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# Displacement Gradients12 - 12.005 Lecture Notes 12...

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12.005 Lecture Notes 12 Displacement Gradients Quantitative description: Suppose ' ( ) '( ) ( ) '( i i i i i i i i D D P x 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 1 2 3 1 2 3 i i i i i j j u u u u du dx dx dx dx x x x x = + + = x 2 Q P u i u i + du i 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 1 1 1 1 1 1 2 3 2 2 2 2 2 1 2 3 3 3 3 3 3 1 2 3 u u u du dx x x x u u u du dx x x x u u u du dx x x x = 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
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