Some Differential Geometric Relations in the Elastic Shell

Abstract

The theory of the elastic shells is one of the most important parts of the theory of solid mechanics. The elastic shell can be described with its middle surface; that is, the three-dimensional elastic shell with equal thickness comprises a series of overlying surfaces like middle surface. In this paper, the differential geometric relations between elastic shell and its middle surface are provided under the curvilinear coordinate systems, which are very important for forming two-dimensional linear and nonlinear elastic shell models. Concretely, the metric tensors, the determinant of metric matrix field, the Christoffel symbols, and Riemann tensors on the three-dimensional elasticity are expressed by those on the two-dimensional middle surface, which are featured by the asymptotic expressions with respect to the variable in the direction of thickness of the shell. Thus, the novelty of this work is that we can further split three-dimensional mechanics equations into two-dimensional variation problems. Finally, two kinds of special shells, hemispherical shell and semicylindrical shell, are provided as the examples.

Experimental Study on Echo Characteristics of Multistatic Large-Scale Underwater Complex Target

“…The study of the complex echo characteristics of underwater targets is primarily based on the theory of elastic spherical shell [1,2], infinite cylindrical shell [3][4][5], and finite cylindrical shell [6][7][8] acoustic scattering. In recent years, significant progress has been made in studies on sound scattering of finite cylindrical shells [9][10][11] with ring ribs and ringribbed plates.…”