Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and provides a new approach to non-local mechanics. In this study, a theoretical consideration on a new fractional non-local model is presented based on existence of fractional strain energy. It has two additional free parameters compared to classical local mechanics: (1) a fractional parameter which controls the strain gradient order in the strain energy relation and makes the model more flexible to describe physical phenomena, and (2) a non-local parameter to consider small scale effects in micron and sub-micron scales. The model has been used to obtain a fractional non-local plate theory. Free vibrations of a rectangular simply supported (S–S–S–S) plate has been investigated and the meaning of different parameters, such as fractional and non-local parameters, has been shown. The non-linear governing equation has been solved by the Galerkin method. A simple form of the governing equation and the numerical solution is an advantage of this fractional non-local model. Furthermore, the fractional nonlocal theory is contrasted with the Eringen nonlocal theory to show that fractional one enables to obtain much wider class of solutions.
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Abdeljawad T (2015) On conformable fractional calculus. J Comput Appl Math 279:57–66
Ahmad WM, El-Khazali R (2007) Fractional-order dynamical models of love. Chaos Solitons Fractals 33(4):1367–1375
Atanackovic TM, Stankovic B (2009) Generalized wave equation in nonlocal elasticity. Acta Mech 208(1):1–10
Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210
Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14(9–10):1487–1498
Cajić MS, Lazarević MP, Šekara TB (2014) Robotic system with viscoelastic element modeled via fractional Zener model. In: Fractional differentiation and its applications (ICFDA), 2014 international conference on. IEEE, pp 1–6
Carpinteri A, Cornetti P, Sapora A, Di Paola M, Zingales M (2009) Fractional calculus in solid mechanics: local versus non-local approach. Phys Scr 2009(T136):014003
Carpinteri A, Cornetti P, Sapora A (2011) A fractional calculus approach to nonlocal elasticity. Eur Phys J Spec Topics 193(1):193–204
Challamel N, Zorica D, Atanacković TM, Spasić DT (2013) On the fractional generalization of Eringen’s nonlocal elasticity for wave propagation. Comptes Rendus Mécanique 341(3):298–303
Chong CM (2002) Experimental investigation and modeling of size effect in elasticity (Doctoral dissertation)
Cuenot S, Frétigny C, Demoustier-Champagne S, Nysten B (2004) Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys Rev B 69(16):165410
Davis GB, Kohandel M, Sivaloganathan S, Tenti G (2006) The constitutive properties of the brain paraenchyma: part 2. Fractional derivative approach. Med Eng Phys 28(5):455–459
De Espındola JJ, da Silva Neto JM, Lopes EM (2005) A generalised fractional derivative approach to viscoelastic material properties measurement. Appl Math Comput 164(2):493–506
Di Paola M, Zingales M (2008) Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int J Solids Struct 45(21):5642–5659
Eringen AC (1966) Linear theory of micropolar elasticity. J Math Mech 15(6):909–923
Eringen AC (1999) Theory of micropolar elasticity. In: Microcontinuum field theories. Springer, New York, pp 101–248
Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248
Eringen AC, Wegner JL (2003) Nonlocal continuum field theories. Appl Mech Rev 56:B20
Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2):475–487
Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323
Katugampola UN (2014) A new fractional derivative with classical properties. arXiv preprint arXiv:1410.6535
Khalil R, Al Horani M, Yousef A, Sababheh M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70
Kilbas A, Srivastava HM, Trujillo JJ (2006) New book: “theory and applications of fractional differential equations”, Elsevier, North-Holland Mathematics Studies, 204. Fract Calculus Appl Anal 9(1):71
Kröner E (1967) Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 3(5):731–742
Lazopoulos KA (2006) Non-local continuum mechanics and fractional calculus. Mech Res Commun 33(6):753–757
Lima MF, Machado JAT, Crisóstomo MM (2007) Experimental signal analysis of robot impacts in a fractional calculus perspective. JACIII 11(9):1079–1085
Lu P, Zhang PQ, Lee HP, Wang CM, Reddy JN (2007) Non-local elastic plate theories. Proc R Soc Lond A Math Phys Eng Sci 463(2088):3225–3240 (The Royal Society)
Ma Q, Clarke DR (1995) Size dependent hardness of silver single crystals. J Mater Res 10(4):853–863
McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15(5):1060
Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124
Ortigueira MD, Machado JAT (2015) What is a fractional derivative? J Comput Phys 293:4–13
Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, vol 198. Academic Press, USA
Rahimi Z, Sumelka W, Yang XJ (2017a) Linear and non-linear free vibration of nano beams based on a new fractional non-local theory. Eng Comput 34(5):1754–1770
Rahimi Z, Sumelka W, Yang XJ (2017b) A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler–Bernoulli beams. Eur Phys J Plus 132(11):479
Rao SS (2007) Vibration of continuous systems. Wiley, New York
Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336
Salvetat JP, Briggs GAD, Bonard JM, Bacsa RR, Kulik AJ, Stöckli T, Burnham NA, Forró L (1999) Elastic and shear moduli of single-walled carbon nanotube ropes. Phys Rev Lett 82(5):944
Sapora A, Cornetti P, Carpinteri A (2013) Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach. Commun Nonlinear Sci Numer Simul 18(1):63–74
Smyshlyaev VP, Fleck NA (1996) The role of strain gradients in the grain size effect for polycrystals. J Mech Phys Solids 44(4):465–495
Stan G, Ciobanu CV, Parthangal PM, Cook RF (2007) Diameter-dependent radial and tangential elastic moduli of ZnO nanowires. Nano Lett 7(12):3691–3697
Sumelka W (2014) Fractional viscoplasticity. Mech Res Commun 56:31–36
Sumelka W, Zaera R, Fernández-Sáez J (2015) A theoretical analysis of the free axial vibration of non-local rods with fractional continuum mechanics. Meccanica 50(9):2309–2323
Torvik PJ, Bagley RL (1984) On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech 51(2):294–298
Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414
This work is supported by the National Science Centre, Poland under Grant no. 2017/27/B/ST8/00351.
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The original version of this article was revised: one of the co-author’s (Ghader Rezazadeh) family name has been corrected.
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Rahimi, Z., Shafiei, S., Sumelka, W. et al. Fractional strain energy and its application to the free vibration analysis of a plate. Microsyst Technol 25, 2229–2238 (2019). https://doi.org/10.1007/s00542-018-4087-8