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unit3_notes - 16.21 Techniques of Structural Analysis and...

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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #3 - Kinematics of deformation Ra´ul Radovitzky February 9, 2005 Figure 1: Kinematics of deformable bodies Deformation described by deformation mapping : x = ϕ ( x ) = x + u (1) 1 We seek to characterize the local state of deformation of the material in a neighborhood of a point P . Consider two points P and Q in the undeformed: P : x = x 1 e 1 + x 2 e 2 + x 3 e 3 = x i e i (2) Q : x + dx = ( x i + dx i ) e i (3) and deformed P : x = ϕ 1 ( x ) e 1 + ϕ 2 ( x ) e 2 + ϕ 3 ( x ) e 3 = ϕ i ( x ) e i (4) Q : x + dx = ϕ i ( x ) + dϕ i e i (5) configurations. In this expression, dx = dϕ i e i (6) Expressing the differentials dϕ i in terms of the partial derivatives of the functions ϕ i ( x j e j ): ∂ϕ 1 ∂ϕ 1 ∂ϕ 1 dϕ 1 = dx 1 + dx 2 + dx 3 , (7) ∂x 1 ∂x 2 ∂x 3 and similarly for dϕ 2 , dϕ 3 , in index notation: ∂ϕ i dϕ i = dx j (8) ∂x j Replacing in equation ( 5 ): ∂ϕ i Q : x + dx = ϕ i + dx j e i (9) ∂x j ∂ϕ i dx i = dx j e i (10) ∂x j We now try to compute the change in length of the segment −→ P Q which deformed into segment −−→ . Undeformed length (to the square): P Q 2 ds 2 = dx = dx · dx = dx i dx i (11) 2 Deformed length (to the square):...
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This note was uploaded on 11/07/2011 for the course AERO 16.21 taught by Professor Dffs during the Fall '10 term at MIT.

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unit3_notes - 16.21 Techniques of Structural Analysis and...

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