Nonlinear Vibration Analysis of FG Nano-Beams in Thermal Environment and Resting on Nonlinear Foundation based on Nonlocal and Strain-Inertia Gradient Theory

Document Type: Original Article

Author

Department of Mechanical Engineering, Islamic Azad University,Borujerd,Iran

Abstract

In present research, nonlinear vibration of functionally graded nano-beams subjected to uniform temperature rise and resting on nonlinear foundation is comprehensively studied. The elastic center can be defined to remove stretching and bending couplings caused by the FG material variation. The small-size effect, playing essential role in the dynamical behavior of nano-beams, is considered here applying strain-inertia gradient and non-local elasticity theory. The governing partial differential equations have been derived based on the Euler-Bernoulli beam theory utilizing the von Karman strain-displacement relations. Subsequently, using the Galerkin method, the governing equations is reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency is then established using the homotopy analysis method.
Finally, the effects of different parameters such as length, nonlinear elastic foundation parameter, thermal loading, non-local parameter and gradient parameters are comprehensively investigated on the FG nano-beams vibration using homotopy analysis method. As the main results, it is observed that by increasing the non-local parameter, the frequency ratio for strain-inertia gradient theory has increasing trend while it has decreasing trend for non-local elasticity theory. Also, the nonlinear natural frequencies obtained using strain-inertia gradient theory are greater than the results of non-local elasticity and classical theory.

Keywords

Main Subjects


[1]   Wessel, J. K., The Handbook of Advanced Materials: Enabling New Designs, John Wiley & Sons, 2004.

[2]   Witvrouw, A., Mehta, A., The Use of Functionally Graded Poly-SiGe Layers for MEMS Applications, In Materials Science forum, Trans Tech Publications, 2005, pp. 255-260.

[3]   Miyamoto, Y., Kaysser, W. A., Rabin, B. H., Kawa saki, A., and Ford, R. G. eds., Functionally Graded Materials: Design, Processing and Applications, Springer Science & Business Media, Vol. 5, 2013.

[4]   Eringen, A. C., On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves, Journal of Applied Physics, Vol. 54, No. 9, 1983, pp. 4703-4710.

[5]   Togun, N., Bağdatlı, S. M., Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory, Mathematical and Computational Applications, Vol. 21, No. 1, 2016, pp. 3.

[6]   Arefi, M., Zenkour, A. M., Analysis of Wave Propagation in a Functionally Graded Nanobeam Resting on Visco-Pasternak’s Foundation, Theoretical and Applied Mechanics Letters, 2017.

[7]   Arefi, M., Zenkour, A. M., A Simplified Shear and Normal Deformations Nonlocal Theory for Bending of Functionally Graded Piezomagnetic Sandwich Nanobeams in Magneto-Thermo-Electric Environment, Journal of Sandwich Structures & Materials, Vol. 18, No. 5, 2016, pp. 624-651.

[8]   Arefi, M., Zenkour, A. M., Thermal Stress and Deformation Analysis of a Size-Dependent Curved Nanobeam Based on Sinusoidal Shear Deformation Theory, Alexandria Engineering Journal, 2017.

[9]   Nazemnezhad, R., Hosseini Hashemi, S., Nonlocal Nonlinear Free Vibration of Functionally Graded Nanobeams, Composite Structures, Vol. 110, 2014, pp. 192-199.

[10]              Ansari, R., Pourashraf, T., and Gholami, R., An Exact Solution for the Nonlinear Forced Vibration of Functionally Graded Nanobeams in Thermal Environment Based on Surface Elasticity Theory, Thin-Walled Structures, Vol. 93, 2015, pp. 169-176.

[11]              Arefi, M., Zenkour, A. M., Transient Analysis of a Three-Layer Microbeam Subjected to Electric Potential, International Journal of Smart and Nano Materials, 2017, pp. 20-40.

[12]              Arefi, M., Zenkour, A. M., Influence of Magneto-Electric Environments on Size-Dependent Bending Results of Three-Layer Piezomagnetic Curved Nanobeam Based on Sinusoidal Shear Deformation Theory, Journal of Sandwich Structures & Materials, 2017, pp. 1099636217723186.

