Lec.15.pptx - 10/12/11 15. FINITE STRAIN &...

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Unformatted text preview: 10/12/11 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 The sequence of deformaIon maRers for large strains 10/12/11 GG303 2 1 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN The sequence of deformaIon does not maRer for small strains 10/12/11 GG303 3 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 ds posiIons dy B Displacements by dx themselves don’t give of l boxes m changes in size or shape Sides but ower iagonals aintain tlheir length, the d change ength 10/12/11 GG303 4 2 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN II The finite strain tensor [E] C DerivaIon of [E] 2 2 2 1 ( ds ) = ( dx ) + ( dy ) ⎡ dx ⎤ 2 2 ( ds ) = ⎡ dx dy ⎤ ⎢ dy ⎥ ⎣ ⎦⎢ ⎥ ⎣ ⎦ 3 4 ⎡ ⎣ ⎡ dx ⎤ ⎢ ⎥ = [ dX ] ⎢ dy ⎦ ⎥ ⎣ T dx dy ⎤ = [ dX ] ⎦ 10/12/11 GG303 5 15. FINITE STRAIN & INFINITESIMAL STRAIN II The finite strain tensor [E] C DerivaIon of [E] 2 2 2 1 ( ds ) = ( dx ) + ( dy ) 5 ( ds )2 = [ dX ]T [ dX ] = [ dX ]T [ I ][ dX ], ⎡1 0⎤ where [ I ] = ⎢ ⎥ ⎣0 1⎦ ⎡ dx ⎤ ⎡ ⎤ 2 dx ′ 2 ds = ⎡ dx dy′ ⎤ ⎢ 2 ( ds ) = ⎡ dx dy ⎤ ⎢ ⎦ ⎢ dy′ ⎥ ⎣ ⎦ ⎢ dy ⎥ 6 ( ′ ) ⎣ ′ ⎥ ⎥ ⎣ ⎦ ⎣ ⎦ T T = [ dX ′ ] [ dX ′ ] 3 ⎡ dx dy ⎤ = [ dX ] ⎣ ⎦ 7 [ dX ′ ] = [ F ][ dX ] From lecture 14: ⎡ dx ⎤ 4 ⎢ dy ⎥ = [ dX ] ⎢ ⎥ ⎣ ⎦ 8 ( ds′ )2 = ⎡[ F ][ dX ]⎤T ⎡[ F ][ dX ]⎤ ⎣ ⎦⎣ ⎦ ⎡[ F ][ dX ]⎤ = [ dX ] [ F ] ⎣ ⎦ T 10/12/11 GG303 T T 6 3 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN C DerivaIon of [E] (cont.) 9 ( ds′ ) − ( ds ) = [ dX ] [ F ] [ F ][ dX ] − [ dX ] [ I ][ dX ] 2 2 T T T 10 ( ds′ ) − ( ds ) = [ dX ] ⎡[ F ] [ F ] − [ I ]⎤ [ dX ] ⎣ ⎦ 2 2 T T 11 {( ds′ ) − ( ds ) } = [ dX ] ⎡[ F ] [ F ] − [ I ]⎤ [ dX ] ⎣ ⎦ 2 T 2 T 2 2 12 1 {( ds′ ) − ( ds ) } = [ dX ] [ E ][ dX ] 2 2 2 10/12/11 T [E] ≡ 1 ⎡[ F ]T [ F ] − [ I ]⎤ ⎦ 2⎣ GG303 7 15. FINITE STRAIN & INFINITESIMAL STRAIN D Meaning of [E] 1 1 {( ds′ )2 − ( ds )2 } = [ dX ]T [ E ][ dX ] 2 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 1 T 2 [ E ] ≡ 2 ⎡[ F ] [ F ] − [ I ]⎤ ⎣ ⎦ 1 3 [ E ] ≡ 2 ⎡[ Ju + I ]T [ Ju + I ] − [ I ]⎤ ⎣ ⎦ But what do the terms of E mean? 10/12/11 GG303 8 4 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN E Expansion of [E] 1 ⎡ ⎢ Ju = ⎢ ⎢ ⎢ ⎢ ⎣ ∂u ∂x ∂v ∂x ⎡⎡ ⎢⎢ 1 ⎢⎢ 2 [ E ] ≡ 2 ⎢ ⎢ ⎢⎢ ⎢⎢ ⎣⎣ ∂u ∂x ∂v ∂x [E] ≡ ∂u ⎤ ⎥ ∂y ⎥ ∂v ⎥ ⎥ ∂y ⎥ ⎦ ⎤ ⎥ ⎡ 1 0 ⎤⎥ +⎢ ⎥⎥ ∂v ⎣ 0 1 ⎦⎥ ∂y ⎥ ⎦ ∂u ∂y 10/12/11 T 1⎡ T J + I ] [ J u + I ] − [ I ]⎤ ⎣[ u ⎦ 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂u ∂x ∂v ∂x ⎤ ⎤ ⎥ ⎥ ⎡ 1 0 ⎤⎥ ⎡ 1 0 ⎤⎥ +⎢ ⎥⎥ − ⎢ ⎥⎥ ∂v ⎣ 0 1 ⎦⎥ ⎣ 0 1 ⎦⎥ ⎥ ∂y ⎥ ⎦ ⎦ ∂u ∂y GG303 9 15. FINITE STRAIN & INFINITESIMAL STRAIN E Expansion of [E] 3 4 ⎡ ⎡ ∂u ∂u ⎢⎢ +1 ∂x ∂y 1⎢ [ E ] ≡ ⎢ ⎢ ∂v ∂v ⎢ 2⎢ +1 ⎢ ⎢ ⎢ ∂x ∂y ⎣⎣ [E] ≡ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ T 1⎡ T J + I ] [ J u + I ] − [ I ]⎤ ⎣[ u ⎦ 2 ⎡ ∂u ∂u +1 ⎢ ∂x ∂y ⎢ ⎢ ∂v ∂v +1 ⎢ ∂y ⎢ ∂x ⎣ ⎤ ⎤ ⎥ ⎥ ⎥ − ⎡ 1 0 ⎤⎥ ⎥ ⎢ 0 1 ⎥⎥ ⎦⎥ ⎥⎣ ⎥ ⎥ ⎦ ⎦ ⎡ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎢⎜ +1 +1 + −1 ⎜ +1 ⎜ ⎟ + + 1⎟ ⎝ ∂x ⎟ ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ∂x ⎟ ⎝ ∂y ⎠ ⎜ ∂x ⎟ ⎝ ∂y ⎠ ⎠ ⎝ ⎠⎜ 1⎢ [E] ≡ ⎢ 2 ⎢ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ +1 + +1 ⎜ ∂y ⎟ ⎜ ∂y ⎟ + ⎜ ∂y + 1⎟ ⎜ ∂y + 1⎟ − 1 ⎢ ⎜ ∂y ⎟ ⎜ ∂x ⎟ ⎜ ∂y ⎟ ⎜ ∂x ⎟ ⎠⎝ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ The meaning of the terms in E is not intuiIve Imagine what the products of mulIple finite deformaIons yield 10/12/11 GG303 10 5 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN F Conversion of [E] to [ε] for small strains Finite strain ⎡ ⎛ ∂u ⎢ 1 ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ + 1⎟ ⎜ + 1⎟ + ⎜ ⎟ ⎜ ⎟ − 1 ⎜ + 1⎟ +⎜ ⎟ +1 ⎜ ⎝ ∂x ⎠ ⎜ ∂y ⎟ ⎝ ∂x ⎠ ⎜ ∂y ⎟ ⎝⎠ ⎝ ⎠ 1 ⎢ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂x ⎠ E] ≡ ⎢ [ 2 ⎢ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂v ⎞ + 1 + ⎜ + 1⎟ ⎜ ⎟⎝ ⎜ ⎟ ⎜ ⎟ + ⎜ + 1⎟ ⎜ + 1⎟ − 1 ⎢ ⎝ ∂y ⎠ ⎜ ∂x ⎟ ⎝ ∂y ⎠ ⎜ ∂x ⎟ ⎠ ⎝⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎝ ∂y ⎠ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ If the displacement derivaIves are <<1, then their products are Iny and can be neglected Infinitesimal ⎡ ∂u ⎛ ∂u ⎞ ∂u ∂v ⎢ ⎛ ⎞ +⎛ ⎞ ⎜ ⎟ +⎛ ⎞ strain ⎜⎟⎜⎟ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎜ ∂x ⎟ ⎝⎠ 1⎢ [ε ] ≡ ⎢ 2 2 ⎢ ⎛ ∂u ⎞ ⎛ ∂v ⎞ + ⎢ ⎜ ∂y ⎟ ⎜ ∂x ⎟ ⎝⎠⎝⎠ ⎣ ⎤ ⎥ ⎥1 T ⎥ = ⎡[ J u ] + [ J u ] ⎤ ⎦ ⎛ ∂v ⎞ ⎛ ∂v ⎞ ⎥ 2 ⎣ ⎜ ∂y ⎟ + ⎜ ∂y ⎟ ⎥ ⎝⎠⎝⎠ ⎦ 10/12/11 GG303 11 15. FINITE STRAIN & INFINITESIMAL STRAIN III DeformaIon paths for finite strain Consider two different deformaIons A DeformaIon 1 ⎡ a1 [F ] = ⎢ 1 ⎢ c1 ⎣ b1 ⎤ ⎥ d1 ⎥ ⎦ B DeformaIon 2 ⎡ a2 [F ] = ⎢ 2 10/12/11 ⎢ c2 ⎣ b2 ⎤ ⎥ d2 ⎥ ⎦ GG303 12 6 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN III DeformaIon paths for finite strain Consider two different deformaIons A DeformaIon 1 ⎡a b ⎤ [ F1 ] = ⎢ c1 d1 ⎥ ⎢ 1 1⎥ ⎣ ⎦ B DeformaIon 2 ⎡ a2 [F ] = ⎢ 2 ⎢ c2 ⎣ 10/12/11 C F2 acts on F1 ⎡ a2 a1 + b2 c1 [ F ][ F ] = ⎢ 2 1 ⎢ c2 a1 + d2 c1 ⎣ a2b1 + b2 d1 ⎤ ⎥ c2b1 + d2 d1 ⎥ ⎦ D F1 acts on F2 ⎡ a1a2 + b1c2 [ F ][ F ] = ⎢ 1 b2 ⎤ ⎥ d2 ⎥ ⎦ 2 ⎢ c1a2 + d1c2 ⎣ a1b2 + b1d2 ⎤ ⎥ c1b1 + d1d2 ⎥ ⎦ E The sequence of finite deformaIons maRers – unless off ­diagonal terms are small GG303 13 15. FINITE STRAIN & INFINITESIMAL STRAIN [F1] ⎡ 2 1 ⎤ ⎡ 1 0 ⎤⎡ 2 1 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ = [ F2 ][ F1 ] ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 2 ⎦ ⎡ x′ ⎤ ⎡ 2 1 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y′ ⎥ ⎣ 0 1 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ 2 1 ⎤ ⎡ 2 1 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 1 ⎦ ⎣ 0 1 ⎦⎣ 0 1 ⎦ [F2] ⎡ 2 2 ⎤ ⎡ 2 1 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ = [ F1 ][ F2 ] ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 2 ⎦ ⎡ x′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ 0 2 ⎦⎣ ⎦ 10/12/11 ⎡ 1 0 ⎤ ⎡ 1 0 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 1 ⎦ GG303 [ F ][ F ] ≠ [ F ][ F ] 2 1 1 2 14 7 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN [F1] ⎡ x′ ⎤ ⎡ 2 1 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ 0 1 ⎦⎣ ⎦ [F1] [F2][F1] ⎡ 2 1 ⎤ ⎡ 2 1 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 1 ⎦ ⎣ 0 1 ⎦⎣ 0 1 ⎦ ⎡ x ′′ ⎤ ⎡ 2 1 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y′′ ⎥ ⎣ 0 2 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ [F2][F1] [F1][F2] [F2] ⎡ x′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y′ ⎥ ⎣ 0 2 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ [F2] 10/12/11 ⎡ 2 1 ⎤ ⎡ 2 1 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 1 ⎦ ⎡ 1 0 ⎤ ⎡ 1 0 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 1 ⎦ ⎡ x ′′′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y′′′ ⎥ ⎣ 0 2 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ [F1][F2] GG303 ⎡ 2 2 ⎤ ⎡ 2 2 ⎤⎡ 1 0 ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ 0 2 ⎦ ⎣ 0 2 ⎦⎣ 0 1 ⎦ [ F ][ F ] ≠ [ F ][ F ] 2 1 1 15 2 15. FINITE STRAIN & INFINITESIMAL STRAIN ⎡ ⎛ ∂u ⎞ ⎛ ∂u ⎞ IV Infinitesimal strain and the ⎢ ⎜ ⎟ +⎜ ⎟ infinitesimal strain tensor [ε] 1 ⎢ ⎝ ∂x ⎠ ⎝ ∂x ⎠ [ε ] ≡ ⎢ A Infinitesimal strain 2 ⎢ ⎛ ∂u ⎞ ⎛ ∂v ⎞ + ⎢ ⎜ ∂y ⎟ ⎜ ∂x ⎟ ⎝⎠⎝⎠ DeformaIon where the ⎣ displacement derivaIves 1⎡ T ⎤ in [Ju] are small relaIve to [ε ] = 2 ⎣[ Ju ] + [ Ju ] ⎦ one so that the products ⎡ ∂u ∂u ⎤ of the derivaIves are very ⎢ ⎥ ∂x ∂y ⎥ J ]= ⎢ small and can be ignored. [ ⎢ ∂v ∂v ⎥ [ J ] ⎢ ⎥ B An approximaIon to finite ⎢ ∂x ∂y ⎦ ⎥ ⎣ strain T u 10/12/11 GG303 u ∂u ⎞ ⎛ ∂v ⎞ ⎤ ⎜ ∂y ⎟ + ⎜ ∂x ⎟ ⎥ ⎝ ⎠ ⎝ ⎠⎥ ⎥ ⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎥ +⎜ ⎟ ⎜ ∂y ⎟ ⎝ ∂y ⎠ ⎥ ⎝⎠ ⎦ ⎡ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ∂u ∂x ∂u ∂y ∂v ∂x ∂v ∂y ⎤ ⎥ ⎥ ∂v ⎥ ∂x ⎥ ⎥ ⎦ 16 8 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN GelaIn Volcano Experiment IV Infinitesimal strain and the infinitesimal strain tensor [ε] (cont.) C Why consider [ε] if it is an approximaIon? 1  Relevant to important geologic deformaIons A Fracture B Earthquake deformaIon C Volcano deformaIon hRp://www.spacegrant.hawaii.edu/class_acts/WebImg/gelaInVolcano.gif 10/12/11 GG303 17 15. FINITE STRAIN & INFINITESIMAL STRAIN C Why consider [ε] if it is an approximaIon? (cont.) 2 Terms of the infinitesimal strain tensor [ε] have clear geometric meaning 3  Can apply principal of superposiIon (addiIon) 4  Infinitesimal deformaIon is essenIally independent of the deformaIon sequence 5 Amenable to sophisIcated mathemaIcal treatment (e.g., elasIcity theory) 6 QuanItaIve predicIve ability GelaIn Volcano Experiment hRp://www.spacegrant.hawaii.edu/class_acts/WebImg/gelaInVolcano.gif 10/12/11 GG303 18 9 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN C Why consider [ε] if it is an approximaIon? (cont.) 7 Infinitesimal strain example ⎡ 1.02 0.01 ⎤ F3 = ⎢ ⎥ ⎣ 0 1.01 ⎦ J u ( 3) ⎡ 0.02 0.01 ⎤ =⎢ ⎥ 0.01 ⎦ ⎣0 [ F ] = [ Ju ] + [ I ] ⎡ 1.01 0 ⎤ F4 = ⎢ ⎥ ⎣ 0 1.02 ⎦ ⎡ 0.01 