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ABSTRACT

Classical vs. intrinsic shell theory

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The International Conference on Computational & Experimental Engineering and Sciences 2011, 18(2), 47-48. https://doi.org/10.3970/icces.2011.018.047

Abstract

Intrinsic methods in elasticity have been introduced in a landmark series of papers by Wei-Zhang Chien in 1944. During the last two decades, Professor Wojciech Pietraszkiewicz and his group have achieved major advances in their analysis from the mechanical and engineering viewpoints, as well as in their actual implementation.
However, it was only in 2004 that their mathematical analysis began to be carried out in earnest, first for three-dimensional elasticity and more recently for elastic shells, by the author and his group.
This presentation will briefly review and discuss various problems as yet unresolved in this approach.
In the classical approach, the main mathematical challenge is to establish that the associated energy has a minimizer. In the linear case, this is achieved through a Korn inequality, which guarantees the positive-definiteness of the associated energy. In the nonlinear case, the situation is on firm grounds in three-dimensional elasticity, thanks to the landmark approach of John Ball in 1977 and his introduction of the notion of polyconvexity. In this direction, the author and his group are currently developing a notion of "polyconvexity on a surface", with the hope of eventually obtaining a satisfactory existence theory for nonlinearly elastic shells.
In the intrinsic approach, the main challenges lie not only in the mathematical analysis, but in effect in the modeling itself. Since the new unknowns are the change of metric and change of curvature tensor fields (instead of the displacement field in the classical approach), the Gauss and Codazzi-Mainardi compatibility equations conditions (or other equivalent equations) must be satisfied by these new unknowns, in order that they indeed correspond to a displacement of the middle surface of the shell. Another challenge is to adequately express boundary conditions of place or of traction in terms of these new unknowns.
In the linear case, we will give in particular an explicit form of the compatibility conditions, as well as an explicit "Cesaro-Volterra integral formula on a surface" for reconstructing a displacement field from the knowledge of these new unknowns. We will also briefly discuss an alternative intrinsic approach, currently explored by the author and his group, which relies on Donati-like compatibility conditions.

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