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 ChinaNSFC 11571275NSFC 11572244Program of Industry in Shaanxi Province2015GY0211. 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 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→,b→∈R3 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=γ0∪γ1 with γ0∩γ1=∅. 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 θ→∈C3(ω¯;R3), such that the two vectors (1)a→αy≔∂αθ→yare linearly independent at all points y∈ω¯. These two vectors thus span the tangent plane to the surface(2)S≔θ→ω¯at the point θ→(y), and the unit vector(3)a→3y≔a→1y×a→2ya→1y×a→2yis normal to S at the point θ→(y). These vectors a→i(y) constitute the covariant basis at the point θ(y), whereas the vectors a→i(y) defined by the relations(4)a→iy·a→jy=δjiconstitute the contravariant basis at the point θ(y), where δji is the Kronecker symbol (note that a→3(y)=a→3(y) and the vector a→α(y) is also in the tangent plane to S at θ→(y)) (cf. Figure 1).

Two-dimensional domain ω and surface S (cf. [2]).

The covariant and contravariant components aαβ and aαβ of the metric tensor of S, the Christoffel symbol Γ∗αβ,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 y∈ω¯ is henceforth dropped): (5)aαβ≔a→α·a→β,aαβ≔a→α·a→β,aαβ=aαβ-1,a→α=aαβa→β,(6)Γ∗αβ,σ≔a→σ·∂αa→β,Γ∗αβσ≔a→σ·∂αa→β,(7)bαβ≔a→3·∂αa→β,bαβ≔aβσbσα,cαβ≔∂αa→3·∂βa→3,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)a≔detaαβ,b≔detbαβ,c≔detcαβ.

Thus, the Riemann tensors on the middle surface S are defined by (cf. [10]) (9)R∗αβγσ≔∂γΓ∗αβσ-∂βΓ∗αγσ+Γ∗αβλΓ∗λγσ-Γ∗αγλΓ∗λβσ.Then, the covariant components of Riemann tensors on S are defined by (10)R∗αβλσ≔aγβR∗αλσγ.

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)Θ→y,ξ=θ→y+ξa→3y,-ε≤ξ≤ε.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=Γ0∪Γ1, where Γ0=θ→(γ0)×[-ε,+ε], Γ1=θ→(γ1)×[-ε,+ε] (cf. [11]). Let x=(xi)=(y1,y2,ξ) denote a generic point in the set Ω¯ε. The mapping Θ→:Ω¯ε→R3 is injective and the three vectors (12)g→ix≔∂iΘ→xare linearly independent at all points x∈Ω¯ε. The vectors g→i(y) are defined by the relations(13)g→ix·g→jx=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 x∈Ω¯ is henceforth dropped): (14)gij≔g→i·g→j,gij≔g→i·g→j,Γij,k≔g→k·∂ig→j,Γijk=gklΓij,l.

The shell Ω^ε with middle surface S (cf. [2]).

The determinant of metric tensor is(15)g≔detgij.

Thus, the Riemann tensors on Θ→(Ω¯ε) are defined by (16)Rikjp≔∂γΓijp-∂βΓikp+ΓijqΓqkp-ΓikqΓqjp.Then, the covariant components of Riemann tensors on Θ→(Ω¯ε) are defined by (17)Riljk≔glpRijkp.

