# Some Differential Geometric Relations in the Elastic Shell.

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 i, j, k, ... take their values in the set {1,2,3}, whereas Greek indices and exponents [alpha], [beta], [gamma], ... 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 [??], [??] [member of] [R.sup.3] are noted by [mathematical expression not reproducible], respectively.

Let [omega] (cf. Figure 1) be an open, bounded, connected subset of [R.sup.2], the boundary [gamma] = [partial derivative][omega] of which is Lipschitz-continuous, and let [gamma] = [[gamma].sub.0] [union] [[gamma].sub.1] with [[gamma].sub.0] [intersection] [[gamma].sub.1] = 0. Let y = ([y.sub.[alpha]]) denote a generic point in the set [bar.[omega]] (i.e., closure of [omega]) and let [[partial derivative].sub.[alpha]] := [partial derivative]/[partial derivative][y.sub.[alpha]]. Let there be given an injective mapping [??] [member of] [C.sup.3]([bar.[omega]]; [R.sup.3]), such that the two vectors

[mathematical expression not reproducible] (1)

are linearly independent at all points y [member of] [[bar.[omega]]. These two vectors thus span the tangent plane to the surface

S := [??] ([bar.[omega]]) (2)

at the point [??](y), and the unit vector

[mathematical expression not reproducible] (3)

is normal to S at the point [??](y). These vectors [[??].sub.i](y) constitute the covariant basis at the point [theta](y), whereas the vectors [[??].sup.i](y) defined by the relations

[mathematical expression not reproducible] (4)

constitute the contravariant basis at the point [theta](y), where [[delta].sup.i.sub.j] is the Kronecker symbol (note that [mathematical expression not reproducible] and the vector [[??].sup.[alpha]](y) is also in the tangent plane to S at [??](y)) (cf. Figure 1).

The covariant and contravariant components [a.sub.[alpha][beta]] and [a.sup.[alpha][beta]] of the metric tensor of S, the Christoffel symbol [mathematical expression not reproducible] on S, the covariant and mixed components [b.sub.[alpha][beta]] and [b.sub.[alpha][beta]] of the curvature tensor of S, and the covariant of the third fundament form on S are then defined as follows (the explicit dependence on the variable y [member of] [bar.[omega]] is henceforth dropped):

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible], (7)

where ([a.sub.[alpha][beta]]) is symmetric and positive-definite matrix field, ([b.sub.[alpha][beta]]) and ([c.sub.[alpha][beta]]) are symmetric matrix fields. The determinants of metric tensor, curvature tensor, and the third fundament form are

[mathematical expression not reproducible]. (8)

Thus, the Riemann tensors on the middle surface S are defined by (cf. [10])

[mathematical expression not reproducible]. (9)

Then, the covariant components of Riemann tensors on S are defined by

[mathematical expression not reproducible]. (10)

Assume that there is a shell [[??].sup.[epsilon]] (cf. Figure 2) with middle surface S = [??]([bar.[omega]]) and whose thickness 2[epsilon] > 0 is arbitrarily small. Hence, for each [epsilon] > 0, the reference configuration of the shell is [mathematical expression not reproducible]; that is,

[mathematical expression not reproducible]. (11)

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 [mathematical expression not reproducible]. The lateral face is [mathematical expression not reproducible] denote a generic point in the set [[bar.[OMEGA].sup.[epsilon]]. The mapping [mathematical expression not reproducible] is injective and the three vectors

[mathematical expression not reproducible] (12)

are linearly independent at all points x [member of] [[bar.[OMEGA].sup.[epsilon]]. The vectors [[??].sup.i] (y) are defined by the relations

[mathematical expression not reproducible]. (13)

These relations constitute the contravariant basis at the point [??](x) [member of] S. The covariant and contravariant components [g.sub.ij] and [g.sup.ij] of the metric tensor of [mathematical expression not reproducible], the Christoffel symbols [mathematical expression not reproducible] are then defined as follows (the explicit dependence on the variable x [member of] [bar.[OMEGA]] is henceforth dropped):

[mathematical expression not reproducible]. (14)

The determinant of metric tensor is

g := det ([g.sub.ij]). (15)

Thus, the Riemann tensors on [mathematical expression not reproducible] are defined by

[mathematical expression not reproducible]. (16)

Then, the covariant components of Riemann tensors on [??]([[bar.[OMEGA].sup.[epsilon]]) are defined by

[R.sub.iljk] := [g.sub.lp] [R.sup.p.sub.ijk]. (17)

2. Main Results

Theorem 1. Assume that there is a shell with middle surface S = [??]([bar.[omega]]) whose thickness 2[epsilon] > 0 is arbitrarily small, where [omega] is open, bounded, and connected in [R.sup.2] with Lipschitz-continuous boundary [gamma] = [partial derivative][omega] and [??] [member of] [C.sup.3]([bar.[omega]]; [R.sup.3]). Hence, for each [epsilon] > 0, the reference configuration of the shell is [??]([bar.[OMEGA]]), where [bar.[OMEGA]] = [bar.[omega]] x [-[epsilon], [epsilon]]; that is,

[mathematical expression not reproducible]. (18)

The metric tensors on [mathematical expression not reproducible], respectively. [b.sub.[alpha][beta]] and [c.sub.[alpha][beta]] are the second and third fundamental forms on [??]([bar.[omega]]). Then, the following differential geometric relations hold:

[mathematical expression not reproducible]. (19)

Proof.

