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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 2.9 An isotropic solid with Young’s modulus E and Poisson’s ratio υ is under a state of hydrostatic stress as given in Problem 2.8. Find the corresponding strain components. Recall: ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ] [ σ σ σ σ ij Solution: (a) Three dimensional stressstrain relations can be expressed as: [ ] ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ × xy xz yz zz yy xx ij xy xz yz zz yy xx a τ τ τ σ σ σ γ γ γ ε ε ε 6 6 , where are elastic compliances. ij a (b) When the material is isotropic, can be obtained individually as: ij a E a a a 1 33 22 11 = = = , E a a a a a a υ − = = = = = = 32 31 21 23 13 12 , G a a a 1 66 55 44 = = = , and others are zero. (c) For a state of hydrostatic stress, we can obtain strain components with matrix multiplication: ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ − − − = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ) 2 1 ( ) 2 1 ( ) 2 1 ( 1 1 1 1 1 1 υ σ υ σ υ σ σ σ σ υ υ υ γ γ γ ε ε ε E E E G G symm G E E E E E E xy xz yz zz yy xx ANS 2.9.1 AAE 352 S08 HW #5 Solutions Mechanics of Aircraft structures C.T. Sun 2.10 For small strains, the volume change V V Δ is identified to be equal to zz YY xx ε ε ε + + . The bulk modulus K of an isotropic solid is defined as the ratio of the average stress and the volume change, i.e., ( ) V V K zz yy xx Δ = + + σ σ σ 3 1 Derive K in terms of E and υ . Solution: (a) Three dimensional stressstrain relations can be expressed as: [ ] ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ × xy xz yz zz yy xx ij xy xz yz zz yy xx a τ τ τ σ σ σ γ γ γ ε ε ε 6 6 , where are elastic compliances....
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This note was uploaded on 04/11/2008 for the course AAE 352 taught by Professor Chen during the Spring '08 term at Purdue.
 Spring '08
 Chen

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