MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2016/1463823 1463823 Research Article Some Differential Geometric Relations in the Elastic Shell http://orcid.org/0000-0003-1110-2566 Shen Xiaoqin 1 Li Haoming 1 Li Kaitai 2 Cao Xiaoshan 1,3 3 Yang Qian 1 Merodio Jose 1 School of Sciences Xi’an University of Technology Xi’an 710054 China xaut.edu.cn 2 School of Mathematics and Statistics Xi’an Jiaotong University Xi’an 710049 China xjtu.edu.cn 3 State Key Laboratory of Transducer Technology Chinese Academy of Sciences Shanghai 200050 China cas.cn 2016 15122016 2016 30 09 2016 07 11 2016 09 11 2016 2016 Copyright © 2016 Xiaoqin Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China NSFC 11571275 NSFC 11572244 Program of Industry in Shaanxi Province 2015GY021
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 , Vekua , and Nikabadze , 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. ). 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 . In what follows, Latin indices and exponents i,j,k, take their values in the set {1,2,3}, whereas Greek indices and exponents α,β,γ, 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 a,bR3 are noted by a·b and a×b, respectively.

Let ω (cf. Figure 1) be an open, bounded, connected subset of R2, the boundary γ=ω of which is Lipschitz-continuous, and let γ=γ0γ1 with γ0γ1=. Let y=(yα) denote a generic point in the set ω¯ (i.e., closure of ω) and let α/yα. Let there be given an injective mapping θC3(ω¯;R3), such that the two vectors (1)aαyαθyare linearly independent at all points yω¯. These two vectors thus span the tangent plane to the surface(2)Sθω¯at the point θ(y), and the unit vector(3)a3ya1y×a2ya1y×a2yis normal to S at the point θ(y). These vectors ai(y) constitute the covariant basis at the point θ(y), whereas the vectors ai(y) defined by the relations(4)aiy·ajy=δjiconstitute the contravariant basis at the point θ(y), where δji is the Kronecker symbol (note that a3(y)=a3(y) and the vector aα(y) is also in the tangent plane to S at θ(y)) (cf. Figure 1).

Two-dimensional domain ω and surface S (cf. ).

The covariant and contravariant components aαβ and aαβ of the metric tensor of S, the Christoffel symbol Γαβ,σ on S, the covariant and mixed components bαβ and bαβ 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 yω¯ is henceforth dropped): (5)aαβaα·aβ,aαβaα·aβ,aαβ=aαβ-1,aα=aαβaβ,(6)Γαβ,σaσ·αaβ,Γαβσaσ·αaβ,(7)bαβa3·αaβ,bαβaβσbσα,cαβαa3·βa3,where (aαβ) is symmetric and positive-definite matrix field, (bαβ) and (cαβ) are symmetric matrix fields. The determinants of metric tensor, curvature tensor, and the third fundament form are(8)adetaαβ,bdetbαβ,cdetcαβ.

Thus, the Riemann tensors on the middle surface S are defined by (cf. ) (9)RαβγσγΓαβσ-βΓαγσ+ΓαβλΓλγσ-ΓαγλΓλβσ.Then, the covariant components of Riemann tensors on S are defined by (10)RαβλσaγβRαλσγ.

Assume that there is a shell Ω^ε (cf. Figure 2) with middle surface S=θ(ω¯) and whose thickness 2ε>0 is arbitrarily small. Hence, for each ε>0, the reference configuration of the shell is Ω^ε=Θ(Ω¯ε), where Ω¯ε=ω¯×[-ε,ε]; that is,(11)Θy,ξ=θy+ξa3y,-εξε.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¯ε) are Γt=Θ(ω×{+ε}) and Γb=Θ(ω×{-ε}). The lateral face is Γl=Γ0Γ1, where Γ0=θ(γ0)×[-ε,+ε], Γ1=θ(γ1)×[-ε,+ε] (cf. ). Let x=(xi)=(y1,y2,ξ) denote a generic point in the set Ω¯ε. The mapping Θ:Ω¯εR3 is injective and the three vectors (12)gixiΘxare linearly independent at all points xΩ¯ε. The vectors gi(y) are defined by the relations(13)gix·gjx=δji.These relations constitute the contravariant basis at the point Θ(x)S. The covariant and contravariant components gij and gij of the metric tensor of Θ(Ω¯ε), the Christoffel symbols Γij,k and Γijk on Θ(Ω¯ε) are then defined as follows (the explicit dependence on the variable xΩ¯ is henceforth dropped): (14)gijgi·gj,gijgi·gj,Γij,kgk·igj,Γijk=gklΓij,l.

The shell Ω^ε with middle surface S (cf. ).

