Abstract
In this work, the size-dependent buckling of functionally graded (FG) Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stress-driven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The stress-driven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions. By employing the eigenvalue method, the critical buckling loads of the beams with different boundary conditions are obtained. The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases. Additionally, the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.
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AUTUMN, K., LIANG, Y. A., HSIEH, S. T., ZESCH, W., CHAN, W. P., KENNY, T. W., FEARING, R., and FULL, R. J. Adhesive force of a single gecko foot-hair. nature, 405, 681–685 (2000)
KOPPERGER, E., LIST, J., MADHIRA, S., ROTHFISCHER, F., LAMB, D. C., and SIMMEL, F. C. A self-assembled nanoscale robotic arm controlled by electric fields. Science, 359, 296–300 (2018)
WANG, Z. L. and SONG, J. H. Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science, 312, 242–246 (2006)
XIANG, R., INOUE, T., ZHENG, Y. J., KUMAMOTO, A., QIAN, Y., SATO, Y., LIU, M., TANG, D. M., GOKHALE, D., GUO, J., HISAMA, K., YOTSUMOTO, S., OGAMOTO, T., ARAI, H., KOBAYASHI, Y., ZHANG, H., HOU, B., ANISIMOV, A., MARUYAMA, M., MIYATA, Y., OKADA, S., CHIASHI, S., LI, Y., KONG, J., KAUPPINEN, E. I., IKUHARA, Y., SUENAGA, K., and MARUYAMA, S. One-dimensional van der Waals heterostructures. Science, 367, 537–542 (2020)
PEISKER, H., MICHELS, J., and GORB, S. N. Evidence for a material gradient in the adhesive tarsal setae of the ladybird beetle Coccinella septempunctata. Nature Communications, 4, 1661 (2013)
ALIBARDI, L. Review: mapping proteins localized in adhesive setae of the tokay gecko and their possible influence on the mechanism of adhesion. Protoplasma, 255, 1785–1797 (2018)
NIX, W. D. and GAO, H. J. Indentation size effects in crystalline materials: a law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 46, 411–425 (1998)
HUANG, Y., ZHANG, F., HWANG, K. C., NIX, W. D., PHARR, G. M., and FENG, G. A model of size effects in nano-indentation. Journal of the Mechanics and Physics of Solids, 54, 1668–1686 (2006)
ZHU, X. W. and LI, L. Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science, 119, 16–28 (2017)
GUO, S., HE, Y. M., LEI, J., LI, Z. K., and LIU, D. B. Individual strain gradient effect on torsional strength of electropolished microscale copper wires. Scripta Materialia, 130, 124–127 (2017)
GREER, J. R., OLIVER, W. C., and NIX, W. D. Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Materialia, 53, 1821–1830 (2005)
LU, L., GUO, X. M., and ZHAO, J. Z. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24 (2017)
ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21, 313–342 (1987)
ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710 (1983)
ERINGEN, A. C. Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1–16 (1972)
GHANNADPOUR, S. A. M., MOHAMMADI, B., and FAZILATI, J. Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96, 584–589 (2013)
KHODABAKHSHI, P. and REDDY, J. N. A unified integro-differential nonlocal model. International Journal of Engineering Science, 95, 60–75 (2015)
SOBHY, M. and ZENKOUR, A. M. Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets on a viscoelastic substrate. Composites Part B: Engineering, 154, 492–506 (2018)
LU, L., GUO, X. M., and ZHAO, J. Z. A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms. International Journal of Engineering Science, 119, 265–277 (2017)
MIRJAVADI, S. S., RABBY, S., SHAFIEI, N., AFSHARI, B. M., and KAZEMI, M. On size-dependent free vibration and thermal buckling of axially functionally graded nanobeams in thermal environment. Applied Physics A-Materials Science Processing, 123, 315 (2017)
BARATI, M. R. and ZENKOUR, A. M. Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Composite Structures, 182, 91–98 (2017)
AL-SHUJAIRI, M. and MOLLAMAHMUTOGLU, C. Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect. Composites Part B: Engineering, 154, 292–312 (2018)
WANG, Y. B., ZHU, X. W., and DAI, H. H. Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Advances, 6, 22 (2016)
ZHANG, P., QING, H., and GAO, C. F. Theoretical analysis for static bending of circular Euler-Bernoulli beam using local and Eringen’s nonlocal integral mixed model. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, 99, 8 (2019)
ZHANG, P., QING, H., and GAO, C. F. Analytical solutions of static bending of curved Timoshenko microbeams using Eringen’s two-phase local/nonlocal integral model. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, 100, 7 (2020)
ROMANO, G., BARRETTA, R., DIACO, M., and DE SCIARRA, F. M. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences, 121, 151–156 (2017)
