Lec.14.pptx - 10/11/11 14. HOMOGENEOUS FINITE STRAIN:...

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Unformatted text preview: 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS I Main Topics A PosiAon, displacement, and differences in posiAon of two points B Chain rule for a funcAon of mulAple variables C Homogenous deformaAon D Examples 10/11/11 GG303 1 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS saUtp.soest.hawaii.edu 10/11/11 GG303 2 1 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x y X1(?) X1’ X2(?) U2 U1 X2’ saUtp.soest.hawaii.edu 10/11/11 GG303 3 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS II PosiAon, displacement, and differences in posiAon of two points A IniAal posiAon vectors: Pt. 1: X1 = x1 + y1 Pt. 2: X2 = x2 + y2 B Final posiAon vectors: Pt. 1’: X1′ = x1′ + y1′ Pt. 2’: X2 ′ = x2 ′ + y2 ′ 10/11/11 GG303 PosiAon Vectors Point 1 moves to Point 1’ Point 2 moves to Point 2’ 4 2 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS II PosiAon, displacement, and differences in posiAon of two points (cont.) C Displacement vectors 1 In terms of posi5ons: Pt.1: U = X − X 1 1′ Displacement Vectors 1 Pt.2: U 2 = X 2 ′ − X 2 2 In terms of components: Pt.1: U1 = u1 + v1 Pt.2: U 2 = u2 + v2 10/11/11 Point 1 moves to Point 1’ Point 2 moves to Point 2’ U2 is displaced, rotated, and “stretched” relaAve to U1. GG303 5 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS II PosiAon, displacement, and differences in posiAon of two points D Difference in posiAons of Points 1 and 2 1 Difference in iniAal posiAons dX = X2 − X1 2 Difference in final posiAons dX′=X2’ − X1’ 10/11/11 Difference in Posi5ons (not displacement vectors) Point 1 moves to Point 1’, not Point 2 Point 2 moves to Point 2’ dX’ is displaced, rotated, and stretched relaAve to dX. GG303 6 3 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS Displacement Gradient Components II PosiAon, displacement, and differences in posiAon of two points E Displacement gradient terms (in a matrix) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂u ∂x ∂v ∂x ∂u ⎤ ⎥ ∂y ⎥ ∂v ⎥ ⎥ ∂y ⎥ ⎦ These describe how the components of U change as the components of X change 10/11/11 Gradient terms combine changes in displacement components with changes in posiAon components GG303 7 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS III Chain rule A FuncAons of two variables z = z ( x, y ); dz = ∂z ∂z dx + dy ∂x ∂y The total change in a funcAon of variables x and y equals its rate of change with respect to x, mulAplied by the change in x, plus its rate of change with respect to y, mulAplied by the change in y. 10/11/11 GG303 8 4 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS III Chain rule A FuncAons of two variables (cont.) x’ = x’(x,y), y ’ = y ’(x,y) u = u(x,y), v = v(x,y) ∂x ′ ∂x ′ dx + dy ∂x ∂y ∂y′ ∂y′ dy′ = dx + dy ∂x ∂y dx ′ = ∂u ∂u dx + dy ∂x ∂y ∂v ∂v dv = dx + dy ∂x ∂y du = 10/11/11 GG303 9 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS Matrix form III Chain rule A Two variables (cont.) ⎡ ⎢ ⎢ ⎣ ∂x ′ ∂x ′ dx + dy ∂x ∂y ∂y′ ∂y′ dy′ = dx + dy ∂x ∂y dx ′ = du = ⎡ ⎢ ⎡ du ⎤ ⎢ ⎢ ⎥=⎢ ⎣ dv ⎦ ⎢ ⎢ ⎣ ∂u ∂u dx + dy ∂x ∂y ∂v ∂v dv = dx + dy ∂x ∂y 10/11/11 ⎡ ∂x ′ ∂x ′ ⎤ ⎢ ⎥ dx ′ ⎤ ⎢ ∂x ∂y ⎥ ⎡ dx ⎤ ⎥= ⎢ ⎥ dy′ ⎥ ⎢ ∂y′ ∂y′ ⎥ ⎢ dy ⎥ ⎦⎢ ⎣ ⎦ ⎥ ⎢ ∂x ∂y ⎥ ⎣ ⎦ [ dX ′ ] = [ F ][ dX ] GG303 ∂u ∂x ∂v ∂x ∂u ⎤ ⎥ ∂y ⎥ ⎡ dx ⎤ ⎢ ⎥ ∂v ⎥ ⎣ dy ⎦ ⎥ ⎥⎢ ∂y ⎥ ⎦ [ dU ] = [ Ju ][ dX ] 10 5 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS III Chain rule B Three variables ⎡ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎤⎢ dx ′ ⎥⎢ dy′ ⎥ = ⎢ ⎢ dz′ ⎥ ⎢ ⎦ ⎢ ⎢ ⎣ ∂x ′ ∂x ∂x ′ ∂y ∂y′ ∂x ∂y′ ∂y ∂z′ ∂x ∂z′ ∂y ⎡ ⎢ ⎢ ⎡ du ⎤ ⎢ ⎢ ⎥=⎢ ⎢ dv ⎥ ⎢ ⎢ dw ⎥ ⎢ ⎣ ⎦ ⎢ ⎢ ⎣ ∂x ′ ⎤ ⎥ ∂z ⎥ ⎡ dx ⎤ ∂y′ ⎥ ⎢ ⎥ ⎥ dy ⎥ ∂z ⎥ ⎢ ⎢ dz ⎥ ⎦ ∂z′ ⎥ ⎣ ⎥ ∂z ⎥ ⎦ [ dX ′ ] = [ F ][ dX ] ∂u ∂y ∂v ∂x ∂v ∂y ∂w ∂x ∂w ∂y ∂u ⎤ ⎥ ∂z ⎥ ⎡ dx ⎤ ∂v ⎥ ⎢ ⎥ ⎥ dy ⎥ ∂z ⎥ ⎢ ⎢ dz ⎥ ⎦ ∂w ⎥ ⎣ ⎥ ∂z ⎥ ⎦ [ dU ] = [ Ju ][ dX ] Final posiAon in terms of iniAal posiAon 10/11/11 ∂u ∂x Displacement in terms of iniAal posiAon GG303 11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS IV Homogeneous strain A EquaAons of homogeneous 2 ­D strain 1 Chain rule ∂x ′ ∂x ′ dx + dy ∂x ∂y ∂y′ ∂y′ dy′ = dx + dy ∂x ∂y dx ′ = du = ∂u ∂u dx + dy ∂x ∂y dv = ∂v ∂v dx + dy ∂x ∂y 10/11/11 1 Scale a At a point, derivaAves have unique (constant) values; equaAons are linear in dx and dy in the neighborhood of the point (e.g., dx’ and dy’ depend on dx and dy raised to the first power. b If the derivaAves do not vary with x or y, (i.e., are constant), then the equaAons are linear in dx and dy no mager how large dx and dy are. This is the condi5on of homogenous strain. c Homogeneous strain applies at a point d Homogeneous strain is applied to “small” regions e DeformaAon in large regions is invariably inhomogeneous (derivaAves vary spaAally) GG303 12 6 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS A EquaAons of homogeneous 2 ­D strain (cont.) 2  Common reformulaAon a For constant derivaAves a, b, c, d, replace dx, dy, dx’ and dy’ by x, y, x’, and y’ (derivaAves are the same for small dx or large x) b Linearity is clarified c Chain rule origin is obscured 10/11/11 ∂x ′ ∂x ′ dx + dy ⇒ ∂x ∂y x ′ = ax + by dx ′ = ∂y′ ∂y′ dx + dy ⇒ ∂x ∂y y′ = cx + dy dy′ = GG303 13 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS A EquaAons of homogeneous 2 ­D strain (cont.) 1 Lagrangian a x’ = ax + by b y’ = cx + dy 2 Eulerian (see derivaAon) a x = Ax’ + By’ b y = Cx’ + Dy’ Final IniAal posiAons posiAons 10/11/11 IniAal posiAons GG303 Final posiAons 14 7 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS DerivaAon of Eulerian equaAons a x’ = ax + by y = (x ’–ax)/b b y’ = cx + by y = (y ’–cx)/d Equate right sides above d (x’ – ax)/b = (y’ – cx)/d e d(x’–ax) = b(y’– cx) f cbx – adx = by’ ­dx’ g x(cb ­ad) = by’ ­dx’ h x = [ ­d/(cb ­ad)] x’ + [b/(cb ­ad)] y’ i x = Ax’ + By’ 10/11/11 j x’ = ax + by x = (x’–by)/a k y’ = cx + by ! !x = (y’–dy)/c Equate right sides above l (y’ – dy)/c = (x’ – by)/a m a(y’ – dy) = c(x’ – by) n cby – ady = cx’ ­ay’ o y(cb ­ad) = cx’ ­ay’ p y = [c/(cb ­ad)]x’ + [ ­a/(cb ­ad)]y’ q y = Cx’ + Dy’ GG303 15 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS IV Homogenous (uniform) deformaAon (cont.) B Straight parallel lines remain straight and parallel (see appendix) C Parallelograms deform into parallelograms in 2 ­D; D Parallelepipeds deform into parallelepipeds in 3 ­D 10/11/11 GG303 16 8 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS IV Homogenous (uniform) deformaAon (cont.) E Circles deform into ellipses in 2 ­D (see appendix); Spheres deform into ellipsoids in 3 ­D F The shape, orientaAon, and rotaAon of the strain ellipse or ellipsoid describe homogeneous deformaAon. G The rotaAon is the angle between the axes of the strain ellipse and their counterparts before any deformaAon occurred (to be elaborated upon later). 