0
Research Articles

Surface Stress Effect on the Vibrational Response of Circular Nanoplates With Various Edge Supports

[+] Author and Article Information
S. Sahmani

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran

1Corresponding author: r_ansari@guilan.ac.ir

Manuscript received April 5, 2012; final manuscript received July 23, 2012; accepted manuscript posted July 27, 2012; published online January 25, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(2), 021021 (Jan 25, 2013) (7 pages) Paper No: JAM-12-1138; doi: 10.1115/1.4007255 History: Received April 05, 2012; Revised July 23, 2012; Accepted July 27, 2012

The classical continuum theory cannot be directly used to describe the behavior of nanostructures because of their size-dependent attribute. Surface stress effect is one of the most important size dependencies of structures at this submicron size, which is due to the high surface to volume ratio of nanoscale domain. In the present study, the nonclassical governing differential equation together with corresponding boundary conditions are derived using Hamilton's principle, into which the surface energies are incorporated through the Gurtin-Murdoch elasticity theory. The model developed herein contains intrinsic length scales to take the size effect into account and is used to analyze the free vibration response of circular nanoplates including surface stress effect. The generalized differential quadrature (GDQ) method is employed to discretize the governing size-dependent differential equation along with simply supported and clamped boundary conditions. The classical and nonclassical frequencies of circular nanoplates with various edge supports and thicknesses are calculated and are compared to each other. It is found that the influence of surface stress can be different for various circumferential mode numbers, boundary conditions, plate thicknesses, and surface elastic constants.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Streitz, F. H., Cammarata, R. C., and Sieradzki, K., 1994, “Surface Stress Effects on Elastic Properties. I. Thin Metal Films,” Phys. Rev. B, 49(15), pp. 10699–10706. [CrossRef]
Dingreville, R., Qu, J., and Cherkaoui, M., 2005, “Surface Free Energy and Its Effects on the Elastic Behavior of Nano-Sized Particles, Wires and Films,” J. Mech. Phys. Solids, 53(8), pp. 1827–1954. [CrossRef]
Yoon, J., Ru, C. Q., and Mioduchowski, A., 2003, “Vibration of an Embedded Multiwall Carbon Nanotube,” Composites Sci. Technol., 63(11), pp. 1533–1542. [CrossRef]
Fu, Y. M., Hong, J. W., and Wang, X. Q., 2006, “Analysis of Nonlinear Vibration for Embedded Carbon Nanotubes,” J. Sound Vib., 296(4–5), pp. 746–756. [CrossRef]
Wang, Q., and Varadan, V. K., 2006, “Wave Characteristics of Carbon Nanotubes,” Int. J. Solids Struct., 43(2), pp. 254–265. [CrossRef]
Liew, K. M., and Wang, Q., 2007, “Analysis of Wave Propagation in Carbon Nanotubes Via Elastic Shell Theories,” Int. J. Eng. Sci., 45(2–8), pp. 227–241. [CrossRef]
Wang, L., Ni, Q., Li, M., and Qian, Q., 2008, “The Thermal Effect on Vibration and Instability of Carbon Nanotubes Conveying Fluid,” Physica E, 40(10), pp. 3179–3182. [CrossRef]
Elishakoff, I., and Pentaras, D., 2009, “Fundamental Natural Frequencies of Double-Walled Carbon Nanotubes,” J. Sound Vib., 322(4–5), pp. 652–664. [CrossRef]
Ansari, R., Hemmatnezhad, M., and Rezapour, J., 2011, “The Thermal Effect on Nonlinear Oscillations of Carbon Nanotubes With Arbitrary Boundary Conditions,” Curr. Appl. Phys.11(3), pp. 692–697. [CrossRef]
Gurtin, M. E., and Murdoch, A. I., 1975, “A Continuum Theory of Elastic Material Surface,” Arch. Rat. Mech. Anal., 57(4), pp. 291–323. [CrossRef]
Gurtin, M. E., and Murdoch, A. I., 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14(6), pp. 431–440. [CrossRef]
Mogilevskaya, S. G., Crouch, S. L., and Stolarski, H. K., 2008, “Multiple Interacting Circular Nano-Inhomogeneities With Surface/Interface Effects,” J. Mech. Phys. Solids, 56(6), pp. 2298–2327. [CrossRef]
Luo, J., and Wang, X., 2009, “On the Anti-Plane Shear of an Elliptic Nano Inhomogeneity,” Eur. J. Mech. A, 28(5), pp. 926–934. [CrossRef]
Gordeliy, E., Mogilevskaya, S. G., and Crouch, S. L., 2009, “Transient Thermal Stresses in a Medium With a Circular Cavity With Surface Effects,” Int. J. Solids Struct., 46(9), pp. 1834–1848. [CrossRef]
Zhao, X. J., and Rajapakse, R. K. N. D., 2009, “Analytical Solutions for a Surface Loaded Isotropic Elastic Layer With Surface Energy Effects,” Int. J. Eng. Sci., 47(11–12), pp. 1433–1444. [CrossRef]
Mogilevskaya, S. G., Crouch, S. L., Grotta, A. L., and Stolarski, H. K., 2010, “The Effects of Surface Elasticity and Surface Tension on the Transverse Overall Elastic Behavior of Unidirectional Nano-Composites,” Composites Sci. Technol., 70(3), pp. 427–434. [CrossRef]
Song, F., and Huang, G. L., 2009, “Modeling of Surface Stress Effects on Bending Behavior of Nanowires: Incremental Deformation Theory,” Phys. Lett. A, 373(43), pp. 3969–3973. [CrossRef]
On, B. B., Altus, E., and Tadmor, E. B., 2010, “Surface Effects in Non-Uniform Nanobeams: Continuum vs. Atomistic Modeling,” Int. J. Solids Struct., 47(9), pp. 1243–1252. [CrossRef]
Wang, G. F., and Feng, X. Q., 2007, “Effects of Surface Stresses on Contact Problems at Nanoscale,” J. Appl. Phys., 101, p. 013510. [CrossRef]
Ansari, R., and Sahmani, S., 2011, “Surface Stress Effects on the Free Vibration Behavior of Nanoplates,” Int. J. Eng. Sci., 49, pp. 1204–1215. [CrossRef]
Ansari, R., and Sahmani, S., 2011, “Bending Behavior and Buckling of Nanobeams Including Surface Stress Effects Corresponding to Different Beam Theories,” Int. J. Eng. Sci., 49, pp. 1244–1255. [CrossRef]
Miller, R. E., and Shenoy, V. B., 2000, “Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnology, 11, pp. 139–147. [CrossRef]
Wang, G. F., and Feng, X. Q., 2007, “Effects of Surface Elasticity and Residual Surface Tension on the Natural Frequency of Microbeams,” Appl. Phys. Lett., 90, p. 231904. [CrossRef]
Shenoy, V. B., 2005, “Atomistic Calculations of Elastic Properties of Metallic fcc Crystal Surfaces,” Phys. Rev. B, 71(9), pp. 1–11. [CrossRef]
Gurtin, M. E., Weissmuller, J., and Larche, F., 1998, “A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philos. Magn. A, 78(5), pp. 1093–1109. [CrossRef]
Weissmuller, J., and Cahn, J. W., 1997, “Mean Stresses in Microstructures Due to Interface Stresses: A Generalization of a Capillary Equation for Solids,” Acta Mater., 45(5), pp. 1899–1906. [CrossRef]
Sharma, P., Ganti, S., and Bhate, N., 2003, “Effect of Surfaces on the Size-Dependent Elastic State of Nano-Inhomogeneities,” Appl. Phys. Lett., 82(4), pp. 535–537. [CrossRef]
Duan, H. L., Wang, J., Huang, Z. P., and Karihaloo, B. L., 2005, “Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities With Interface Stress,” J. Mech. Phys. Solids, 53(7), pp. 1574–1596. [CrossRef]
Sharma, P., and Ganti, S., 2004, “Size-Dependent Eshelby's Tensor for Embedded Nano-Inclusions Incorporating Surface/Interface Energies,” ASME J. Appl. Mech., 71(5), pp. 663–671. [CrossRef]
Lu, L., He, L. H., Lee, H. P., and Lu, C., 2006, “Thin Plate Theory Including Surface Effects,” Int. J. Solids Struct., 43(16), pp. 4631–4647. [CrossRef]
Gurtin, M. E., and Murdoch, A., 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Rat. Mech. Anal., 57, 291–323. [CrossRef]
Shu, C., 2000, Differential Quadrature and Its Application in Engineering, Springer, London.

