A new shear deformation micro-beam model based on a nonlocal elasticity theory
DOI:
https://doi.org/10.38152/bjtv7n4-049Abstract
In this article, a nonlocal refined shear deformation beam theory is proposed to analyze the free vibration behavior of FG nanobeams. The formulation is based on the nonlocal differential constitutive equations of Eringen that accounts for small scale effects. Material properties vary continuously through thickness according to a power law distribution. The study is performed within the framework of a new parametric shear deformation theory which provides parabolic transverse shear strains across the thickness direction and hence, does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the nanobeam are satisfied rigorously. Equations of motion of FG nanobeam are derived from both the nonlocal differential constitutive relations of Eringen and Hamilton’s principle. The system of differential equations is solved using the Navier solution technique. The parametric study is conducted to examine the effects of volume fraction index, length scale parameter and length ratio on the free frequencies of vibration of FG thin and thick nanobeams.
References
Yamanouchi M, Koizumi M, Shiota I. Proceedings of the First International Symposium on Functionally Gradient Materials Japan1990.
Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally graded materials: Design, processing and applications. United States: Kluwer Academic Publishers, Hingham, MA (US), 1999.
Shen H-S, Wang Z-X. Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates. Composite Structures. 2012;94:2197-208.
Palani V, Swain A. Nonlinear vibration analysis of composite and functionally graded material shell structures: A literature review from 2013 to 2023. International Journal of Non-Linear Mechanics. 2025;168:104939.
Yadav S, Liu S, Singh RK, Sharma AK, Rawat P. A state-of-art review on functionally graded materials (FGMs) manufactured by 3D printing techniques: Advantages, existing challenges, and future scope. Journal of Manufacturing Processes. 2024;131:2051-72.
Li Y, Feng Z, Hao L, Huang L, Xin C, Wang Y, et al. A Review on Functionally Graded Materials and Structures via Additive Manufacturing: From Multi-Scale Design to Versatile Functional Properties. Advanced Materials Technologies. 2020;5:1900981.
Reddy JN. A Simple Higher-Order Theory for Laminated Composite Plates. ASME Journal of Applied Mechanics. 1984;51:745–52.
Mahi A, Adda Bedia EA, Tounsi A. A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling. 2015;39:2489-508.
Mohammadi H. An isogeometric formulation for free vibration and buckling analyses of FG graphene platelets-reinforced porous beams based on a two-variable shear deformation theory. International Journal of Computational Materials Science and Engineering. 2024:2450012.
Liu X, Pagani A, Carrera E, Liu X. Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels. Thin-Walled Structures. 2024;198:111707.
Bouakkaz K, Djedid IK, Draiche K, Tounsi A, Hussain M. Mechanical behaviour of advanced composite beams via a simple quasi-3D integral higher-order beam theory. Advances in Materials Research. 2024;13:335-53.
Mellal F, Bennai R, Avcar M, Nebab M, Atmane HA. On the vibration and buckling behaviors of porous FG beams resting on variable elastic foundation utilizing higher-order shear deformation theory. Acta Mechanica. 2023;234:3955-77.
Zozulya VV, Saez A. A high-order theory of a thermoelastic beams and its application to the MEMS/NEMS analysis and simulations. Archive of Applied Mechanics. 2016;86:1255-72.
Batra RC, Porfiri M, Spinello D. Review of modeling electrostatically actuated microelectromechanical systems. Smart Materials and Structures. 2007;16:R23-R31.
Sharma B, Kumar M, Sharma A. Recent advances in micro- and bio- electromechanical system architectures for energy efficient chemiresistors. Microelectronic Engineering. 2024;288:112168.
Kurmendra, Agarwal S. MEMS Switch Realities: Addressing Challenges and Pioneering Solutions. Micromachines. 2024;15:556.
Kumar A, Varghese A, Kalra D, Raunak A, Jaiverdhan, Prasad M, et al. MEMS-based piezoresistive and capacitive microphones: A review on materials and methods. Materials Science in Semiconductor Processing. 2024;169:107879.
Lam DCC, Yang F, Chong ACM, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids. 2003;51:1477-508.
Chong ACM, Yang F, Lam DCC, Tong P. Torsion and bending of micron-scaled structures. Journal of Materials Research. 2001;16:1052-8.
Mindlin RD, Tiersten HF. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis. 1962;11:415-48.
Koiter WT. Couple stresses in the theory of elasticity, I and II. Proceedings , Koninklijke Nederlandse Akademie van Wetenschappen, Series B1964. p. 17–44.
Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics. 1983;54:4703.
Eringen AC. Nonlocal polar elastic continua. International Journal of Engineering Science. 1972;10:1-16.
Fleck NA, Hutchinson JW. A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids. 2001;49:2245-71.
Ramezani S. A shear deformation micro-plate model based on the most general form of strain gradient elasticity. International Journal of Mechanical Sciences. 2012;57:34-42.
Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures. 2002;39:2731–43.
Gurtin M, Ian Murdoch A. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis. 1975;57:291-323.
Gurtin ME, Ian Murdoch A. Surface stress in solids. International Journal of Solids and Structures. 1978;14:431-40.
Mawphlang BRKLL, Patra PK. Study of the large bending behavior of CNTs using LDTM and nonlocal elasticity theory. International Journal of Non-Linear Mechanics. 2024;166:104828.
Feng J, Yu H, Ma S, Hao S, Wu R. Axial vibration characteristics of carbon nanotube-based mass sensors containing nanoparticles using nonlocal elasticity theory. Physica B: Condensed Matter. 2022;634:413804.
Darban H, Luciano R, Caporale A, Basista M. Modeling of buckling of nanobeams embedded in elastic medium by local-nonlocal stress-driven gradient elasticity theory. Composite Structures. 2022;297:115907.
Soltani M, Atoufi F, Mohri F, Dimitri R, Tornabene F. Nonlocal elasticity theory for lateral stability analysis of tapered thin-walled nanobeams with axially varying materials. Thin-Walled Structures. 2021;159:107268.
Sayyad AS, Ghugal YM. Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen's Nonlocal Theory. International Journal of Applied Mechanics. 2020;12.
Reddy JN. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science. 2007;45:288-307.
Ke L-L, Wang Y-S, Wang Z-D. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Composite Structures. 2012;94:2038-47.
Thai H-T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science. 2012;52:56-64.
Nazemnezhad R, Hosseini-Hashemi S. Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures. 2014;110:192-9.
Rahmani O, Pedram O. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science. 2014;77:55-70.
Eltaher MA, Emam SA, Mahmoud FF. Free vibration analysis of functionally graded size-dependent nanobeams. Applied Mathematics and Computation. 2012;218:7406-20.
