Chapter 8 Differential Geometry of a Surface
8.1
Introduction
A shell is a thin layer of material. Consequently, the kinematics of a shell are intimately related to the geometry of surfaces, the boundary surfaces, or some intermediate surface. In most practical theories, the differential geometry of one reference surface completely determines the strain components throughout the thickness. Our study is limited to those aspects of the geometry which are essential to a full understanding of these theories of shells.
8.2
Base Vectors and Metric Tensors of the Surface
Let us direct our attention to the surface θ3 = 0 of Figure 8.1. The position vector to a point P on that surface is r(θ1 , θ2 , 0) ≡ 0r(θ1 , θ2 ).
(8.1)
Let θ3 be the distance along the normal to our reference surface (θ3 = 0) ˆ 3 denote the unit normal vector at the point P . Then the position and let a vector to an arbitrary point Q is ˆ 3 (θ1 , θ2 ). r(θ1 , θ2 , θ3 ) = 0r(θ1 , θ2 ) + θ3 a
(8.2)
Observe that θα (α = 1, 2) are arbitrary coordinates of the surface, while θ3 is a special coordinate, the distance along the normal. © 2003 by CRC Press LLC
Figure 8.1 Position and coordinates of a surface
The tangent base vectors of our coordinate system follow from (2.5) and (8.2): ˆ 3,α , g α = 0r ,α + θ3 a
ˆ 3, g3 = a
(8.3a, b)
where the Greek indices have the range 1, 2. Let us denote the tangent vectors at the reference surface as follows: ai (θ1 , θ2 ) ≡ g i (θ1 , θ2 , 0).
(8.4)
It follows from (8.3a, b) and (8.4) that ˆ 3,α , g α = aα + θ3 a aα = 0r ,α ,
ˆ 3. g3 = a
(8.5) (8.6a, b)
In the manner of (2.6), we define the reciprocal base vector ai which is © 2003 by CRC Press LLC
Figure 8.2 Tangent and normal base vectors at the reference surface (θ3 = 0) normal to the θi surface: ai (θ1 , θ2 ) ≡ g i (θ1 , θ2 , 0), ai · aj = δji .
(8.7) (8.8)
Since the θ3 is distance along the normal to the reference surface θ3 = 0, ˆ 3 = g3 = a ˆ 3. a3 = a
(8.9a, b)
At each point of the surface we have two basic triads, ai and ai , where aα ˆ 3 is the unit normal. and aα are tangent to the surface and a3 = a3 = a The base vector aα is tangent to the θα line, and the vector aα is normal to the θα line as shown in Figure 8.2. The vectors aα can be expressed as a linear combination of aα and vice versa: aα = aαβ aβ ,
aα = aαβ aβ .
(8.10a, b)
These linear relations are a subcase of (2.7) and (2.8), and the coefficients aαβ and aαβ are the components of the contravariant and covariant metric © 2003 by CRC Press LLC
tensors of our surface coordinates θα . Observe that, aαβ (θ1 , θ2 ) = g αβ (θ1 , θ2 , 0),
aαβ (θ1 , θ2 ) = gαβ (θ1 , θ2 , 0).
(8.11a, b)
From (8.8) and (8.10a, b), it follows that aαβ = aα · aβ ,
aαβ aβγ = δγα .
aαβ = aα · aβ ,
(8.12a, b), (8.13)
Let |aαβ | ≡ a = g(θ1 , θ2 , 0).
(8.14a, b)
It follows from (8.2) that a11 =
a22 , a
a22 =
a11 , a
|aαβ | =
a12 = −
a12 , a
1 . a
(8.15a, c)
(8.16)
The vectors aa , aa and the tensors aαβ , aαβ play a role in the two-dimensional subspace (surface) as the vectors g a , g a and the tensors g αβ , gαβ in the three-dimensional space.
