Dual Extremum Principles in Finite Deformation Theory with Applications to Post-Buckling Analysis of Extended Nonlinear Beam Model

The critical points of the generalized complementary energy variational principles are clarified. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoff type stress only) is constructed. We prove that the well-known generalized Hellinger-Reissner’s energy L(u, s) is a saddle point functional if and only is the Gao-Strang gap function is positive. In this case, the system is stable and the minimum potential energy principle is equivalent to a unique maximum dual variational principle. However, if this gap function is negative, then L(u, s) is a so-called ∂-critical point functional. In this case, the system has two extremum complementary principles. An interesting triality theorem for nonconvex variational problem is discovered, which can be used to study nonlinear bifurcation problems, phase transitions, variational inequality, and other things.