Introduction. Despite several significant contributions, the complete nonlinear bending theory of elastic shells which would include fully general and nonlinear constitutive equations and strain measures is not as yet available. The earliest general investigations are by Synge and Chien [1] and Chien [2] who evidently were the first to adopt the intrinsic approach, thus avoiding direct reference to displacements. These papers of Synge and Chien, which are remarkable for their generality,

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... linear constitutive relations.1 More recently, an exact theory of strain for shells and rods, based on the concept of oriented bodies of E. and F. Cosserat, was formulated by Ericksen and Truesdell [4] who, however, did not consider the problem of constitutive equations.2 Other developments which also employ linear constitutive relations are founded under the Kirchhoff hypothesis and often contain other approximations. Among these we mention the incomplete treatment of Novozhilov [5;, E. Reissner's [6, 7] formulation of axisymmetric deformation of shells of revolution, and the more general works of Sanders [8] and Leonard [9]. Both Sanders [8] and Leonard [9] independently have, through an heuristic argument, proposed the differences of the first fundamental forms and of the second fundamental forms between the deformed and the undeformed middle surface of the shell as measures of deformation in "extension" and "bending," respectively; also, after postulating the existence of a two-dimensional strain energy function for shells dependent only on these two measures of deformation,3 they have derived constitutive equations by means of the generalized Hooke's law for the case of finite displacement and infinitesimal strains. Beginning with the three-dimensional field equations, the present paper is concerned with an exact, complete, and fully general nonlinear theory of elastic shells founded under the Kirchhoff hypothesis.4 Since the fully general equations of equilibrium in terms of stress and couple resultants and the appropriate boundary conditions are well known,5 the main task confronting us is the determination of suitable strain measures and the derivation of constitutive equations.