2. Main ResultsTheorem 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 θ→∈C3(ω¯;R3). Hence, for each ε>0, the reference configuration of the shell is Θ→(Ω¯), where Ω¯=ω¯×[-ε,ε]; that is, (18)Θ→y,ξ=θ→y+ξa→3y.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=g→α·g→β=∂αΘ→·∂βΘ→=∂αθ→+ξa→3·∂βθ→+ξa→3=∂αθ→·∂βθ→+∂αθ→·∂βξa→3+∂αξa→3·∂βθ→+∂αξa→3·∂βξa→3=a→α·a→β+ξa→α·∂βa→3+ξ∂αa→3·a→β+ξ2∂αa→3·∂βa→3.Submitting (1) and (5)–(7) into (20), based on the symmetry of bαβ, we have (21)gαβ=aαβ-2ξbαβ+ξ2cαβ,(22)g3α=g→3·g→α=∂3Θ→·∂αΘ→=∂3θ→+ξa→3·∂αθ→+ξa→3=a→3·∂αθ→+ξa→3=a→3·∂αθ→+a→3·∂αξa→3=a→3·a→α+ξa→3·∂αa→3.From the definition of a3, we know (23)a→3·a→α=0,a→3·a→3=1.Then,(24)∂αa→3·a→3=2a→3·∂αa→3=0.Thus, (25)a→3·∂αa→3=0.Submitting (23)-(25) into (22), we get (26)g3α=0.Similarly, (27)gα3=0,g33=g→3·g→3=∂3Θ→·∂3Θ→=∂3θ→+ξa→3·∂3θ→+ξa→3=∂3θ→·∂3θ→+∂3θ→·∂3ξa→3+∂3ξa→3·∂3θ→+∂3ξa→3·∂3ξa→3=0+0+0+a→3·a→3=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.

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

Theorem 3.

Under the assumptions of Theorem 1, let Γij,k and Γ∗αβ,γ be the Christoffel symbols on Θ→(y,ξ) and θ→(ω¯), respectively. Then, the following formulae hold: (31)Γαβ,σ=Γ∗αβ,σ+ξa→σ·∂αβa→3+ξ∂σa→3·∂αa→β+ξ2∂σa→3·∂αβa→3,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=g→σ·∂αg→β=∂σΘ→·∂α∂βΘ→=∂σθ→+ξa→3·∂αβθ→+ξa→3=∂σθ→·∂αβθ→+∂σθ→·∂αβξa→3+∂σξa→3·∂αβθ→+∂σξa→3·∂αβξa→3=Γ∗αβ,σ+ξa→σ·∂αβa→3+ξ∂σa→3·∂αa→β+ξ2∂σa→3·∂αβa→3,(33)Γαβ,3=g→3·∂αg→β=∂3Θ→·∂α∂βΘ→=∂3θ→+ξa→3·∂αβθ→+ξa→3=a→3·∂αβθ→+ξa→3=a→3·∂αβθ→+ξa→3·∂αβa→3.Since (a→3,a→3)=1, we have (34)∂αβa→3,a→3=∂α∂βa→3,a→3=∂α2a→3·∂βa→3=2∂αa→3·∂βa→3+2a→3·∂αβa→3=0.Thus, (35)a→3·∂αβa→3=-∂αa→3·∂βa→3.Submitting (35) and (7) into (33), we get (36)Γαβ,3=bαβ-ξcαβ,Γα3,σ=g→σ·∂αg→3=∂σΘ→·∂α∂3Θ→=∂σθ→+ξa→3·∂α∂3θ→+ξa→3=∂σθ→+ξa→3·∂αa→3=∂σθ→·∂αa→3+ξ∂σa→3·∂αa→3=-bασ+ξcασ.Similarly, (37)Γ3α,σ=-bασ+ξcασ,Γ33,α=g→α·∂3g→3=∂αΘ→·∂3∂3Θ→=∂αθ→+ξa→3·∂3∂3θ→+ξa→3=∂αθ→+ξa→3·∂3a→3=0,Γα3,3=g→3·∂αg→3=∂3Θ→·∂α∂3Θ→=∂3θ→+ξa→3·∂α∂3θ→+ξa→3=a→3·∂αa→3=0,Γ33,3=g→3·∂3g→3=∂3Θ→·∂3∂3Θ→=∂3θ→+ξa→3·∂3∂3θ→+ξa→3=a→3·∂3a→3=0.

Thus, the Christoffel symbols Γijk and Γ∗αβσ 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=gστΓ∗αβ,τ+ξa→τ·∂αβa→3+ξ∂τa→3·∂αa→β+ξ2∂τa→3·∂αβa→3,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 R∗αγβσ,R∗αλγβ 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)R∗αγβσ=bαβbγσ-bαγbβσ,R∗αλγβ=bαβbλγ-bαγbβλ,α,β,σ,γ,λ=1,2.

