# assn7 - 1.050 Deformation and Strain Tensor(HW#7 Due MIT...

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1.050: Deformation and Strain Tensor (HW#7) Due: October 31, 2007 MIT – 1.050 (Engineering Mechanics I) Fall 2007 Instructor: Markus J. BUEHLER Team Building and Team Work: We strongly encourage you to form Homework teams of three students. Each team only submits one solution for correction. We expect true team work, i.e. one where everybody contributes equally to the result. This is testified by the team members signing at the end of the team copy a written declaration that "the undersigned have equally contributed to the homework". Ideally, each student will work first individually through the homework set. The team then meets and discusses questions, difficulties and solutions, and eventually, meets with the TA or the instructor. Important: Specify all resources you use for your solution. The following set of exercises is designed to familiarize you with deformation and strain measurements. 1. Derivation: Show the derivation of: J = det F for the special case of a volume change in only two directions. From the lecture notes, we derived that the deformed volume d Ω d is related to the original volume d Ω 0 by: d Ω d = ( F dX r 1 )⋅( F dX r 2 × F dX r 3 )= det F [ dX r 1 ⋅( dX r 2 × dX r 3 )]= Jd Ω 0 where d Ω 0 = dX r 1 ⋅( dX r 2 × dX r 3 ) . For the 2-D case, the deformation gradient F and the material vectors are: F 11 F 12 0 a 1 a 2 0 F = F 21 F 22 0 ; X r 1 = b 1 ; X r 2 = b 2 ; X r 3 = 0 0 0 1 0 0 1 ⎣ ⎦

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Due: Wednesday – October 31, 2007 (In class) Page 2 of 9 2. A ‘Simple’ Shear Exercise : In this exercise, we will investigate the deformation of a cube by simple shear. The cube has sides of unit-length. The magnitude of the simple shear is given on the sketch below. e r 2 2 a 1 = L 1 e r 3 e r 1 = L The displacement field for the simple shear is defined by: ξ r ( x , y , z ) = ( 2 ay ) e r 1 Where y is the distance in the e r 2 -axis. a. FINITE DEFORMATION THEORY: From finite deformation theory (refer to manuscript and lecture notes), the following results are obtained: Change in volume: d Ω t d Ω 0 = 0 d Ω 0 2 Change in surface oriented by N r = e r 1 : da = 1 + 4 a dA r Maximum dilation: λ () = 2 a 2 + 2 a 4 + a 2 + 1 1 u I Distortion in plane ( e r 1 × e r 2 ) : θ ( e r 1 , e r 2 ) = sin 1 1 + 2 a 4 a 2 For this part of the exercise, use your geometry skills only to calculate: i. The change in volume ( d Ω t d Ω 0 ) / d Ω 0 ii. The change in surface da for the surface oriented by N r = e r 1 θ ( e r 1 , e r 2 )
Due: Wednesday – October 31, 2007 (In class) Page 3 of 9 b. INFINITESIMAL (LINEAR) DEFORMATION THEORY:

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## This note was uploaded on 11/29/2011 for the course CIVIL 1.018j taught by Professor Markusbuehler during the Fall '08 term at MIT.

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assn7 - 1.050 Deformation and Strain Tensor(HW#7 Due MIT...

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