8.3
Products of the Base Vectors
In accordance with (2.24a, b) and (8.14a, b), we define √ eαβ (θ1 , θ2 ) ≡ eαβ3 (θ1 , θ2 , 0) = a αβ3 , αβ3 e αβ (θ1 , θ2 ) ≡ eαβ3 (θ1 , θ2 , 0) = √ . a
(8.17a, b) (8.17c, d)
Following (2.26a, b), we have ˆ 3, aα × aβ = eαβ a
ˆ 3, aα × aβ = e αβ a
(8.18a, b)
ˆ 3 × aα = eαβ aβ , a
ˆ 3 × aα = e αβ aβ . a
(8.18c, d)
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8.4
Derivatives of the Base Vectors
In accordance with (2.46a, b), we define the Christoffel symbols for the surface coordinates: Γαβγ (θ1 , θ2 ) ≡ aγ · aα,β = Γαβγ (θ1 , θ2 , 0),
(8.19a, b)
Γγαβ (θ1 , θ2 ) ≡ aγ · aα,β = Γγαβ (θ1 , θ2 , 0).
(8.20a, b)
Here, the overbar ( ) signifies that the symbol is evaluated at the reference surface θ3 = 0. If the derivatives are continuous (the surface is smooth), then it follows from (8.6a) and the definitions (8.19a) and (8.20a) that Γγαβ = Γγβα .
Γαβγ = Γβαγ ,
(8.21a, b)
Also, from (8.2b) and (8.19a) we have Γαβγ = 12 (aαγ,β + aβγ,α − aαβ,γ ).
(8.22)
Recall (8.8), (8.9a, b), and (8.10a, b): viz., aα · aβ = δβα ,
aα = aαβ aβ ,
ˆ 3 · aα = a ˆ 3 · aα = 0, a
aα = aαβ aβ , ˆ3 · a ˆ 3 = 1. a
(8.23a–c) (8.24a–c)
From (8.23a) and the definition (8.20a), we obtain Γγαβ = −aα · aγ,β .
(8.25)
In view of (8.23b, c) and the definitions (8.19a) and (8.20a), we see that Γαβγ = aγη Γηαβ ,
Γηαβ = aηγ Γαβγ .
(8.26a, b)
Normal components of the derivatives aα,β are denoted as follows: ˆ 3 · aα,β = Γαβ3 (θ1 θ2 , 0), bαβ (θ1 , θ2 ) ≡ a
(8.27a, b)
1 2 1 2 ˆ 3 · aα,β = −Γα bα β (θ , θ ) ≡ a 3β (θ θ , 0).
(8.28a, b)
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From (8.27a, b), (8.28a, b), and (8.24a–c) we obtain ˆ 3,β , bαβ = bβα = −aα · a α ˆ 3,β . bα β = −a · a
(8.29a, b) (8.30)
According to (8.23b, c), (8.29b), and (8.30), we have bαβ = aαγ bγβ ,
αγ bα β = a bγβ .
(8.31a, b)
The components bαβ and baβ constitute associated surface tensors which we call curvature tensors. Finally, it follows from (8.24c) that ˆ 3,a = 0, ˆ3 · a a
(8.32a)
and, consequently, Γ3a3 = Γ33a = 0.
(8.32b, c)
In accordance with (8.19a), (8.20a), (8.23b), (8.26b), and (8.27a), we have ˆ 3, aα,β = Γγαβ aγ + bαβ a ˆ 3. = Γαβγ aγ + bαβ a
(8.33a) (8.33b)
Likewise, according to (8.25) and (8.28a) γ α ˆ 3. aα,β = −Γα βγ a + bβ a
(8.34)
Equations (8.33a, b) and (8.34) can also be derived by setting θ3 = 0 in (2.45a, b) and (2.49) and employing the aforementioned formulas. Equations (8.33a, b) and (8.34) are Gauss’ formulas for the derivatives of the tangent vectors. Observe that the Christoffel symbols Γγαβ or Γαβγ determine the components (of aα,β and aα,β ) tangent to the surface while the components bαβ or bα β are normal. ˆ 3 follow from (8.29a, b) and (8.30) The derivatives of the normal a ˆ 3,β = −bαβ aα = −bα a β aα . Equations (8.35a, b) are known as Weingarten’s formulas. © 2003 by CRC Press LLC
(8.35a, b)
8.5
Metric Tensor of the Three-Dimensional Space
Returning to the base vector g α of (8.5) and employing (8.29b) and (8.30), we have g a = aα − θ3 bβα aβ ,
(8.36a)
= aα − θ3 bαβ aβ .