10/11/11 GG303 17 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS IV Homogenous (uniform) deformaAon (cont.) H Lagrangian equaAons for posiAon 1 x ′ = ax + by y′ = cx + dy I Lagrangian equaAons for displacement 1 u = x′ − x = ( a − 1) x + by v = y′ − y = cx + ( d − 1) y 2 ⎡ u ⎤ = ⎡ a − 1 b ⎤ ⎡ x ⎤ ⎢ ⎥ 2 ⎡ ⎢ x′ ⎤ ⎡ a b ⎤ ⎡ x ⎤ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎦ ⎢ ⎥ ⎣ ′ ⎥ ⎣ c d ⎦⎣ ⎦ ⎢ ⎥ ⎣v⎦ 3 3 [ X ′ ] = [ F ][ X ] ⎢ ⎣ ⎡e = ⎢ ⎢g ⎣ c U = [ J u ][ X ] ⎥ d −1 ⎦⎣ y ⎦ ⎢ ⎥ f ⎤⎡ x ⎤ ⎥⎢ ⎥ h ⎥⎢ y ⎥ ⎦ ⎦⎣ ⎡1 0⎤ ⎥ ⎣0 1⎦ Ju= Jacobian matrix for displacements 4 [ J ] = [ F ] − [ I ], where I = ⎢ u F = DeformaAon gradient matrix 10/11/11 GG303 18 9 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x ′ = 1x + 0 y V Examples A No deformaAon y′ = 0 x + 1y u = 0x + 0y v = 0x + 0y PosiAon transformaAons (Lagrangian) Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ 0 1 ⎦⎣ ⎦ ⎡ u ⎤ ⎡ 0 0 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ v ⎦ ⎣ 0 0 ⎦⎢ y ⎦ ⎣ PosiAon transformaAons (matrix form) ⎡1 0⎤ ⎢ ⎥ ⎣0 1⎦ DeformaAon gradient tensor F 10/11/11 Displacement equaAons (matrix form) ⎡0 0⎤ ⎢ ⎥ ⎣0 0⎦ Displacement gradient tensor Ju GG303 19 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x ′ = 1x + 0 y + cx V Examples B Rigid body rotaAon y′ = 0 x + 1y + cy PosiAon transformaAons (Lagrangian) ⎡ x ′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎡ cx ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ + ⎢ c ⎥ ⎥ ⎢y ⎥ ⎢ ⎥⎢ ⎣ ′ ⎦ ⎣ 0 1 ⎦⎣ ⎦ ⎣ y⎦ PosiAon transformaAons (matrix form) ⎡1 0⎤ ⎢ ⎥ ⎣0 1⎦ DeformaAon gradient tensor F 10/11/11 GG303 u = 0x + 0y v = 0x + 0y Displacement equaAons (Lagrangian) ⎡ u ⎤ ⎡ 0 0 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎣ v ⎦ ⎣ 0 0 ⎦⎢ y ⎥ ⎣ ⎦ Displacement equaAons (matrix form) ⎡0 0⎤ ⎢ ⎥ ⎣0 0⎦ Displacement gradient tensor Ju 20 10 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS Example C: Rigid body rotaAon V Examples C Rigid body rotaAon u = ( cos 60° − 1) x − ( sin 60° ) y x ′ = ( cos 60° ) x − ( sin 60° ) y v = ( sin 60° ) x + ( cos 60° − 1) y y′ = ( sin 60° ) x + ( cos 60° ) y PosiAon transformaAons (Lagrangian) ⎡ x ′ ⎤ ⎡ cos 60° − sin 60° ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ sin 60° cos 60° ⎦ ⎣ ⎦ Displacement equaAons (Lagrangian) ⎡ x ′ ⎤ ⎡ cos 60° − 1 − sin 60° ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ cos 60° − 1 ⎦ ⎣ y ⎦ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ sin 60° PosiAon transformaAons (matrix form) ⎡ cos 60° − sin 60° ⎤ ⎢ ⎥ ⎣ sin 60° cos 60° ⎦ DeformaAon gradient tensor F 10/11/11 Displacement equaAons (matrix form) ⎡ cos 60° − 1 − sin 60° ⎤ ⎢ ⎥ cos 60° − 1 ⎦ ⎣ sin 60° Displacement gradient tensor Ju GG303 21 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS V Examples D Uniaxial shortening x ′ = 1x + 0 y y′ = 0 x + 0.5 y u = 0x + 0y v = 0 x − 0.5 y PosiAon