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REVIEW ARTICLES

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

[+] Author and Article Information
David Yang Gao

Divison of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg VA 24061

Appl. Mech. Rev 50(11S), S64-S71 (Nov 01, 1997) doi:10.1115/1.3101852 History: Online April 20, 2009

Abstract

The critical points of the generalized complementary energy variation 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, and a complete duality theory in geometric nonlinear system is established. We prove that the well-known generalized Hellenger-Reissner’s energy L(u ,s ) is a saddle point functional if and only if 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 trinity 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 effects 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 unilaterial buckling state is controled by a minimum complementary principle. However, the maximum complementary principle controls the phase transitions.

Copyright © 1997 by American Society of Mechanical Engineers
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