0 ⎤ Ju ( 4 ) = ⎢ ⎥ 0.02 ⎦ ⎣0 Sequence results can be obtained regardless of the order of events, but also by superposiIon to a high degree of accuracy [J ] = [F] − [I ] u ⎡ 1.01 0 ⎤ ⎡ 1.02 0.01 ⎤ ⎡ 1.0302 0.0100 ⎤ ⎥⎢ ⎥=⎢ ⎥ ⎣ 0 1.02 ⎦ ⎣ 0 1.01 ⎦ ⎣ 0.0000 1.0302 ⎦ [ F ][ F ] = ⎢ 4 3 ⎡ 1.02 0.01 ⎤ ⎡ 1.01 0 ⎤ ⎡ 1.0302 0.0101 ⎤ ⎥⎢ ⎥=⎢ ⎥ ⎣ 0 1.01 ⎦ ⎣ 0 1.02 ⎦ ⎣ 0.0000 1.0302 ⎦ [ F ][ F ] = ⎢ 3 4 ⎡ 1.0300 0.0100 ⎤ ⎥ ⎣ 0.0000 1.0300 ⎦ [F ] = ⎢ 3+ 4 ⎡ 0.02 0.01 ⎤ ⎡ 0.01 0 ⎤ ⎡ 0.0300 0.0100 ⎤ ⎡ J u ( 3) ⎤ + ⎡ J u ( 4 ) ⎤ = ⎢ ⎥+⎢ ⎥=⎢ ⎥ ⎣ ⎦⎣ ⎦ 0.01 ⎦ ⎣ 0 0.02 ⎦ ⎣ 0.0000 0.0300 ⎦ ⎣0 10/12/11 GG303 19 15. FINITE STRAIN & INFINITESIMAL STRAIN IV Infinitesimal strain and the infinitesimal strain tensor [ε] D Taylor series expansion We seek U2 given U1 and dX 1 ⎛ ∂u ∂u ⎞ u2 = u1 + du = u1 + ⎜ dx + dy + … ⎝ ∂x ⎠ ∂y ⎟ 2 ⎛ ∂v ∂v ⎞ v2 = v1 + dv = v1 + ⎜ dx + dy⎟ + … ⎝ ∂x ∂y ⎠ 3 10/12/11 ⎡ ∂u ⎡ u2 ⎤ ⎡ u1 ⎤ ⎢ ∂x ⎢ ⎥=⎢ ⎥+⎢ ⎢ v2 ⎥ ⎢ v1 ⎥ ⎢ ∂v ⎣ ⎦⎣ ⎦⎢ ⎢ ∂x ⎣ ∂u ⎤ ⎥ ∂y ⎥ ⎡ dx ⎤ ⎢ ⎥ + … ⇒ [U 2 ] ≈ [U1 ] + [ dU1 ] = [U1 ] + [ J u ][ dX ] ∂v ⎥ ⎣ dy ⎦ ⎢ ⎥ ⎥ ∂y ⎥ ⎦ GG303 20 10 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN D Taylor series expansion (cont.) Now split [Ju] into two matrices: the infinitesimal strain matrix [ε] and the anI ­symmetric rotaIon matrix [ω] ⎡ ⎢ [ Ju ] = ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ T Ju ] + [ Ju ] 1⎢ [ =⎢ [ε ] = 2 2 ⎢ ⎢ ⎣ ∂u ∂u + ∂x ∂x ∂v ∂u + ∂x ∂y ∂u ∂x ∂v ∂x ⎡ ∂u ∂u ⎤ ⎥ ⎢ ∂y ⎥ ∂x T [ J u ] = ⎢ ⎢ ∂u ∂v ⎥ ⎥ ⎢ ∂y ⎥ ⎢ ∂y ⎣ ⎦ ∂v ∂x ∂v ∂y ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ∂u ∂v ⎤ + 0 ⎥ ⎢ T ∂y ∂x ⎥ Ju ] − [ Ju ] 1⎢ [ [ω ] = =⎢ ∂v ∂v ⎥ 2 2 ∂v ∂u + − ⎥ ⎢ ∂y ∂y ⎥ ⎢ ∂x ∂y ⎦ ⎣ ∂u ∂v ⎤ − ⎥ ∂y ∂x ⎥ ⎥ 0 ⎥ ⎥ ⎦ [ε ] + [ω ] = [ Ju ] 10/12/11 GG303 21 15. FINITE STRAIN & INFINITESIMAL STRAIN IV Infinitesimal strain and the infinitesimal strain tensor [ε] D Taylor series expansion [U ] ≈ [U ] + [ J ][ dX ] = [U ] + ⎡[ε ] + [ω ]⎤ [ dX ] ⎣ ⎦ 2 1 u 1 The deformaIon can be decomposed into a translaIon, a strain, and a rotaIon 10/12/11 GG303 22 11 10/12/11 15. FINITE STRAIN & INFINITESIMAL STRAIN 10/12/11 GG303 23 15. FINITE STRAIN & INFINITESIMAL STRAIN 10/12/11 GG303 24 12 ...
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