MCEN 5023/ASEN 5012 Chapter 4 Stress and Strain Tensors – Deformation and Strain Fall, 2006 Deformation and Strain Displacement & Deformation Displacement: A vector or the magnitude of a vector from the initial position to a subsequent position assumed by a body. Deformation: An alteration of shape, as by pressure or stress. Example: Case 1 Case 2 Time 0 Case 3 Time t Deformation and Strain Deformation and Strain Strain characterizes a deformation Example: 1D strain L0 L L − L0 εI = L0 Deformation and Strain Kinematics of Continuous Body Time t Time 0 xi ai a 2 x2 x3 a3 x1 a1 Time 0: Undeformed configuration Reference (initial) configuration Material configuration Time t: Deformed configuration Current configuration Spatial configuration Deformation and Strain Kinematics of Continuous Body xi = xi (a1 , a2 , a3 , t ) OR, due to continuous body ai = ai ( x1 , x2 , x3 , t ) Lagrangian Description: The motion is described by the material coordinate and time t. Eulerian Description: The motion is described by the spatial coordinate and time t. Deformation and Strain Lagrangian vs. Eulerian Lagrangian Eulerian xi = xi (a1 , a2 , a3 , t ) ai = ai ( x1 , x2 , x3 , t ) (Tacking a material point) (Monitoring a spatial point) a 2 x2 The spatial coordinates of this material point change with time. t=0 t=t1 t=t2 Different material points pass this spatial point x1 a1 Deformation and Strain Lagrangian vs. Eulerian Lagrangian Tracking a material point. Material point is fixed but the spatial coordinates have to be updated. Good for constitutive model Solid Mechanics Eulerian Tracking a spatial point. Spatial coordinates are fixed but Material points keep changing. Not good for constitutive model. Fluid Mechanics Solid Mechanics Deformation and Strain Kinematics of Continuous Body Time t Time 0 ai ui xi a 2 x2 x3 a3 x1 a1 Using undeformed configuration as reference: ui (a1 , a2 , a3 ) = xi (a1 , a2 , a3 ) − ai Using deformed configuration as reference: ui ( x1 , x2 , x3 ) = xi − ai ( x1 , x2 , x3 ) Deformation and Strain Measure the deformation Time t Time 0 ai ui P xi P0 Q Q0 a 2 x2 P0 = {a1, a2 , a3 } x3 a3 x1 a1 Q0 = {a1 + da1, a2 + da2 , a3 + d a3 } P = {x1, x2 , x3 } Q = {x1 + dx1, x2 + dx2 , x3 + dx3 } Deformation and Strain Measure the deformation Deformation and Strain Measure the deformation Deformation and Strain Strain Tensor: ⎞ 1 ⎛⎜ ∂xk ∂xk Eij = − δ ij ⎟ ⎟ 2 ⎜⎝ ∂ai ∂a j ⎠ Green Strain ∂ak ∂ak ⎞⎟ 1 ⎛⎜ eij = δ ij − 2 ⎜⎝ ∂xi ∂x j ⎟⎠ Almansi Strain Deformation and Strain Strain Tensor: 1 ⎛⎜ ∂ui ∂u j ∂uk ∂uk ⎞⎟ Green Strain Eij = + + 2 ⎜⎝ ∂a j ∂ai ∂ai ∂a j ⎟⎠ 1 ⎛⎜ ∂ui ∂u j ∂uk ∂uk ⎞⎟ eij = + − ⎜ 2 ⎝ ∂x j ∂xi ∂xi ∂x j ⎟⎠ Almansi Strain Applicable to both small and finite (large) deformation. Deformation and Strain Physical Explanations of Strain Tensor Time t Time 0 P0 da a 2 x2 x3 a3 x1 a1 Q0 ui P dx Q Deformation and Strain Physical Explanations of Strain Tensor Time t Time 0 P0 da a2 x2 x3 a3 x1 a1 Q0 ui P dx Q Deformation and Strain Physical Explanations of Strain Tensor Time t Time 0 n v a2 x2 x3 a3 x1 a1 ui v’ n’ Deformation and Strain ∂ui ∂ui If << 1 << 1 ∂a j ∂x j small deformation The quadratic term in Green strain and Almansi strain can be neglected. 1 ⎛⎜ ∂ui ∂u j ⎞⎟ Eij = + ⎜ 2 ⎝ ∂a j ∂ai ⎟⎠ 1 ⎛⎜ ∂ui ∂u j ⎞⎟ eij = + ⎜ 2 ⎝ ∂x j ∂xi ⎟⎠ Also, in small deformation, the distinction between Lagrangian and Eulerian disappears. 