On the construction of extended problems and related functionals for general nonlinear equations

Starting from existing methods for the symmetrisation of general nonlinear, nonpotential operators (Tonti, Int. J. Engng. Sci. 22 (11-12) (1984) 1343-1371; Auchmuty, Nonlinear Anal. Theory Methods: Appl. 12 (5) (1988) 531-564) this work discusses some alternative formulations and illustrates some significant implications of such methods, which should make them more suited to practical application. Further, a new class of the so-called "extended" functionals is proposed, much simpler to construct than the preceding ones. Even if the definition of the functionals requires the doubling of the unknown functions, the new unknowns have a precise physical meaning in the solution or the problem, which may help in the actual solution process. The application of the new method is illustrated by means of two examples in the field of continuum mechanics: the nonassociated elastic-plastic rate constitutive equations, and the nonlinear continuum dynamics equations with initial conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.