Kyuichiro Washizu

A Note on the Principle of Stationary Complementary Energy in Nonlinear Elasticity

ABSTRACT There have been recurring efforts to formulate a principle of stationary complementary energy in nonlinear theory of elasticity, namely a principle, of which the functional as well as subsidiary conditions are expressed purely in terms of stresses only. Among various efforts made so far in answering to this challenging problem, two approaches may be mentioned typical, both of which start from the principle of stationary potential energy. In the first approach, the principle of stationary potential energy is generalized by the use of Piola stress tensor and displacement-gradient tensor [1]. If it were possible to invert the relations between Piola stress tensor and the displacement-gradient tensor explicitly, the generalized principle could lead to obtaining the principle of stationary complementary energy. It has been found, however, that this inversion is extremely difficult. In the second approach, the formulation is made by the use of polar decomposition theorem of the Jacobian [2], It involves engineering strain tensor and its conjugate stress tensor which are to be regarded as functions of Piola stress tensor and material rotation. Thus, the functional is expressed in terms of the rigid body rotations as well as Piola stress tensor. The topic of this presentation will be confined to the first approach. It is shown that although the explicit inversion is extremely difficult in general, it has been found possible for several special problems: Large deflection problem of a beam, buckling problem of a column, problem of the so-called elastica, Karmans large deflection problem of a flat plate and Marguerres large deflection problem of a thin shallow shell. It will be shown in these formulations that the functional as well as the subsidiary conditions are expressed purely in terms of generalized stress resultants, and that the subsidiary conditions are given in linear forms while the complementary energy functions are given in nonlinear forms with respect to these generalized stress resultants. It is added that Refs. [3] and [4] are among recent contributions to this field of study, it is also added that the essential part of this presentation is contained in Ref. [5].

XXXIV – A Note on the Principle of Stationary Complementary Energy in Non-linear Elasticity

The present paper is a note on Zubovs approach to a problem of formulating a principle of stationary complementary energy in nonlinear theory of elasticity. It has been well recognized that if it were possible to invert the relations between Piola stress tensors and displacement-gradient tensors explicitly, the generalized principle of potential energy could lead to obtaining the principle of stationary complementary energy, and that this inversion is generally extremely difficult, however. The present paper shows that the explicit inversions have been found possible for two special problems, namely, problems of the so-called elastica and Marguerres thin shallow shell theory.