(8.36b)
The component gαβ is obtained according to (2.9) with the aid of (8.2b), (8.31a), and (8.36a, b): gαβ = aαβ − 2θ3 bαβ + (θ3 )2 bαγ bγβ .
(8.37a)
From (8.6b), (8.24a–c), and (8.36a), we have gα3 = 0,
g33 = 1.
(8.37b, c)
According to (2.12) and (8.37a–c), the components g αβ are rational functions of θ3 while g a3 = 0 and g 33 = 1.
8.6
Fundamental Forms
An incremental change of position on the surface is accompanied by a ˆ 3 ; the change in the position vector 0r and a change in the normal vector a first-order differentials are: d 0r = 0r ,α dθα = aa dθα ,
(8.38)
ˆ 3,a dθa = −bαβ aβ dθa . dˆ a3 = a
(8.39)
From (8.38) and (8.39), we can form three scalar products, namely d 0r · d 0r ≡ d 0 2 = aαβ dθα dθβ ,
(8.40)
d 0r · da3 = −bαβ dθα dθβ ,
(8.41)
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Figure 8.3 Curvature and torsion: orthogonal curves c1 and c2 on a surface dˆ a3 · dˆ a3 = aγη bηα bγβ dθα dθβ .
(8.42)
Equations (8.40) to (8.42) are known as the first, second , and third fundamental forms, respectively. The coefficients of the first quadratic form are components of the metric tensor of the surface coordinates; these components serve to measure distances on the surface. Now, let us examine the role of the coefficients bαβ appearing in the second quadratic form.
8.7
Curvature and Torsion
Let c1 and c2 denote orthogonal curves on the surface of Figure 8.3 and let 1 and 2 denote arc lengths along curves c1 and c2 , respectively. At each point of curve c1 , we may construct a triad of orthonormal vectors ˆ 1 is tangent to c1 , ˆ 3 is normal to the surface, λ as shown in Figure 8.3; a ˆ and λ2 is tangent to c2 . ˆ 3 reNow, consider the motion of this triad as it advances along c1 , a ˆ 1 tangent to c1 . The tangent vector maining normal to the surface, and λ ˆ 1 turns about a ˆ 2 ; that is, ˆ 3 and about λ λ
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ˆ1 dλ ˆ 2. ˆ 3 + σ12 λ = κ11 a d1
(8.43)
ˆ 1 because This derivative has no component in the direction λ d ˆ ˆ (λ1 · λ1 ) = 0. d1 ˆ 1 is determined The normal curvature of the surface in the direction of λ ˆ 1 turns about λ ˆ 2 , toward the direction of a ˆ 3 ; the by the rate at which λ ˆ 1 is normal curvature in the direction of λ ˆ3 · κ11 ≡ a
ˆ1 dλ dˆ a3 ˆ =− · λ1 . d1 d1
(8.44a, b)
The component σ12 is known as the geodesic curvature: ˆ2 · σ12 = λ
ˆ1 dλ . d1
(8.45)
ˆ 2 turns about a ˆ 3 and As the triad advances along c1 , the tangent vector λ ˆ about λ1 ˆ2 dλ ˆ 1 + τ21 a ˆ 3. = −σ21 λ (8.46) d1 ˆ 2 and λ ˆ 1 , respectively, and noting the Multiplying (8.43) and (8.46) by λ ˆ ˆ orthogonality (λ1 · λ2 = 0), we have ˆ ˆ ˆ 1 · dλ2 = −λ ˆ 2 · dλ1 , λ d1 d1 σ12 = σ21 .