[13]              Arefi, M., Zenkour, A. M., Transient Sinusoidal Shear Deformation Formulation of a Size-Dependent Three-Layer Piezo-Magnetic Curved Nanobeam, Acta Mechanica, Vol. 228, No. 10, 2017, pp. 3657-3674.

[14]              Arefi, M., Zenkour, A. M., Size-Dependent Vibration and Bending Analyses of the Piezomagnetic Three-Layer Nanobeams, Applied Physics A, Vol. 123, No. 3, 2017, pp. 202.

[15]              Arefi, M., Zenkour, A. M., Size-Dependent Electro-Elastic Analysis of a Sandwich Microbeam Based on Higher-Order Sinusoidal Shear Deformation Theory and Strain Gradient Theory, Journal of Intelligent Material Systems and Structures, 2017, pp. 1045389X17733333.

[16]              Arefi, M., Zenkour, A. M., Vibration and Bending Analysis of a Sandwich Microbeam with Two Integrated Piezo-Magnetic Face-Sheets, Composite Structures, Vol. 1, No. 159, 2017, pp. 479-490.

[17]              Arefi, M., Zenkour, A. M., Wave Propagation Analysis of a Functionally Graded Magneto-Electro-Elastic Nanobeam Rest on Visco-Pasternak Foundation, Mechanics Research Communications, Vol. 79, 2017, pp. 51-62.

[18]              Arefi, M., Pourjamshidian, M., and Arani, A. G., Application of Nonlocal Strain Gradient Theory and Various Shear Deformation Theories to Nonlinear Vibration Analysis of Sandwich Nano-Beam with FG-CNTRCs Face-Sheets in Electro-Thermal Environment, Applied Physics A, Vol. 123, No. 5, 2017, pp. 323.

[19]              Liao, S., Series Solution of Large Deformation of a Beam with Arbitrary Variable Cross Section Under an Axial load, The ANZIAM Journal, Vol. 51, No. 1, 2009, pp. 10-33.

[20]              Liao, S., Series Solution of Nonlinear Eigenvalue Problems by Means of the Homotopy Analysis Method, Nonlinear Analysis: Real World Applications, Vol. 10, No. 4, 2009, pp. 2455-2470.

[21]              Shahlaei Far, S., Nabarrete, A., and Balthazar, J. M., Nonlinear Vibrations of Cantilever Timoshenko Beams: a Homotopy Analysis, Latin American Journal of Solids and Structures, Vol. 10, 2016, pp. 1866-1877.

[22]              Askes, H., Aifantis, E. C., Gradient Elasticity in Statics and Dynamics: An Overview of Formulations, Length Scale Identification Procedures, Finite Element Implementations and New Results, International Journal of Solids and Structures, Vol. 48, No. 13, 2011, pp. 1962-1990.

[23]              Daneshmand, F., Rafiei, M., Mohebpour, S. R., and Heshmati, M., Stress and Strain-Inertia Gradient Elasticity in Free Vibration Analysis of Single Walled Carbon Nanotubes with First Order Shear Deformation Shell Theory, Applied Mathematical Modelling, Vol. 37, No. 16, 2013, pp. 7983-8003.

[24]              Karličić, D., Kozić, P., and Pavlović, R., Flexural Vibration and Buckling Analysis of Single-Walled Carbon Nanotubes Using Different Gradient Elasticity Theories Based on Reddy and Huu-Tai Formulations, Journal of Theoretical and Applied Mechanics, 2015, pp. 53.

[25]              Eringen, A. C., Nonlocal Continuum Field Theories, Springer Science & Business Media, 2002.

[26]              Pradhan, S. C., Murmu, T., Small Scale Effect on the Buckling of Single-Layered Graphene Sheets Under Biaxial Compression Via Nonlocal Continuum Mechanics, Computational Materials Science, Vol. 47, No. 1, 2009, pp. 268-274.

[27]              Murmu, T., Pradhan, S. C., Buckling of Biaxially Compressed Orthotropic Plates at Small Scales, Mechanics Research Communications, Vol. 36, No. 8, 2009, pp. 933-938.

