Stress and Strain Tensors – Deformation and Strain
MCEN 5023/ASEN 5012 Chapter 4
Deformation and Strain
Displacement & Deformation
Deformation: An alteration of shape, as by pressure or stress.
Example:Displacement: A vector or the magnitude of a vector from the initial
position to a subsequent position assumed by a body.
Time 0
Time t
Case 1
Case 2
Deformation and Strain
Deformation and Strain
Strain characterizes a deformation
Example: 1D strain L0 L 0 0L
L
L
I−
=
ε
Deformation and Strain
Kinematics of Continuous Body
2
x
1x
3x
a
1 2a
3a
Time 0 Undeformed configurationReference (initial) configuration Material configuration Time 0: Deformed configuration Current configuration Spatial configuration Time t: i
a
Time t ix
Deformation and Strain
Kinematics of Continuous Body
)
,
,
,
(
a
1a
2a
3t
x
x
i=
iOR, due to continuous body
)
,
,
,
(
x
1x
2x
3t
a
a
i=
iLagrangian Description:
Eulerian Description:
The motion is described by the material
coordinate and time t.
The motion is described by the spatial
coordinate and time t.
Deformation and Strain
)
,
,
,
(
a
1a
2a
3t
x
x
i=
ia
i=
a
i(
x
1,
x
2,
x
3,
t
)
Lagrangian
Eulerian
2x
1x
1a
2a
t=0 t=t1 t=t2(Tacking a material point) (Monitoring a spatial point)
The spatial coordinates of this material point change with time. Different material points pass this spatial point
Deformation and Strain
Lagrangian vs. Eulerian
Lagrangian
Eulerian
Solid Mechanics Fluid Mechanics
Solid Mechanics Tracking a material point. Tracking a spatial point.
Spatial coordinates are fixed but Material points keep changing. Material point is fixed but the spatial
coordinates have to be updated.
Deformation and Strain
Kinematics of Continuous Body
2
x
1x
3x
a
1 2a
3a
Time 0 Time t ia
x
i iu
Using undeformed configuration as reference:
i i
i
a
a
a
x
a
a
a
a
u
(
1,
2,
3)
=
(
1,
2,
3)
−
Using deformed configuration as reference:
)
,
,
(
)
,
,
(
x
1x
2x
3x
a
x
1x
2x
3u
i=
i−
iDeformation and Strain
Measure the deformation
2
x
1x
3x
a
1 2a
3a
Time 0 Time t iu
P0 Q0 P Q ia
ix
{
1 2 3}
0a
,
a
,
a
P
=
{
1 1 2 2 3 3}
0a
d
a
,
a
d
a
,
a
d
a
Q
=
+
+
+
{
x
1,
x
2,
x
3}
P
=
{
x
1dx
1,
x
2dx
2,
x
3dx
3}
Q
=
+
+
+
Deformation and Strain
Deformation and Strain
Deformation and Strain
Strain Tensor:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
∂
∂
∂
∂
=
ij j k i k ija
x
a
x
E
δ
2
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−
=
j k i k ij ijx
a
x
a
e
δ
2
1
Green Strain Almansi StrainDeformation and Strain
Strain Tensor:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
j k i k i j j i ija
u
a
u
a
u
a
u
E
2
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
+
∂
∂
=
j k i k i j j i ijx
u
x
u
x
u
x
u
e
2
1
Green Strain
Almansi Strain
Deformation and Strain
Physical Explanations of Strain Tensor
2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u
a
d
x
d
P0 Q0 P QDeformation and Strain
Physical Explanations of Strain Tensor
2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u a d x d P0 Q0 P Q
Deformation and Strain
Physical Explanations of Strain Tensor
2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u v’ v n n’
Deformation and Strain
If
1
<<
1
∂
∂
<<
∂
∂
j i j ix
u
a
u
small deformation
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
i j j i ija
u
a
u
E
2
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
i j j i ijx
u
x
u
e
2
1
The quadratic term in Green strain and Almansi strain can be neglected.
