Vector Transformation in Two Dimensions
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In three dimensions, the vector transformation rule is written as
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where .
In two dimensions, this transformation rule is the familiar
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In matrix form,
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Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.
Tensor Transformation in Two Dimensions
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In three dimensions, the second-order tensor transformation rule is written as
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where .
The Cauchy stress is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as
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In matrix form,
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Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.
Tensor Transformation in two Dimensions, the intrinsic approach
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Four order tensor
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