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16100lecture4_cg

16100lecture4_cg - Coordination Transformations for Strain...

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Coordination Transformations for Strain & Stress Rates To keep the presentation as simple as possible, we will look at purely two-dimensional stress-strain rates. Given an original coordinate system ( x , y ) and a rotated system ( y x ˆ, ˆ ) as shown below: x x' y y' θ Recall that the strain rates in the x-y coordinate system are: u 1  ∂ u v v ε = ε = ε = xx xy yy x 2 y + x y Or, in index notation: = ε ij 1 2 x u i j + u x i j Also, we note that the unit vectors for the rotated axes are: K i K ˆ = cos θ i K + sin θ j K K K ˆ j = − sin θ i + cos θ j Thus, the location of a point in ( y x ˆ, ˆ ) is: cos θ sin θ   x ˆ x =   y ˆ y sin θ cos θ   Similarly, the velocity components are related by:   cos θ sin θ    u ˆ u =      v ˆ v   sin θ cos θ    For differential changes, we also have cos θ sin θ   dx ˆ dx =   dy ˆ dy sin θ cos θ  
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