MCEN 5023/ASEN 5012
Chapter 3
Stress and Strain Tensors – Stress
Fall, 2006
1
Stress Tensor
Traction
P1
ΔF
v
Traction:
Area: ΔS
u
Su
X3
ΔF
T = lim
ΔS →0 ΔS
v
P2
Β
Pn
X2
X1
Traction is a vector, whose direction and magnitude
depend on how the surface ΔS is obtained, or the
direction of ΔS.
2
Stress Tensor
Traction and Stress Tensor
X3
X3
X2
X2
X1
X1
To better visualize, we move
these six faces to form a cube.
Six faces passing a material point.
σ 23
e2
e2
T
σ 21
T = σ 21e1 + σ 22e 2 + σ 23e3
σ 22
X3
X2
X1
Stress at a material point
3
Stress Tensor
Traction and Stress Tensor
σ 33
e1
σ 32
σ 31
T = σ 11e1 + σ 12e 2 + σ 13e 3
σ 23
e2
σ 13
σ 12
X3
σ 11
σ 21
σ 22
T = σ 21e1 + σ 22e 2 + σ 23e 3
e3
T = σ 31e1 + σ 32e 2 + σ 33e 3
ei
T = σ ij e j
X2
X1
σ ij
represents the stress state of a material point.
4
Stress Tensor
Stress Tensor
σ 33
σ = σ ij e i ⊗ e j
σ 32
σ 31
⎡σ 11 σ 12 σ 13 ⎤
= ⎢⎢σ 21 σ 22 σ 23 ⎥⎥
⎢⎣σ 31 σ 32 σ 33 ⎥⎦
σ 23
σ 13
σ 12
X3
σ 21
σ 22
σ 11
X2
X1
Normal Stresses:
σ 11 σ 22 σ 33
Shear Stresses:
σ 12 σ 21 σ 23 σ 32 σ 13 σ 31
σ ij = σ ji
5
Stress Tensor
Stress Tensor: Sign Rules
The normal of a surface:
Positive surface:
σ 33
σ 32
σ 31
σ 12
σ 21
n
σ 13
σ 22
X1
n
σ 13
σ 12
σ 21
σ 22
σ 31
σ 32
X2
σ 23
σ 11
σ 23
X3
σ 11
Negative surface:
σ 33
6
Stress Tensor
Traction and Stress Tensor
P1
v
Traction: T = Δlim
S →0
ΔF
v
P2
Area: ΔS
Su
X3
Β
Pn
ΔF
ΔS
Stress Tensor
True Stress: if both force and area are measured
in deformed configuration.
Nominal Stress: if area is measured in
undeformed configuration.
X2
X1
Undeformed
Deformed
A0
F
F
A
L
L0
F
Nominal Stress:
A0
Small deformation: A ≈ A0
F
True Stress:
A
7
Stress Tensor
Traction on an arbitrary surface
v
T
ΔS1
v
n1
X2
θ
σ 11
X1
σ 12
σ 21
n2
ΔS v
ΔS 2
σ 22
8
Stress Tensor
v
Traction on an arbitrary surface
X2
T
ΔS1
v
n1
θ
σ 11
X1
σ 12
σ 21
n2
ΔS v
ΔS 2
σ 22
9
Stress Tensor
Traction on an arbitrary surface
In general 3D case:
v
T j = σ ij v i
v
T = vσ
Cauchy’s Formula
Cauchy’s formula ensures that it is necessary and sufficient
to use a stress tensor, σ, to describe traction on any surface.