transformaAons (Lagrangian) Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 1 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y′ ⎥ ⎣ 0 0.5 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ ⎡u⎤ ⎡0 0 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎣ v ⎦ ⎣ 0 −0.5 ⎦ ⎢ y ⎥ ⎣ ⎦ PosiAon transformaAons (matrix form) Displacement equaAons (matrix form) ⎡1 0 ⎤ ⎢ ⎥ ⎣ 0 0.5 ⎦ DeformaAon gradient tensor F 10/11/11 ⎡0 0⎤ ⎢ ⎥ 0 −0.5 ⎦ ⎣ Displacement gradient tensor Ju GG303 22 11 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS Example E: DilaAon x′ = 2 x + 0 y V Examples E DilaAon y′ = 0 x + 2 y u = 1x + 0 y v = 0 x + 1y PosiAon transformaAons (Lagrangian) Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 2 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ 0 2 ⎦⎣ ⎦ ⎡ u ⎤ ⎡ 1 0 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ v ⎦ ⎣ 0 1 ⎦⎢ y ⎦ ⎣ PosiAon transformaAons (matrix form) ⎡2 0⎤ ⎢ ⎥ ⎣0 2⎦ DeformaAon gradient tensor F 10/11/11 Displacement equaAons (matrix form) ⎡1 0⎤ ⎢ ⎥ ⎣0 1⎦ Displacement gradient tensor Ju GG303 23 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x′ = 2 x + 0 y y′ = 0 x + 0.5 y v = 0 x − 0.5 y PosiAon transformaAons (Lagrangian) Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 2 0 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ y ⎥ ⎣ 0 0.5 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ ⎡u⎤ ⎡1 0 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎣ v ⎦ ⎣ 0 −0.5 ⎦ ⎢ y ⎥ ⎣ ⎦ PosiAon transformaAons (matrix form) V Examples F Pure shear strain (biaxial strain, no dilaAon u = 1x + 0 y Displacement equaAons (matrix form) ⎡2 0 ⎤ ⎢ ⎥ ⎣ 0 0.5 ⎦ DeformaAon gradient tensor F 10/11/11 ⎡1 0⎤ ⎢ ⎥ ⎣ 0 −0.5 ⎦ Displacement gradient tensor Ju GG303 24 12 10/11/11 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x ′ = 1x + 2 y V Examples G Simple shear strain y′ = 0 x + 1y u = 0x + 2y v = 0x + 0y PosiAon transformaAons (Lagrangian) Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 1 2 ⎤ ⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ y ⎥ ⎢y ⎥ ⎢ ⎥ ⎣ ′ ⎦ ⎣ 0 1 ⎦⎣ ⎦ ⎡ u ⎤ ⎡ 0 2 ⎤⎡ x ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ v ⎦ ⎣ 0 0 ⎦⎢ y ⎦ ⎣ PosiAon transformaAons (matrix form) Displacement equaAons (matrix form) ⎡1 2⎤ ⎢ ⎥ ⎣0 1⎦ DeformaAon gradient tensor F 10/11/11 ⎡0 2⎤ ⎢ ⎥ ⎣0 0⎦ Displacement gradient tensor Ju GG303 25 14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS x ′ = 2 x + 1y y′ = 0 x + 0.5 y v = 0 x − 0.5 y PosiAon transformaAons (Lagrangian) V Examples H General deformaAon (plane strain) u = 1x + 1y Displacement equaAons (Lagrangian) ⎡ x′ ⎤ ⎡ 2 1 ⎤⎡ x ⎤ ⎡ u ⎤ ⎡ 1 1 ⎤⎡ x ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎥⎢ ⎥=⎢ ⎥⎢ ⎢ y′ ⎥ ⎣ 0 −0.5 ⎦ ⎢ y ⎥ ⎣ v ⎦ ⎣ 0 −0.5 ⎦ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ PosiAon transformaAons (matrix form) Displacement equaAons (matrix form) ⎡2 1⎤ ⎢ ⎥ ⎣ 0 −0.5 ⎦ DeformaAon gradient tensor F 10/11/11 ⎡1 1⎤ ⎢ ⎥ ⎣ 0 −0.5 ⎦ Displacement gradient tensor Ju GG303 26 13 ...
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This note was uploaded on 12/05/2011 for the course GEOLOGY 300 taught by Professor Stephenmartel during the Fall '11 term at University of Hawaii, Manoa.

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