Finite deformation beam models and triality theory in dynamical post-buckling analysis.

*(English)*Zbl 1068.74569Summary: Two new finitely deformed dynamical beam models are established for study on nonlinear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola-Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that, if the gap function introduced by Y. Gao and G. Strang [Q. Appl. Math. 47, No. 3, 487–504 (1989; Zbl 0691.73012)] in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for nonlinear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the nonlinear dynamical systems. A pair of dual Duffing equations are obtained.

##### MSC:

74H60 | Dynamical bifurcation of solutions to dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74H45 | Vibrations in dynamical problems in solid mechanics |