stress tensors in Different Coordinate Systems5

stress tensors in Different Coordinate Systems5 - 12.005...

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12.005 Lecture Notes 5 Quantities in Different Coordinate Systems How to express quantities in different coordinate systems? Direction cosine ij is cosine of angle φ ij between primed axis i and unprimed axis j. If 'and represent unit vectors that are the axes of two coordinate systems with the same origin, they are related by the equation ˆ i x j x ˆ where α ij is the cosine of the angle between the primed axis and the unprimed axis . For example, ' ˆ i x j x ˆ 12 is the cosine of the angle between and . ' ˆ 1 x 2 ˆ x ij represents a 9- component matrix called the transformation matrix. Unlike the stress tensor, it is not symmetric ( ij ji ). x 3 x 3 P ' x 2 ' x 1 ' α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 x 1 x 2 x 3 x 2 ' x 3 ' x 1 ' x 2 x 1 φ 23 φ 12 φ 11 Direction Cosines Axis Figure by MIT OCW. 33 ii j j j j i j j j1 ˆˆ x' x, x x == =α→ α ∑∑ Figure 5.1
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= 33 32 31 23 22 21 13 12 11 α ij In matix equations, the transformation law is written = 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1 ' ' ' x x x x x x The inverse transformation law is written ' ˆ ˆ j ji i x x = Consider the following transformation of coordinates: x 2 x 1 x 3 , x 3 x 2 ' x 1 30 o ' ' Figure by MIT OCW. Figure 5.1a
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° ° ° ° = 0 0 0 0 30 120 0 60 30 ij α The explicit transformation equations are 2 1 2 2 1 1 ˆ 30 cos ˆ 120 cos ' ˆ ˆ 60 cos ˆ 30 cos ' ˆ x x x x x x ° + ° = ° + ° = Since and are both unit length, these equations are easy to verify from the picture. ' ˆ i x j x ˆ First-order tensors First-order tensors or vectors have two components in 2D coordinates and three components in 3D coordinates. They transform according to the same laws as coordinate axes because coordinate axes are themselves vectors. If u j is a vector in the coordinate system and u j x ˆ i is a vector in the coordinate system, then the following equations describe their transformation: ' ˆ i x ' ' j ji i j ij i u u u u = = Note that ij is positive if the angle is measured counterclockwise from to . It is negative if the angle is measured clockwise.
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stress tensors in Different Coordinate Systems5 - 12.005...

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