Also, in small deformation, the distinction between Lagrangian and Eulerian disappears.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
i j j i ij ijx
u
x
u
e
E
2
1
Deformation and Strain
1 1 11 11x
u
E
e
∂
∂
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
i j j i ij ijx
u
x
u
e
E
2
1
Cauchy’s infinitesimal strain tensor
2 2 22 22
x
u
E
e
∂
∂
=
=
3 3 33 33x
u
E
e
∂
∂
=
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
2 1 1 2 12 122
1
x
u
x
u
E
e
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
3 1 1 3 13 132
1
x
u
x
u
E
e
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
3 2 2 3 23 232
1
x
u
x
u
E
e
Deformation and Strain
If
1
<<
1
∂
∂
<<
∂
∂
j i j ix
u
a
u
small deformation
Note:
small deformation
In most of the cases,1
1
<<
∂
∂
<<
∂
∂
j i j ix
u
a
u
But,Deformation and Strain
Engineering StrainsCoordinates:
x, y, z
Displacements:
u, v, w
Normal strains: 11e
x
u
x∂
=
∂
=
ε
22e
y
v
y∂
=
∂
=
ε
33e
z
w
z∂
=
∂
=
ε
Deformation and Strain
Engineering Strains
Shear Strains:
122
e
x
v
y
u
xy∂
=
∂
+
∂
∂
=
γ
232
e
y
w
z
v
yz∂
=
∂
+
∂
∂
=
γ
132
e
x
w
z
u
xz∂
=
∂
+
∂
∂
=
γ
Deformation and Strain
Deformation and Strain
Cauchy’s Shear Strain and Engineering Shear Strains
B
A
C
A
’
B
’
C
’
x
1x
2dx
2dx
1Deformation and Strain
Cauchy’s Shear Strain and Engineering Shear Strains
B
A
C
A
’
B
’
C
’
x
1x
2dx
2dx
1θ
1θ
2y
u
x
u
∂
∂
=
∂
∂
=
2 1 1θ
x
v
x
u
∂
∂
=
∂
∂
=
1 2 2θ
2 1θ
θ
γ
xy=
+
(
1 2)
122
1
θ
+
θ
=
e
Deformation and Strain
Cauchy’s Shear Strain and Engineering Shear Strains
[ ]
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
z yz xz yz y xy xz xy xe
e
e
e
e
e
e
e
e
e
ε
γ
γ
γ
ε
γ
γ
γ
ε
2
1
2
1
2
1
2
1
2
1
2
1
33 23 13 23 22 12 13 12 11Tensor
Not a tensor!!!
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
z yz xz yz y xy xz xy xε
γ
γ
γ
ε
γ
γ
γ
ε
Engineering Strain
Deformation and Strain
Transformation of Coordinate System
In general
ij
jk
ik
ij
e
e
′
=
β
β
Deformation and Strain
Transformation of Coordinate System – 2D
2
X
′
1X
′
1e
′
2e
′
2 X 1 Xe
1 2e
θ
(
)
(
2 2)
12 11 22 12 22 2 12 11 2 22 22 2 12 11 2 11sin
cos
cos
sin
cos
cos
sin
2
sin
sin
cos
sin
2
cos
e
e
e
e
e
e
e
e
e
e
e
e
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
−
+
−
=
′
+
−
=
′
+
+
=
′
Deformation and Strain
Deformation and Strain
Deformation and Strain
Strain Deviations Mean Strain3
3
1 33 22 11 0θ
=
+
+
=
e
e
e
e
Strain deviation tensor
e
e
I
0
e
−
=
′
ij ij ije
e
e
′
=
−
0δ
Octahedral Shear Strain
(
) (
) (
)
(
2)
31 2 23 2 12 2 11 33 2 33 22 2 22 11 06
3
2
e
e
e
e
e
e
e
e
e
−
+
−
+
−
+
+
+
=
γ
Deformation and Strain
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
i j j i ij ijx
u
x
u
e
E
2
1
Determine Displacement Fields from Strains
Deformation and Strain
2 1 1 1x
3
x
x
u
+
=
∂
∂
2 1 2 1x
x
u
=
∂
∂
The strain fields are inconsistent because
3 1 2 1 2 1 1 2 = ∂ ∂ ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ x x u x u x 1 2 1 1 2 2 1 1 2x x x u x u x ∂ ∂ = ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ 2 1 1 2 1 2 1 2
x
x
u
x
x
u
∂
∂
∂
≠
∂
∂
∂
Deformation and Strain
Compatibility of Strain Fields
A B C D Undeformed A B C D
Compatible strain fields
A B
C D C’
Incompatible strain fields
A
B C
D C’
Deformation and Strain
Integrability Condition In general(
1 2)
1 1f
x
,
x
x
u
=
∂
∂
(
)
2 1 2 1g
x
,
x
x
u
=
∂
∂
Integrability condition ( Compatibility of strain fields )
1 2
x
g
x
f
∂
∂
=
∂
∂
2 1 1 2 1 2 1 2x
x
u
x
x
u
∂
∂
∂
=
∂
∂
∂
Deformation and Strain
Compatibility of Strain Fields
1 1 11
x
u
e
∂
∂
=
2 2 22x
u
e
∂
∂
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
=
2 1 1 2 122
1
x
u
x
u
e
Deformation and Strain
Compatibility of Strain Fields
0
, , , ,kl+
kl ij−
ik jl−
jl ik=
ije
e
e
e
St. Venant Equations of Compatibility
Totally 81 equations, but only 6 are essential.
12 , 12 11 , 22 22 , 11