Elastic Strain Tensor Q How do we know

Elastic Strain Tensor ? Q: How do we know if strain occurs? A: Distance between two points change after deform Strain = du/dx strain tensor e ij

Outline 1. Definition Strain Tensor 2. Rotation and Strain Matrices 3. Examples

Examples of Comp. (dvol. ) & Shear Strains (dangle) Angle between any two perp. lines doesn’t change n. D Angle between any two perp. lines does change Q: How do we know if strain occurs? A: Distance between two points change after deform

Simple Strain du dx Strain = du/dx du=change in relative distance between two neighbor poins after deform

Simple Strain du dx Strain = du/dx du=change in relative location between two neighbor poins after deform

Strain Tensor du dx Two pts x volume and x+dxand before deformation Deform track two pts u(x+dx) du dx u(x) x du=change in relative locations between two neighbor points after deform x’+dx’ We need to determine change in |dx| after deformation, that is |dx|2 - |dx’|2 > 0 indicates strain dx’ = dx+du dx x’ Let’s strain, rotate, translate 2 points Relative neighbor location before (after): dx (dx’) Displacement vector: u(x) = x’- x Deformation vector: du(x) = dx’- dx 2 2 Original length squared: dl = dx + dx = dxi dx i 1 3 2 Deformed length squared: 2 dl ‘ 2= dx’i dx’ = (dui +dxi ) i = dui + dxi + 2 dui dxi 2 Length change: dl’ 2 - dl = dui + 2 dui dxi (1) Q: How do we know if strain occurs? A: Distance between two points change after deform

Physical Meaning of Strain Tensor du dx u(x+dx) Two pts x volume and x+dxand before deformation Deform track two pts x’+dx’ du dx u(x) x 2 dx’ = dx+du dx x’ 2 Length change: dl’ - dl = 2 dui dxi + dui (1) u(x+dx)i -u(x)i Substitute 2 dui dxi = dui dxj dxi + duj dxi dx j 2 dx i 2 Length change: dl’ - dl = into equation (1) Ignore (why? ) (du i + du j + dukdu k )dx idx j dx i dx j (2) Q: How do we know if strain occurs? A: Distance between two points change after deform ~ Strain tensor

Strain Tensor Summary du dx Deformation is the change in shape and/or size of a continuum body after it undergoes a displacement between an initial or undeformed configuration , at time 0 , and a current or deformed configuration at the current time t. Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body, Si dxi Sj Eij dxj Displacement vector=absolute change in position after deform: Deformation vector= relative change in displacement of two neighboring points after deform: 3 u(x+dx)-u(x)= i i i du = S du dx = S dx i du=Deformation vector after deform x+dx Time t=0 y i j dx + j=1 j u(x+dx) u(x) x j=1 3 Time t=later x’ What is physical meaning of dx. TEdx ? (see eqn 2) What is physical meaning of dx. TEEdx ? Symmetric strain tensor e ij Rotation tensor w ij j

Outline 1. Definition Strain Tensor 2. Rotation and Strain Matrices 3. Examples

Strain Tensor Matrix du dx Two pts x volume and x+dxand before deformation Deform track two pts u(x+dx) du dx x u(x) dx’ = dx+du dx Displacement vector: u(x) = x’- x x’+dx’ x’ ; u(x+dx) = x’+dx’- x- dx u(x+dx)= u(x) + du du dx + du dudy + du dudz dx dx dy dz v(x+dx) = v(x) + dv dx + dv dvdy + dvdvdz dy dx dy dz w(x+dx) = w(x) +dw dw dx + dwdw dy + dwdw dzdz dx dy dz

Strain Tensor Matrix Two pts x volume and x+dxand before deformation Deform volume and track twoptspts u(x+dx) du dx x u(x) dx’ = dx+du dx Displacement vector: u(x) = x’- x x’+dx’ x’ ; u(x+dx) = x’+dx’- x- dx u(x+dx)= u(x) + du du dx + du dudy + du dudz dx dx dy dz v(x+dx) = v(x) + dv dx + dv dvdy + dvdvdz dy dx dy dz w(x+dx) = w(x) +dw dw dx + dwdw dy + dwdw dzdz dx dy dz Decompose matrix into symmetric and antisymmetric parts e+W

Strain Tensor (Special Cases) e =0 If only rotation occurs =tan. J W=0 If only strain occurs

e Principal Strain Directions 33 e 11 Similarity Transform: Rotation matrix so that strain matrix is diagonalized No shear strains in this coordinate system! x. T e e e 11 NT N 12 21 22 31 32` e e e T 13 T x x = N N N 23 e T 33` 11 0 0 e 0 0 22 0 0 e Nx 33 Geometric Interpretation 2 2 2 ax + by + cxy = cnst describes an ellipse 2 ex + fy = cnst describes an ellipse with no crossterms x x 2 1 2 2 rotation x x 1 (x x x ) e 1 2 3 11 e e 21 31 e e e 12 22 32` e e e 13 x x = cnst x 1 2 23 33` 3 1 x T e e e x ][N] x = cnst 0 0 [N (x x x ) [N ][N] e (x x x ) e x ee e 0 x e 0 ee 1 T 1 2 11 3 12 13 11 1 2 3 21 22 23 22 31 32` 1 2 2 3 3 33`

Principal Strain Directions x x 2 2 x x 1 1 Zero shear strains for above coord. system Shear strains for above coord. system

Strain Tensor Summary du dx Deformation is the change in shape and/or size of a continuum body after it undergoes a displacement between an initial or undeformed configuration , at time 0, and a current or deformed configuration at the current time t. Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body, Displacement vector=absolute change in position after deform: u(r 0 ) = r-r 0 Deformation vector= relative change in displacement of two neighboring points after deform: 3 u(x+dx)-u(x)= i i i du = S du dx = S dx i Deformation Displacementvector after deform x+dx Time t=0 y i j dx + j=1 j u(x+dx) u(x) x j=1 3 x’ Time t=later Symmetric strain tensor e ij Rotation tensor w ij j