The determinant of metric tensor is(15)gdetgij.

Thus, the Riemann tensors on Θ(Ω¯ε) are defined by (16)RikjpγΓijp-βΓikp+ΓijqΓqkp-ΓikqΓqjp.Then, the covariant components of Riemann tensors on Θ(Ω¯ε) are defined by (17)RiljkglpRijkp.

2. Main Results Theorem 1.

Assume that there is a shell with middle surface S=θ(ω¯) whose thickness 2ε>0 is arbitrarily small, where ω is open, bounded, and connected in R2 with Lipschitz-continuous boundary γ=ω and θC3(ω¯;R3). Hence, for each ε>0, the reference configuration of the shell is Θ(Ω¯), where Ω¯=ω¯×[-ε,ε]; that is, (18)Θy,ξ=θy+ξa3y.The metric tensors on Θ(y,ξ) and θ(ω¯ε) are gij and aαβ, respectively. bαβ and cαβ are the second and third fundamental forms on θ(ω¯). Then, the following differential geometric relations hold: (19)gαβ=aαβ-2ξbαβ+ξ2cαβ,gα3=g3α=0,g33=1,α,β=1,2,ξ-ε,ε.

Proof.

(20)gαβ=gα·gβ=αΘ·βΘ=αθ+ξa3·βθ+ξa3=αθ·βθ+αθ·βξa3+αξa3·βθ+αξa3·βξa3=aα·aβ+ξaα·βa3+ξαa3·aβ+ξ2αa3·βa3.Submitting (1) and (5)–(7) into (20), based on the symmetry of bαβ, we have (21)gαβ=aαβ-2ξbαβ+ξ2cαβ,(22)g3α=g3·gα=3Θ·αΘ=3θ+ξa3·αθ+ξa3=a3·αθ+ξa3=a3·αθ+a3·αξa3=a3·aα+ξa3·αa3.From the definition of a3, we know (23)a3·aα=0,a3·a3=1.Then,(24)αa3·a3=2a3·αa3=0.Thus, (25)a3·αa3=0.Submitting (23)-(25) into (22), we get (26)g3α=0.Similarly, (27)gα3=0,g33=g3·g3=3Θ·3Θ=3θ+ξa3·3θ+ξa3=3θ·3θ+3θ·3ξa3+3ξa3·3θ+3ξa3·3ξa3=0+0+0+a3·a3=1.

Since (gij)=(gij)-1, the contravariant components of gij should be expressed as follows.

Theorem 2.

Under the assumptions of Theorem 1, let gij be the contravariant components of the metric tensors on Θ(y,ξ). Then, the following formulae hold:(28)g11=g-1a22-2ξb22+ξ2c22,g12=g21=-g-1a12-2ξb12+ξ2c12,g22=g-1a11-2ξb11+ξ2c11,gα3=g3α=0,g33=1,where g=det(gij)=(a11-2ξb11+ξ2c11)(a22-2ξb22+ξ2c22)-(a12-2ξb12+ξ2c12)2.

Proof.

(29) g i j - 1 = g α β 0 0 g 33 - 1 = g α β - 1 0 0 g 33 - 1 , where(30)gαβ-1=g11g12g21g22-1=g-1g22-g12-g21g11.

Since (gij)=(gij)-1, formula (28) can be derived easily.

Theorem 3.

Under the assumptions of Theorem 1, let Γij,k and Γαβ,γ be the Christoffel symbols on Θ(y,ξ) and θ(ω¯), respectively. Then, the following formulae hold: (31)Γαβ,σ=Γαβ,σ+ξaσ·αβa3+ξσa3·αaβ+ξ2σa3·αβa3,Γαβ,3=bαβ-ξcαβ,Γα3,σ=Γ3α,σ=-bασ+ξcασ,Γ33,α=Γα3,3=Γ3α,3=Γ33,3=0,α,β,σ=1,2.

Proof.

