EN0175-13

EN0175-13 - EN0175 10 17 06 Summary of elementary strain...

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Unformatted text preview: EN0175 10 / 17 / 06 Summary of elementary strain concepts and their generalizations to 3D tensors: 1D 3D ( ) 1 1 − = − = = λ δ ε l l l l ( ) I U − ( ) 1 2 1 − − = − = = λ δ ε l l l l ( ) 1 − − V I ( ) ( ) 1 2 1 2 2 2 2 2 3 − = − = λ ε l l l ( ) I C − 2 1 ( ) ( ) 2 2 2 2 4 1 2 1 2 − − = − = λ ε l l l ( ) 1 2 1 − − B I For small strain: ( ) ( ) ( ) ( ) ε ε ε ε ε = = = = 4 3 2 1 ( ) ( ) 1 1 2 1 2 1 − − − = − = − = − = B I I C V I I U ε ( ) i j j i ij u u , , 2 1 + = ε or ( ) u u T v v ∇ + ∇ = 2 1 ε 6 equations: 1 1 11 x u ∂ ∂ = ε , 2 2 22 x u ∂ ∂ = ε , 3 3 33 x u ∂ ∂ = ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = 1 2 2 1 12 2 1 x u x u ε , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = 1 3 3 1 13 2 1 x u x u ε , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = 2 3 3 2 23 2 1 x u x u ε 1 equation: x u ∂ ∂ = ε For small strain and small rotation, the rigid body rotation part is analyzed as follows: ω + = I R where ω is defined as the small rotation tensor. ( ) ( ) u I F I I I U R v ∇ + = = + + ≅ + + = ω ε ε ω ( ) ( ) u u u u u u T T v v v v v v ∇ − ∇ = ∇ + ∇ − ∇ = − ∇ = 2 1 2 1 ε ω ω ω − = T (antisymmetric tensor) 1 EN0175 10 / 17 / 06 Alternative way of deriving small strain tensor: 1 d...
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This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.

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EN0175-13 - EN0175 10 17 06 Summary of elementary strain...

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