(8.47a) (8.47b)
ˆ 1 is determined by the rate The torsion of the surface in the direction of λ ˆ ˆ ˆ 3 ; the torsion in the at which λ2 turns about λ1 , toward the direction of a ˆ 1 is direction of λ ˆ3 · τ21 ≡ a
ˆ2 dλ dˆ a3 ˆ =− · λ2 . d1 d1
(8.48a, b)
ˆ1 The torsion τ21 is also called the geodesic torsion in the direction of λ (see, for example, A. J. McConnell [138] and I. S. Sokolnikoff [139]). © 2003 by CRC Press LLC
To express the normal curvature and torsion in other terms we note that α ˆ = ∂θ a ≡ λα a , λ 1 α 1 ∂1 α
(8.49a, b)
α ˆ = ∂θ a ≡ λα a . λ 2 α 2 ∂2 α
(8.50a, b)
With the aid of (8.33a), we obtain from (8.49a) ˆ1 dλ β µ α β ˆ 3. = λµ1,β λβ1 + λα 1 λ1 Γαβ aµ + λ1 λ1 bαβ a d1 Then, according to (8.44a) β κ11 = λα 1 λ1 bαβ .
(8.51a)
ˆ 2 is Likewise, the normal curvature in the direction of λ β κ22 = λα 2 λ2 bαβ .
(8.51b)
To obtain an expression for the torsion τ21 from (8.48a), we require the derivative of (8.50b). By means of (8.33a), we obtain ˆ2 dλ β µ α β ˆ 3. = λµ2,β λβ1 + λα 2 λ1 Γαβ aµ + λ2 λ1 bαβ a d1 Then, according to (8.48a) β τ21 = λα 2 λ1 bαβ .
(8.52)
Observe that τ12 = τ21 ; ˆ 2 is the same. ˆ 1 and λ that is, the torsion in the orthogonal directions of λ In accordance with (8.18c), ˆ =a ˆ = λα e aβ , ˆ3 × λ λ 1 αβ 2 1 β ˆ ×a ˆ =λ ˆ 3 = λα λ 2 eβα a , 1 2
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or, according to (8.49a, b) and (8.50), βγ λγ2 = λα 1 eαβ a ,
(8.53a)
βγ λγ1 = λα 2 eβα a .
(8.53b)
In view of (8.53a, b), equation (8.52) has the alternative forms β τ12 = λα 1 λ2 bαβ ,
(8.54a)
µ βη = λα eµη bαβ , 1 λ1 a
(8.54b)
= λβ2 λµ2 aαη eηµ bαβ .
(8.54c)
ˆ = λα aα is given by a form The normal curvature κ in the direction of λ like (8.51a), namely, (8.55) κ = λα λβ bαβ . Now, let us seek the directions in which the normal curvature κ has extremal values. The conditions for an extremal value are ∂κ = 0. ∂λα Additionally, λα must satisfy the auxiliary (normality) condition, λα λβ aαβ = 1.
(8.56)
The directions are given by the solution of the linear equations: (bαβ − κaαβ )λα = 0.
(8.57)
However, equations (8.57) have a nontrivial solution if, and only if, the determinant of the coefficients vanishes; that is, |bαβ − κaαβ | = 0,
(8.58a)
a αµ βη e e (bαβ − κaαβ )(bµη − κaµη ) = 0, 2
(8.58b)
or, in expanded form,
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or βη α γ 1 κ2 − bα bβ bη = 0. α κ + 2 eαγ e
(8.58c)
The coefficients in the quadratic equation (8.58c) play an important role in the differential geometry of surfaces. One is known as the Gaussian curvature: βη α γ 1 k ≡ |bα bβ bη = b11 b22 − b12 b21 . (8.59) β | = 2 eαγ e The mean curvature is defined as follows: h ≡ 12 bα α.
(8.60)
With the notations of (8.59) and (8.60), equation (8.58c) has the form: κ2 − 2 hκ + k = 0.
