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4020-F2011-HW5Solutions

4020-F2011-HW5Solutions - MSE 4020 HW5-SOLUTION Problem 1(a...

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1 MSE 4020 HW5-SOLUTION 10/03/2011 Problem 1 (a) x axis corresponds to index 1, and y axis corresponds to index 2. Thus, only ! 12 " 0 Thus, ! 11 ! 22 ! 33 ! 23 ! 13 ! 12 " # $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' = 0 0 0 0 0 ! 12 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' = C 11 C 12 C 12 0 0 0 C 12 C 11 C 12 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 " # $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ( 11 ( 22 ( 33 ) 23 ) 13 ) 12 " # $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' * ! 12 = C 44 + ) 12 * G = ! 12 ) 12 = C 44 (b) Stress and state in unrotated coordinate system, ! ! = 0 ! 12 0 ! 12 0 0 0 0 0 " # $ $ $ $ % & ' ' ' ' , ! ( = 0 ) 2 0 ) 2 0 0 0 0 0 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' After transformation to the new coordinate system, it becomes, ! ! ' = " ! !" # 1 = 2 2 2 2 0 # 2 2 2 2 0 0 0 1 $ % & & & & & & & & ' ( ) ) ) ) ) ) ) ) 0 ! 12 0 ! 12 0 0 0 0 0 $ % & & & & ' ( ) ) ) ) 2 2 # 2 2 0 2 2 2 2 0 0 0 1 $ % & & & & & & & & ' ( ) ) ) ) ) ) ) ) = #! 12 0 0 0 ! 12 0 0 0 0 $ % & & & & ' ( ) ) ) ) Similarly for the strain after transformation,
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2 ! ' = " !" # 1 = 2 2 2 2 0 # 2 2 2 2 0 0 0 1 $ % & & & & & & & & ' ( ) ) ) ) ) ) ) ) 0 * 2 0 * 2 0 0 0 0 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) 2 2 # 2 2 0 2 2 2 2 0 0 0 1 $ % & & & & & & & & ' ( ) ) ) ) ) ) ) ) = # * 2 0 0 0 * 2 0 0 0 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) In the new coordinate system, using Nye notation gives that !" 12 " 12 0 0 0 0 # $ % % % % % % % % & ' ( ( ( ( ( ( ( ( == C 11 C 12 C 12 0 0 0 C 12 C 11 C 12 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 # $ % % % % % % % % % & ' ( ( ( ( ( ( ( ( ( ! ) 2 ) 2 0 0 0 0 # $ % % % % % % % % % % & ' ( ( ( ( ( ( ( ( ( ( * " 12 = C 12 + ! ) 2 , - . / 0 1 + C 11 + ) 2 = )+ C 11 ! C 12 2 * G = " 12 ) = C 11 ! C 12 2 (c) If the material is isotropic, then the Shear modulus obtained from (a) and (b) should be consistent with each other, i.e., C 11 ! C 12 2 = C 44
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3 Problem 2 (a) Using 9 × 9 matrix, the stress and strain have the following relation: ! 11 ! 22 ! 33 ! 23 ! 13 ! 12 ! 23 ! 13 ! 12 " # $ $ $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' ' ' = 0 0 0 0 0 ! 12 0 0 ! 12 " # $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' = C 11 C 12 C 12 0 0 0 0 0 0 C 12 C 11 C 12 0 0 0 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 0 0 0 C 44 0 0 C 44 0 0 0 0 0 0 C 44 0 0 C 44 0 0 0 0 0 0 C 44 0 0 C 44 0 0 0 C 44 0 0 C 44 0 0 0 0 0 0 C 44 0 0 C 44 0 0 0 0 0 0 C 44 0 0 C 44 " # $ $ $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ( 11 ( 22 ( 33 ( 23 ( 13 ( 12 ( 23 ( 13 ( 12 " # $ $ $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ) !
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