On Nonlinear Theory of Rigid-Flexible Shells Without the Kirchhoff Hypotheses

Abstract

As a development and a specification of some variants of nonlinear shell theory suggested in [1,3-5,7,9,11], we deduce the equations of shell mechanics without using the Kirchhoff hypotheses. The longitudinal deformation of normal fibres is taken into account without increasing the order of differential equilibrium equations. The transversal shears are approximated according to a linear law. Using the generalized forces and moments, the equilibrium equations are reduced to a canonical form, i.e. as they look in linear theory. It is shown that a nonlinear boundary problem of shell mechanics if the boundary conditions are expressed exclusively in terms of the displacements, is generally incorrect in sense of Lagrange principle,

• 1. Suppose that the radius-vector

  $$ \mathop R\limits^ \circ (\alpha ,\xi ) = \mathop r\limits^ \circ (\alpha ) + \xi \mathop n\limits^ \circ (\alpha ),\alpha = ({\alpha ^1},{\alpha ^2}) \in \mathop \Omega \limits^ \circ ,\xi \in [ - \frac{1}{2}\mathop h\limits^ \circ ,\frac{1}{2}\mathop h\limits^ \circ ], $$

  ((1.1))

after deformation of the shell turns into

$$ R(\alpha ,\xi ) = r(\alpha ) + {\lambda _\zeta }(\alpha )[\xi + \frac{1}{2}{\xi ^2}{\kappa _\zeta }(\alpha )]n(\alpha ) + \xi {\psi _\beta }(\alpha ){r^\beta }(\alpha ), $$

((1.2))

where $r = \mathop r\limits^ \circ + u,\,u = {u^\alpha }\mathop {{r_\alpha }}\limits^ \circ + w\mathop n\limits^ \circ ;$ the quantities characterizing the initial(undeformed) configuration are denoted by “○”