We consider the dynamical evolution of a thin rod described by an appropriately scaled wave equation of nonlinear elasticity. Under the assumption of well-prepared initial data and external forces, we prove that a solution exists for arbitrarily large times, if the diameter of the cross section is chosen sufficiently small. The scaling regime is such that the limiting equations are linear.
H.Abels, M.G.Mora and S.Müller, Large time existence for thin vibrating plates, Comm. Partial Differential Equations36(12) (2011), 2062–2102. doi:10.1080/03605302.2011.618209.
2.
H.Abels, M.G.Mora and S.Müller, The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity, Calc. Var. Partial Differential Equations41(1–2) (2011), 241–259. doi:10.1007/s00526-010-0360-0.
3.
T.Ameismeier, Thin vibrating rods: Γ-convergence, large time existence and first order asymptotics, PhD thesis, University Regensburg, 2021. doi:10.5283/epub.46123.
4.
S.S.Antman, Nonlinear Problems of Elasticity, 2nd edn, Vol. 107, Springer-Verlag, Berlin–New York, 2005.
5.
M.Giaquinta and L.Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edn, Vol. 11, Edizioni della Normale, Pisa, 2012.
M.Lecumberry and S.Müller, Stability of slender bodies under compression and validity of the von Kármán theory, Arch. Ration. Mech. Anal.193(2) (2009), 255–310. doi:10.1007/s00205-009-0232-y.
9.
J.-L.Lions and E.Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I–III, Die Grundlehren der mathematischen Wissenschaften, Vol. 181, Springer-Verlag, New York–Heidelberg, 1972, Translated from the French by P. Kenneth.
10.
W.McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
11.
M.G.Mora and S.Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. Partial Differential Equations18(3) (2003), 287–305. doi:10.1007/s00526-003-0204-2.
12.
M.G.Mora and S.Müller, A nonlinear model for inextensible rods as a low energy Γ-limit of three-dimensional nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire21(3) (2004), 271–293. doi:10.1016/j.anihpc.2003.08.001.
13.
Y.Qin and P.-F.Yao, The time-dependent von Kármán shell equation as a limit of three-dimensional nonlinear elasticity, J. Syst. Sci. Complex.34(2) (2021), 465–482. doi:10.1007/s11424-020-9146-4.
14.
L.Scardia, The nonlinear bending-torsion theory for curved rods as Γ-limit of three-dimensional elasticity, Asymptot. Anal.47(3–4) (2006), 317–343.
15.
L.Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence, Proc. Roy. Soc. Edinburgh Sect. A139(5) (2009), 1037–1070. doi:10.1017/S0308210507000194.
16.
J.Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