10
Stress Tensor
Transformation of Coordinates
e1′
e1′
T
T
e′1
n1
n1
σ 11
σ 11
X2
e′2
e′1
σ 12
X1
σ 21
n2
σ 22
X2
σ 12
X1
σ 21
n2
σ 22
11
Stress Tensor
Transformation of Coordinates
e1′
T
n1
e′2
e′1
σ 11
X2
σ 12
X1
σ 21
n2
σ 22
12
Stress Tensor
X2
Transformation of Coordinates
e1
T
n1
X 2′
e2
e1′
X1
e ′2
σ 21
σ 11
X2
σ 12
e ′1
σ 22
e ′1
e ′2
X 1′
σ 22′
n2
σ 11′
σ 12′
σ 21′
X1
Previous slide
e1′
T k = β1i β kjσ ij
σ 11
σ 12
σ 12
σ 11
σ 11′
σ 12′
σ 21′
σ 22′
σ 21
σ 22
13
Stress Tensor
Transformation of Coordinates
σ ij′ = β ik β jlσ kl
X
e2
X 2′
2
X 1′
e ′1
e ′2
X
1
e1
β ik = e′i • e k = e k • e′i = cos(θ (e′i , e k ))
14
15
Stress Tensor
Transformation of Coordinates
σ ij′ = β ik β jlσ kl
16
Stress Tensor
Transformation of Coordinates – 2D
X
e2
X 2′
e ′2
2
X 1′
e ′1
θ
X
1
e1
17
Stress Tensor
Transformation of Coordinates – 2D
18
Stress Tensor
Transformation of Coordinates – 2D Mohr’s Circle
σ 11′ =
(σ 11 + σ 22 ) + (σ 11 − σ 22 ) cos 2θ + σ
2
2
Shear Stress
12 sin 2θ
σ 11 + σ 22
σ 11 − σ 22
2
2
σ 12′ = −
2
CD =
12
cos 2θ
σ 11 − σ 22
2
DE = σ 12
σ 11
σ 11 − σ 22
σ 22
2
cos (2θ )
D
A
(σ 11 − σ 22 ) sin 2θ + σ
2
σ 12
2θ
Normal stress
2θ
C
B
⎛ σ − σ 22 ⎞
2
CE = ⎜ 11
⎟ + σ 12
2
⎝
⎠
2θ
E
σ 12
19
Stress Tensor
Transformation of Coordinates – 2D Mohr’s Circle
X 2′
X
2
X 1′
Shear Stress
θ
σ 12
X
1
σ 12′
σ 11 + σ 22
σ 11
2
′
σ 22
σ 12′
σ 11′
Normal stress
σ 22
2θ
σ 12
20
Stress Tensor
Transformation of Coordinates – 2D Mohr’s Circle
Steps to construct and use Mohr’s circle
1. On the normal stress-shear stress plot, find
the following three points:
(σ11, -σ12)
(σ22, σ12)
( (σ11 + σ22)/2, 0)
2. Draw a circle based on these three points.
Shear Stress
σ 12
σ 12′
( (σ11 + σ22)/2, 0) as center, and (σ11, -σ12)
and (σ22, σ12) as two points on the circle.
3. Rotate the line connecting points (σ11, -σ12)
and (σ22, σ12) by 2θ.
4. The newly obtained two points on the circle
are
(σ’11, -σ’12) and (σ’22, σ12’)
σ 11 + σ 22
′ σ 22
σ 22
σ 12′
2
σ 11 σ 11′
Normal
stress
2θ
σ 12
For any two points on the Mohr’s
circle, if the line connecting them
passes the center, they are stress
components under the same
coordinate system.
21
Stress Tensor
Observations from 2D Mohr’s Circle
Shear Stress
⎛ σ − σ 22 ⎞
2
CE = ⎜ 11
⎟ + σ 12
2
⎝
⎠
2
τ max
σ 11 + σ 22
2
σ2
Normal stress
σ1
C
E
⎛ σ 11 + σ 22 ⎞
⎛ σ − σ 22 ⎞
2
⎟ + ⎜ 11
⎟ + σ 12
2
2
⎝
⎠
⎝
⎠
2
σ1 = ⎜
⎛ σ + σ 22 ⎞
⎛ σ − σ 22 ⎞
2
σ 2 = ⎜ 11
⎟ − ⎜ 11
⎟ + σ 12
2
2
⎝
⎠
⎝
⎠
2
⎛ σ − σ 22 ⎞
2
= ⎜ 11
⎟ + σ 12
2
⎝
⎠
2
τ max
22
Stress Tensor
Observations from 2D Mohr’s Circle
Normal stress
1.
2.
The maximum normal stress and minimum
normal stress occur under the same
coordinate system.
Under the coordinate system where the
maximum/minimum normal stresses are
reached, there is no shear stress.
Shear Stress
Normal
stress
Shear stress
1.
2.
The maximum shear stress is reached by
rotating the coordinate system by 45 degree
from the principle directions.
At maximum shear stress, the two normal
3.
Pure Shear, if σ11=σ22=0, or σ1 = -σ2.
stresses are σ11=σ22=(σ1 +σ2)/2
23
Stress Tensor
Observations from 2D Mohr’s Circle
F
F
F
F
Pure Shear
24
Stress Tensor
Observations from 2D Mohr’s Circle
F
F
25