Displacement Gradients12 - 12.005 Lecture Notes 12...

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12.005 Lecture Notes 12 Displacement Gradients Quantitative description: Suppose ' () ' ( ) ' ( ii i i DD Px P x u Q x dx Q x dx u du →+ +→ ++ + ) i Figure 12.1 Suppose: The deformation is continuous. The first derivative i j u x are continuous and very small. Then chain rule 123 12 3 ii i i ij j uuu u du dx dx dx dx xx x x ∂∂∂ =++= ∂∂ x 2 Q P u i u i + d P' D' D Q' x 1 x 3 Figure by MIT OCW.

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Note: i j u x relates two vectors and is therefore a second rank tensor. and i du dx j 111 11 123 222 22 333 33 uuu du dx xxx du dx du dx ⎡⎤ ∂∂∂ ⎢⎥ = ⎣⎦ For the Ventura Basin results shown, for the 2-D solution, for the stations HOPP-HAPY- SNP, 6 0.2 0.45 10 0.08 0.48 i j u x each year Note: We have no sensitivity to rigid body translations 0 i j u x ⎛⎞ ⎜⎟ ⎝⎠ . What about rotations? Figure 12.2 α α x 2 B A A' B' x 1 Figure by MIT OCW.
2 1 1 2 12 12 21 tan 1 2 u x u x uu xx α ω =− → − = ⎛⎞

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This note was uploaded on 10/25/2010 for the course MIT Geodynamic taught by Professor Ywn during the Fall '10 term at MIT.

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Displacement Gradients12 - 12.005 Lecture Notes 12...

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