Vectors_Tensors_19_Curved_Geometries

Vectors_Tensors_19_Curved_Geometries - Section 1.19 1.19...

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Section 1.19 Solid Mechanics Part III Kelly 175 1.19 Curvilinear Coordinates: Curved Geometries In this section is examined the special case of a two-dimensional curved surface. 1.19.1 Monoclinic Coordinate Systems Base Vectors A curved surface can be defined using two covariant base vectors 2 1 , a a , with the third base vector, 3 a , everywhere of unit size and normal to the other two, Fig. 1.19.1 These base vectors form a monoclinic reference frame, that is, only one of the angles between the base vectors is not necessarily a right angle. Figure 1.19.1: Geometry of the Curved Surface In what follows, in the index notation, Greek letters such as β α , take values 1 and 2; as before, Latin letters take values from 1. .3. Since 3 3 a a = and 0 3 3 = = a a a , 0 3 3 = = a a a (1.19.1) the determinant of metric coefficients is 1 0 0 0 0 22 21 12 11 2 g g g g J = , 1 0 0 0 0 1 22 21 12 11 2 g g g g J = (1.19.2) The Cross Product Particularising the results of §1.16.5, define the surface permutation symbol to be the triple scalar product 1 a 2 a 3 a 1 Θ 2 Θ
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Section 1.19 Solid Mechanics Part III Kelly 176 g e g e 1 , 3 3 αβ β α ε = × = × a a a a a a (1.19.3) where = is the Cartesian permutation symbol, 1 12 + = , 1 21 = , and zero otherwise, with ημ βα η μ μη δ e e e e e e = = = , (1.19.4) From 1.19.3, 3 3 a a a a a a e e = × = × (1.19.5) and so g 2 1 3 a a a × = (1.19.6) The cross product of surface vectors, that is, vectors with component in the normal ( 3 g ) direction zero, can be written as 3 2 1 2 1 3 3 2 1 2 1 3 1 a a a a v u v v u u g v u e v v u u g v u e = = = = × (1.19.7) The Metric and Surface elements Considering a line element lying within the surface, so that 0 3 = Θ , the metric for the surface is () ( ) ( ) Θ Θ = Θ Θ = = Δ d d g d d d d s a a s s 2 (1.19.8) which is in this context known as the first fundamental form of the surface . Similarly, from 1.16.35, a surface element is given by 2 1 ΔΘ ΔΘ = Δ g S (1.5.9) Christoffel Symbols The Christoffel symbols can be simplified as follows. A differentiation of 1 3 3 = a a leads to
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Section 1.19 Solid Mechanics Part III Kelly 177 3 , 3 3 , 3 a a a a = α (1.19.10) so that, from Eqn 1.18.6, 0 33 3 3 = Γ = Γ (1.19.11) Further, since 0 / 3 3 = Θ a , 0 , 0 333 33 = Γ = Γ (1.19.12) These last two equations imply that the ijk Γ vanish whenever two or more of the subscripts are 3. Next, differentiate 1.19.1 to get β a a a a = , 3 3 , , a a a a = , 3 3 , (1.19.13) and Eqns. 1.18.6 now lead to βα αβ 3 3 3 3 Γ = Γ = Γ = Γ (1.19.14) From 1.18.8, using 1.19.11, 0 3 3 33 3 3 3 3 3 3 3 33 3 3 3 = Γ = Γ + Γ = Γ Γ = Γ + Γ = Γ γ αβγ g g g g (1.19.15) and, similarly { Problem 1} 0 3 33 33 3 3 = Γ = Γ = Γ (1.19.16) 1.19.2 The Curvature Tensor In this section is introduced a tensor which, with the metric coefficients, completely describes the surface. First, although the base vector 3 a maintains unit length, its direction changes as a function of the coordinates 2 1 , Θ Θ , and its derivative is, from 1.18.2 or 1.18.5 (and using 1.19.15) a a a 3 3 3 Γ = Γ = Θ k k , a a a 3 3 3 Γ = Γ = Θ k k (1.19.17) Define now the curvature tensor K to have the covariant components K , through
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Section 1.19 Solid Mechanics Part III Kelly 178 β αβ α a a K
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

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Vectors_Tensors_19_Curved_Geometries - Section 1.19 1.19...

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