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.
1. 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. [6–9]). 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 , whereas Greek indices and exponents take their values in the set . In addition, the repeated index summation convention is systematically used. The Euclidean scalar product and the exterior product of are noted by and , respectively.
Let (cf. Figure 1) be an open, bounded, connected subset of , the boundary of which is Lipschitz-continuous, and let with . Let denote a generic point in the set (i.e., closure of ) and let . Let there be given an injective mapping , such that the two vectors are linearly independent at all points . These two vectors thus span the tangent plane to the surfaceat the point , and the unit vectoris normal to at the point . These vectors constitute the covariant basis at the point , whereas the vectors defined by the relationsconstitute the contravariant basis at the point , where is the Kronecker symbol (note that 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 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 is symmetric and positive-definite matrix field, and are symmetric matrix fields. The determinants of metric tensor, curvature tensor, and the third fundament form are
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 (cf. Figure 2) with middle surface and whose thickness is arbitrarily small. Hence, for each , the reference configuration of the shell is , where ; that 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 are and . The lateral face is , where , (cf. [11]). Let denote a generic point in the set The mapping is injective and the three vectors are linearly independent at all points . The vectors are defined by the relationsThese relations constitute the contravariant basis at the point . The covariant and contravariant components and of the metric tensor of , the Christoffel symbols and on are then defined as follows (the explicit dependence on the variable is henceforth dropped):
The determinant of metric tensor is
Thus, the Riemann tensors on are defined by Then, the covariant components of Riemann tensors on are defined by
2. Main Results
Theorem 1. Assume that there is a shell with middle surface whose thickness is arbitrarily small, where is open, bounded, and connected in with Lipschitz-continuous boundary and . Hence, for each , the reference configuration of the shell is , where ; that is, The metric tensors on and are and , respectively. and are the second and third fundamental forms on . Then, the following differential geometric relations hold:
Proof. Submitting (1) and (5)–(7) into (20), based on the symmetry of , we have From the definition of , we know Then,Thus, Submitting (23)-(25) into (22), we get Similarly,
Since , the contravariant components of should be expressed as follows.
Theorem 2. Under the assumptions of Theorem 1, let be the contravariant components of the metric tensors on . Then, the following formulae hold:where .
Proof. whereSince , formula (28) can be derived easily.
Theorem 3. Under the assumptions of Theorem 1, let and be the Christoffel symbols on and , respectively. Then, the following formulae hold:
Proof. Since , we have Thus, Submitting (35) and (7) into (33), we get Similarly,
Thus, the Christoffel symbols and have similar relations.
Theorem 4. Under the assumptions of Theorem 1, let be the Christoffel symbols on . Then, the following formulae hold:
Proof. Because of (13), we have Thus, formula (38) can be derived easily from the results of Theorems 2 and 3.
Theorem 5. Under the assumptions of Theorem 1, let and be the Riemann tensors on and , respectively. Then, the following formulae hold:
Proof. As we all know, formula (40) has been proven by Ciarlet in [12] (cf. Theorem 1.6-1). We only should prove formula (41).
From Gaussian formula of coordinate systems (cf. [7]), we have Submitting into (42), we have Submitting (42) into (43), we have Similarly,Because of , we can deduce by (44)-(45) that Since and are linearly independent, we have Thus, formula (41) has been proven.
3. 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 m is the radius of the middle surface , is longitude, and is colatitude. The thickness of the middle surface is 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: The Riemann tensors on are as follows:
Hence, for each , the reference configuration of the shell with middle surface is () where .
Therefore, the covariant and contravariant components of the metric tensor on are given by The Christoffel symbols on are as follows:
The Riemann tensors on are as follows:
3.2. 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 m is a constant, , and . The thickness of the middle surface is where is the semithickness.
Then, Therefore, the covariant and contravariant components of the metric tensor on are given by
Then,Thus, The Christoffel symbols on areThe Riemann tensors on are as follows:
Hence, for each , the reference configuration of the shell with middle surface is () where .
So, the covariant and contravariant components of the metric tensor on are given by The Christoffel symbols on are as follows: The Riemann tensors on are as follows:
Competing Interests
There are no competing interests regarding this paper.
Acknowledgments
This paper is supported by National Natural Science Foundation of China (NSFC 11571275, NSFC 11572244) and Program of Industry in Shaanxi Province (2015GY021).