12.005 Lecture Notes 12
Displacement Gradients
Quantitative description:
Suppose
D → D'
P( xi ) → P '( xi + ui )
Q( xi + dxi ) → Q '( xi + dxi + ui + dui )
x3
D'
D
du i Q'
+
ui
Q
P
ui
P'
x2
x1
Figure 12.1
Figure by MIT OCW.
Suppose:
The deformation is continuous.
∂u
The first derivative i are continuous and very small.
∂x j
Then chain rule ⇒
∂u
∂u
∂u
∂u
dui = i dx1 + i dx2 + i dx3 = i dx j
∂x1
∂x2
∂x3
∂x j
Note:
∂ui
relates two vectors dui and dx j and is therefore a second rank tensor.
∂x j
⎡ ∂u
⎡ du1 ⎤ ⎢ 1
⎢
⎥ ⎢ ∂x1
⎢
⎥ ⎢ ∂u
⎢ du2 ⎥ = ⎢ 2
⎢
⎥ ⎢ ∂x1
⎢
⎥ ⎢
⎢⎣ du3 ⎥⎦ ⎢ ∂u3
⎣ ∂x1
∂u1
∂x2
∂u2
∂x2
∂u3
∂x2
∂u1 ⎤
⎥ ⎡ dx1 ⎤
∂x3 ⎥ ⎢ ⎥
∂u2 ⎥ ⎢ ⎥
⎥ ⎢ dx2 ⎥
∂x3 ⎥ ⎢ ⎥
∂u3 ⎥ ⎢⎢ ⎥⎥
⎥ dx
∂x3 ⎦ ⎣ 3 ⎦
For the Ventura Basin results shown, for the 2-D solution, for the stations HOPP-HAPYSNP,
∂ui ⎡ 0.2 0.45 ⎤
=⎢
× 10−6 each year
⎥
∂x j ⎣ 0.08 −0.48⎦
⎛ ∂u
⎞
Note: We have no sensitivity to rigid body translations ⎜ i ≡ 0 ⎟ .
⎜ ∂x
⎟
⎝ j
⎠
What about rotations?
x2
B
B'
α
α
A
A'
x1
Figure 12.2
Figure by MIT OCW.
∂u2
= − tan α → − α
∂x1
∂u1
=α
∂x2
1 ⎛ ∂u1 ∂u2 ⎞
−
⎟ ≡ ω12
2 ⎝ ∂x2 ∂x1 ⎠
α= ⎜
So part of this displacement gradient tensor just gives rigid body rotation.
Rewriting
∂ui 1 ⎛ ∂ui ∂u j ⎞ 1 ⎛ ∂ui ∂u j ⎞
= ⎜
+
−
⎟+ ⎜
⎟
∂x j 2 ⎜⎝ ∂x j ∂xi ⎟⎠ 2 ⎜⎝ ∂x j ∂xi ⎟⎠
where
∂ui ∂u j
+
= ε ij This is strain. It is symmetric.
∂x j ∂xi
⇒ six independent components
∂ui ∂u j
−
= ωij This is rotation. It is antisymmetric.
∂x j ∂xi
⇒ three independent components
For Ventura Basin, each year
⎡0.20
0.26 ⎤
⎡ 0
0.19 ⎤
0 ⎥⎦
× 10−6
ε ij = ⎢
⎥
⎣0.26 −0.48⎦
ωij = ⎢
⎣ −0.19
Interpretation of ε ij
x3
dx2
Before
After
dx2 + du2
Figure 12.3
Figure by MIT OCW.
For example,
1 ⎛ ∂u2 ∂u2 ⎞ ∂u2
+
⎟=
2 ⎝ ∂x2 ∂x2 ⎠ ∂x2
ε 22 = ⎜
⎛ ∂u ⎞
dx2 + du2 = ⎜1 + 2 ⎟ dx2 = (1 + ε 22 ) dx2
⎝ ∂x2 ⎠
Ventura Basin shows
N-S direction shortening
E-W direction lengthening
x2
Change in shape
x3
x3
Q'
Q
φ''
P'
P
x2
Figure 12.4
Figure by MIT OCW.
φ ' tan φ ' =
∂u3
∂x2
φ '' tan φ '' =
∂u2
∂x3
φ '+ φ '' =
∂u3 ∂u2
+
= 2ε 23
∂x2 ∂x3
where ε 23 is ½ distortion of x2 , x3 axes.
2ε 23 ≡ γ 23 It is called “engineering” strain.
φ'
x2