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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 2.15 Write the strain energy density expression in terms of stress components by using (2.95) for isotropic solids and show that the Poisson’s ratio is bounded by 1 and 0.5. Solution: From equation (2.95), we have the strain energy density: } ]{ [ } { 2 1 } ]{ [ } { 2 1 σ σ ε ε a c W T T = = (2.15.1) For isotropic material, the stressstrain relationship can be expressed in terms of elastic constants or elastic compliances: [ ] ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ × xy xz yz zz yy xx ij xy xz yz zz yy xx c γ γ γ ε ε ε τ τ τ σ σ σ 6 6 , where are elastic constants. ij c G c c c 2 33 22 11 + = = = λ , λ = = = = = = 32 31 21 23 13 12 c c c c c c , G c c c = = = 66 55 44 , and the rest are zero. where ( ) ( ) υ υ υ λ 2 1 1 − + = E and ( ) υ + = 1 2 E G [ ] ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ × 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 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 the rest are zero. The strain energy density in terms of stress components can be derived from equation (2.15.1) as 2.15.1 Mechanics of Aircraft structures C.T. Sun { } [ ] )} )( 1 ( 2 ) ( 2 { 2 1 2 1 2 2 2 2 2 2 6 6 yz xz xy zz yy zz xx yy xx zz yy xx xy xz yz zz yy xx ij xy xz yz zz yy xx E a W τ τ τ υ σ σ σ σ σ σ υ σ σ σ τ τ τ σ σ σ τ τ τ σ σ σ + + + + + + − + + = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = × If we choose to use the principal directions as the coordianate system, then , 1 xx σ σ = , 2 yy σ σ = , 3 zz σ σ = and , where yz xz xy = = = τ τ τ 3 , 2 , 1...
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This note was uploaded on 01/18/2012 for the course AAE 352 taught by Professor Chen during the Fall '08 term at Purdue.
 Fall '08
 Chen

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