Elastic Strain Tensor
?
Q: How do we know if strain occurs? A: Distance between two points change after deform
strain
tensor
Strain = du/dx
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
nD
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)
x’+dx’
du
dx
u(x)
x
dx’
dx
du=change in relative locations
between two neighbor points after deform
We need to determine change
in |dx| after deformation, that is
|dx|2 - |dx’|2 > 0 indicates strain
= dx+du
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
2
2
Original length squared: dl = dx + dx + dx = dxi dx i
1
3
2
Deformed length squared:
Length change:
dl ‘2= dx’i dx’i = (dui +dxi )
= dui dui + dxi dxi + 2 dui dxi
2
2
2
dl’ - dl = dui dui + 2dui 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)
= dx+du
x’
x
2
dx’
dx
2
Length change: dl’ - dl = 2dui dxi + dui dui
Substitute 2dui dxi = dui dxj dxi + duj dxi dx j
dx j
2
dx i
2
Length change: dl’ - dl =
(1)
into equation (1)
Ignore (why?)
(du i + du j + dukdu k )dx idx j
dx j dx i 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,
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
du = S du dx = S
dx
i
du=Deformation vector
after deform
x+dx
Time t=0
y
i
j=1
dx
+
j
j=1
j
u(x+dx)
u(x)
x
3
x’
What is physical meaning
of dxTEdx ? (see eqn 2)
What is physical meaning
of dxTEEdx ?
Time t=later
e
Symmetric strain
tensor
ij
w
Rotation
tensor
ij
j
Pure Shear
dx
dy
y
du
x
dv
1. Given the undeformed (dashed line) and deformed (solid line) square, measure
distances in the diagram by ruler & fill in the values in the 2nd column to the right.
2. Now determine the values for the 2x2 strain tensor matrix shown above.
3. Using the definition of change in squared length below, determine deformation
values for (dx,dy)=(1,0) and for (dx,dy=(0,1).
ignore
2
2
Length change: dl’ - dl =
(du i + du j + dukdu k )dx idx j
dx j dx i dx i dx 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)
x’+dx’
du
dx
u(x)
dx’
dx
x
Displacement vector: u(x) = x’- x
= dx+du
x’
; u(x+dx) = x’+dx’- x- dx
u(x+dx)
u(x+dx)= u(x) + du
du dx + du
dudy + du
dudz dx
dx
dy
dz
dx
dy
dz
v(x+dx) = v(x) + dv dx + dv
dvdy + dvdvdz dy
dx
dy
dz
w(x+dx) = w(x)
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
Deform
volume
and
track
track
two
twopts
pts
u(x+dx)
x’+dx’
du
dx
u(x)
dx’
dx
x
Displacement vector: u(x) = x’- x
= dx+du
x’
; u(x+dx) = x’+dx’- x- dx
u(x+dx)
u(x+dx)= u(x) + du
du dx + du
dudy + du
dudz dx
dx
dy
dz
dx
dy
dz
v(x+dx) = v(x) + dv dx + dv
dvdy + dvdvdz dy
dx
dy
dz
w(x+dx) = w(x)
w(x) +dw
dw dx +
dwdw dy +
dwdw dzdz
dx
dy
dz
Decompose matrix into symmetric
and antisymmetric parts
e+W
e
Principal Strain Directions
e
Similarity Transform: Rotation matrix so that strain matrix is diagonalized
33
11
No shear strains in this coordinate system!
e
e
e
e
e
e
11
N
12
21
22
31
32`
e
e
e
13
e
T
N=
23
33`
0
11
0
e
0
0
0
0
22
e
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
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
e
e
e
T
x ][N] x = cnst
0
0 [N
(x x x ) [N ][N] e
x
(x x x ) e e
x
e e0 x
0
e0 e0 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`
33`
Principal Strain Directions
x
x
2
x
2
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(r0 ) = r-r0points after deform:
Deformation vector= relative change in displacement of two neighboring
3
u(x+dx)-u(x)=
i
i
du = S du dx = S
dx
i
Deformation
Displacementvector
vector
after deform
x+dx
Time t=0
y
i
j=1
dx
+
j
j=1
j
u(x+dx)
u(x)
x
3
x’
Time t=later
e
Symmetric strain
tensor
ij
w
Rotation
tensor
ij
j
Outline
1. Definition Strain Tensor
2. Rotation and Strain Matrices
3. Examples
Strain Tensor (Special Cases)