(8.61)
˜ α and λ ˜ β the corLet κ ˜ 11 and κ ˜ 22 be the two real roots of (8.61) and λ 1 2 β ˜ responding solutions of (8.57). Then, multiplying (8.57) by λ1 , likewise by ˜ β , and recalling equation (8.56), we obtain λ 2 ˜αλ ˜β ˜α ˜β κ ˜ 11 = κ ˜ 11 aαβ λ 1 1 = bαβ λ1 λ1 , ˜αλ ˜β ˜α ˜β κ ˜ 22 = κ ˜ 22 aαβ λ 2 2 = bαβ λ2 λ2 . ˜ 22 are the extremal values of the normal curvaThe two real roots κ ˜ 11 and κ ture; they are called the principal curvatures. The corresponding directions are called the principal directions. Also, from (8.57), we have ˜αλ ˜β ˜ a λ ˜α ˜β bαβ λ 1 2 −κ 11 αβ 1 λ2 = 0, ˜αλ ˜β ˜ a λ ˜α ˜β bαβ λ 2 1 −κ 22 αβ 2 λ1 = 0. However, bαβ and aαβ are symmetric; consequently, the difference of the last two equations is ˜αλ ˜β (˜ κ11 − κ ˜ 22 )aαβ λ 1 2 = 0. ˜αλ ˜β ˜ 22 , then aαβ λ If the principal curvatures are distinct, i.e, κ ˜ 11 = κ 1 2 = 0; that is, the principal directions are orthogonal. © 2003 by CRC Press LLC
Since κ ˜ 11 and κ ˜ 22 are the roots of (8.61), ˜ 22 ) = κ2 − 2 hκ + k = 0. (κ − κ ˜ 11 )(κ − κ It follows that ˜ 22 , k=κ ˜ 11 κ
(8.62)
h = 12 (˜ κ11 + κ ˜ 22 ).
(8.63)
Consider the directions in which the torsion vanishes: according to equation (8.54a), the condition of vanishing torsion is β bαβ λα 1 λ2 = 0,
with the supplementary conditions β α β aαβ λα 1 λ1 = aαβ λ2 λ2 = 1, β aαβ λα 1 λ2 = 0.
In view of (8.57), and the orthogonality of the principal directions, the principal directions fulfill the foregoing conditions when κ ˜ 11 = κ ˜ 22 . If κ ˜ 11 = κ ˜ 22 at a point of a surface, then the normal curvature is the same in all directions and the torsion vanishes in all directions; such a point is known as an umbilic of the surface. At an umbilic bαβ = κaαβ , where κ is the normal curvature in all directions. If c1 and c2 are (orthogonal) coordinate lines and if the coordinates are arc lengths 1 and 2 , then λ11 = λ22 = 1, λ12 = λ21 = 0, and, according to (8.51a, b) and (8.54a), κ11 ≡
1 = b11 = b11 , r1
κ22 ≡
1 = b22 = b22 , r2
τ21 = τ12 = b12 = b12 = b21 . Here, r1 and r2 are the principal radii of curvature; these are the radii of the ˆ 1) arcs traced by the intersections of the surface with the planes of (ˆ a3 , λ ˆ 2 ). and (ˆ a3 , λ © 2003 by CRC Press LLC
Figure 8.4 Volume and area differentials
8.8
Volume and Area Differentials
An elemental area ds of an arbitrary θ3 surface is delineated by θα lines as shown in Figure 8.4. The corresponding θα surfaces contain faces of the elemental volume dv. The edges of the elemental area ds0 of the reference surface are contained in the same θα surfaces. An edge of the elemental volume approaches the vector dr i = r ,i dθi = g i dθi .
(8.64a, b)
The volume dv is expressed as follows: dv = dr 3 · (dr 1 × dr 2 ) = g 3 · (g 1 × g 2 ) dθ1 dθ2 dθ3 .
(8.65)
The area ds is given by the expression: ˆ 3 · (dr 1 × dr 2 ) ds = a ˆ 3 · (g 1 × g 2 ) dθ1 dθ2 . =a © 2003 by CRC Press LLC
(8.66)
The corresponding area ds0 on the reference surface has the form: ˆ 3 · (a1 × a2 ) dθ1 dθ2 . ds0 = a
(8.67)
In accordance with (2.23a) and (8.3b), we recall that ˆ 3 · (g 1 × g 2 ) = g 3 · (g 1 × g 2 ) = a
√
g.
(8.68a, b)
Likewise, according to (8.17a) and (8.18a), ˆ 3 · (a1 × a2 ) = a
√
a.
(8.69)
In view of (8.68a, b) and (8.69), equations (8.65) to (8.67) take the forms: dv = ds = ds0 =
√ √ √
g dθ1 dθ2 dθ3 ,
(8.70)
g dθ1 dθ2 ,
(8.71)
a dθ1 dθ2 .
