Mechanistic Models for Nanobeams with Surface Stress Effects
Publication: Journal of Engineering Mechanics
Volume 144, Issue 11
Abstract
In this paper, a mechanistic model for nanobeams with surface energy effects is developed by using a variational formulation. This work is motivated by the unusual response of nanocantilevers predicted by models based on the Young-Laplace equation for surface stress. The governing equation and boundary conditions derived from the variational methods are compared with the governing equations and boundary conditions used in the Young-Laplace models and other formulations. A key difference in the shear force boundary condition is noted. Analytical solutions for simply supported, cantilevered, and fixed-fixed beams are reexamined. It is shown that the unusual behavior of nanocantilevers predicted by the Young-Laplace models is due to the shear force boundary condition used. The current formulation leads to consistent solutions for beams under different boundary conditions.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the Thailand Research Fund (TRF). The authors acknowledge discussions with Dr. E. de Mesquita Neto (University of Campinas, Brazil) and Dr. A. Phani (University of British Columbia, Canada).
References
Bao, M. H. 2005. Analysis and design principles of MEMS devices. Amsterdam, Netherlands: Elsevier.
Cammarata, R. C. 1997. “Surface and interface stress effects on interfacial and nanostructured materials.” Mater. Sci. Eng. A 237 (2): 180–184. https://doi.org/10.1016/S0921-5093(97)00128-7.
Challamel, N., and I. Elishakoff. 2012. “Surface stress effects may induce softening: Euler-Bernoulli and Timoshenko buckling solutions.” Physica E 44 (9): 1862–1867. https://doi.org/10.1016/j.physe.2012.05.019.
Chen, C. Q., Y. Shi, Y. S. Zhang, J. Zhu, and Y. J. Yan. 2006. “Size dependence of Young's modulus in ZnO nanowires.” Phy. Rev. Lett. 96 (7): 075505.
Craighead, H. G. 2000. “Nanoelectromechanical Systems.” Science 290 (5496): 1532–1535. https://doi.org/10.1126/science.290.5496.1532.
Gavan, K. B., H. J. R. Westra, W. J. Venstra, E. W. J. M. van der Drift, and H. S. J. van der Zant. 2009. “Size-dependent effective Young’s modulus of silicon nitride cantilevers.” Appl. Phys. Lett. 94 (23): 233108. https://doi.org/10.1063/1.3152772.
Gere, J. M., and S. P. Timoshenko. 1991. Mechanics of materials. London: Chapman & Hall.
Gurtin, M. E., and A. I. Murdoch. 1975a. “A continuum theory of elastic material surfaces.” Arch. Ration. Mech. Anal. 57 (4): 291–323. https://doi.org/10.1007/BF00261375.
Gurtin, M. E., and A. I. Murdoch. 1975b. “Addenda to our paper A continuum theory of elastic material surfaces.” Arch. Ration. Mech. Anal. 59 (4): 389–390. https://doi.org/10.1007/BF00250426.
Gurtin, M. E., and A. I. Murdoch. 1978. “Surface stress in solids.” Int. J. Solids Struct. 14 (6): 431–440. https://doi.org/10.1016/0020-7683(78)90008-2.
He, J., and C. M. Lilley. 2008. “Surface effect on the elastic behavior of static bending nanowires.” Nano Lett. 8 (7): 1798–1802. https://doi.org/10.1021/nl0733233.
He, J., and C. M. Lilley. 2009. “The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation.” Comput. Mech. 44 (3): 395–403. https://doi.org/10.1007/s00466-009-0380-9.
Jia, N., Y. Yao, Y. Yang, and S. Chen. 2017. “Size effect in the bending of a Timoshenko nanobeam.” Acta Mech. 228 (6): 2363–2375. https://doi.org/10.1007/s00707-017-1835-2.
Liu, C., and R. K. N. D. Rajapakse. 2010. “Surface energy incorporated continuum models for static and dynamic response of nanoscale beams.” IEEE Trans. Nanotechnol. 42 (6): 422–434. https://doi.org/10.1080/07408170903394348.
