14. HOMOGENEOUS FINITE STRAIN

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1 I Main Topics A vectors B vectors C Infinitesimal differences in posi>on D Infinitesimal differences in displacement E Chain rule for a func>on of mul>ple variables F Gradient tensors and matri representa>on G Homogenous (uniform) deforma>on H Eamples 10/12/15 GG303 1 sattp.soest.hawaii.edu 10/12/15 GG

2 X 1 (?) X 2 (?) X 1 U 1 U 2 X 2 sattp.soest.hawaii.edu 10/12/15 GG303 3 II vectors A Ini>al posi>on vectors: Vectors Pt.1:! X 1 =! 1 +! 1 Pt.2 :! X 2 =! 2 +! 2 B Final posi>on vectors: Pt.1' : X! 1 =! 1 +! 1 Pt.2' : X! 2 =! 2 +! 2 Point 1 moves to Point 1 Point 2 moves to Point 2 10/12/15 GG

3 III vectors A In terms of posi>ons: Vectors Pt.1: U! 1 = X! 1 X! 1 Pt.2 : U! 2 = X! 2 X! 2 B vector components: Pt.1:! U 1 =! u 1 +! v 1 Pt.2 :! U 2 =! u 2 +! v 2 Point 1 moves to Point 1 Point 2 moves to Point 2 U 2 is displaced, rotated, and stretched rela>ve to U 1. 10/12/15 GG303 5 III Infinitesimal differences in posi>on A Difference in ini>al posi>ons (Pts. 1 and 2) Difference in Posi4ons (not displacement vectors) d X! = X! 2 X! 1 B Difference in final posi>ons (Pts. 1 and 2 ) d X! = X! 2 X! 1 Point 1 moves to Point 1, not Point 2 Point 2 moves to Point 2 dx is displaced, rotated, and stretched rela>ve to dx. 10/12/15 GG

4 IV Infinitesimal differences in displacement A Difference in displacement Differences in d U! = U! 2 U! 1 B Components of difference in displacement d! U = d! u + d! v Point 1 moves to Point 1, not Point 2 Point 2 moves to Point 2 dx is displaced, rotated, and stretched rela>ve to dx. 10/12/15 GG303 7 V Chain rule A Func>ons of two variables z = z(, ); dz = z z d + d The total change in a func>on of variables and equals its rate of change with respect to, mul>plied b the change in, plus its rate of change with respect to, mul>plied b the change in. 10/12/15 GG

5 V Chain rule A Func>ons of two variables (cont.) = (,), = (,) u = u(,), v = v(,) d = d + d d = d + d du = u u d + d dv = d + d 10/12/15 GG303 9 V Chain rule B 2D matri representa>on d = d + d d = d + d du = u u d + d dv = d + d d d = 10/12/15 GG du dv [ d X ] = F = du Matri form u [ ][ dx] u [ ] = [ J u ][ dx] d d d d 5

6 V Chain rule C 3D matri representa>on d d = d z z z [ d X ] = F z z z z d d dz du dv dw = u w u w u z z w z [ ][ dx] [ du ] = [ J u ][ dx] d d dz Difference in final posi>on in terms of difference in ini>al posi>on Difference in displacement in terms of difference in ini>al posi>on 10/12/15 GG VI Gradient tensors and matri representa>on A Deforma>on gradient tensor [F]: relates differences in ini>al posi>on to differences in final posi>on B gradient tensor [J u ]: relates differences in ini>al posi>on to differences in displacement d d = Differences in final posi>on du dv [ d X ] = F = du Matri form (2D) u [ ][ dx] u d d d d Differences in ini>al posi>on Differences Differences in displacement in ini>al [ ] = [ J 10/12/15 GG303 u ][ dx] posi>on 12 6

7 VII Homogeneous (uniform) deforma>on A 2D Equa>ons 2 Scale 1 Chain rule a At a point, deriva>ves have unique (constant) values; are linear in d and d in the neighborhood of the d = d + d point (e.g., d and d depend on d and d raised to the first power. d = d + b If the deriva>ves do not var with or, d (i.e., are constant as a func>on of posi>on), then the are linear in d and d no mager how large d and d are. This is the condi4on of homogenous du = u u d + d strain. c Homogeneous strain applies at a point d Homogeneous strain is applied to small dv = regions d + d e Deforma>on in large regions is invariabl inhomogeneous (deriva>ves var spa>all) 10/12/15 GG VII Homogeneous (uniform) deforma>on (cont.) B Common reformula>on 1 For constant deriva>ves a, b, c, d, one replaces d, d, d and d b,,, and (deriva>ves are the same for small d or large ) 2 This dis>nc>on is seldom stated! 3 Linearit is clarified 4 Chain rule origin is obscured d = d + d = a + b d = d + d = c + d 10/12/15 GG

