1. Eﬀect of surface stress change on the stiﬀness of
cantilever plates
Michael Lachut
Supervisor: Prof. John E. Sader
Department of Mathematics and Statistics
University of Melbourne
June 25, 2013
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
1 / 27
2. Micromechanical devices
Human Hair
a
b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
2 / 27
3. Resonant frequency
Consider the resonant frequency of a resonator
ω=
1
2π
k
m
Spring constant k assumed constant
The resonators mass m changes due to molecular absorption
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
3 / 27
4. Resonant frequency
Consider the resonant frequency of a resonator
ω=
1
2π
k
m
Spring constant k assumed constant
The resonators mass m changes due to molecular absorption
BUT!!
Miniaturization to the micro- and nanoscale enhances surface eﬀects
Particulary, surface stress shown to aﬀect cantilever stiﬀness
Mass measurements potentially ambiguous
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
3 / 27
5. Axial-force model
x3
σ s+
x2
x1
σs−
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
4 / 27
6. Axial-force model
x3
σ s+
x2
x1
σs−
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
4 / 27
7. Lagowski model
ω = ω0 1 − γ
L2
h3
σ
1/2
, where γ = (1 − ν 2 )/E
Lagowski et. al, Appl. Phys. Lett. 26 (1975)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
5 / 27
8. Invalidity of the axial-force model
+
σs δ(x3 − h/2)
x3
σ(b)
x1
h
−
σs δ(x3 + h/2)
Gurtin et. al, Appl. Phys. Lett. 29 (1976)
h/2
T =b
σ (b) + σ(x3 ) dx3 = 0
−h/2
+
−
where σ(x3 ) = σs δ(x3 − h/2) + σs δ(x3 + h/2)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
6 / 27
9. Invalidity of the axial-force model
+
σs δ(x3 − h/2)
x3
σ(b)
x1
h
−
σs δ(x3 + h/2)
Gurtin et. al, Appl. Phys. Lett. 29 (1976)
h/2
T =b
σ (b) + σ(x3 ) dx3 = 0
−h/2
+
−
where σ(x3 ) = σs δ(x3 − h/2) + σs δ(x3 + h/2)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
6 / 27
10. Recent references still using the axial-force model
G. Y. Chen et al., J. Appl. Phys., 77, 3618, (1995)
Y. Zhang et al., J. Appl. Phys. D, 37, 2140, (2004)
A. W. McFarland et al., Appl. Phys. Lett., 87, 053505, (2005)
J. Dorignac et al., Phys. Rev. Lett., 97, 186105, (2006)
K. S. Hwang et al., Phys. Rev. Lett., 89, 173905, (2006)
G. F. Wang et al., Appl. Phys. Lett., 90, 231904, (2007)
S. Zaitsev et al., Sens. Actu. A, 179, 237, (2012)
Y. Zhang, Sens. Actu. A, 194, 169, (2013)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
7 / 27
11. Surface stress load on an unrestrained plate
FREE PLATE
Before Surface Stress Load
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
8 / 27
12. Surface stress load on an unrestrained plate
FREE PLATE
T
σs
After Surface Stress Load
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
8 / 27
13. Surface stress load on an unrestrained plate
FREE PLATE
T
σs
T
u1 (x1 , x2 ) = − (1−ν)σs x1
Eh
T
u2 (x1 , x2 ) = − (1−ν)σs x2
Eh
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
8 / 27
14. Problem decomposition
Decompose into two Sub-Problems
σs
σs
T
T
u 2 = σ x2
Free plate has no stiﬀness eﬀect
Stiﬀness of original problem given by that of the clamp loaded
problem
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
9 / 27
15. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
16. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
17. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
18. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
19. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
If σ << O(h / b)2
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
<<
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
20. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
If σ << O(h / b)2
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
<<
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
21. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
If σ << O(h / b)2
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
>>
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
22. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Relative change in eﬀective rigidity (for x1 < O[b])
Deﬀ
−1∼O σ
¯
D0
b
h
