G.Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson, Paris, 1990.
8.
P.G.Ciarlet and P.Destuynder, A justification of the two-dimensional linear plate model, J. Méc. Paris18(2) (1979), 315–344.
9.
B.Dacarogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Vol. 78, Springer-Verlag, Berlin, 1989.
10.
G.Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl., Vol. 8, Birkhäuser, 1993.
11.
I.Ekeland and R.Temam, Analyse Convexe et Problèmes Variationnels, Dunod, Paris, 1974.
12.
E.El Bachari, Modélisation d’une jonction en élasticité non linéaire par Γ-convergence, C. R. Acad. Paris I325 (1997), 1241–1246. doi:10.1016/S0764-4442(97)83561-6.
13.
R.Ferreira and E.Zappale, Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity, arXiv:1712.02598v1 [math.AP].
14.
G.Friesecke, R.D.James and S.Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math.55 (2002), 1461–1506. doi:10.1002/cpa.10048.
15.
G.Friesecke, R.D.James and S.Müller, A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence, Arch. Ration. Mech. Anal.180 (2006), 183–236. doi:10.1007/s00205-005-0400-7.
16.
A.Gaudiello, B.Gustafsson, C.Lefter and J.Mossino, Asymptotic analysis of a class of minimization problems in a thin multidomain, Calc. Var. Partial Differential Equations15(2) (2002), 181–201. doi:10.1007/s005260100114.
17.
A.Gaudiello and R.Hadiji, Junction of one-dimensional minimization problems involving valued maps, Adv. Differential Equations13(9–10) (2008), 935–958.
18.
A.Gaudiello and R.Hadiji, Ferromagnetic thin multi-structures, J. Differential Equations257 (2014), 1591–1622. doi:10.1016/j.jde.2014.05.015.
19.
A.Gaudiello and K.Hamdache, A reduced model for the polarization in a ferroelectric thin wire, NoDEA Nonlinear Differential Equations Appl.22(6) (2015), 1883–1896. doi:10.1007/s00030-015-0348-8.
20.
A.Gaudiello, R.Monneau, J.Mossino, F.Murat and A.Sili, Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var.13(3) (2007), 419–457. doi:10.1051/cocv:2007036.
21.
G.Griso, Asymptotic behavior of structures made of plates, Anal. Appl.3(4) (2005), 325–356. doi:10.1142/S0219530505000613.
22.
G.Griso, Asymptotic behavior of structures made of curved rods, Anal. Appl.6(1) (2008), 11–12. doi:10.1142/S0219530508001031.
23.
H.Le Dret, Model of a junction between two rods, J. Math. Pures. Appl.68 (1989), 365–397.
24.
H.Le Dret, Problèmes Variationnels dans Le Multi-Domaines: Modélisation des Jonctions et Applications, Rech. Appl. Math., Vol. 19, Masson, Paris, 1991.
25.
H.Le Dret and A.Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl.74 (1995), 549–578.
26.
H.Le Dret and A.Raoult, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear Sci.6 (1996), 59–84. doi:10.1007/BF02433810.
27.
M.G.Mora and S.Muller, Convergence of equilibria of three-dimensional thin elastic beams, Proc. Roy. Soc. Edinburgh138A (2008), 873–896. doi:10.1017/S0308210506001120.
28.
A.Raoult, Non-polyconvexity of the stored energy function of a Saint Venant–Kirchhoff material, Aplikace Matematiky6 (1986), 417–419.
29.
V.V.Slastikov and C.Sonnenberg, Reduced models for ferromagnetic nanowires, J. Appl. Math.77 (2012), 220–235.
30.
J.Tambaca and I.Velčić, Derivation of the nonlinear bending-torsion model for a junction of elastic rods, Proc. Roy. Soc. Edinburgh142A (2012), 633–664. doi:10.1017/S0308210510000491.
31.
I.Velčić, Nonlinear weakly curved rod by Γ-convergence, J. Elasticity108 (2012), 125–150. doi:10.1007/s10659-011-9358-x.