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  • 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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