Complementary-Energy Methods for Geometrically Non-linear Structural Models: An Overview and Recent Developments in the Analysis of Frames

Boundary-value problems in solid mechanics are often addressed, from both theoretical and numerical points of view, by resorting to displacement/rotation-based variational formulations. For conservative problems, such formulations may be constructed on the basis of the Principle of Stationary Total Potential Energy. Small deformation problems have a unique solution and, as a consequence, their corresponding total potential energies are globally convex. In this case, under the so-called Legendre transform, the total potential energy can be transformed into a globally concave total complementary energy only expressed in terms of stress variables. However, large deformation problems have, in general, for the same boundary conditions, multiple solutions. As a result, their associated total potential energies are globally non-convex. Notwithstanding, the Principle of Stationary Total Potential Energy can still be regarded as a minimum principle, only involving displacement/rotation fields. The existence of a maximum complementary energy principle defined in a truly dual form has been subject of discussion since the first contribution made by Hellinger in 1914. This paper provides a survey of the complementary energy principles and also accounts for the evolution of the complementary-energy based finite element models for geometrically non-linear solid/structural models proposed in the literature over the last 60 years, giving special emphasis to the complementary-energy based methods developed within the framework of the geometrically exact Reissner-Simo beam theory for the analysis of structural frames.