A new variational principle for finite elastic displacements

Abstract

Variational principles for finite elastic displacements have been formulated in terms of Green strain and Kirchhoff-Trefftz stress tensors. The first is a functional of the displacement field only and implies stationarity of the total potential. The second is a canonical principle, in the sense of Friedrichs [1], involving both stresses and displacements and generalizing Reissner's principle [5]. In contrast with the geometrically linearized elasticity theory, it cannot be reduced to a complementary energy principle involving equilibrium stresses only [9,10]. The paper discusses the Levinson[9] and Zubov[11] formulation in terms of displacement gradients and the Piola stress tensor, which, however interesting from a theoretical viewpoint, does not appear suitable for practical applications. A new set of variational principles, of displacement, canonical or complementary energy types, is found to derive from the use of the polar decomposition theorem of the jacobian. It involves the engineering strain tensor and its conjugate stress tensor is to be regarded as a function of the Piola tensor and the material rotation. The complementary energy formulation is discussed in terms of first order stress functions. The presence of the rotational degrees of freedom opens the possibility of discretizing the rotational equilibrium equations in approximate solutions.