CH.2. DEFORMATION AND STRAIN Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Deformation Gradient Tensor Material Deformation Gradient Tensor Lecture 2 Lecture 3 Inverse (Spatial) Deformation Gradient Tensor Displacements Lecture 4 Displacement Gradient Tensors Strain Tensors Green-Lagrange or Material Strain Tensor Lecture 5 Euler-Almansi or Spatial Strain Tensor Variation of Distances Stretch Lecture 6 Unit elongation Variation of Angles Lecture 7 2 Overview (cont’d) Physical interpretation of the Strain Tensors Lecture 8 Material Strain Tensor, E Spatial Strain Tensor, e Lecture 9 Polar Decomposition Lecture 10 Volume Variation Lecture 11 Area Variation Lecture 12 Volumetric Strain Lecture 13 Infinitesimal Strain Infinitesimal Strain Theory Strain Tensors Stretch and Unit Elongation Lecture 14 Physical Interpretation of Infinitesimal Strains Engineering Strains Variation of Angles 3 Overview (cont’d) Infinitesimal Strain (cont’d) Polar Decomposition Lecture 15 Volumetric Strain Strain Rate Spatial Velocity Gradient Tensor Lecture 16 Strain Rate Tensor and Rotation Rate Tensor or Spin Tensor Physical Interpretation of the Tensors Material Derivatives Lecture 17 Other Coordinate Systems Cylindrical Coordinates Lecture 18 Spherical Coordinates 4 2.1 Introduction Ch.2. Deformation and Strain 5 Deformation Deformation: transformation of a body from a reference configuration to a current configuration. Focus on the relative movement of a given particle w.r.t. the particles in its neighbourhood (at differential level). It includes changes of size and shape. 6 2.2 Deformation Gradient Tensors Ch.2. Deformation and Strain 7 Continuous Medium in Movement Ω0: non-deformed (or reference) Ω or Ωt: deformed (or present) configuration, at reference time t0. configuration, at present time t. X : Position vector of a particle at x : Position vector of the same particle reference time. at present time. ϕ ()X,t t t0 Q Reference or dX Ωt Q’ non-deformed dx P’ Ω0 P X x Present or deformed 8 Fundamental Equation of Deformation The Equations of Motion: not xii= ϕ ( XXXt123,,,)(= xXXXti123 ,,,){} i∈ 1,2,3 not x=ϕ ()() X,,tt = xX Differentiating w.r.t. X : not ∂xti ()X, dxi = dXj= F ij (){}X, t dXj i , j ∈ 1, 2, 3 ∂X j ∂xX(),t not Fundamental equation dx= ⋅= d X FX(), td ⋅ X ∂X of deformation (material) deformation gradient tensor 9 Material Deformation Gradient Tensor The (material) deformation gradient tensor: ∂ REMARK ∂ X1 not The material Nabla operator ∂ = ⊗∇ ∂ ∇= F() X,tt xX ( ,) is defined as: ∇≡ eˆ ∂X ∂X i 2 not i ∂ ∂xi F= ij,∈{} 1, 2, 3 ∂X ij ∂ 3 X j ∂∂∂xxx 111 ∂∂∂ XXX123 x1 ∂∂∂∂∂∂xxx =⊗=∇ = 222 []Fxx2 ∂∂∂XXX123 ∂∂∂ XXX 123 x 3 ∂∂∂xxx = x = ∇T 333 ∂∂∂XXX123 F(X,t): is a primary measure of deformation characterizes the variation of relative placements in the neighbourhood of a material point (particle). dx= FX(), td ⋅ X 10 Inverse (spatial) Deformation Gradient