Section 1.13 Solid Mechanics Part III Kelly 1101.13 Coordinate Transformation of Tensor Components It has been seen in §1.5.2 that the transformation equations for the components of a vector are jijiuQu′=, where Qis the transformation matrix. Note that these ijQ’s are not the components of a tensor– these sQij'are mapping the components of a vector onto the components of the same vectorin a second coordinate system – a (second-order) tensor, in general, maps one vector onto a different vector. The equation jijiuQu′=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 ieand ie′. It has been seen in the context of vectors that, Eqn. 1.5.4, ),cos(jiijjixxQ′≡=′⋅ee. (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 differentvectors (nine different vectors in all). It is interesting that the relationship 1.13.1 can also be derived as follows: jijjijijjiiQeeeeeeeIee′=′⋅′=′⊗′==)()((1.13.2) Dotting each side here with ke′then gives 1.13.1. Eqn. 1.13.2, together with the corresponding inverse relations, read jijiQee′=, jjiiQee=′(1.13.3) Note that the components of the transformation matrix Qare 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, nmpqnqmpnnqmmppqqppqjiijTQQQQTTTeeeeeeee⊗′=⊗′=′⊗′′≡⊗(1.13.4)
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