(8.72)
In accordance with (8.17c, d), equation (8.68b) has the alternative form √
g=
√
ˆ 3 · (g α × g β ). a 12 e αβ a
(8.73a)
Substituting (8.36a) into (8.73a) and employing (8.18a), we obtain √
g=
√
a 12 e αβ eµη (δαµ − θ3 bµα )(δβη − θ3 bηβ ).
(8.73b)
With the expressions (8.59) and (8.60) for the Gaussian and mean curvatures, k and h, equation (8.73b) is reduced to the form: √
g=
√
a [1 − 2 hθ3 + k(θ3 )2 ].
It follows from (8.70) to (8.72) and (8.73b) that g dv ds = 1 − 2 hθ3 + = = k(θ3 )2 . 3 ds0 ds0 dθ a
(8.73c)
(8.74a–c)
The area ds1 of the θ1 face of the elemental volume is obtained from the expression: ˆ1 · (dr 2 × dr 3 ), ds1 = e © 2003 by CRC Press LLC
ˆ1 denotes the unit normal to ds1 . Since e ˆ1 = g 1 / where e
g 11 , we have
g 1j ds1 = g j · (g 2 × g 3 ) dθ2 dθ3 g 11 =
g 11 g dθ2 dθ3 .
(8.75a)
Likewise, ds2 =
8.9
g 22 g dθ1 dθ3 .
(8.75b)
Vectors, Derivatives, and Covariant Derivatives
A vector V can be expressed as a linear combination of the base vectors ai or ai : ˆ3 V = V α aα + V 3 a ˆ 3. = Vα aα + V 3 a It follows from (8.10a, b) that the components Vα and V α are associated as follows: V α = aαβ Vβ , Vα = aαβ V β . With the aid of (8.33a, b) to (8.35a, b) we can express the partial derivatives V,α in the alternative forms: a3 , V,α = (V µ,α + V β Γµβα − V 3 bµα )aµ + (V 3,α + V β bαβ )ˆ
(8.76a)
V,α = (Vµ,α − Vβ Γβµα − V 3 bµα )aµ + (V 3,α + Vβ bβα )ˆ a3 .
(8.76b)
The covariant derivatives of the components V α and Vα are defined in the manner of (2.50a, b) and (2.51a, b); namely,
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V α ||β ≡ V α,β + V µ Γα µβ ,
(8.77a)
V α ||β ≡ Vα,β − Vµ Γµαβ .
(8.77b)
The double bar signifies covariant differentiation with respect to the surface rather than the three-dimensional space; note the difference: V α |β ≡ V α,β + V i Γα iβ . In the latter, the repeated Latin index implies a sum of three terms; moreover, the components need not be evaluated at a particular surface. With the notations of (8.77a, b), equations (8.76a, b) assume the forms V,α = (V µ ||α − V 3 bµα )aµ + (V 3,α + V β bβα )ˆ a3 ,
(8.78a)
V,α = (V µ ||α − V 3 bµα )aµ + (V 3,α + Vβ bβα )ˆ a3 .
(8.78b)
From equation (2.53), we infer that √ ∂ a √ λ = a Γλα . ∂θα
(8.79a)
Finally, by means of the result (8.79a), we obtain √ ∂ √ a V α = a V α ||α . α ∂θ
8.10
(8.79b)
Surface Tensors
A quantity such as F αβ··· (θ1 , θ2 ),
Pαβ··· (θ1 , θ2 ),
α··· 1 2 or Tβ··· (θ , θ ),
(8.80a–c)
is the component of a contravariant, covariant, or mixed surface tensor, respectively, if the component F αβ··· (θ¯1 , θ¯2 ),
P αβ··· (θ¯1 , θ¯2 ),
¯1 ¯2 or T α··· β··· (θ , θ ),
(8.81a–c)
in another coordinate system θ¯α is obtained from (8.80a–c) by a linear transformation of the types (2.55), (2.57), and (2.58), with Greek indices in place of Latin indices. © 2003 by CRC Press LLC
A surface invariant has no free indices; for example, A = T αβ Sαβ . From the definitions, it follows that the base vectors aα and aα transform as components of covariant and contravariant tensors and, therefore, from (8.2a, b) it follows that aαβ and aαβ are components of surface tensors, covariant and contravariant, respectively. Likewise, from (8.27a) and (8.28a), it follows that bαβ and bα β are components of covariant and mixed surface tensors. Furthermore, eαβ and e αβ are components of associated surface tensors, covariant and contravariant, e αβ = aαγ aβη eγη ,
eαβ = aαγ aβη e γη .