(32)Γαβ,σ=gσ·αgβ=σΘ·αβΘ=σθ+ξa3·αβθ+ξa3=σθ·αβθ+σθ·αβξa3+σξa3·αβθ+σξa3·αβξa3=Γαβ,σ+ξaσ·αβa3+ξσa3·αaβ+ξ2σa3·αβa3,(33)Γαβ,3=g3·αgβ=3Θ·αβΘ=3θ+ξa3·αβθ+ξa3=a3·αβθ+ξa3=a3·αβθ+ξa3·αβa3.Since (a3,a3)=1, we have (34)αβa3,a3=αβa3,a3=α2a3·βa3=2αa3·βa3+2a3·αβa3=0.Thus, (35)a3·αβa3=-αa3·βa3.Submitting (35) and (7) into (33), we get (36)Γαβ,3=bαβ-ξcαβ,Γα3,σ=gσ·αg3=σΘ·α3Θ=σθ+ξa3·α3θ+ξa3=σθ+ξa3·αa3=σθ·αa3+ξσa3·αa3=-bασ+ξcασ.Similarly, (37)Γ3α,σ=-bασ+ξcασ,Γ33,α=gα·3g3=αΘ·33Θ=αθ+ξa3·33θ+ξa3=αθ+ξa3·3a3=0,Γα3,3=g3·αg3=3Θ·α3Θ=3θ+ξa3·α3θ+ξa3=a3·αa3=0,Γ33,3=g3·3g3=3Θ·33Θ=3θ+ξa3·33θ+ξa3=a3·3a3=0.

Thus, the Christoffel symbols Γijk and Γαβσ have similar relations.

Theorem 4.

Under the assumptions of Theorem 1, let Γijk be the Christoffel symbols on Θ(y,ξ). Then, the following formulae hold: (38)Γαβσ=gστΓαβ,τ+ξaτ·αβa3+ξτa3·αaβ+ξ2τa3·αβa3,Γαβ3=bαβ-ξcαβ,Γα3σ=Γ3ασ=gστ-bατ+ξcατ,Γ33α=Γα33=Γ3α3=Γ333=0,α,β,σ=1,2.

Proof.

Because of (13), we have (39)Γαβσ=gσlΓαβ,l=gστΓαβ,τ+gσ3Γαβ,3=gστΓαβ,τ,Γαβ3=g3lΓαβ,l=g31Γαβ,1+g32Γαβ,2+g33Γαβ,3=Γαβ,3,Γα3σ=Γ3ασ=gσlΓα3,l=gστΓα3,τ+gσ3Γα3,3=gστΓα3,τ,Γ33α=gαlΓ33,l=gατΓ33,τ+gα3Γ33,3=0,Γα33=Γ3α3=g3lΓα3,l=g3τΓα3,τ+g33Γα3,3=Γα3,3,Γ333=g3lΓ33,l=g3τΓ33,τ+g33Γ33,3=Γ33,3.

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

Theorem 5.

Under the assumptions of Theorem 1, let Rikjp,Rilkj and Rαγβσ,Rαλγβ be the Riemann tensors on Θ(y,ξ) and θ(y), respectively. Then, the following formulae hold: (40)Rikjp=0,Rilkj=0,i,j,k,p,l=1,2,3;(41)Rαγβσ=bαβbγσ-bαγbβσ,Rαλγβ=bαβbλγ-bαγbβλ,α,β,σ,γ,λ=1,2.

Proof.

As we all know, formula (40) has been proven by Ciarlet in  (cf. Theorem  1.6-1). We only should prove formula (41).

From Gaussian formula of coordinate systems (cf. ), we have (42)βaα=βaα·aλaλ+βaα·a3a3=Γαβλaλ+bαβa3.Submitting λa3=-bλσaσ into (42), we have (43)βγaα=γΓαβλaλ+Γαβλγaλ+γbαβa3+bαβγa3.Submitting (42) into (43), we have (44)βγaα=γΓαβσ+ΓαβλΓλγσ-bαβbγσaσ+Γαβλbγλ+γbαβa3.

Similarly,(45)γβaα=βΓαγσ+ΓαγλΓλβσ-bαγbβσaσ+Γαγλbβλ+βbαγa3.

Because of βγaα=γβaα, we can deduce by (44)-(45) that (46)0=γΓαβσ-βΓαγσ+ΓαβλΓλγσ-ΓαγλΓλβσ-bαβbγσ+bαγbβσaσ+Γαβλbγλ-Γαγλbβλ+γbαβ-βbαγa3.Since aσ and a3 are linearly independent, we have (47)γΓαβσ-βΓαγσ+ΓαβλΓλγσ-ΓαγλΓλβσ=bαβbγσ-bαγbβσ,γbαβ-Γαγλbβλ=βbαγ-Γαβλbγλ.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 θ(ω¯) defined by (48)θy1,y2=rcosy1siny2,rsiny1siny2,rcosy2,where r=1 m is the radius of the middle surface S, 0y12π is longitude, and 0y2π/2 is colatitude. The thickness of the middle surface S is 2ε where ε is the semithickness.

Middle surface of hemispherical shell.

