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Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 1, The shallow shells

Hamdouni A., Millet O.

Archives of Mechanics

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2003

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Vol. 55, nr 2

135-175

EN

The purpose of this paper is to construct a classification of asymptotic shell models (inferred from the non linear three-dimensional elasticity) with respect to the applied forces and to the geometrical data. To do this, we use a constructive approach based on a dimensional analysis of the nonlinear three-dimensional equilibrium equations, which naturally gives rise to the appearance of dimensionless numbers characterizing the applied forces and the geometry of the shell. In order to limit our study to one-scale problems, these dimensionless numbers are expressed in terms of to the relative thickness ... of the shell, which is considered as the perturbation parameter. This leads, on the one hand, to distinguish shallow shells from strongly curved shells which have a different asymptotic behaviour, and on the other hand, to fix the applied force level. For each of these two classes of shells, using the usual asymptotic method, we propose a complete classification of two-dimensional shell models based on decreasing force levels, from severe to low. In the first part of this paper, we present the classification for shallow shells. We obtain successively the nonlinear membrane model, another membrane model, Koiter's non linear shallow shell model, and the linear Novozhilov-Donnell one, respectively for severe, high, moderate and low forces.

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Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 2, The strongly curved shells

Hamdouni A., Millet O.

Archives of Mechanics

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2003

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Vol. 55, nr 2

177-219

EN

In the first part of this paper we have deduced a classification of asymptotic shallow shell models with respect to the level of applied forces, from the non-linear three-dimensional elasticity. We have used a constructive approach based on a dimensional analysis of the non-linear three-dimensional equilibrium equations, which naturally makes appear dimensionless numbers characterizing the applied forces (... and ...) and the geometry of the shell (... and C). To limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness ... of the shell, considered as the perturbation parameter. In the first part, we have studied the case of shallow shells corresponding to C=.... In the second part of this paper, we will study the case of strongly curved shells for which C=.... The classification that we obtain is then more complex. It depends not only on the force levels, but also on the existence of inextensional displacements which keep invariant the metric of the middle surface of the shell.

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Dimensional analysis and asymptotic expansions of the equilibrium equations in nonlinear elasticity. Cz. 1, The membrane model

Millet O., Hamdouni A., Cimetiere A.

Archives of Mechanics

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1998

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Vol. 50, nr 6

953-973

EN

In this paper, we develop a new asymptotic constructive approach in nonlinear plate theory. The dimensional analysis of the three-dimensional equilibrium equations naturally leads to dimensionless numbers which reflect the geometry of the structure and the magnitude of forces. These numbers also define the domain of validity of the two-dimensional models which will later be obtained by asymptotic expansions. For nonlinear plates, we prove that the two-dimensional models we obtain by asymptotic expansions are determined by the magnitude of the forces applied. In this first part, we consider a plate subjected to large loads. In this case, we prove that the nonlinear plate model we obtain by asymptotic expansions is a membrane model. In the second part of this article, we will consider a plate subjected to smaller applied forces.

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Dimensional analysis and asymptotic expansions of the equilibrium equations in nonlinear elasticity. Cz. 2, The two-dimensional von Karman model

Millet O., Hamdouni A., Cimetiere A.

Archives of Mechanics

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1998

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Vol. 50, nr 6

975-1001

EN

In the second part of this article, which is a continuation of [8] which dealt with a plate subjected to large loads, we consider a plate subjected to moderate applied forces level within the framework of nonlinear elasticity. We then apply the new constructive approach developed in the first part which needs no a priori assumption. For these moderate forces, we prove that the two-dimensional model we obtain by asymptotic expansions is the von Karman one. Finally the two-dimensional stress field in the plate is deduced from the three-dimensional constitutive equations without any a priori assumption.