[mathematical expression not reproducible]. (20)

Submitting (1) and (5)-(7) into (20), based on the symmetry of [b.sub.[alpha][beta]], we have

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible]. (22)

the definition of [a.sub.3], we know

[mathematical expression not reproducible]. (23)

Then,

[mathematical expression not reproducible]. (24)

Thus,

[mathematical expression not reproducible]. (25)

Submitting (23)-(25) into (22), we get

[g.sub.3[alpha]] = 0. (26)

Similarly,

[mathematical expression not reproducible]. (27)

Since ([g.sup.ij]) = [([g.sub.ij]).sup.-1], the contravariant components of [g.sup.ij] should be expressed as follows.

Theorem 2. Under the assumptions of Theorem 1, let [g.sup.ij] be the contravariant components of the metric tensors on [??](y, [xi]). Then, the following formulae hold:

[mathematical expression not reproducible], (28)

where [mathematical expression not reproducible].

Proof.

[mathematical expression not reproducible], (29)

where

[mathematical expression not reproducible]. (30)

Since [([g.sup.ij]) = [([g.sub.ij]).sup.-1], formula (28) can be derived easily.

Theorem 3. Under the assumptions of Theorem 1, let [[GAMMA].sub.ij,k] and [[??].sub.[alpha][beta],[gamma]] be the Christoffel symbols on [mathematical expression not reproducible], respectively. Then, the following formulae hold:

[mathematical expression not reproducible], (31)

Proof.

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible]. (33)

Since [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (34)

Thus,

[mathematical expression not reproducible]. (35)

Submitting (35) and (7) into (33), we get

[mathematical expression not reproducible]. (36)

Similarly,

[mathematical expression not reproducible]. (37)

Thus, the Christoffel symbols [mathematical expression not reproducible] have similar relations.

Theorem 4. Under the assumptions of Theorem 1, let [[GAMMA].sup.k.sub.ij] be the Christoffel symbols on [??](y,[xi]). Then, the following formulae hold:

[mathematical expression not reproducible], (38)

Proof. Because of (13), we have

[mathematical expression not reproducible]. (39)

Thus, formula (38) can be derived easily from the results of Theorems 2 and 3.

Theorem 5. Under the assumptions of Theorem 1, let [mathematical expression not reproducible] be the Riemann tensors on [mathematical expression not reproducible], respectively. Then, the following formulae hold:

[mathematical expression not reproducible], (40)

[mathematical expression not reproducible]. (41)

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

[mathematical expression not reproducible]. (42)

Submitting [mathematical expression not reproducible] into (42), we have

[mathematical expression not reproducible]. (43)

Submitting (42) into (43), we have

[mathematical expression not reproducible]. (44)

Similarly,

[mathematical expression not reproducible]. (45)

Because of [mathematical expression not reproducible], we can deduce by (44)-(45) that

[mathematical expression not reproducible]. (46)

Since [mathematical expression not reproducible] are linearly independent, we have

[mathematical expression not reproducible]. (47)

Thus, formula (41) has been proven.

3. Examples

3.1. Hemispherical Shell. Assume that the middle surface S of shell is a hemispherical surface (see Figure 3) whose reference equation is given by the mapping [??]([bar.[omega]]) defined by

[mathematical expression not reproducible], (48)

where r = 1 m is the radius of the middle surface S, 0 [less than or equal to] [[gamma].sub.1] [less than or equal to] 2[pi] is longitude, and 0 [less than or equal to] [y.sub.2] [less than or equal to] [pi]/2 is colatitude. The thickness of the middle surface S is 2[epsilon] where [epsilon] is the semithickness.

Then,

[mathematical expression not reproducible], (49)

Hence, the covariant and contravariant components of the metric tensor on S are given by

[mathematical expression not reproducible]. (50)

Then,

[mathematical expression not reproducible]. (51)

Thus,

[mathematical expression not reproducible]. (52)

The Christoffel symbols on S are as follows:

[mathematical expression not reproducible]. (53)

The Riemann tensors on S are as follows:

[mathematical expression not reproducible]. (54)

Hence, for each [epsilon] > 0, the reference configuration of the shell with middle surface [mathematical expression not reproducible]

[mathematical expression not reproducible], (55)

where -[epsilon] [less than or equal to] [xi] [less than or equal to] [epsilon].