8 VII Homogeneous (uniform) deforma>on (cont.) C Lagrangian a = a + b b = c + d D Eulerian (see deriva>on) a = A + B b = C + D Final posi>ons Ini>al posi>ons Ini>al posi>ons Final posi>ons 10/12/15 GG Deriva>on of Eulerian a = a + b = ( a)/b b = c + d = ( c)/d Equate right sides above d ( a)/b = ( c)/d e d( a) = b( c) f cb ad = b - d g (cb- ad) = b - d h = [- d/(cb- ad)] + [b/(cb- ad)] i = A + B Eulerian Start with Lagrangian j = a + b = ( b)/a k = c + d!! = ( d)/c Equate right sides above l ( d)/c = ( b)/a m a( d) = c( b) n cb ad = c - a o (cb- ad) = c - a p = [c/(cb- ad)] + [- a/(cb- ad)] q = C + D Bogom line: if final posi>ons can be epressed as a linear combina>on of ini>al posi>ons, then the ini>al posi>ons can be epressed as a linear combina>on of ini>al posi>ons 10/12/15 GG

9 VII Homogenous (uniform) deforma>on (cont.) B Straight parallel lines remain straight and parallel (see appendi) C Parallelograms deform into parallelograms in 2- D; D Parallelepipeds deform into parallelepipeds in 3- D 10/12/15 GG VII Homogenous (uniform) deforma>on (cont.) E Circles deform into ellipses in 2- D (see appendi); Spheres deform into ellipsoids in 3- D F The shape, orienta>on, and rota>on of the strain (stretch) ellipse or ellipsoid describe homogeneous deforma>on. G The rota>on is the angle between the aes of the strain ellipse and their counterparts before an deforma>on occurred (to be elaborated upon later). 10/12/15 GG

10 VII Homogenous (uniform) deforma>on (cont.) H Lagrangian for posi>on 1 = a + b 2 3 = c + d = a b c d [ X ] = [ F] [ X] F = Deforma>on gradient matri I Lagrangian for displacement 1 u = = ( a 1) + b v = = c + ( d 1) 2 a 1 b 10/12/15 GG u v = U = J u e = g [ ][ X] c d 1 f h 1 0 [ J u ] = [ F] [ I ], where I = 0 1 J u = Jacobian matri for displacements VIII Eamples A No strain = = = (matri form) Deforma>on gradient tensor F v = u v u = = (matri form) gradient tensor J u 10/12/15 GG

11 VIII Eamples B Rigid bod transla>on = c = c = (matri form) c c Deforma>on gradient tensor F u = c v = c u v = + (matri form) gradient tensor J u c c 10/12/15 GG Eample C: Rigid bod rota>on VIII Eamples C Rigid bod rota>on ( ) ( sin 60 ) = cos60 = ( sin60 ) + ( cos60 ) u = ( cos60 1) ( sin 60 ) v = ( sin 60 ) + ( cos60 1) = cos60 sin 60 sin 60 cos60 (matri form) cos60 sin 60 sin60 cos60 Deforma>on gradient tensor F = cos60 1 sin60 sin60 cos60 1 (matri form) cos60 1 sin60 sin60 cos60 1 gradient tensor J u 10/12/15 GG

12 VIII Eamples D Uniaial shortening = = u = v = = u v = (matri form) Deforma>on gradient tensor F (matri form) gradient tensor J u 10/12/15 GG Eample E: Dila>on VIII Eamples E Dila>on = = u = v = = u v = (matri form) Deforma>on gradient tensor F (matri form) gradient tensor J u 10/12/15 GG

13 VIII Eamples F Pure shear strain (biaial strain, no dila>on = = u = v = = u v = (matri form) Deforma>on gradient tensor F (matri form) gradient tensor J u 10/12/15 GG VIII Eamples G Simple shear strain = = u = v = = u v = 0 2 (matri form) Deforma>on gradient tensor F (matri form) 0 2 gradient tensor J u 10/12/15 GG

14 VIII Eamples H General deforma>on (plane strain) = = = (matri form) Deforma>on gradient tensor F u v v = = (matri form) u = gradient tensor J u 10/12/15 GG Appendi: Proof that under homogenous 2D strain that an ellipse transforms into an ellipse The equa>on of an ellipse centered at the origin is a 2 + 2e + d 2 = 1 10/12/15 GG

15 15 Appendi: Proof that under homogenous 2D strain that an ellipse transforms into an ellipse The equa>on of an ellipse centered at the origin is This equa>on can be wrigen in matri form as 10/12/15 GG a b c d = 1 a + b c + d = 1 a 2 + b + c + d 2 = 1 a 2 + b + c ( ) + d 2 = 1 a 2 + 2e + d 2 = 1 Lepng (b+c) = 2e ields this 10/12/15 GG a b c d = 1 = A B C D A B C D T a b c d A B C D = 1 T A B C D T a b c d A B C D = 1 A C B D a b c d A B C D = 1 a 1 b 1 c 1 d 1 = 1 These have the same form, so both are of an ellipse. So an ellipse transforms to an ellipse under homogeneous finite strain. All constants Constants