FERNANDEZ-SAEZ, J., ZAERA, R., LOYA, J. A., and REDDY, J. N. Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. International Journal of Engineering Science, 99, 107–116 (2016)
JIANG, P., QING, H., and GAO, C. F. Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model. Applied Mathematics and Mechanics (English Edition), 41(2), 207–232 (2020) https://doi.org/10.1007/s10483-020-2569-6
ROMANO, G. and BARRETTA, R. Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, 14–27 (2017)
BARRETTA, R., FABBROCINO, F., LUCIANO, R., DE SCIARRA, F. M., and RUTA, G. Buckling loads of nano-beams in stress-driven nonlocal elasticity. Mechanics of Advanced Materials and Structures, 27, 869–875 (2020)
BARRETTA, R., FAGHIDIAN, S. A., and LUCIANO, R. Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mechanics of Advanced Materials and Structures, 26, 1307–1315 (2019)
BARRETTA, R., FAGHIDIAN, S. A., and DE SCIARRA, F. M. Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. International Journal of Engineering Science, 136, 38–52 (2019)
SEDIGHI, H. M. and MALIKAN, M. Stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment. Physica Scripta, 95, 055218 (2020)
ZHANG, P., QING, H., and GAO, C. F. Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model. Composite Structures, 245, 112362 (2020)
DARBAN, H., FABBROCINO, F., FEO, L., and LUCIANO, R. Size-dependent buckling analysis of nanobeams resting on two-parameter elastic foundation through stress-driven nonlocal elasticity model. Mechanics of Advanced Materials and Structures, 28, 2408–2416 (2020)
BIAN, P. L. and QING, H. Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model. Applied Mathematics and Mechanics (English Edition), 42(3), 425–440 (2021) https://doi.org/10.1007/s10483-021-2708-9
MA, Y. B. Size-dependent thermal conductivity in nanosystems based on non-Fourier heat transfer. Applied Physics Letters, 101, 211905 (2012)
DONG, Y., CAO, B. Y., and GUO, Z. Y. Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics. Physica E: Low-dimensional Systems and Nanostructures, 56, 256–262 (2014)
YU, Y. J., LI, C. L., XUE, Z. N., and TIAN, X. G. The dilemma of hyperbolic heat conduction and its settlement by incorporating spatially nonlocal effect at nanoscale. Physics Letters A, 380, 255–261 (2016)
YU, Y. J., TIAN, X. G., and LIU, X. R. Size-dependent generalized thermoelasticity using Eringen’s nonlocal model. European Journal of Mechanics-A/Solids, 51, 96–106 (2015)
YU, Y. J., XUE, Z. N., LI, C. L., and TIAN, X. G. Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity. Composite Structures, 146, 108–113 (2016)
LEI, J., HE, Y. M., LI, Z. K., GUO, S., and LIU, D. B. Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams. Journal of Thermal Stresses, 42, 526–539 (2019)
BARATI, M. R. and ZENKOUR, A. Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments. Mechanics of Advanced Materials and Structures, 25, 669–680 (2018)
LEI, J., HE, Y. M., GUO, S., LI, Z. K., and LIU, D. B. Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM. Journal of Thermal Stresses, 40, 665–689 (2017)
SARKAR, N. Thermoelastic responses of a finite rod due to nonlocal heat conduction. Acta Mechanica, 231, 947–955 (2020)
SINGH, P. and YADAVA, R. D. S. Effect of surface stress on resonance frequency of microcantilever sensors. IEEE Sensors Journal, 18, 7529–7536 (2018)
WU, J. Z. and ZHANG, N. H. Clamped-end effect on static detection signals of DNA-microcantilever. Applied Mathematics and Mechanics (English Edition), 42(10), 1423–1438 (2021) https://doi.org/10.1007/s10483-021-2780-6
WU, J. K. About beam (in Chinese). Mechanics in Engineering, 30, 106–109 (2008)
XI, Y. Y., LYU, Q., ZHANG, N. H., and WU, J. Z. Thermal-induced snap-through buckling of simply-supported functionally graded beams. Applied Mathematics and Mechanics (English Edition), 41(12), 1821–1832 (2020) https://doi.org/10.1007/s10483-020-2696-7
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Citation: XU, C., LI, Y., LU, M. Y., and DAI, Z. D. Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity. Applied Mathematics and Mechanics (English Edition), 43(3), 355–370 (2022) https://doi.org/10.1007/s10483-022-2828-5
Project supported by the National Natural Science Foundation of China (Nos. 51435008 and 51705247) and the China Postdoctoral Science Foundation (No. 2020M671474)
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Xu, C., Li, Y., Lu, M. et al. Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity. Appl. Math. Mech.-Engl. Ed. 43, 355–370 (2022). https://doi.org/10.1007/s10483-022-2828-5
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DOI: https://doi.org/10.1007/s10483-022-2828-5
Key words
- size effect
- stress-driven nonlocal model
- constitutive boundary condition
- nonlocal heat conduction
- functionally graded (FG) beam
- buckling load