Tensor The inverse Equations of Motion: not −1 Xii= ϕ ()(){} xxxt123,,,= Xi xxxt12 ,,,3 i∈ 1,2,3 not =ϕ −1 = X()() x ,,tt Xx Differentiating w.r.t. x : not ∂Xti ()X, −1 dX i = dx j= F ij (){}x,t dxj i , j ∈ 1, 2, 3 ∂x j ∂Xx(),t not dX= ⋅= d xF−1 () x, td ⋅ x ∂x Inverse (spatial) deformation gradient tensor 11 Inverse (spatial) Deformation Gradient Tensor − The spatial (or inverse) deformation gradient tensor: dXF=1 ( x, td) ⋅ x ∂ −1 REMARK F( x,tt) ≡ Xx (,) ⊗∇ x The spatial Nabla 1 ∂ −1 X i operator is defined as: ∂ Fij = ij, ∈{ 1, 2, 3} [∇=] ∂ x j ∂ x2 ∇≡ ˆ ei ∂ ∂∂∂XXX ∂xi 111 T ∂∂∂ x = ∇ xxx123 3 X1 ∂∂∂∂∂∂XXX −1 =⊗=∇ = 222 FX[ ] X 2 ∂∂∂xxx123 ∂ x 1 ∂ x 2 ∂ x 3 X 3 ∂∂∂XXX333 -1 = X F (x,t): ∂∂∂xxx123 is a primary measure of deformation characterizes the variation of relative placements in the neighbourhood of a spatial point. It is not the spatial description of the material deformation gradient tensor 12 Properties of the Deformation Gradients The spatial deformation gradient tensor is the inverse of the material deformation gradient tensor: ∂∂xXik ∂ x i −−11 = =δij FF ⋅ = F ⋅= F 1 ∂∂Xxkj ∂ x j If F is not dependent on the space coordinates, FX ( ,) tt ≡ F () the deformation is said to be homogeneous. Every part of the solid body deforms as the whole does. The associated motion is called affine. ∂x −1 If there is no motion, xX = and F = = F = 1 . ∂X 13 Example Compute the deformation gradient and inverse deformation gradient tensors for a motion equation with Cartesian components given by, X+ Yt2 []x =Yt()1 + t Ze Using the results obtained, check that FF ⋅= − 1 1 . 14 X+ Yt2 []x =Yt()1 + Example - Solution t Ze The Cartesian components of the deformation gradient tensor are, X+ Y2 t 12Yt 0 T ∂∂∂ FX(),t=⊗≡ x∇∇[] x =Yt()1 + , ,= 01 +t 0 ∂∂∂XYZ tt Ze 00e The Cartesian components of the inverse motion equation will be given by, yt2 Xx= − 2 2 yt ()1+ t 10 + (1t ) −1 y []Xx=ϕ (),tY = = FXx()( ,tt ),= f() x , t = 0 1 + t 0 1+ t −t t Z= ze fx(),t 00e 15 Example - Solution The Cartesian components of the inverse deformation gradient tensor are, 2yt − 102 (1+ t) −1 1 Fx( ,0t) = 0 1+ t −t 00e And it is verified that FF ⋅= − 1 1 : 2yt 22yt yt − −+ 2 yt 12 01 220 10(1+t) (11++tt) ( ) (1+ t ) 100 −1 11 + t FF⋅=0 1 +t 0 ⋅ 0 0= 0 0= 010 =[1] 11++tt 0 0 et 001 −−t tt 00e 0 0 ee 16 2.3 Displacements Ch.2. Deformation and Strain 17 Displacements Displacement: relative position of a particle, in its current (deformed) configuration at time t, with respect to its position in the initial (undeformed) configuration. Displacement field: displacement of all the particles in the continuous medium. Material description (Lagrangian form): t UX ()(),,tt= xX − X t0 Ω Uii()()XX,, tx=−∈ tX i i{}1, 2, 3 P U=u P’ Spatial description (Eulerian form): Ω0 X x ux()(),,tt= x − Xx ui()(){}xx, t=−∈ xX ii, t i 1, 2, 3 18 Displacement Gradient Tensor UX()(),,tt= xX − X Taking partial