Some Differential Geometric Relations in the Elastic Shell

Xiaoqin Shen,Qian Yang,Haoming Li,Kaitai Li,Xiaoshan Cao

doi:10.1155/2016/1463823

Xiaoqin Shen, Qian Yang + Show 3 more

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https://doi.org/10.1155/2016/1463823

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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.

Highlights

In [1, 2], differential geometric formulae of three-dimensional (3D) domains and two-dimensional (2D) surface are defined in curvilinear ordinates, respectively
Assume that there is a shell with middle surface S = θ⃗(ω) whose thickness 2ε > 0 is arbitrarily small, where ω is open, bounded, and connected in R2 with Lipschitz-continuous boundary γ = ∂ω and θ⃗ ∈ C3(ω; R3)
Under the assumptions of Theorem 1, let gij be the contravariant components of the metric tensors on Θ⃗ (y, ξ)

Summary

Introduction

In [1, 2], differential geometric formulae of three-dimensional (3D) domains and two-dimensional (2D) surface are defined in curvilinear ordinates, respectively. The Riemann tensors on the middle surface S are defined by (cf [10]) The covariant and contravariant components gij and gij of the metric tensor of Θ⃗ (Ωε), the Christoffel symbols Γij,k and Γikj on Θ⃗ (Ωε) are defined as follows (the explicit dependence on the variable x ∈ Ω is dropped): gij fl g⃗i ⋅ g⃗j, gij fl g⃗i ⋅ g⃗j, (14)

Results

Conclusion