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