On a Variational Theorem in Elasticity

Abstract

1. Introduction. There are in the theory of elasticity two well known variational theorems or principles, one of them being for displacements and the other one being for stresses [1]. The former is also called the principle of minimum potential energy while the latter is often referred to as the principle of minimum complementary energy. With reference to the differential equations of the theory of elasticity, which consist of equations of equilibrium for the components of stress and of the stress-displacement relations, a possible characterisation of the difference between the two variational theorems is as follows. In the theorem for displacements the stress-displacement relations are taken as equations of definition for the components of stress in terms of appropriate displacement derivatives and the variational equation is equivalent to the system of differential equations of equilibrium. In the theorem for stresses the differential equations of equilibrium serve to restrict the class of admissible stress variations and the variational equation is equivalent to the system of stress-displacement relations.Both variational principles have been found valuable for the purpose of obtaining approximate solutions of boundary value problems. When viewed in the above light these approximate solutions are such that part of the complete system of differential equations (either the stress-displacement relations or the equations of equilibrium) is satisfied exactly while the remaining equations are S'atisfied approximately only.It is natural to ask whether it might not be possible to use the calculus of variations for the purpose of obtaining approximate solutions in such a manner that there is no preferential treatment for either one of the two kinds of differential equations which occur in the theory. In what follows this question is answered in the affirmative. A variational problem will be formulated which has both the equation of equilibrium and the stress-displacement relations as appropriate Euler equations.It is also shown that application of the theorem to a problem which had previously been treated by means of the method of complementary energy leads to the results obtained by the complementary-energy method in a manner which represents some simplification over the earlier derivations.

On variational methods in finite and incremental elastic deformation problems with discontinuous fields

“…Prange's work, however, remained unknown, since it was not published. Thus it appears that the first general variational statement of the kind mentioned above was published by Reissner [8]. In 1955, Hu [9] published a paper in which he developed variational methods for small strain problems, which involved arbitrary variations of the displacement, the stress, and the strain fields.…”

Enhanced layerwise model for laminates with imperfect interfaces – Part 1: Equations and theoretical validation

“…The governing equations may be obtained by the Reissner's variational method [5,6] or the principle of virtual displacements. As compared to single layer theories, layerwise models represent a more natural method to calculate interfacial stresses and capture specific aspects of interfaces in laminates [7][8][9][10] in order to predict delamination or to take into account edge effects.…”

“…However, the study of stress concentrations or stress controlled phenomena could be more natural, more convenient with a direct description of stress fields. A noteworthy work has been made in this way by Pagano [17] who used Reissner´s variational mixed formulation [5] and a stress field approximation to obtain an efficient model. The stress field selection verifies the continuity conditions across the interfaces of the multilayer.…”

Enhanced layerwise model for laminates with imperfect interfaces – Part 2: Experimental validation and failure prediction

“…In these models, an approximation of displacements or stresses and displacements through the thickness of each layer is proposed. The Reissner's variational method [13] or the principle of virtual displacements is commonly used to obtain the constitutive equations. Another type of layerwise model is the local model developed by Pagano [14].…”