On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527–3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.