Figures

Grahic Jump Location
Fig. 1

Schematic of a circular nanoplate with upper and lower thin skin layers carrying surface effects: Kinematic parameters, coordinate system, and geometry

Grahic Jump Location
Fig. 2

Variation of frequency ratio with thickness of nanoplates corresponding to various values of radius to thickness ratio (m=0)

Grahic Jump Location
Fig. 3

Variation of frequency ratio with thickness of nanoplates corresponding to various values of radius to thickness ratio (m=1)

Grahic Jump Location
Fig. 4

Variation of frequency ratio with thickness of nanoplates corresponding to various values of radius to thickness ratio (m=2)

Grahic Jump Location
Fig. 5

Variation of frequency ratio with thickness of nanoplates corresponding to various values of radius to thickness ratio (m=3)

Grahic Jump Location
Fig. 6

Variation of frequency ratio with circumferential wave number of nanoplates corresponding to various values of radius to thickness ratio (h=2 nm)

Grahic Jump Location
Fig. 7

Variation of frequency ratio with radius to thickness ratio of nanoplates corresponding to various circumferential wave numbers (h=1 nm)

Grahic Jump Location
Fig. 8

Variation of frequency ratio with thickness of nanoplate corresponding to different values of 2μs+λs with the assumption of τs=ρs=0 and m=2

Grahic Jump Location
Fig. 9

Variation of frequency ratio with thickness of nanoplate corresponding to different values of ρs with the assumption of 2μs+λs=τs=0 and m=2

Grahic Jump Location
Fig. 10

Variation of frequency ratio with thickness of nanoplate corresponding to different values of τs with the assumption of 2μs+λs=ρs=0 and m=2

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In