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
23. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Relative change in eﬀective rigidity (for x1 < o[b])
Deﬀ
−1∼O σ
¯
D0
b
L
b
h
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
24. Scaling analysis of clamp loaded problem
σ
Nij = 0
Nij = 0
x 1< o(b)
x 1 > o(b)
Consider the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Relative change in eﬀective rigidity (for x1 < o[b])
Deﬀ
−1∼O σ
¯
D0
b
L
b
h
2
b
L
b
h
2
Relative frequency shift
∆ω
= φω (ν)¯
σ
ω0
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
10 / 27
25. Results for ∆ω/ω0 when L/b = 25/3
0.004
L / b = 25/3
0.002
∆ω
ω0
0
−0.002
Increasing Poisson's Ratio ν
−0.004
−2
−1
0
2
σ (b / h)
1
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
2
11 / 27
26. Final result for the relative change in eﬀective stiﬀness
Relative change in resonant frequency
∆ω
= −0.042ν σ
¯
ω0
b
L
b
h
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
12 / 27
27. Final result for the relative change in eﬀective stiﬀness
Relative change in resonant frequency
∆ω
= −0.042ν σ
¯
ω0
b
L
b
h
2
Relative change in Eﬀective Spring Constant
∆keﬀ
= −0.063ν σ
¯
k0
b
L
b
h
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
12 / 27
28. Problem decomposition for V-shaped cantilevers
Decompose into two Sub-Problems
FREE
d
ORIGINAL
T
σs
d
T
σs
L
u 2 = σ x2
c
CLAMP LOADED
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
13 / 27
29. Normalized Frequency Shift
V-shape cantilever stiﬀness vs. Rectangular cantilever
stiﬀness
V-shape
0.002
Decreasing Poisson’s Ratio
0.0015
0.001
0.0005
Increasing Poisson’s Ratio
0
−0.0005
Rectangle
0.05
0.1
0.15
0.2
0.25
0.3
d/c
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
14 / 27
30. Physical Mechanisms
(a)
(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
15 / 27
31. Practical implications
Silicon Nitride Devices (Si3 N 4)
Cantilever
Device
Dimensions
T
σs
−1
∆ω / ω0
(µ m)
(Nm )
Rectangular
499, 97, 0.8
(L, b,h)
+
− 60
− 0.01
+
V-shape
180, 180,18,0.8
(L, c,d,h)
+
−1
+
− 2.1x 10
−3
Stress Effect: ∆ω / ω0 = − 0.042νσ(b / L)(b / h) 2 b / h >> 1
Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ
b/h~3
P. Li, Z. You and T.Cui,et. al, Phys. Rev. Lett.101, (2012) (2012)
Karabalin Appl. Phys. Lett. 108, 093111,
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
16 / 27
32. Practical implications
Silicon Nitride Devices (Si3 N 4)
Cantilever
Device
Dimensions
T
σs
−1
∆ω / ω0
(µ m)
(Nm )
Rectangular
499, 97, 0.8
(L, b,h)
+
− 60
− 0.01
+
V-shape
180, 180,18,0.8
(L, c,d,h)
+
−1
+
− 2.1x 10
−3
Stress Effect: ∆ω / ω0 = − 0.042νσ(b / L)(b / h) 2 b / h >> 1
Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ
b/h~3
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Observed frequency shifts remain unexplained
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
16 / 27
33. Practical implications
Silicon Nitride Devices (Si3 N 4)
Cantilever
Device
Dimensions
T
σs
−1
∆ω / ω0
(µ m)
(Nm )
Rectangular
499, 97, 0.8
(L, b,h)
+
− 60
− 0.01
+
V-shape
180, 180,18,0.8
(L, c,d,h)
+
−1
+
− 2.1x 10
−3
Stress Effect: ∆ω / ω0 = − 0.042νσ(b / L)(b / h) 2 b / h >> 1
Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ
b/h~3
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Observed frequency shifts remain unexplained
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
16 / 27
34. Practical implications
Silicon Nitride Devices (Si3 N 4)
Cantilever
Device
Dimensions
T
σs
−1
∆ω / ω0
(µ m)
(Nm )
Rectangular
499, 97, 0.8
(L, b,h)
+
− 60
− 0.01
+
V-shape
180, 180,18,0.8
(L, c,d,h)
+
−1
+
− 2.1x 10
−3
Stress Effect: ∆ω / ω0 = − 0.042νσ(b / L)(b / h) 2 b / h >> 1
Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ
b/h~3
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Observed frequency shifts remain unexplained
Karabalin et. al, Phys. Rev. Lett. 108, (2012)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
16 / 27
35. Introduction to buckling
Column
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
17 / 27
36. Introduction to buckling
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
17 / 27
37. Introduction to buckling
What load will buckle a
cantilever plate?
