Finite Rotations and Complementary Extremum Principles

The equations of nonlinear elasticity can be written in terms of the following conjugate quantities (BUFLER [ 1 ] ) : (a) Deformation (or displacement) gradients and 1.Piola-Kirchhoff stresses, (b) Green’s deformations (or strains) and 2.Piola-Kirch-hoff stresses, (c) stretches (or extensions) and Biot stresses. In connection with the complementary energy theorem in nonlinear elasticity frequently the first group of variables is used, see f.i. KOITER [2]. In this case (a), however, the “stress and strain quantities” (deformation gradients) are not objective and the inversion of the necessarily nonlinear constitutive equations generally yields severe difficulties (OGDEN [3]) which do not appear if restriction is made to moderate rotations (STUMPF [4]). In the formulations (b) and (c), on the other hand, objective quantities are involved and a linear material behaviour (Kirchhoff and semilinear material respectively) can be used. Formulation (c) seems to be advantageous for two further reasons: Firstly the stretches (or extensions) and the conjugate stresses (called engineering stresses by F. DE VEUBEKE [5]) are to be considered as “natural quantities” [3] — indeed these ones are frequently used for large deformation problems of plates, membranes and shells, and the assumption of a convex strain energy density with respect to the extensions is physically meaningful — and secondly the rotations are taken into account explicitely. Furthermore the associated most general variational principle does allow independent variations of the displacements, stretches, stresses, reactive forces and rotations the corresponding Euler equations representing the force equilibrium (including the statical boundary conditions and the reactive forces), the constitutive equations, the kinematical field equations, the kinematical boundary conditions and the moment equilibrium, see REISSNER [6] and BUFLER [7] [8].