[28]              Li, L., Hu, Y., Nonlinear Bending and Free Vibration Analyses of Nonlocal Strain Gradient Beams Made of Functionally Graded Material, International Journal of Engineering Science, Vol. 107, 2016, pp. 77-97.

[29]              Reddy, J. N., Theory and Analysis of Elastic Plates and Shells, CRC Press, 2006.

[30]              Ventsel, E., Krauthammer, T., Thin Plates and Shells: Theory: Analysis, and Applications, CRC Press, 2001.

[31]              Wang, C. M., Reddy, J. N., and Lee, K. H. eds., Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier, 2000.

[32]              Ansari, R., Gholami, R., and Rouhi, H., Size-Dependent Nonlinear Forced Vibration Analysis of Magneto-Electro-Thermo-Elastic Timoshenko Nanobeams Based Upon the Nonlocal Elasticity Theory, Composite Structures, Vol. 126, 2015, pp. 216-226.

[33]              Esfahani, S. E., Kiani, Y., Komijani, M., and Eslami, M. R., Vibration of a Temperature-Dependent Thermally Pre/Postbuckled FGM Beam Over a Nonlinear Hardening Elastic Foundation, Journal of Applied Mechanics, Vol. 81, No. 1, 2014, pp. 011004.

[34]              Raju, I. S., Rao, G. V., and Raju, K. K., Effect of Longitudinal or Inplane Deformation and Inertia on the Large Amplitude Flexural Vibrations of Slender Beams and Thin Plates, Journal of Sound and Vibration, Vol. 49, No. 3, 1976, pp. 415-422.

[35]              Mindlin, R. D., Second Gradient of Strain and Surface-Tension in Linear Elasticity, International Journal of Solids and Structures, Vol. 1, No. 4, 1965, pp. 417-438.

[36]              Fleck, N. A. and Hutchinson, J. W., A Reformulation of Strain Gradient Plasticity, Journal of the Mechanics and Physics of Solids, Vol. 49, No. 10, 2001, pp. 2245-2271.

[37]              Li, L., Hu, Y., Buckling Analysis of Size-Dependent Nonlinear Beams Based on a Nonlocal Strain Gradient Theory, International Journal of Engineering Science, Vol. 97, 2015, pp. 84-94.

[38]              Shokrieh, M. M., Zibaei, I., Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-Beams by Considering Boundary Conditions Effect, Latin American Journal of Solids and Structures, Vol. 12, No. 12, 2015, pp. 2208-2230.

[39]              Motallebi, A. A., Poorjamshidian, M., and Sheikhi, J., Vibration Analysis of a Nonlinear Beam Under Axial Force by Homotopy Analysis Method, Journal of Solid Mechanics, Vol. 6, No. 3, 2014, pp. 289-98.

[40]              Liao, S., Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, 2003.

[41]              Pirbodaghi, T., Ahmadian, M. T., and Fesanghary, M., On the Homotopy Analysis Method for Non-Linear Vibration of Beams, Mechanics Research Communications, Vol. 36, No. 2, 2009, pp. 143-148.

[42]              Thomson, W., Theory of Vibration with Applications, CRC Press, 1996.

[43]              Nayfeh, A. H., Introduction to Perturbation Techniques, John Wiley & Sons, 1981.

[44]              Mathai, A. M., Haubold, H. J., Special Functions for Applied Scientists, Vol. 4, New York, Springer, 2008.

[45]              Singh, G., Sharma, A. K., and Rao, G. V., Large-Amplitude Free Vibrations of Beams—a Discussion on Various Formulations and Assumptions, Journal of Sound and Vibration, Vol. 142, No. 1, 1990, pp. 77-85.

[46]              Askes, H., Aifantis, E. C., Gradient Elasticity and Flexural Wave Dispersion in Carbon Nanotubes, Physical Review B, Vol. 80, No. 19, 2009, pp. 195412.

[47]              Fallah, A., Aghdam, M. M., Nonlinear Free Vibration and Post-Buckling Analysis of Functionally Graded Beams on Nonlinear Elastic Foundation, European Journal of Mechanics-A/Solids, Vol. 30, No. 4, 2011, pp. 571-583.