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 (42)∂βa→α=∂βa→α·a→λa→λ+∂βa→α·a→3a→3=Γ∗αβλa→λ+bαβa→3.Submitting ∂λa→3=-bλσa→σ into (42), we have (43)∂βγa→α=∂γΓ∗αβλa→λ+Γ∗αβλ∂γa→λ+∂γbαβa→3+bαβ∂γa→3.Submitting (42) into (43), we have (44)∂βγa→α=∂γΓ∗αβσ+Γ∗αβλΓ∗λγσ-bαβbγσa→σ+Γ∗αβλbγλ+∂γbαβa→3.

Because of ∂βγa→α=∂γβa→α, we can deduce by (44)-(45) that (46)0=∂γΓ∗αβσ-∂βΓ∗αγσ+Γ∗αβλΓ∗λγσ-Γ∗αγλΓ∗λβσ-bαβbγσ+bαγbβσa→σ+Γ∗αβλbγλ-Γ∗αγλbβλ+∂γbαβ-∂βbαγa→3.Since a→σ and a→3 are linearly independent, we have (47)∂γΓ∗αβσ-∂βΓ∗αγσ+Γ∗αβλΓ∗λγσ-Γ∗αγλΓ∗λβσ=bαβbγσ-bαγbβσ,∂γbαβ-Γ∗αγλbβλ=∂βbαγ-Γ∗αβλbγλ.Thus, formula (41) has been proven.

3. Examples3.1. 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)θ→y1,y2=rcosy1siny2,rsiny1siny2,rcosy2,where r=1 m is the radius of the middle surface S, 0≤y1≤2π is longitude, and 0≤y2≤π/2 is colatitude. The thickness of the middle surface S is 2ε where ε is the semithickness.

Middle surface of hemispherical shell.

Then, (49)a→1=∂1θ→=-rsiny1siny2,rcosy1siny2,0,a→2=∂2θ→=rcosy1cosy2,rsiny1cosy2,-rsiny2,∂1a→1=-rcosy1siny2,-rsiny1siny2,0,∂1a→2=∂2a→1=-rsiny1cosy2,rcosy1cosy2,0,∂2a→2=-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)a→3=a→1×a→2a→1×a→2=cosy1siny2,siny1siny2,cosy2,∂1a→3=-siny1siny2,cosy1siny2,0,∂2a→3=cosy1cosy2,siny1cosy2,-siny2,∂11a→3=-cosy1siny2,-siny1siny2,0,∂12a→3=∂21a→3=-siny1cosy2,cosy1cosy2,0,∂22a→3=-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)Γ∗11,2=-r2siny2cosy2,Γ∗12,1=Γ∗21,1=r2siny2cosy2,other Γ∗αβ,γ=0,Γ∗121=Γ∗211=coty2,Γ∗112=-siny2cosy2,other Γ∗βγα=0.The Riemann tensors on S are as follows: (54)R∗1212=R∗2121=-r2sin2y2,R∗1221=R∗2112=r2sin2y2,other R∗αλσβ=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)Θ→y,ξ=θ→y+ξa→3y,where -ε≤ξ≤ε.

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)θ→y1,y2=rcosy1,rsiny1,y2,where r=1 m is a constant, 0≤y1≤π, and 0≤y2≤h(h=3m). The thickness of the middle surface S is 2ε where ε is the semithickness.

Middle surface of semicylindrical shell.

Then, (60)a→1=∂1θ→=-rsiny1,rcosy1,0,a→2=∂2θ→=0,0,1,∂1a→1=-rcosy1,-rsiny1,0,∂1a→2=∂2a→1=0,0,0,∂2a→2=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)a→3=a→1×a→2a→1×a→2=cosy1,siny1,0,∂1a→3=-siny1,cosy1,0,∂2a→3=0,0,0,∂11a→3=-cosy1,-siny1,0,∂12a→3=∂21a→3=0,0,0,∂22a→3=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)Γ∗αβ,γ=0,Γ∗βγα=0.The Riemann tensors on S are as follows: (65)R∗αγβσ=0,R∗αλγβ=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)Θ→y,ξ=θ→y+ξa→3y,where -ε≤ξ≤ε.

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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