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

Abstract

Two new finitely deformed dynamical beam models are established for serious study on non-linear 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 Gao and Strang in 1989 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 non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.