Then, (49)a1=1θ=-rsiny1siny2,rcosy1siny2,0,a2=2θ=rcosy1cosy2,rsiny1cosy2,-rsiny2,1a1=-rcosy1siny2,-rsiny1siny2,0,1a2=2a1=-rsiny1cosy2,rcosy1cosy2,0,2a2=-rcosy1siny2,-rsiny1siny2,-rcosy2.Hence, the covariant and contravariant components of the metric tensor on S are given by (50)aαβ=r2sin2y200r2,aαβ=1r2sin2y2001r2.

Then, (51)a3=a1×a2a1×a2=cosy1siny2,siny1siny2,cosy2,1a3=-siny1siny2,cosy1siny2,0,2a3=cosy1cosy2,siny1cosy2,-siny2,11a3=-cosy1siny2,-siny1siny2,0,12a3=21a3=-siny1cosy2,cosy1cosy2,0,22a3=-cosy1cosy2,-siny1siny2,-cosy2,bαβ=-rsin2y200-r,cαβ=sin2y2001.Thus,(52)a=detaαβ=r4sin2y2,b=detbαβ=r2sin2y2,c=detcαβ=sin2y2.The Christoffel symbols on S are as follows: (53)Γ11,2=-r2siny2cosy2,Γ12,1=Γ21,1=r2siny2cosy2,other  Γαβ,γ=0,Γ121=Γ211=coty2,Γ112=-siny2cosy2,other  Γβγα=0.The Riemann tensors on S are as follows: (54)R1212=R2121=-r2sin2y2,R1221=R2112=r2sin2y2,other  Rαλσβ=0.

Hence, for each ε>0, the reference configuration of the shell with middle surface S=θ(ω¯) is Θ(Ω¯) (Ω¯=ω¯×[-ε,ε]) (55)Θy,ξ=θy+ξa3y,where -εξε.

Therefore, the covariant and contravariant components of the metric tensor on Θ(Ω¯) are given by (56)gij=r+ξ2sin2y2000r+ξ20001,gij=1r+ξ2sin2y20001r+ξ20001.The Christoffel symbols on Θ(Ω¯) are as follows: (57)Γ11,2=-r+ξ2siny2cosy2,Γ11,3=-r+ξsin2y2,Γ12,1=Γ21,1=r+ξ2siny2cosy2,Γ13,1=ξ-rsin2y2,Γ22,3=-r+ξ,Γ23,2=ξ-r,other  Γij,k=0,Γ121=Γ211=coty2,Γ131=Γ311=1r+ξ,Γ112=-siny2cosy2,Γ232=Γ322=1r+ξ,Γ113=-r+ξsin2y2,Γ223=-r+ξ,other  Γijk=0.

The Riemann tensors on Θ(Ω¯) are as follows: (58)Rikjp=0,Rilkj=0,i,j,k,p,l=1,2,3.

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 θ(ω¯) defined by (59)θy1,y2=rcosy1,rsiny1,y2,where r=1 m is a constant, 0y1π, and 0y2h(h=3m). The thickness of the middle surface S is 2ε where ε is the semithickness.

Middle surface of semicylindrical shell.

Then, (60)a1=1θ=-rsiny1,rcosy1,0,a2=2θ=0,0,1,1a1=-rcosy1,-rsiny1,0,1a2=2a1=0,0,0,2a2=0,0,0.Therefore, the covariant and contravariant components of the metric tensor on S are given by (61)aαβ=r2001,aαβ=r-2001.

Then,(62)a3=a1×a2a1×a2=cosy1,siny1,0,1a3=-siny1,cosy1,0,2a3=0,0,0,11a3=-cosy1,-siny1,0,12a3=21a3=0,0,0,22a3=0,0,0,bαβ=-r000,cαβ=1000.Thus, (63)a=detaαβ=r2,b=detbαβ=r2,c=detcαβ=1.The Christoffel symbols on S are(64)Γαβ,γ=0,Γβγα=0.The Riemann tensors on S are as follows: (65)Rαγβσ=0,Rαλγβ=0,α,β,σ,γ,λ=1,2.

Hence, for each ε>0, the reference configuration of the shell with middle surface S=θ(ω¯) is Θ(Ω¯) (Ω¯=ω¯×[-ε,ε]) (66)Θy,ξ=θy+ξa3y,where -εξε.

So, the covariant and contravariant components of the metric tensor on Θ(Ω¯) are given by (67)gij=r+ξ200010001,gij=1r+ξ200010001.The Christoffel symbols on Θ(Ω¯) are as follows: (68)Γ11,3=-r+ξ,Γ13,1=r+ξ,other  Γij,k=0,Γ131=Γ311=1r+ξ,Γ113=-r+ξ,other  Γijk=0.The Riemann tensors on Θ(Ω¯) are as follows: (69)Rikjp=0,Rilkj=0,i,j,k,p,l=1,2,3.

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).

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