EN0175-07

EN0175-07 - EN0175 09 / 26 / 06 Review on coordinate...

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EN0175 09 / 26 / 06 Review on coordinate transformation (change of basis) for tensors. 1 e v 2 e v 3 e v ' 2 e v ' 1 e v ' 3 e v u v ' ' p p i i e u e u u v v v = = ( ) ' ' ' p i i p p e e u e u u v v v v = = ( ) i q q i i e e u e u u v v v v = = ' ' Combining the above 2 equations yields ( )( ) ip iq q p i q i q p Q Q u e e e e u u ' ' ' ' ' = = v v v v T h e r e f o r e pq ip iq Q Q δ Similarly, we can show ij jp ip Q Q (Hint: using ( )( ) jp ip j p i p j j i Q Q u e e e e u u = = ' ' v v v v ) In matrix form: I Q Q T = , I Q Q T = Such matrices/tensors are called orthogonal matrices/tensors. Example: Transformation from 2D Cartesian coordinate to 2D Polar coordinate. 1 e v 2 e v r e v r θ e v 1
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EN0175 09 / 26 / 06 = θθ cos sin sin cos Q , = = 1 0 0 1 cos sin sin cos cos sin sin cos T Q Q Chap 3. Stress in a solid Continuum ––– Continuous media ignoring the atomic and other discreteness of matters.
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EN0175-07 - EN0175 09 / 26 / 06 Review on coordinate...

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