Abstract
Investigated herein is the static bending of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. To the best of the researchers’ knowledge, in the literature, there is no study carried out into gradient elasticity theory for bending analysis of bi-directional strain-gradient Euler–Bernoulli (BDSGEB) nanostructures with arbitrary functions. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The transfer matrix for beams is given analytically. For an initial value problem, the transport matrix is unique. The diagrams of the solutions for different end conditions and various values of the parameters are given and the results are discussed.
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References
Banerjee JR, Ananthapuvirajah A (2018) Free vibration of functionally graded beams and frameworks using the dynamic stiffness method. J Sound Vib 422:34–47
Ben-Oumrane S, Abedlouahed T, Ismail M, Mohamed BB, Mustapha M, El Abbas AB (2009) A theoretical analysis of flexional bending of Al/Al\(_{2}\)O\(_{3}\) S-FGM thick beams. Comput Mater Sci 44(4):1344–1350
Bhangale RK, Ganesan N (2006) Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core. J Sound Vib 295(1):294–316
Birman V (2014) Mechanics and energy absorption of a functionally graded cylinder subjected to axial loading. Int J Eng Sci 78:18–26
Chen WR, Chang H (2017) Closed-form solutions for free vibration frequencies of functionally graded Euler–Bernoulli beams. Mech Compos Mater 53(1):79–98
Elishakoff I, Candan S (2001) Apparently first closed-form solution for vibrating: inhomogeneous beams. Int J Solids Struct 38(19):3411–3441
Exadaktylos GE, Vardoulakis I (2001) Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics. Tectonophysics 335(1):81–109
Golmakani ME, Vahabi H (2017) Nonlocal buckling analysis of functionally graded annular nanoplates in an elastic medium with various boundary conditions. Microsyst Technol 23:3613–3628
Kiani K (2017) In-plane vibration and instability of nanorotors made from functionally graded materials accounting for surface energy effect. Microsyst Technol 23(10):4853–4869
Koizumi M (1997) FGM activities in Japan. Compos Part B Eng 28(1):1–4 (Use of composites multi-phased and functionally graded materials)
Krner E (1963) On the physical reality of torque stresses in continuum mechanics. Int J Eng Sci 1(2):261–278
Lee JW, Lee JY (2017) Free vibration analysis of functionally graded Bernoulli–Euler beams using an exact transfer matrix expression. Int J Mech Sci 122:1–17
Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78
Nejad MZ, Hadi A (2016) Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams. Int J Eng Sci 106:1–9
Nejad MZ, Rastgoo A, Hadi A (2014) Exact elasto-plastic analysis of rotating disks made of functionally graded materials. Int J Eng Sci 85:47–57
Ozturk A, Gulgec M (2011) Elastic–plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation. Int J Eng Sci 49(10):1047–1061
Papargyri-Beskou S, Tsepoura KG, Polyzos D, Beskos DE (2003) Bending and stability analysis of gradient elastic beams. Int J Solids Struct 40(2):385–400
Reddy JN, El-Borgi S (2014) Eringen’s nonlocal theories of beams accounting for moderate rotations. Int J Eng Sci 82:159–177
Sankar BV (2001) An elasticity solution for functionally graded beams. Compos Sci Technol 61(5):689–696
Tang C, Alici G (2011) Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: I. Experimental determination of length-scale factors. J Phys D Appl Phys 44(33):335501
Tiersten HF, Bleustein JL (1974) Generalized elastic continua. In: Herrmann G (ed) R.D. Mindlin and applied mechanics. Pergamon, New York, pp 67–103
Uzun B, Yaylı MÖ (2019) Finite element model of functionally graded nanobeam for free vibration analysis. Int J Eng Appl Sci 11(2):387–400
Vardoulakis IG, Sulem J (1995) Bifurcation analysis in geomechanics, 1st edn. CRC Press
Wang L, Haiyan H (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71:195412
Ziegler T, Kraft T (2014) Functionally graded materials with a soft surface for improved indentation resistance: layout and corresponding design principles. Comput Mater Sci 86:88–92
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The work reported here is supported by the Alexander von Humboldt Foundation.
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Çelik, M., Artan, R. An investigation of static bending of a bi-directional strain-gradient Euler–Bernoulli nano-beams with the method of initial values. Microsyst Technol 26, 2921–2929 (2020). https://doi.org/10.1007/s00542-020-04926-2
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DOI: https://doi.org/10.1007/s00542-020-04926-2