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An asymptotic analysis method for the linearly shell theory

Authors:

LI Kaitai ZHANG Wenling HUANG Aixiang

Institution:

College of Sciences, Xi'an Jiaotong University, Xi'an 710049, China

Abstract:

In this paper, we consider a linearly elastic shell, i.e. a three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions U ^KT(x, ξ) in terms of U:

$\left\{ \begin{gathered} U^{KT} (x,\xi )\,\,\,\,\,\,: = U(x) + \prod _1 U\xi + \prod _2 U\xi ^2 , \hfill \\ \prod _1 U\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - a^{\alpha \beta } \mathop \nabla \limits^* _\beta U^3 \vec e_\alpha - \lambda _0 \gamma _0 (U)\vec n, \hfill \\ \prod _2 U\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{1}{{\lambda + \mu }}\mathop \nabla \limits^* _\beta (a_ + ^{\alpha \beta \lambda \sigma } \gamma _{\lambda \sigma } (U)) - b^{\alpha \beta } \mathop \nabla \limits^* _\beta U^3 } \right)\vec e_\alpha \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{1}{2}\lambda _0 (\rho _0^{KT} (U) - (1 + \lambda _0 )H\gamma _0 (U) - 2\beta _0 (U))\vec n\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered} \right.$

. We also provide the error estimates between our model and the three-dimensional displacement vector field:

$\parallel u - U^{KT} \parallel _{\,1,\hat \Omega } \leqslant C \in ^r ,\,\,\,\,\,r = 3/2, an elliptic membrane,\,\,\,\,\,r = 1/2, a general membrane,$

where C is a constant dependent only upon the data ∥u∥_3,Ω,∥U ^KT∥_3,Ω, $\parallel u\parallel _{\,3,\Omega } ,\parallel U^{KT} \parallel _{\,3,\Omega } ,\vec \theta$ .

Keywords:

linear elastic shell asymptotic expansion method

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