Abstract
Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.
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References
Asghari M, Taati E (2013) A size-dependent model for functionally graded micro-plates for mechanical analyses. J Vib Control 19:1614–1632
Cardoso JB, Sousa LG, Castro JA, Valido AJ (2002) Optimal design of laminated composite beam structures. Struct Multidiscip Optim 24(3):205–211
Corne DW, Knowles JD, Oates MJ (2000, September) The Pareto envelope-based selection algorithm for multiobjective optimization. In: International conference on parallel problem solving from nature pp 839–848
Deb K, Pratap A, Agarwal S, Meyarivan TAMT (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197
Dems K (1980) Multiparameter shape optimization of elastic bars in torsion. Int J Numer Methods Eng 15(10):1517–1539
Dems K, Plaut RH (1990) Design of beams, plates and their elastic foundations for uniform foundation pressure. Struct Optim 2(4):213–222
Eringen AC (1968) Theory of micropolar elasticity. In: Leibowitz H (ed) Fracture vol. 2. Academic Press, New York, pp 621–629
Farokhi H, Païdoussis MP, Misra AK (2016) A new nonlinear model for analyzing the behaviour of carbon nanotube-based resonators. J Sound Vib 378:56–75
Ghayesh MH, Amabili M, Farokhi H (2013) Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int J Eng Sci 71:1–14
Ghayesh MH, Farokhi H, Gholipour A (2017) Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams. Int J Mech Sci 122:370–383
Gholipour A, Farokhi H, Ghayesh MH (2015) In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dyn 79(3):1771–1785
Kocatürk T, Şimşek M, Akbaş ŞD (2011) Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material. Sci Eng Compos Mater 18(1–2):21–34
Kruusing A (2000) Analysis and optimization of loaded cantilever beam microactuators. Smart Mater Struct 9(2):186–196
Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508
Lee J, Haftka RT, Griffin OH Jr, Watson LT, Sensmeier MD (1994) Detecting delaminations in a composite beam using anti-optimization. Structural and Multidisciplinary Optimization 8(2–3):93–100
Liu S, An X, Jia H (2008) Topology optimization of beam cross-section considering warping deformation. Struct Multidiscip Optim 35(5):403–411
McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060–1067
Molaei M, Ahmadian MT, Taati E (2014a) Effect of thermal wave propagation on thermoelastic behavior of functionally graded materials in a slab symmetrically surface heated using analytical modeling. Compos Part B 60:413–422
Molaei M, Taati E, Basirat H (2014b) Optimization of functionally graded materials in the slab symmetrically surface heated using transient analytical solution. J Therm Stresses 37:137–159
Murata T, Ishibuchi H, Tanaka H (1996) Multi-objective genetic algorithm and its applications to flowshop scheduling. Comput Ind Eng 30(4):957–968
Pelesko JA, Bernstein DH (2003) Modeling MEMS and NEMS, Chapter 4 & 7. Chapman & Hall/CRC, Boca Raton
Polajnar M, Drazumeric R, Kosel F (2013) Geometry optimization of a slender cantilever beam subjected to lateral buckling. Struct Multidiscip Optim 47(6):809–819
rey Horn J, Nafpliotis N, Goldberg DE (1993) Multiobjective optimization using the niched pareto genetic algorithm. Illigal Rep 93005:61801–62296
Shi JX, Shimoda M (2015) Interface shape optimization of designing functionally graded sandwich structures. Compos Struct 125:88–95
Şimşek M (2015) Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method. Compos Struct 131:207–214
Şimşek M, Aydın M (2017) Size-dependent forced vibration of an imperfect functionally graded (FG) microplate with porosities subjected to a moving load using the modified couple stress theory. Compos Struct 160:408–421
Sivanagendra P, Ananthasuresh GK (2009) Size optimization of a cantilever beam under deformation-dependent loads with application to wheat stalks. Struct Multidiscip Optim 39(3):327–336
Sorokin SV, Nielsen JB, Olhoff N (2001a) Analysis and optimization of energy flows in structures composed of beam elements–Part I: problem formulation and solution technique. Struct Multidiscip Optim 22(1):3–11
Sorokin SV, Nielsen JB, Olhoff N (2001b) Analysis and optimization of energy flows in structures composed of beam elements–Part II: examples and discussion. Struct Multidiscip Optim 22(1):12–23
Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248
Taati E (2016) Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates. Int J Eng Sci 100:45–60
Taati E, Nikfar M, Ahmadian MT (2012) Formulation for static behavior of the viscoelastic Euler-Bernoulli micro-beam based on the modified couple stress theory. Proceedings of the ASME international mechanical engineering congress & exposition November 9–15 Houston
Taati E, Molaei M, Basirat H (2014a) Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mech 25:1823–1842
Taati E, Molaei M, Reddy JN (2014b) Size-dependent generalized thermoelasticity model for Timoshenko micro-beams based on strain gradient and non-Fourier heat conduction theories. Compos Struct 116:595–611
Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414
Witvrouw A, Mehta A (2005) The use of functionally graded poly-SiGe layers for MEMS applications. Mater Sci Forum 492:255–260
Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271
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Taati, E., Sina, N. Multi-objective optimization of functionally graded materials, thickness and aspect ratio in micro-beams embedded in an elastic medium. Struct Multidisc Optim 58, 265–285 (2018). https://doi.org/10.1007/s00158-017-1895-x
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DOI: https://doi.org/10.1007/s00158-017-1895-x