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Unformatted text preview: Section 2.12 Solid Mechanics Part III Kelly 298 2.12 Pull Back, Push Forward and Lie Time Derivatives 2.12.1 PushForward and PullBack The concepts of pullback and pushforward have a number of uses, in particular they will be used to define the Lie derivative further below. Vectors Consider a vector V given in terms of the reference configuration base vectors: i i i i V V G G V = = (2.12.1) The pushforward of V , ( ) V * χ , is defined to be the vector with the same components, but with respect to the current configuration base vectors. The pushforward of a vector depends on the type of components; the symbol b is used for covariant components i V and the symbol # for contravariant components i V . Thus, using 2.10.8, ( ) ( ) FV FG g V V F G F g V = = = = = = − − i i i i i i i i b V V V V # * T T * χ χ . (2.12.2) A special case is the push forward of a line element in the reference configuration, Eqn. 2.10.7, ( ) x g X d d d i i = Θ = # * χ . (2.12.3) which is consistent with the fact that, with convected coordinates, the line element X d has the same coordinates with respect to the reference configuration basis as does x d with respect to the current configuration basis. Similarly, consider a vector v given in terms of the current configuration basis: i i i i v v g g v = = (2.12.4) The pullback of v , ( ) v 1 * − χ , is defined to be the vector with components i v (or i v ) with respect to the reference configuration base vectors i G (or i G ). Thus, using 2.10.8, ( ) ( ) v F g F G v v F g F G v 1 1 # 1 * T T 1 * − − − = = = = = = i i i i i i i i b v v v v χ χ . (2.12.5) Section 2.12 Solid Mechanics Part III Kelly 299 and, for a line element in the current configuration, ( ) X x F G x d d dx d i i = = = − − 1 # 1 * χ . (2.12.6) Note that a pushforward and pullback applied successively to a vector with the same component type will result in the initial vector. From the above, for two material vectors U and V and two spatial vectors u and v , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b b b b v u v u v u V U V U V U 1 * # 1 * # 1 * 1 * * # * # * * − − − − ⋅ = ⋅ = ⋅ ⋅ = ⋅ = ⋅ χ χ χ χ χ χ χ χ (2.12.7) Tensors Consider a material tensor A : j i j i j i i j j i ij j i ij A A A A G G G G G G G G A ⊗ = ⊗ = ⊗ = ⊗ = ⋅ ⋅ (2.12.8) As for the vector, the pushforward of A , ( ) A * χ , is defined to be the tensor with the same components, but with respect to the deformed base vectors. Thus, using 2.10.8, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T T T / * 1 T \ * T # * 1 T T T * AF F FG G F g g A FAF G F FG g g A FAF FG FG g g A AF F G F G F g g A − − ⋅ ⋅ − − ⋅ ⋅ − − − − = ⊗ = ⊗ = = ⊗ = ⊗ = = ⊗ = ⊗ = = ⊗ = ⊗ = j i j i j i j i j i i j j i i j j i ij j i ij j i ij j i ij b A A A A A A A A χ χ χ χ . (2.12.9) Similarly, consider a spatial tensor a : j i j i j i i j j i ij j i ij a a a a g g g g g g g g a ⊗ = ⊗ = ⊗ = ⊗ = ⋅ ⋅ (2.12.10) The pullback is ( ) ( )...
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 Fall '11
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