σs
L
T
b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
17 / 27
38. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
39. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
40. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
41. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Balance
Nij = 0
k 0
σ
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
42. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
Nij = 0
D
Nij = 0
k 0
x 1< o(b)
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
43. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2
∂xi ∂xi
∂2w
∂xj ∂xj
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
Nij = 0
D
x 1< o(b)
k
0
x 1 > o(b)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
44. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2w
∂xj ∂xj
∂2
∂xi ∂xi
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
Nij = 0
D
x 1< o(b)
k
0
x 1 > o(b)
Critical strain load scales by
σcr ∼ O
¯
h
b
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
45. Scaling analysis for the critical strain load
Consider again the thin plate equation
D0
∂2w
∂xj ∂xj
∂2
∂xi ∂xi
− Nij
∂2w
=q
∂xi ∂xj
Balance
σ
k
Nij = 0
D
x 1< o(b)
0
x 1 > o(b)
Critical strain load scales by
σcr ∼ O
¯
2
h
b
General form of the critical strain load
σcr = ψ(ν)
¯
h
b
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
18 / 27
46. Independence of aspect ratio (L/b)
0
b / h = 48
−0.2
k
−1
k0
−0.4
−0.6
Increasing L / b
−0.8
−1.0
−2
L / b = 2, 4, 8, 16
−1
0
1
2
−2
σ (x 10 )
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
19 / 27
47. Dependence on thickness-to-width ratio (b/h)
80
60
40
(b / h)2 σcr
20
Increasing ν
0
−20
ν = 0, 0.25, 0.49
−40
−60
0
0.01
0.02
0.03
0.04
0.05
h/b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
20 / 27
48. Expressions for the positive and negative critical strain
loads
Positive strain load
(+)
σcr
¯
= 63.91 1 − 0.92ν + 0.63ν
2
h
b
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
21 / 27
49. Expressions for the positive and negative critical strain
loads
Positive strain load
(+)
σcr
¯
= 63.91 1 − 0.92ν + 0.63ν
2
h
b
2
Negative strain load
(−)
σcr
¯
= −38.49 1 + 0.17ν + 1.95ν
2
h
b
2
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
21 / 27
50. Buckled mode shapes
Positive strain loads (σ > 0)
Negative strain loads (σ < 0)
0.05
0.05
0
W
0
−0.05
W
−0.1
−0.05
−0.1
−0.15
−0.15
0.5
x1
b
0
0.5
0.5
x1
b
x2
b
0
0.5
1.0 −0.5
1.0 −0.5
0
W
x2
b
0
W
−0.5
−1.0
0.5
1
−1.0
0.5
1
2
x1
b
−0.5
2
0
x2
b
3
4
−0.5
x1
b
0
x2
b
3
4
−0.5
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
22 / 27
51. Buckled mode shapes
Positive strain loads (σ > 0)
Negative strain loads (σ < 0)
0.05
0.05
0
W
0
−0.05
W
−0.1
−0.05
−0.1
−0.15
−0.15
0.5
x1
b
0
0.5
0.5
x1
b
x2
b
0
0.5
1.0 −0.5
1.0 −0.5
0
W
x2
b
0
W
−0.5
−1.0
0.5
1
−1.0
0.5
1
2
x1
b
−0.5
2
0
x2
b
3
4
−0.5
x1
b
0
x2
b
3
4
−0.5
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
22 / 27
52. Mechanical pressure compared to buckled mode shapes
Normalized Mechanical Pressure: P (× 10−2)
10
5
2.5
0.4
Region 1
-40
0.2
x2 / b
0
-30
-20
-10
Positive strain load
-2.5 Region 2 0
−0.2
-40
Region 1
−0.4
0
10
0.4
0.2
2.5
5
0.6
0.8
1.5
1
0.5
0.4
ν = 0.25
0
0.2
0
ν = 0.25
-2
-4
-6 -8 -10 -12
x2 / b
-0.2
-2
0
1
2
3
0.2
-2
-4
-6
-8
-10
-12
x2 / b
-14
Buckled Mode Shape: σ < 0 (× 10−2)
Buckled Mode Shape: σ > 0 (× 10−2)
0.4
1
x1 / b
(a)
0
-0.2
0
0
(b)
0.2
-2
-0.4
0.6
x1 / b
0.8
1
0
(c)
0.2
0.4
-14
-0.4
0.5
1
1.5
0.4
0.6
0.8
1
x1 / b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
23 / 27
53. Mechanical pressure compared to buckled mode shapes
Normalized Mechanical Pressure: P (× 10−2)
10
5
2.5
0.4
Region 1
-40
0.2
x2 / b
0
-30
-20
-10
Positive strain load
-2.5 Region 2 0
−0.2
-40
Region 1
−0.4
0
10
0.4
0.2
2.5
5
0.6
0.8
1.5
1
0.5
0.4
ν = 0.25
0
0.2
0
ν = 0.25
-2
-4
-6 -8 -10 -12
