Dual Extremum Principles in Finite Deformation

The critical points of the generalized complementary energy variational principles are clariied. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoo 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. In order to study the shear eeects in frictional post-buckling problems, a new second order 2-D nonlinear beam model is developed. Its total potential is a double-well energy. A stability criterion for post-buckling analysis is proposed, which shows that the minimax complementary principle controls a stable buckling state. The unilateral buckling state is controlled by a minimum complementary principle. However, the maximum complementary principle controls the phase transitions.