With the aid of (8.17c, d) and (8.79a), we obtain for the covariant derivative: e αβ ||γ = 0.
Two surface tensors are said to be associated tensors if each component of one is a particular linear combination of the components of the other. The combination is formed by multiplying components of the tensor with components of the metric tensor and summing. For example, T α· γβ and Tαµη are components of associated tensors related as follows: T α· γβ = aγµ aβη Tαµη
⇐⇒
Tαµη = aµγ aηβ T α· γβ .
Observe that these components of the associated tensors are referred to the same surface coordinates. Also notice that a vacant position is marked by a dot. If the tensor is completely symmetric; that is Tαβγ = Tβαγ = Tαγβ = Tγβα , then the positions need not be marked. The quantities bαβ and bα β of (8.31a, b) are components of associated symmetric tensors, covariant and mixed. Forming an associated tensor is often termed raising, or lowering, indices. © 2003 by CRC Press LLC
Figure 8.5 Green’s theorem: surface s bounded by a curve c
8.11
Green’s Theorem (Partial Integration) for a Surface
The analysis of shells by energy principles often involves integrals of the form I≡
s
Aα B,α ds =
s
Aα B,α
√
a dθ1 dθ2 ,
(8.82a, b)
wherein integration extends over a surface s (e.g., the midsurface or a reference surface of the shell) bounded by a curve c; Aα and B are functions of the surface coordinates. We assume that the surface s and curve c are smooth and that Aα , B, and derivatives are continuous. If, in subsequent developments, the surface in question may be the deformed surface of a shell, then the minuscules (s), (c), and (a) may be replaced by (S), (C), and (A), respectively. ˆ 1 the normal to c tangent to ˆ 2 denote the unit tangent to c, and λ Let λ s. Let x2 denote arc length along c and x1 the arc length along a curve normal to c, as shown in Figure 8.5. © 2003 by CRC Press LLC
Now consider one term of (8.82a, b), namely, I1 =
s
√ ( a A1 B,1 dθ1 ) dθ2 .
(8.83a)
Integrating (8.83a) with respect to θ1 , we obtain I1 ≡
√
c
a A1 B dθ2 −
s
√ ( a A1 ),1 B dθ1 dθ2 .
(8.83b)
According to (8.50a) and (8.53b), λ22 ≡
ˆ ∂θ2 ˆ = λ1√· a1 . = a2 · λ 2 2 ∂x a
On the curve c, we have θα = θα (x2 ), and, therefore, dθ2 =
ˆ ·a λ 1 √ 1 dx2 . a
Consequently, (8.83b) takes the form: I1 ≡
c
ˆ · a ) dx2 − A1 B(λ 1 1
√ ( a A1 ),1 B dθ1 dθ2 .
(8.83c)
√ ( a A2 ),2 B dθ1 dθ2 .
(8.84)
1 √ √ ( a Aα ),α B ds. a
(8.85)
s
Likewise, I2 ≡
s
= c
√ ( a A2 B,2 dθ2 ) dθ1
2
ˆ · a ) dx2 − A B(λ 1 2
s
It follows that I= c
ˆ · a ) dx2 − Aα B(λ 1 α
s
In physical problems I is an invariant, Aα transforms as the component of a contravariant surface tensor, and B is invariant. © 2003 by CRC Press LLC
Figure 8.6 Position of particles A and B on the surface s0
8.12
Equations of Gauss and Codazzi
Because a shell is thin, the position of any particle is often identified with a neighboring point on a reference surface. Indeed, in most theories‡ an approximation serves to express the three-dimensional field r(θ1 , θ2 , θ3 ) explicitly in terms of the two-dimensional field 0r(θ1 , θ2 ), the position vector of particles on the reference surface. We are concerned about the existence of the vector 0r(θ1 , θ2 ). Since a rigid-body motion is irrelevant to a discussion of deformation, let us assume that the position vector A of one particle at A(a1 , a2 ) of a surface s0 is known. Now, let us set out to compute the position vector 1 2 1 2 0r(b , b ) of an arbitrary particle at B(b , b ): The relative position of particles A and B in Figure 8.6 is 0r
−A =
B
dr = A
B
= A
‡A
B
A
aα dθα =
r ,α dθα
B
A
aα d(θα − bα ).
prominent example is the Kirchhoff-Love theory (see Chapter 10).
