Some Differential Geometric Relations in the Elastic Shell

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.


Introduction
In [1,2], differential geometric formulae of three-dimensional (3D) domains and two-dimensional (2D) surface are defined in curvilinear ordinates, respectively.Besides, there are some scientists, such as Pobedrya [3], Vekua [4], and Nikabadze [5], who have some contributions in this field.In this paper, we assume that the three-dimensional elastic shell with equal thickness comprises a series of overlying surfaces like middle surface.Thus, the differential geometric relations between 3D elasticity and 2D middle surface are provided which are very important for forming 2D shell model from 3D equations (cf.[6][7][8][9]).Concretely, the metric tensor, the determinant of metric matrix field, the Christoffel symbols, and Riemann tensors on the 3D domain are expressed by those on the 2D surface, which are featured by the asymptotic expressions with respect to the variable in the direction of thickness of the shell.In Section 3, two kinds of special shells, that is, hemispherical shell and semicylindrical shell, are provided as the examples.
In this section, we mainly introduce some notations.Our notations are essentially borrowed from [2].In what follows, Latin indices and exponents , , , . . .take their values in the set {1, 2, 3}, whereas Greek indices and exponents , , , . . .take their values in the set {1, 2}.In addition, the repeated index summation convention is systematically used.The Euclidean scalar product and the exterior product of ⃗ , ⃗  ∈ R 3 are noted by ⃗  ⋅ ⃗  and ⃗  × ⃗ , respectively.Let  (cf. Figure 1) be an open, bounded, connected subset of R 2 , the boundary  =  of which is Lipschitz-continuous, and let  =  0 ∪  1 with  0 ∩  1 = 0. Let  = (  ) denote a generic point in the set  (i.e., closure of ) and let   fl /  .Let there be given an injective mapping ⃗  ∈  3 (; R 3 ), such that the two vectors are linearly independent at all points  ∈ .These two vectors thus span the tangent plane to the surface at the point ⃗ (), and the unit vector Figure 1: Two-dimensional domain  and surface  (cf.[2]).
is normal to  at the point ⃗ ().These vectors ⃗   () constitute the covariant basis at the point (), whereas the vectors ⃗   () defined by the relations constitute the contravariant basis at the point (), where    is the Kronecker symbol (note that ⃗  3 () = ⃗  3 () and the vector ⃗   () is also in the tangent plane to  at ⃗ ()) (cf. Figure 1).The covariant and contravariant components   and   of the metric tensor of , the Christoffel symbol * Γ , on , the covariant and mixed components   and    of the curvature tensor of , and the covariant of the third fundament form on  are then defined as follows (the explicit dependence on the variable  ∈  is henceforth dropped): where Thus, the Riemann tensors on the middle surface  are defined by (cf.[10]) Then, the covariant components of Riemann tensors on  are defined by * Assume that there is a shell Ω (cf. Figure 2) with middle surface  = ⃗ () and whose thickness 2 > 0 is arbitrarily small.Hence, for each  > 0, the reference configuration of the shell is In this sense, the 3D elastic shell with equal thickness comprises a series of overlying surfaces like middle surface.The top and bottom faces of ⃗ Θ(Ω  ) are Γ  = ⃗ Θ( × {+}) and The mapping ⃗ Θ : Ω  →  3 is injective and the three vectors are linearly independent at all points  ∈ Ω  .The vectors ⃗   () are defined by the relations These relations constitute the contravariant basis at the point ⃗ Θ() ∈ .The covariant and contravariant components   and   of the metric tensor of ⃗ Θ(Ω  ), the Christoffel symbols Γ , and Γ   on ⃗ Θ(Ω  ) are then defined as follows (the explicit dependence on the variable  ∈ Ω is henceforth dropped): The determinant of metric tensor is Thus, the Riemann tensors on ⃗ Θ(Ω  ) are defined by Then, the covariant components of Riemann tensors on ⃗ Θ(Ω  ) are defined by

Examples
3.1.Hemispherical Shell.Assume that the middle surface  of shell is a hemispherical surface (see Figure 3) whose reference equation is given by the mapping ⃗ () defined by where  = 1 m is the radius of the middle surface , 0 ≤  1 ≤ 2 is longitude, and 0 ≤  2 ≤ /2 is colatitude.The thickness of the middle surface  is 2 where  is the semithickness. Then, Hence, the covariant and contravariant components of the metric tensor on  are given by Then, Thus, The Christoffel symbols on  are as follows: * where − ≤  ≤ .Therefore, the covariant and contravariant components of the metric tensor on ⃗ Θ(Ω) are given by The Christoffel symbols on ⃗ Θ(Ω) are as follows:

Semicylindrical Shell.
Assume that the middle surface  of shell is a semicylindrical surface (see Figure 4) whose reference equation is given by the mapping ⃗ () defined by where  = 1 m is a constant, 0 ≤  1 ≤ , and 0 ≤  2 ≤ ℎ (ℎ = 3 m).The thickness of the middle surface  is 2 where  is the semithickness.Then,  (69)
) is symmetric and positive-definite matrix field, (  ) and (  ) are symmetric matrix fields.