Abstract
The deviation from the classical elastic characteristics induced by the free surface energy can be considerable for nanostructures due to the high surface to volume ratio. Consequently, this type of size dependency should be accounted for in the mechanical behaviors of nanoscale structures. In the current investigation, the influence of free surface energy on the nonlinear primary resonance of silicon nanoshells under soft harmonic external excitation is studied. In order to obtain more accurate results, the interaction between the first, third, and fifth symmetric vibration modes with the main oscillation mode is taken into consideration. Through the implementation of the Gurtin-Murdoch theory of elasticity into the classical shell theory, a size-dependent shell model is developed incorporating the effect of surface free energy. With the aid of the variational approach, the governing differential equations of motion including both of the cubic and quadratic nonlinearities are derived. Thereafter, the multi-time-scale method is used to achieve an analytical solution for the nonlinear size-dependent problem. The frequency-response and amplitude-response of the soft harmonic excited nanoshells are presented corresponding to different values of shell thickness and surface elastic constants as well as various vibration mode interactions. It is depicted that through consideration of the interaction between the higher symmetric vibration modes and the main oscillation mode, the hardening response of nanoshell changes to the softening one. This pattern is observed corresponding to both of the positive and negative values of the surface elastic constants and the surface residual stress.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
TOGUN, N. and BAGDATLI, S. M. Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory. Composites Part B: Engineering, 97, 255–262 (2016)
BORNASSI, S. and HADDADPOUR, H. Nonlocal vibration and pull-in instability analysis of electrostatic carbon-nanotube based NEMS devices. Sensors and Actuators A: Physical, 266, 185–196 (2017)
GUO, J., CHEN, J., and PAN, E. Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory. Physica E, 87, 98–106 (2017)
LI, C., LIU, J. J., CHENG, M., and FAN, X. L. Modeling of nonlinear vibration of graphene sheets using a meshfree method based on nonlocal elasticity theory. Composites Part B: Engineering, 116, 153–169 (2017)
ZHANG, L. W., ZHANG, Y., and LIEW, K. M. Modeling of nonlinear vibration of graphene sheets using a meshfree method based on nonlocal elasticity theory. Applied Mathematical Modelling, 49, 691–704 (2017)
LU, L., GUO, X., and ZHAO, J. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24 (2017)
LIU, J. C., ZHANG, Y. Q., and FAN, L. F. Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between. Physics Letters A, 381, 1228–1235 (2017)
ZHANG, H., WANG, C. M., and CHALLAMEL, N. Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Composite Structures, 165, 148–159 (2017)
YANG, Z. and HE, D. Vibration and buckling of orthotropic functionally graded micro-plates on the basis of a re-modified couple stress theory. Results in Physics, 7, 3778–3787 (2017)
FANG, J., GU, J., and WANG, H. Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences, 136, 188–199 (2017)
APUZZO, A., BARRETTA, R., FAGHIDIAN, S. A., LUCIANO, R., and MORATTI DE SCIARRA, F. Free vibrations of elastic beams by modified nonlocal strain gradient theory. International Journal of Engineering Science, 133, 99–108 (2018)
KIANI, K. and PAKDAMAN, H. Nonlocal vibrations and potential instability of monolayers from double-walled carbon nanotubes subjected to temperature gradients. International Journal of Mechanical Sciences, 144, 576–599 (2018)
WANG, J., SHEN, H., ZHANG, B., LIU, J., and ZHANG, Y. Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory. Physica E, 101, 85–93 (2018)
THANH, C. L., PHUNG-VAN, P., THAI, C. H., NGUYEN-XUAN, N., and ABDEL WAHAB, M. Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory. Composite Structures, 184, 633–649 (2018)
SAHMANI, S., AGHDAM, M. M., and RABCZUK, T. Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Composite Structures, 198, 51–62 (2018)
SAHMANI, S., AGHDAM, M. M., and RABCZUK, T. A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Materials Research Express, 5, 045048 (2018)
SAHMANI, S., AGHDAM, M. M., and RABCZUK, T. Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Composite Structures, 186, 68–78 (2018)
WANG, X. Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Applied Mathematics Letters, 77, 94–100 (2018)
SHEN, J. P., WANG, P. Y., LI, C., and WANG, Y. Y. New observations on transverse dynamics of microtubules based on nonlocal strain gradient theory. Composite Structures, 225, 111036 (2019)