derivatives of U w.r.t. X : ∂Ut(XX ,) ∂∂ xt ( ,) X def =−∈ ii i Uii()()XX,, tx tX i i{}1, 2, 3 = − =−=FJijδ ij ij ∂X ∂∂ XX j jj ∂U δ J==−∈i Fδ ij,{} 1, 2, 3 Fij ij ij ∂X ij ij j Material Displacement def Gradient Tensor JX()(),,tt= UX ⊗=−∇ F 1 Taking partial derivatives of u w.r.t. x : ux()(),,tt= x − Xx def ∂uti(,)xx ∂∂ x ii Xt (,) −1 = − =−δδijFj ij −= ij ij u()(){}xx, t=−∈ xX, t i 1, 2, 3 ∂∂∂xxxjjj i ii δ −1 ij Fij ∂u − j==−∈i δ F1 ij,{} 1, 2, 3 ij ∂x ij ij Spatial Displacement Gradient j def Tensor −1 jx()(),,tt= ux ⊗=∇ 1 − F ∂x − REMARK If motion is a pure shifting: xX( ,)t=+ X U () t ⇒=== F1 F1 and j == J 0 . ∂X 19 2.4 Strain Tensors Ch.2. Deformation and Strain 20 Strain Tensors F characterizes changes of relative placements during motion but is not a suitable measure of deformation for engineering purposes: It is not null when no changes of distances and angles take place, e.g., in rigid-body motions. Strain is a normalized measure of deformation which characterizes the changes of distances and angles between particles. It reduces to zero when there is no change of distances and angles between particles. 21 Strain Tensors Consider FX(),t t t0 Q ddxF= ⋅ X = Ω dxi F ij dX j dS dX dx Q’ ds -1 ddXF= ⋅ x Ω P P’ 0 X −1 x dXi = Fij dx j where dS is the length of segment dX : dS= dXX ⋅ d and ds is the length of segment dx : ds= dxx ⋅ d 22 Strain Tensors -1 ddXF= ⋅ x ddxF= ⋅ X −1 dXi = Fij dx j dxi= F ij dX j dS= dXX ⋅ d ds= dxx ⋅ d One can write: 2 TT ()ds= dxx ⋅ d =[][][][] d x ⋅ d x = FX ⋅ d ⋅ FX ⋅ d = d XFFX ⋅T ⋅⋅ d 2 = = = = T ()ds dxk dx k F ki dX i F kj dX j dX i F ki F kj dX j dX i F ik F kj dX j REMARK not 2 T −11T − −−T 1The convention ()dS=⋅= dXXX d[][] d ⋅ d XFx = ⋅ d ⋅ Fx ⋅ d =⋅⋅⋅ d xFFx d not −−1 T T () •=• () 2 = = −1 − 1 = −−11 = −T −1 ()dS dXk dX k F ki dx i F kj dx j dx i F ki F kj dx j dx i F ik F kj dx j is used. 23 Green-Lagrange Strain Tensor 2 T 2 ()ds= dXF ⋅ ⋅⋅ F d X ()dS= dXX ⋅ d Subtracting: 22 ()()ds− dS =⋅⋅⋅−⋅=⋅⋅⋅−⋅⋅=⋅ dXFTT F d X d X d X d XF F d X d X11 d X d X() F T ⋅−⋅= Fd X2 d XE ⋅⋅ d X def = 2E The Green-Lagrange or Material Strain Tensor is defined: 1 EX(),t =() FT ⋅− F 1 2 1 Eij ()X, t=−∈() FFki kjδ ij i, j {} 1, 2, 3 2 E is symmetrical: 11TT 1 ETT=() FF ⋅−1 = FTT ⋅() F − 11 T =() FF T ⋅− = E 22( ) 2 Eij= E ji ij, ∈{} 1, 2, 3 24 Euler-Almansi Strain Tensor 2 = ⋅ 2 =⋅⋅⋅−−T 1 ()ds dxx d ()dS dxF F d x Subtracting: 22 −−TT11−− ()()ds− dS =⋅−⋅ dx d x d xF ⋅ F ⋅=⋅⋅−⋅ d x d x1 d x d xF ⋅ F ⋅= d x def = 2e =dx ⋅()1 − F−−T ⋅ F1 ⋅d x =2 dd xe ⋅⋅ x def = 2e The Euler-Almansi or Spatial Strain Tensor is defined: 1 −−T 1 ex(),t =()1 −⋅ F F 2 1 −−11 eij ()x, t=−∈(δij F ki F kj ) ij,{} 1, 2, 3 2 11−−T −TT − 1−− e is symmetrical: eTT=()11 −⋅ FF11 = T −()() F ⋅ F T =() 1 −⋅ FF T1 = e 22( ) 2 eij= e ji ij, ∈{} 1, 2, 3 25 Particularities of the Strain Tensors The Green-Lagrange and the Euler-Almansi Strain Tensors are different tensors.