x2 / b
-0.2
-2
0
1
2
3
0.2
-2
-4
-6
-8
-10
-12
x2 / b
-14
Buckled Mode Shape: σ < 0 (× 10−2)
Buckled Mode Shape: σ > 0 (× 10−2)
0.4
1
x1 / b
(a)
0
-0.2
0
0
(b)
0.2
-2
-0.4
0.6
x1 / b
0.8
1
0
(c)
0.2
0.4
-14
-0.4
0.5
1
1.5
0.4
0.6
0.8
1
x1 / b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
23 / 27
54. Mechanical pressure compared to buckled mode shapes
Normalized Mechanical Pressure: P (× 10−2)
−10
−5
−2.5
0.4
Region 1
40
0.2
x2 / b
0
30
20
10
Negative strain load
2.5 Region 2 0
−0.2
40
Region 1
−0.4
0
−10
0.4
0.2
−2.5
−5
0.6
0.8
1.5
1
0.5
0.4
ν = 0.25
0
0.2
0
ν = 0.25
-2
-4
-6 -8 -10 -12
x2 / b
-0.2
-2
0
1
2
3
0.2
-2
-4
-6
-8
-10
-12
x2 / b
-14
Buckled Mode Shape: σ < 0 (× 10−2)
Buckled Mode Shape: σ > 0 (× 10−2)
0.4
1
x1 / b
(a)
0
-0.2
0
0
(b)
0.2
-2
-0.4
0.6
x1 / b
0.8
1
0
(c)
0.2
0.4
-14
-0.4
0.5
1
1.5
0.4
0.6
0.8
1
x1 / b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
23 / 27
55. Mechanical pressure compared to buckled mode shapes
Normalized Mechanical Pressure: P (× 10−2)
−10
−5
−2.5
0.4
Region 1
40
0.2
x2 / b
0
30
20
10
Negative strain load
2.5 Region 2 0
−0.2
40
Region 1
−0.4
0
−10
0.4
0.2
−2.5
−5
0.6
0.8
1.5
1
0.5
0.4
ν = 0.25
0
0.2
0
ν = 0.25
-2
-4
-6 -8 -10 -12
x2 / b
-0.2
-2
0
1
2
3
0.2
-2
-4
-6
-8
-10
-12
x2 / b
-14
Buckled Mode Shape: σ < 0 (× 10−2)
Buckled Mode Shape: σ > 0 (× 10−2)
0.4
1
x1 / b
(a)
0
-0.2
0
0
(b)
0.2
-2
-0.4
0.6
x1 / b
0.8
1
0
(c)
0.2
0.4
-14
-0.4
0.5
1
1.5
0.4
0.6
0.8
1
x1 / b
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
23 / 27
56. Practical implications
Silicon Nitride: 30 × 12 × 0.09 µm 3 (L× b × h)
×
×
Graphene (2-layer): 3.2 × 0.8 × 0.0006 µm 3 (L× b × h)
Cantilever
Material
Si3 N 4
T
σs > 0(< 0)
(Nm− 1 )
98.7 (−78.5)
Graphene 0.0324 (−0.0276)
∆ T > 0 (< 0)
(K)
4010 (unphysical)
8.9 (−7.6)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
24 / 27
57. Practical implications
Silicon Nitride: 30 × 12 × 0.09 µm 3 (L× b × h)
×
×
Graphene (2-layer): 3.2 × 0.8 × 0.0006 µm 3 (L× b × h)
Cantilever
Material
Si3 N 4
T
σs > 0(< 0)
(Nm− 1 )
98.7 (−78.5)
Graphene 0.0324 (−0.0276)
∆ T > 0 (< 0)
(K)
4010 (unphysical)
8.9 (−7.6)
Graphene cantilever stability is marginal!!
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
24 / 27
58. Practical implications
Silicon Nitride: 30 × 12 × 0.09 µm 3 (L× b × h)
×
×
Graphene (2-layer): 3.2 × 0.8 × 0.0006 µm 3 (L× b × h)
Cantilever
Material
Si3 N 4
T
σs > 0(< 0)
∆ T > 0 (< 0)
(Nm− 1 )
98.7 (−78.5)
Graphene 0.0324 (−0.0276)
(K)
4010 (unphysical)
8.9 (−7.6)
Li et. al, Appl. Phys. Lett. 101, (2012)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
24 / 27
59. Conclusion
Positive/negative surface stress loads induce a negative/positive
change in stiﬀness
Practical V-shaped cantilevers more sensitive to surface stress changes
Observed frequency shifts not due to surface stress
Buckling driven by compressive in-plane stresses
Novel graphene cantilevers buckle under surface stress and thermal
loads
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
25 / 27
60. Publications
M. J. Lachut and J. E. Sader, Phys. Rev. Lett. 99, 206102 (2007)
M. J. Lachut and J. E. Sader, Appl. Phys. Lett. 95, 193505 (2009)
M. J. Lachut and J. E. Sader, Phys. Rev. B. 85, 085440 (2012)
M. J. Lachut and J. E. Sader, J. Appl. Phys. 113, 024501 (2013)
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
26 / 27
61. Acknowledgements
Thanks to:
Family and Friends for their emotional support
Esteemed colleagues for their insightful comments
Conﬁrmation panel for their support, and
Supervisor Prof. John E. Sader for being an inspirational mentor
Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics of cantilever plates 25, 2013
Eﬀect of surface stress change on the stiﬀness University of Melbourne)
June
27 / 27
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