Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects
- 69 Downloads
In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton’s principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.
Key wordsnanobeam nonlocal strain gradient theory two-step perturbation method deep postbuckling
Chinese Library ClassificationO344.1
2010 Mathematics Subject Classification74H55
Unable to display preview. Download preview PDF.
- KULATHUNGA, D. D. T. K., ANG, K. K., and REDDY, J. N. Accurate modeling of buckling of single- and double-walled carbon nanotubes based on shell theories. Journal of Physics: Condensed Matter, 21, 435301 (2009)Google Scholar
- PENG, X. W., GUO, X. M., LIU, L., and WU, B. J. Scale effects on nonlocal buckling analysis of bilayer composite plates under non-uniform uniaxial loads. Applied Mathematics and Mechanics (English Edition), 36(1), 1–10 (2015) https://doi.org/10.1007/s10483-015-1900-7MathSciNetCrossRefzbMATHGoogle Scholar
- GHORBANPOUR-ARANI, A., KOLAHDOUZAN, F., and ABDOLLAHIAN, M. Nonlocal buck- ling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory. Applied Mathematics and Mechanics (English Edition), 39(4), 529–546 (2018) https://doi.org/ 10.1007/s10483-018-2319-8MathSciNetCrossRefGoogle Scholar
- WANG, B., DENG, Z. C., and ZHANG, K. Nonlinear vibration of embedded single-walled car- bon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory. Applied Mathematics and Mechanics (English Edition), 34(3), 269–280 (2013) https://doi.org/10.1007/s10483-013-1669-8MathSciNetCrossRefzbMATHGoogle Scholar
- MOHAMMADIMEHR, M., FARAHI, M. J., and ALIMIRZAEI, S. Vibration and wave propaga- tion analysis of twisted micro-beam using strain gradient theory. Applied Mathematics and Mechanics (English Edition), 37(10), 1375–1392 (2016) https://doi.org/10.1007/s10483-016-2138-9MathSciNetCrossRefzbMATHGoogle Scholar
- SAHMANI, S. and FATTAHI, A. M. Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Applied Mathematics and Mechanics (English Edition), 39(4), 561–580 (2018) https://doi.org/10.1007/s10483-018-2321-8MathSciNetCrossRefzbMATHGoogle Scholar
- TIMOSHENKO, S. P. and GERE, J. M. Theory of Elastic Stability, McGraw-Hill Book Company, New York (1961)Google Scholar