HW5 Solutions

HW5 Solutions - Mechanics of Aircraft structures C.T. Sun...

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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 Poissons ratio is bounded by -1 and 0.5. Solution: From equation (2.95), we have the strain energy density: } ]{ [ } { 2 1 } ]{ [ } { 2 1 a c W T T = = (2.15.1) For isotropic material, the stress-strain 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 6 6 , where are elastic constants. ij c G c c c 2 33 22 11 + = = = , = = = = = = 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 ( ) ( ) 2 1 1 + = E and ( ) + = 1 2 E G [ ] = xy xz yz zz yy xx ij xy xz yz zz yy xx a 6 6 , where are elastic compliances. ij a E a a a 1 33 22 11 = = = , E a a a a a a = = = = = = 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 + + + + + + + + = = If we choose to use the principal directions as the coordianate system, then , 1 xx = , 2 yy = , 3 zz = and , where yz xz xy = = = 3 , 2 , 1...
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HW5 Solutions - Mechanics of Aircraft structures C.T. Sun...

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