Therefore, the covariant and contravariant components of the metric tensor on [??]([bar.[OMEGA]]) are given by

[mathematical expression not reproducible]. (56)

The Christoffel symbols on [??]([bar.[OMEGA]]) are as follows:

[mathematical expression not reproducible],

other [[GAMMA].sub.ij,k] = 0,

[mathematical expression not reproducible]. (57)

The Riemann tensors on [??]([bar.[OMEGA]) are as follows:

[mathematical expression not reproducible]. (58)

3.2. Semicylindrical Shell. Assume that the middle surface S of shell is a semicylindrical surface (see Figure 4) whose reference equation is given by the mapping [??]([bar.[omega]]) defined by

[??]([y.sub.1], [y.sub.2]) = (r cos [y.sub.1], r sin [y.sub.1], [y.sub.2]), (59)

where r = 1 misa constant, 0 [less than or equal to] [[gamma].sub.1] [less than or equal to] [pi], and 0 [less than or equal to] [y.sub.2] [less than or equal to] h (h = 3 m). The thickness of the middle surface S is 2e where e is the semithickness.

Then,

[mathematical expression not reproducible]. (60)

Therefore, the covariant and contravariant components of the metric tensor on S are given by

[mathematical expression not reproducible]. (61)

Then,

[mathematical expression not reproducible]. (62)

Thus,

[mathematical expression not reproducible]. (63)

The Christoffel symbols on S are

[mathematical expression not reproducible]. (64)

The Riemann tensors on S are as follows:

[mathematical expression not reproducible]. (65)

Hence, for each [epsilon] > 0, the reference configuration of the shell with middle surface [mathematical expression not reproducible]

[mathematical expression not reproducible], (66)

where -[epsilon] [less than or equal to] [xi] [less than or equal to] [epsilon].

So, the covariant and contravariant components of the metric tensor on [??]([bar.[OMEGA]]) are given by

[mathematical expression not reproducible]. (67)

The Christoffel symbols on [??]([bar.[OMEGA]]) are as follows:

[mathematical expression not reproducible]. (68)

The Riemann tensors on [??]([bar.[OMEGA]]) are as follows:

[mathematical expression not reproducible]. (69)

Competing Interests

There are no competing interests regarding this paper.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (NSFC 11571275, NSFC11572244) and Program of Industry in Shaanxi Province (2015GY021).

References

[1] P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Heidelberg, Germany, 2005.

[2] P. G. Ciarlet, Mathematical Elasticity, vol. 3 of Theory of Shells, North-Holland, 2000.

[3] B. E. Pobedrya, Lectures on Tensor Analysis, MGU, Moscow, Russia, 1986 (Russian).

[4] I. N. Vekua, The Basics of Tensor Analysis and Theory of Covariants, Nauka, Moscow, Russia, 1978 (Russian).

[5] M. U. Nikabadze, "On some problems of tensor calculus. I," Journal of Mathematical Sciences, vol. 161, no. 5, pp. 668-697, 2009.

[6] W. T. Koiter, "A consistent first approximation in the general theory of thin elastic shells," in Proceedings of the IUTAM Symposium on the Theory of Thin Elastic Shells, pp. 12-33, Delft, The Netherlands, August 1959.

[7] W. T. Koiter, "On the foundations of the linear theory of thin elastic shells," Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen B, vol. 73, pp. 169-195, 1970.

[8] K. Li and X. Shen, "A dimensional splitting method for the linearly elastic shell," International Journal of Computer Mathematics, vol. 84, no. 6, pp. 807-824, 2007.

[9] X. Shen, K. Li, and Y. Ming, "Asymptotic expansions of stress tensor for linearly elastic shell," Applied Mathematical Modelling, vol. 37, no. 16-17, pp. 7964-7972, 2013.

[10] K. Li and A. Huang, Tensor Analysis and Its Applications, Chinese Scientific Press, 2004 (Chinese).

[11] L. Xiao, "Justification of two-dimensional nonlinear dynamic shell equations of Koiter's type," Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 3, pp. 383-395, 2005.

[12] P. G. Ciarlet, Differential Geometry: Theory and Applications, Higher Education Press, Shanghai, China, 2007.

http://dx.doi.org/10.1155/2016/1463823

Xiaoqin Shen, (1) Haoming Li,1 Kaitai Li, (2) Xiaoshan Cao, (1,3) and Qian Yang (1)

(1) School of Sciences, Xi'an University of Technology, Xi'an 710054, China

(2) School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

(3) State Key Laboratory of Transducer Technology Chinese Academy of Sciences, Shanghai 200050, China

Correspondence should be addressed to Xiaoqin Shen; xqshen@xaut.edu.cn

Received 30 September 2016; Revised 7 November 2016; Accepted 9 November 2016

Caption: Figure 1: Two-dimensional domain u and surface S (cf. [2]).

Caption: Figure 2: The shell [[??].sup.[epsilon]] with middle surface S (cf. [2])

Caption: Figure 3: Middle surface of hemispherical shell.

Caption: Figure 4: Middle surface of semicylindrical shell.