TANG, H., LI, L., HU, Y., MENG, W., and DUAN, K. Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Structures, 137, 377–391 (2019)
JALAEI, M. H., GHORBANPOUR-ARANI, A., and NGUYEN-XUAN, H. Investigation of thermal and magnetic field effects on the dynamic instability of FG Timoshenko nanobeam employing nonlocal strain gradient theory. International Journal of Mechanical Sciences, 161, 105043 (2019)
SAHMANI, S. and SAFAEI, B. Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin-Walled Structures, 140, 342–356 (2019)
SAHMANI, S. and SAFAEI, B. Nonlocal strain gradient nonlinear resonance of bi-directional functionally graded composite micro/nano-beams under periodic soft excitation. Thin-Walled Structures, 143, 106226 (2019)
JALAEI, M. H. and CIVALEK, O. A nonlocal strain gradient refined plate theory for dynamic instability of embedded graphene sheet including thermal effects. Composite Structures, 220, 209–220 (2019)
ZHANG, B., SHEN, H., LIU, J., WANG, Y., and ZHANG, Y. Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects. Applied Mathematics and Mechanics (English Edition), 40(4), 515–548 (2019) https://doi.org/10.1007/s10483-0192482-9
SAHMANI, S., FATTAHI, A. M., and AHMED, N. A. Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Engineering with Computers, 35, 1173–1189 (2019)
WANG, Y., LIU, Y., and ZU, J. W. Nonlinear free vibration of piezoelectric cylindrical nanoshells. Applied Mathematics and Mechanics (English Edition), 40(5), 601–620 (2019) https://doi.org/10.1007/s10483-019-2476-6
GURTIN, M. E. and MURDOCH, A. I. A continuum theory of elastic material surface. Archive for Rational Mechanics and Analysis, 57, 291–323 (1975)
GURTIN, M. E. and MURDOCH, A. I. Surface stress in solids. International Journal of Solids and Structures, 14, 431–440 (1978)
WANG, G. F. and FENG, X. Q. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Applied Physics Letters, 90, 231904 (2007)
LUO, J. and XIAO, Z. M. Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity. International Journal of Engineering Science, 47, 883–893 (2009)
ZHAO, X. J. and RAJAPAKSE, R. K. N. D. Analytical solutions for a surface loaded isotropic elastic layer with surface energy effects. International Journal of Engineering Science, 47, 1433–1444 (2009)
WANG, Z. Q., ZHAO, Y. P., and HUANG, Z. P. The effects of surface tension on the elastic properties of nano structures. International Journal of Engineering Science, 48, 140–150 (2010)
CHIU, M. S. and CHEN, T. Bending and resonance behavior of nanowires based on Timoshenko beam theory with high-order surface stress effects. Physica E, 54, 149–156 (2013)
SHAAT, M., MAHMOUD, F. F., GAO, X. L., and FAHEEM, A. F. Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. International Journal of Mechanical Sciences, 79, 31–37 (2014)
SAHMANI, S., BAHRAMI, M., AGHDAM, M. M., and ANSARI, R. Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams. Composite Structures, 118, 149–158 (2014)
SAHMANI, S. and AGHDAM, M. M. Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments. Archives of Civil and Mechanical Engineering, 17, 623–638 (2017)
LU, L., GUO, X., and ZHAO, J. On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy. International Journal of Engineering Science, 124, 24–40 (2018)
SUN, J., WANG, Z., ZHOU, Z., XU, X., and LIM, C. W. Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model. Applied Mathematical Modelling, 59, 341–356 (2018)
LU, L., GUO, X., and ZHAO, J. A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Applied Mathematical Modelling, 68, 583–602 (2019)
SARAFRAZ, A., SAHMANI, S., and AGHDAM, M. M. Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects. Applied Mathematical Modelling, 66, 195–226 (2019)
AMABILI, M., PELLICANO, F., and PAIDOUSSISI M. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid, part II: large-amplitude vibrations without flow. Journal of Sound and Vibration, 228, 1103–1124 (1999)
AMABILI, M. and PAIDOUSSIS, M. P. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Applied Mechanics Reviews, 56, 349–381 (2003)
MILLER, R. E. and SHENOY, V. B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 11, 139–147 (2000)
ZHU, R., PAN, E., CHUNG, P. W., CAI, X., LIEW, K. M., and BULDUM, A. Atomistic calculation of elastic moduli in strained silicon. Semiconductor Science and Technology, 21, 906–911 (2006)
ZEIGHAMPOUR, H. and TADI BENI, Y. Cylindrical thin-shell model based on modified strain gradient theory. International Journal of Engineering Science, 78, 27–47 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sarafraz, A., Sahmani, S. & Aghdam, M.M. Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory. Appl. Math. Mech.-Engl. Ed. 41, 233–260 (2020). https://doi.org/10.1007/s10483-020-2564-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-020-2564-5