12.005 Lecture Notes 13
Measurement of Displacement Gradient Tensor
Techniques of measurement of displacement gradient tensor:
• Leveling
• Triangulation
• Trilateration
• Very Long Baseline Interferometry (VLBI)
• Global Positioning System (GPS)
Leveling
Telescope
(level)
Benchmark
Triangulation
Benchmark
θ
Protractor
Telescope
Corner
Cubes
l2
Trilateration
l1
2li = c∆ti
Laser
Quaser
VLBI
∆x= c∆t
GPS (Global Positioning System)
Figure 13.1
Figure by MIT OCW.
Sensitivity:
technique
angle
distance
height
orientation
Yes
leveling
triangulation
Yes
trilateration
Yes
Yes
VLBI
Yes
Yes
Yes
Yes
GPS
Yes
Yes
Yes
Yes
(
)
Consider plane strain ε 33 ≡ 0 and a four-benchmark network
x2
x1
Consider the following strain tensors:
⎡ε 0 0⎤
⎢
⎥
Case 1. ⎢0 ε 0⎥
⎢0 0 0⎥
⎣
⎦
⎡ε 0 0⎤
⎢
⎥
Case 2. ⎢0 −ε 0⎥
⎢0 0 0⎥
⎣
⎦
⎡0 ε 0⎤
⎢
⎥
Case 3. ⎢ε 0 0⎥
⎢0 0 0⎥
⎣
⎦
⎡2ε 0 0⎤
⎢
⎥
Case 4. ⎢ 0 0 0⎥
⎢ 0 0 0⎥
⎣
⎦
Can they be observed using triangulation?
Can they be observed using trilateration?
Which angles would change?
Which line lengths would change?
What would the change in line length be?
ε is a tensor, so ε ' = αεα T , just as for stress
⎡ cosθ sin θ 0⎤
⎢
⎥
α = ⎢−sin θ cosθ 0⎥
⎢ 0
0
1⎥⎦
⎣
x2
x2'
x1'
Θ
Figure 13.2
Figure by MIT OCW.
x1
Evaluating this for plane strain is straightforward:
ε11 ' =
ε11 + ε 22
ε11 − ε 22
cos 2θ + ε12 sin 2θ
2
2
ε +ε
ε −ε
ε 22 ' = 11 22 − 11 22 cos 2θ − ε12 sin 2θ
2
2
⎛ε − ε ⎞
ε12 ' = − ⎜ 11 22 ⎟sin 2θ + ε12 cos 2θ
⎝ 2 ⎠
+
Note: Shear strains are tough to measure.
These relationships are useful in relating longitudinal strains
δl
l
δl
l
to the total strain tensor.
can be easily measured using trilateration (10’s km scale), strain gauge (10’s mm
scale).
For example, “delta rosetle”
x2
x1'
x1''
30o
60o
Coils
30o
60o
60o
x1
Figure 13.3
Figure by MIT OCW.
Equilateral triangle of transducers records elongations in x1, x1’, and x1’’directions.
These can be related to ε .
Summary of infinitesimal strain:
B
A
B'
A'
C
C'
Figure 13.4
Figure by MIT OCW.
The above figure shows translation and rotation.
δl
Both length change and angle change δθ depend on orientations.
l
δl
>0
Solid lines
l
δl
Dashed lines
<0
l
∠BAC decreases
∠ABC increases
Strain tensor:
⎡ε
ε
ε ⎤
⎢ 11 12 13 ⎥
ε ij = ⎢ε12 ε 22 ε 23 ⎥
⎥
⎢
⎣ε13 ε 23 ε 33 ⎦
diagonal ⇒
δl
l
off diagonal ∼
for lines along axes
δθ
2
for deformation of axes
The strain tensor depends on the orientation of the coordinate system.
Geologic Strain indicators – pebbles elliptical