We present explicit formulas for the shallow shell model consisting of a couple of a wave equation and a plate equation, where the middle surface is viewed as a natural manifold with the induced metric from the classical Euclidean space of three dimensions. The Green formula for the shallow is given by the displacement field which expresses the relationship between the interior and the boundary. Next, the ellipticity of the strain energy for the shallow shell is studied under some curvature assumptions on the middle surface. Finally, the motion equations for shallow shells are obtained in terms of the displacement field as an unknown. The new ingredients in these formulas are that they take a form which is not described by a coordinate patch to provide the shell theory with the modern geometry.
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Key Laboratory of Control and SystemsInstitute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of Sciences, Beijing 100080, China;
