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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
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
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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