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Lec.15 - 15 FINITE STRAIN INFINITESIMAL STRAIN(AT A POINT I...

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10/12/11 1 15. FINITE STRAIN & INFINITESIMAL STRAIN (AT A POINT) I Main Topics A The finite strain tensor [E] B DeformaIon paths for finite strain C Infinitesimal strain and the infinitesimal strain tensor ε 10/12/11 GG303 1 15. FINITE STRAIN & INFINITESIMAL STRAIN 10/12/11 GG303 2 The sequence of deformaIon maRers for large strains
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10/12/11 2 15. FINITE STRAIN & INFINITESIMAL STRAIN 10/12/11 GG303 3 The sequence of deformaIon does not maRer for small strains 15. FINITE STRAIN & INFINITESIMAL STRAIN II The finite strain tensor [E] A Used to find the changes in the squares of distances (ds) 2 between points in a deformed body based on differences in their iniIal posiIons B Displacements by themselves don’t give changes in size or shape 10/12/11 GG303 4 Sides of lower boxes maintain their length, but the diagonals change length dy dx ds
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10/12/11 3 15. FINITE STRAIN & INFINITESIMAL STRAIN II The finite strain tensor [E] C DerivaIon of [E] 1 2 3 4 10/12/11 GG303 5 dx dy = dX [ ] ds ( ) 2 = dx ( ) 2 + dy ( ) 2 ds ( ) 2 = dx dy dx dy dx dy = dX [ ] T 15. FINITE STRAIN & INFINITESIMAL STRAIN II The finite strain tensor [E] C DerivaIon of [E] 1 2 3 4 10/12/11 GG303 6 5 6 7 8 dx dy = dX [ ] ds ( ) 2 = dx ( ) 2 + dy ( ) 2 ds ( ) 2 = dX [ ] T dX [ ] = dX [ ] T I [ ] dX [ ] , where I [ ] = 1 0 0 1 d s ( ) 2 = d x d y d x d y = d X [ ] T d X [ ] ds ( ) 2 = dx dy dx dy dx dy = dX [ ] T d X [ ] = F [ ] dX [ ] From lecture 14 : d s ( ) 2 = F [ ] dX [ ] T F [ ] dX [ ] F [ ] dX [ ] T = dX [ ] T F [ ] T
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10/12/11 4 15. FINITE STRAIN & INFINITESIMAL STRAIN C DerivaIon of [E] (cont.) 9 10 11 12 10/12/11 GG303 7 d s ( ) 2 ds ( ) 2 = dX [ ] T F [ ] T F [ ] dX [ ] dX [ ] T I [ ] dX [ ] d s ( ) 2 ds ( ) 2 = dX [ ] T F [ ] T F [ ] I [ ] dX [ ] d s ( ) 2 ds ( ) 2 { } 2 = dX [ ] T F [ ] T F [ ] I [ ] dX [ ] 2 1 2 d s ( ) 2 ds ( ) 2 { } = dX [ ] T E [ ] dX [ ] E [ ] 1 2 F [ ] T F [ ] I [ ] 15. FINITE STRAIN & INFINITESIMAL STRAIN D Meaning of [E] 1 2 3 But what do the terms of E mean? 10/12/11 GG303 8 1 2 d s ( ) 2 ds ( ) 2 { } = dX [ ] T E [ ] dX [ ] E [ ] 1 2 F [ ] T F [ ] I [ ] E [ ] 1 2 J u + I [ ] T J u + I [ ] I [ ] Given E and dX (the difference in iniIal posiIons) of points), then one can find the difference in the squares of the lengths of lines connecIng the points before and afer deformaIon
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10/12/11 5 15. FINITE STRAIN & INFINITESIMAL STRAIN E Expansion of [E] 1 2 10/12/11 GG303 9 J u = u x u y v x v y E [ ] 1 2 J u + I [ ] T J u + I [ ] I [ ] E [ ] 1 2 u x u y v x v y + 1 0 0 1 T u x u y v x v y + 1 0 0 1 1 0 0 1 15. FINITE STRAIN & INFINITESIMAL STRAIN E Expansion of [E] 3 4 10/12/11 GG303 10 E [ ] 1 2 J u + I [ ]
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