1 ⎛⎜ ∂ui ∂u j ⎞⎟ Eij = eij = + ⎜ 2 ⎝ ∂x j ∂xi ⎟⎠ Cauchy’s infinitesimal strain tensor Deformation and Strain Cauchy’s infinitesimal strain tensor 1 ⎛⎜ ∂ui ∂u j ⎞⎟ Eij = eij = + ⎜ 2 ⎝ ∂x j ∂xi ⎟⎠ ∂u1 e11 = E11 = ∂x1 1 ⎛ ∂u2 ∂u1 ⎞ ⎟⎟ + e12 = E12 = ⎜⎜ 2 ⎝ ∂x1 ∂x2 ⎠ ∂u2 e22 = E22 = ∂x2 1 ⎛ ∂u3 ∂u1 ⎞ ⎟⎟ e13 = E13 = ⎜⎜ + 2 ⎝ ∂x1 ∂x3 ⎠ ∂u3 e33 = E33 = ∂x3 1 ⎛ ∂u3 ∂u2 ⎞ ⎟⎟ e23 = E23 = ⎜⎜ + 2 ⎝ ∂x2 ∂x3 ⎠ Deformation and Strain ∂ui ∂ui If << 1 << 1 ∂a j ∂x j small deformation Note: In most of the cases, ∂ui << 1 ∂a j But, ∂ui << 1 ∂x j small deformation Deformation and Strain Engineering Strains Coordinates: x, y, z Displacements: u, v, w Normal strains: ∂u εx = = e11 ∂x ∂v ε y = = e22 ∂y ∂w εz = = e33 ∂z Deformation and Strain Engineering Strains Shear Strains: γ xy ∂u ∂v = + = 2e12 ∂y ∂x γ yz ∂v ∂w = + = 2e23 ∂z ∂y γ xz ∂u ∂w = + = 2e13 ∂z ∂x Deformation and Strain Stretches at small deformation Deformation and Strain Cauchy’s Shear Strain and Engineering Shear Strains A’ x2 A C’ B’ dx2 B dx1 C x1 Deformation and Strain Cauchy’s Shear Strain and Engineering Shear Strains ∂u1 ∂u θ1 = = ∂x2 ∂y A’ x2 θ1 γ xy = θ1 + θ 2 A dx2 B θ2 C’ B’ dx1 ∂u2 ∂v θ2 = = ∂x1 ∂x C x1 1 e12 = (θ1 + θ 2 ) 2 Deformation and Strain Cauchy’s Shear Strain and Engineering Shear Strains ⎡e11 [e] = ⎢⎢e12 ⎢⎣e13 e12 e22 e23 ⎡ εx ⎢ e13 ⎤ ⎢1 ⎥ e23 ⎥ = ⎢ γ xy ⎢2 e33 ⎥⎦ ⎢ 1 γ ⎢⎣ 2 xz ⎡εx ⎢ Engineering Strain ⎢γ xy ⎢γ xz ⎣ 1 γ xy 2 εy 1 γ yz 2 γ xy γ xz ⎤ ⎥ ε y γ yz ⎥ γ yz ε z ⎥⎦ 1 ⎤ γ xz ⎥ 2 1 ⎥ γ yz ⎥ 2 ⎥ ε z ⎥⎥ ⎦ Tensor Not a tensor!!! Deformation and Strain Transformation of Coordinate System In general eij′ = β ik β jk eij Deformation and Strain Transformation of Coordinate System – 2D X e2 X 2′ e ′2 2 X 1′ e ′1 θ X 1 e1 ′ = cos 2 θ e11 + 2 sin θ cosθ e12 + sin 2 θ e22 e11 ′ = sin 2 θ e11 − 2 sin θ cosθ e12 + cos 2 θ e22 e22 ( ) ′ = sin θ cos(e22 − e11 ) + cos 2 θ − sin 2 θ e12 e12 Deformation and Strain Transformation of Coordinate System – 2D Mohr Circle Deformation and Strain Strain Invariants Deformation and Strain Strain Deviations Mean Strain e11 + e22 + e33 θ1 e0 = = 3 3 Strain deviation tensor e′ = e − e0 I eij′ = eij − e0δ ij Octahedral Shear Strain γ0 = 2 3 (e11 − e22 )2 + (e22 − e33 )2 + (e33 − e11 )2 + 6(e122 + e232 + e312 ) Deformation and Strain Determine Displacement Fields from Strains 1 ⎛⎜ ∂ui ∂u j ⎞⎟ Eij = eij = + ⎜ 2 ⎝ ∂x j ∂xi ⎟⎠ Questions: Can the displacements be determined uniquely? Deformation and Strain Determine Displacement Fields from Strains ∂u1 = x1 + 3x2 ∂x1 ∂u1 = x12 ∂x2 The strain fields are inconsistent because ∂ ⎛ ∂u1 ⎞ ∂ 2u1 ⎟⎟ = ⎜⎜ =3 ∂x2 ⎝ ∂x1 ⎠ ∂x2 ∂x1 ∂ ⎛ ∂u1 ⎞ ∂ 2u1 ⎟⎟ = ⎜⎜ = 2x1 ∂x1 ⎝ ∂x2 ⎠ ∂x1∂x2 ∂ 2u1 ∂ 2u1 ≠ ∂x2 ∂x1 ∂x1∂x2 Deformation and Strain Compatibility of Strain Fields B A B C D C Compatible strain fields A D B C’ B C Undeformed A D Incompatible strain fields C C’ A D Deformation and Strain Integrability Condition In general ∂u1 = f ( x1 , x2 ) ∂x1 ∂u1 = g ( x1 , x2 ) ∂x2 Integrability condition ( Compatibility of strain fields ) ∂ 2u1 ∂ 2u1 = ∂x2 ∂x1 ∂x1∂x2 ∂f ∂g = ∂x2 ∂x1 Integration of strain fields yields unique displacement components. Deformation and Strain Compatibility of Strain Fields ∂u1 e11 = ∂x1 ∂u2 e22 = ∂x2 1 ⎛ ∂u2 ∂u1 ⎞ ⎟⎟ e12 = ⎜⎜ + 2 ⎝ ∂x1 ∂x2 ⎠ Deformation and Strain Compatibility of Strain Fields eij ,kl + ekl ,ij − eik , jl − e jl ,ik = 0 St. Venant Equations of Compatibility Totally 81 equations, but only 6 are essential. e11, 22 + e22,11 = 2e12,12 e11, 23 = −e23,11 + e12,13 + e13,12 e22,33 + e33, 22 = 2e23, 23 e22,13 = −e13, 22 + e21, 23 + e23, 21 e11,33 + e33,11 = 2e13,13 e33,12 = −e12,33 + e31,32 + e32,31