Geometric nonlinearity: potential energy, complementary energy, and the gap function

Abstract

Abstract. Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function, we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

On modeling and global solutions for d.c. optimization problems by canonical duality theory

“…This theory was developed originally from Gao and Strang's work in nonconvex mechanics [21] and has been applied successfully for solving a large class of challenging problems in both nonconvex analysis/mechancis and global optimization, such as phase transitions in solids [23], post-buckling of large deformed beam [38], nonconvex polynomial minimization problems with box and integer constraints [13,15,18], Boolean and multiple integer programming [6,40], fractional programming [7], mixed integer programming [20], polynomial optimization [14], high-order polynomial with log-sum-exp problem [3]. A comprehensive review on this theory and breakthrough from recent challenges are given in [19].…”

Dual extremum principles for geometrically exact finite strain beams

“…This nomenclature was introduced first by Gao and Strang in the general context of geometric nonlinearity [14].…”

“…with L r c a quadratic operator of vector d. It will be shown that, under this assumption, two dual extremum principles can be formulated following Gao and Strang's approach [14].…”

“…According to Gao and Strang (1989) [14], in the framework of the finite elasticity theory, characterized by a quadratic geometrical operator, both global and local extremality conditions depend on a so-called complementary gap functional, a non-zero quantity due to the nonconvexity of the strain energy with respect to the displacement variables. Furthermore, the HRE can only be regarded as a saddle-type functional if, and only if, the gap functional is positive for any statically admissible stress field.…”

“…Two distinct approaches will be followed: one relies on Noble-Sewell's theory and the other has its basis on Gao-Strang's theory [14]. While the latter approach is restricted to quadratic geometrical (also called compatibility) operators, being valid therefore only for moderate large deflections, the former is valid for any magnitude of the displacements, rotations and strains.…”