Lu, P., L. H. He, H. P. Lee, and C. Lu. 2006. “Thin plate theory including surface effects.” Int. J. Solids Struct. 43 (16): 4631–4647. https://doi.org/10.1016/j.ijsolstr.2005.07.036.
Miller, R. E., and V. B. Shenoy. 2000. “Size dependent elastic properties of structural elements.” Nanotechnology 11 (3): 139–147. https://doi.org/10.1088/0957-4484/11/3/301.
Nilsson, S. G., X. Borrise, and L. Montelius. 2004. “Size effect on Young’s modulus of thin chromium cantilevers.” Appl. Phys. Lett. 85 (16): 3555–3557. https://doi.org/10.1063/1.1807945.
Nilsson, S. G., E. L. Sarwe, and L. Montelius. 2003. “Fabrication and mechanical characterization of ultrashort nanocantilevers.” Appl. Phys. Lett. 83: 990–993.
Qiao, L., and X. Zheng. 2013. “Effect of surface stress on the stiffness of micro/nanocantilevers: Nanowire elastic modulus measured by nano-scale tensile and vibrational techniques.” J. Appl. Phys. 113 (1): 013508. https://doi.org/10.1063/1.4772649.
Sapsathiarn, Y., and R. K. N. D. Rajapakse. 2012. “A model for large deflections of nanobeams and experimental comparison.” IEEE Trans. Nanotechnol. 11 (2): 247–254. https://doi.org/10.1109/TNANO.2011.2160457.
Sapsathiarn, Y., and R. K. N. D. Rajapakse. 2013. “Finite-element modeling of circular nanoplates.” J. Nanomech. Micromech. 3: 59–66. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000056.
Sapsathiarn, Y., and R. K. N. D. Rajapakse. 2017. “Static and dynamic analyses of nanoscale rectangular plates incorporating surface energy.” Acta Mech. 228 (8): 2849–2863. https://doi.org/10.1007/s00707-015-1521-1.
Shenoy, V. B. 2005. “Atomistic calculations of elastic properties of metallic fcc crystal surfaces.” Phys. Rev. B 71 (9): 094104. https://doi.org/10.1103/PhysRevB.71.094104.
Song, F., G. L. Huang, H. S. Park, and X. N. Liu. 2011. “A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses.” Int. J. Solids Struct. 48 (14–15): 2154–2163. https://doi.org/10.1016/j.ijsolstr.2011.03.021.
Wang, G. F., and X. Q. Feng. 2009. “Surface effects on buckling of nanowires under uniaxial compression.” Appl. Phys. Lett. 94 (14): 141913.
Washizu, K. 1982. Variational methods in elasticity and plasticity. 2nd ed. New York: Pergamon.
Wong, E. W., P. E. Sheehan, and C. M. Lieber. 1997. “Nanobeam mechanics: Elasticity, strength and toughness of nanorods and nanotubes.” Science 277 (5334): 1971–1975. https://doi.org/10.1126/science.277.5334.1971.
Yakobson, B. I. 2003. “Nanomechanics.” In Handbook of nanoscience, engineering and technology, 848–878. Boca Raton, FL: CRC Press.
Yao, Y., and S. H. Chen. 2016. “Surface effect in the bending of nanowires.” Mech. Mater. 100 (Sep): 12–21. https://doi.org/10.1016/j.mechmat.2016.06.005.
Zhang, Y., Q. Ren, and Y. P. Zhao. 2004. “Modelling analysis of surface stress on a rectangular cantilever beam.” J. Phys. D: Appl. Phys. 37 (15): 2140–2145. https://doi.org/10.1088/0022-3727/37/15/014.
Information & Authors
Information
Published In
Copyright
©2018 American Society of Civil Engineers.
History
Received: Apr 30, 2018
Accepted: Apr 30, 2018
Published online: Aug 24, 2018
Published in print: Nov 1, 2018
Discussion open until: Jan 24, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.