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An integration by parts results in α α 0r − A = (b − a )aα ]A +
B
A
(bα − θα )aα,β dθβ .
By means of (8.33a), this assumes the form: 0r−A
α
α
= (b −a )aα ]A +
B
A
ˆ 3 (bα −θα )bαβ ] dθβ . (8.86) [aγ (bα −θα )Γγαβ + a
The integral has the form
B
I= A
Fβ dθβ .
The integral exists in a simply connected region (no holes in the surface) independently of the path of integration if, and only if,‡ ∂Fβ ∂Fµ = . ∂θµ ∂θβ
(8.87)
In view of (8.33a), (8.34), and the symmetry of Γγαβ and bαβ , condition (8.87) takes the form
bα − θ α
Γηαβ,µ − Γηαµ,β + Γηγµ Γγαβ − Γηγβ Γγαµ − bηµ bαβ + bηβ bαµ aη
ˆ 3 = . + bγµ Γγαβ − bγβ Γγαµ + bαβ,µ − bαµ,β a However, the condition must be satisfied at every point of the surface. Consequently, the vector in brackets must vanish; this requires that each component vanish, that is Rη· αµβ ≡ Γηαβ,µ − Γηαµ,β + Γηγµ Γγαβ − Γηγβ Γγαµ = bηµ bαβ − bηβ bαµ ,
(8.88)
‡ If the surface contain holes, then an additional condition is needed to ensure the continuity about each hole.
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and
bαβ,µ − bγβ Γγαµ = bαµ,β − bγµ Γγαβ .
(8.89)
Equation (8.88) is equivalent to aλη Rη· αµβ ≡ Rλαµβ = bλµ bαβ − bλβ bαµ .
(8.90a)
Moreover, in view of the symmetry of Γγαβ and bαβ , equation (8.90a) is an identity except in the case: R1212 = R2121 = −R2112 = −R1221 .
(8.90b)
In other words, equation (8.90a) represents one condition for the existence of 0r − A; that condition can be restated as follows: 1 4
e λα e µβ Rλαµβ =
1 4
e λα e µβ (bλµ bαβ − bλβ bαµ ).
According to (8.59), the right side is the Gaussian curvature. Therefore, this condition takes the form 1 4
e λα e µβ Rλαµβ = k.
(8.91)
Equation (8.91) is the Gauss equation of the surface. If we introduce the covariant derivative of the tensor bαβ , namely, bαβ ||µ = bαβ,µ − bγβ Γγαµ − bαγ Γγβµ , then (8.89) takes the form bαβ ||µ = bαµ ||β .
(8.92)
Equation (8.92) represents either of the two equations, namely, b11 ||2 = b12 ||1 ,
b21 ||2 = b22 ||1 .
(8.93a, b)
These are the Codazzi equations of the surface. The Gauss and Codazzi equations (8.91) and (8.93a, b) are the necessary and sufficient conditions for the existence of the position vector 0r − A. They play an important role in the subsequent discussion of compatibility in the deformation of shells. © 2003 by CRC Press LLC
Using the expressions for the covariant derivative [equation (8.77a, b)] and the previously derived equations (8.88) and (8.91), we obtain for the second covariant derivatives of the components Vα (or V α ) of any vector V (θ1 , θ2 ): Vα ||βγ − Vα ||γβ = Rµαβγ V µ , = (bµβ bαγ − bµγ bαβ )V µ ,
(8.94) (8.95)
V1 ||12 − V1 ||21 = − k V 2,
(8.96)
V2 ||12 − V2 ||21 = k V 1.
(8.97)
It follows from equations (8.96) and (8.97) that the order of covariant differentiation is immaterial, i.e., . Vα ||βγ = Vα ||γβ , if k 1, i.e, as one approaches a plane surface.
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