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On the dynamics of elastic strips. (English) Zbl 1053.74026

Summary: The dynamics of elastic strips, i.e., long thin rods with noncircular cross-section, is analyzed by studying the solutions of Kirchhoff equations. First, it is shown that if a naturally straight strip is deformed into a helix, the only equilibrium helical configurations are those with no internal twist and whose principal bending direction is either along the normal or the binormal. Second, the linear stability of a straight twisted strip under tension is analyzed, showing the possibility of both pitchfork and Hopf bifurcations depending on external and geometric constraints. Third, nonlinear amplitude equations are derived describing the dynamics close to the different bifurcation regimes. Finally, special analytical solutions to these equations are used to describe the buckling of strips. In particular, finite-length solutions with a variety of boundary conditions are considered.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74H55 Stability of dynamical problems in solid mechanics
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