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Vectors_Tensors_13_Coordinate_Transformation_Tensors

Vectors_Tensors_13_Coordinate_Transformation_Tensors -...

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Section 1.13 Solid Mechanics Part III Kelly 110 1.13 Coordinate Transformation of Tensor Components It has been seen in §1.5.2 that the transformation equations for the components of a vector are j ij i u Q u = , where [ ] Q is the transformation matrix. Note that these ij Q ’s are not the components of a tensor – these s Q ij ' are mapping the components of a vector onto the components of the same vector in a second coordinate system – a (second-order) tensor, in general, maps one vector onto a different vector. The equation j ij i u Q u = is in matrix element form, and is not to be confused with the index notation for vectors and tensors. 1.13.1 Relationship between Base Vectors Consider two coordinate systems with base vectors i e and i e . It has been seen in the context of vectors that, Eqn. 1.5.4, ) , cos( j i ij j i x x Q = e e . (1.13.1) Recal that the i ’s and j ’s here are not referring to the three different components of a vector, but to different vectors (nine different vectors in all). It is interesting that the relationship 1.13.1 can also be derived as follows: j ij j i j i j j i i Q e e e e e e e Ie e = = = = ) ( ) ( (1.13.2) Dotting each side here with k e then gives 1.13.1. Eqn. 1.13.2, together with the corresponding inverse relations, read j ij i Q e e = , j ji i Q e e = (1.13.3) Note that the components of the transformation matrix [ ] Q are the components of the change of basis tensor 1.10.24-25. 1.13.2 Tensor Transformation Rule As with vectors, the components of a (second-order) tensor will change under a change of coordinate system. In this case, using 1.13.3, n m pq nq mp n nq m mp pq q p pq j i ij T Q Q Q Q T T T e e e e e e e e = = (1.13.4)

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