Abstract
A generalized non-local stress–strain gradient theory is presented using fractional calculus. The proposed theory includes as a special case: the classical theory; the non-local strain gradient theory; the Eringen non-local theory; the strain gradient theory; the general Eringen non-local theory; and the general strain gradient theory. This new formulation is therefore more comprehensive and more complete to model physical phenomena. Its application has been shown in free vibration, buckling and bending of simply supported (S–S) nano-beams. The non-linear governing equations have been solved by the Galerkin method. Furthermore the effects of different (additional) model parameters like: the length scale parameter; the non-local parameter; and different orders (integer and non-integer) of strain and stress gradients have been shown.
Similar content being viewed by others
References
Agrawal, R., Peng, B., Gdoutos, E.E., Espinosa, H.D.: Elasticity size effects in ZnO nanowires—a combined experimental-computational approach. Nano Lett. 8(11), 3668–3674 (2008)
Al-Smadi, M., Freihat, A., Khalil, H., Momani, S., Ali Khan, R.: Numerical multistep approach for solving fractional partial differential equations. Int. J. Comput. Methods 14(03), 1750029 (2017)
Aydogdu, M.: A general non-local beam theory: its application to nanobeam bending, buckling and vibration. Phys. E 41(9), 1651–1655 (2009)
Bhrawy, A.H., Alofi, A.S.: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 26(1), 25–31 (2013)
Cao, G., Chen, X.: Energy analysis of size-dependent elastic properties of ZnO nanofilms using atomistic simulations. Phys. Rev. B 76(16), 165407 (2007)
Carpinteri, A., Cornetti, P., Sapora, A.: Nonlocal elasticity: an approach based on fractional calculus. Meccanica 49(11), 2551–2569 (2014)
Challamel, N., Zorica, D., Atanacković, T.M., Spasić, D.T.: On the fractional generalization of Eringenʼs non-local elasticity for wave propagation. Comptes Rendus Mécanique 341(3), 298–303 (2013)
D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math Appl. 66(7), 1245–1260 (2013)
da Graça Marcos, M., Duarte, F.B., Machado, J.T.: Fractional dynamics in the trajectory control of redundant manipulators. Commun. Nonlinear Sci. Numer. Simul. 13(9), 1836–1844 (2008)
Diao, J., Gall, K., Dunn, M.L., Zimmerman, J.A.: Atomistic simulations of the yielding of gold nanowires. Acta Mater. 54(3), 643–653 (2006)
Eringen, A.C.: Non-local polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
Eringen, A.C.: On differential equations of non-local elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)
Failla, G., Santini, A., Zingales, M.: A non-local two-dimensional foundation model. Arch. Appl. Mech. 83(2), 253–272 (2013)
Faraji Oskouie, M., Ansari, R., Rouhi, H.: Bending analysis of functionally graded nanobeams based on the fractional non-local continuum theory by the variational legendre spectral collocation method. Meccanica 53(4), 1115–1130 (2018)
Hadjesfandiari, A. R., Dargush, G. F.: Foundations of consistent couple stress theory. arXiv preprint arXiv:1509.06299 (2015)
Hilfer, R.: Applications of fractional calculus in physics. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)
Jing, G.Y., Duan, H., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., Wang, J.X., Yu, D.P.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73(23), 235409 (2006)
Khaniki, H.B., Hosseini-Hashemi, S., Nezamabadi, A.: Buckling analysis of nonuniform non-local strain gradient beams using generalized differential quadrature method. Alex. Eng. J. 57(3), 1361–1368 (2018)
Lazopoulos, K.A.: On bending of strain gradient elastic micro-plates. Mech. Res. Commun. 36(7), 777–783 (2009)
Li, L., Hu, Y.: Buckling analysis of size-dependent nonlinear beams based on a non-local strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)
Li, X., Bhushan, B., Takashima, K., Baek, C.W., Kim, Y.K.: Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97(1–4), 481–494 (2003)
Li, L., Li, X., Hu, Y.: Free vibration analysis of non-local strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)
Liebold, C., Müller, W.H.: Applications of strain gradient theories to the size effect in submicro-structures incl. experimental analysis of elastic material parameters. Bull. TICMI 19(1), 45–55 (2015)
Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order non-local elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)
Lu, L., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the non-local strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017)
Malara, G., Spanos, P.D.: Nonlinear random vibrations of plates endowed with fractional derivative elements. Probab. Eng. Mech. (2017). https://doi.org/10.1016/j.probengmech.2017.06.002
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
Olsson, P.A., Melin, S., Persson, C.: Atomistic simulations of tensile and bending properties of single-crystal bcc iron nano-beams. Phys. Rev. B 76(22), 224112 (2007)
Rahimi, Z., Rezazadeh, G., Sumelka, W., Yang, X.J.: A study of critical point instability of micro and nano beams under a distributed variable-pressure force in the framework of the inhomogeneous non-linear non-local theory. Arch. Mech. 69(6), 413–433 (2017a)
Rahimi, Z., Sumelka, W., Yang, X.J.: Linear and non-linear free vibration of nano beams based on a new fractional non-local theory. Eng. Comput. 34(5), 1754–1770 (2017b)
Rahimi, Z., Rezazadeh, G., Sadeghian, H.: Study on the size dependent effective Young modulus by EPI method based on modified couple stress theory. Microsyst. Technol. 24(7), 2983–2989 (2018)
Rahimi, Z., Sumelka, W., Shafiei, S.: The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model. Technical Sciences, Bulletin of the Polish Academy of Sciences (2018b)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer. Algorithm 74(1), 223–245 (2017)
Rashidi, H., Rahimi, Z., Sumelka, W.: Effects of the slip boundary condition on dynamics and pull-in instability of carbon nanotubes conveying fluid. Microfluid. Nanofluid 22(11), 131 (2018)
Ray, S. S., Atangana, A., Oukouomi Noutchie, S. C., Kurulay, M., Bildik, N., Kilicman, A.: Editorial: Fractional calculus and its applications in applied mathematics and other sciences. Math. Probl. Eng. (2014). https://doi.org/10.1155/2014/849395
Reddy, J.N.: Non-local theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007)
Sadeghian, H., Yang, C.K., Goosen, J.F.L., Van Der Drift, E., Bossche, A., French, P.J., Van Keulen, F.: Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability. Appl. Phys. Lett. 94(22), 221903 (2009)
Sapora, A., Cornetti, P., Chiaia, B., Lenzi, E.K., Evangelista, L.R.: Non-local diffusion in porous media: a spatial fractional approach. J. Eng. Mech. 143(5), D4016007 (2017)
Secer, A., Alkan, S., Akinlar, M.A., Bayram, M.: Sinc–Galerkin method for approximate solutions of fractional order boundary value problems. Bound. Value Probl. 2013(1), 1 (2013)
Shah, F.A., Abass, R., Debnath, L.: Numerical solution of fractional differential equations using Haar wavelet operational matrix method. Int. J. Appl. Comput. Math. 3(3), 2423–2445 (2017)
Sumelka, W., Blaszczyk, T., Liebold, C.: Fractional Euler–Bernoulli beams: theory, numerical study and experimental validation. Eur. J. Mech. A/Solids 54, 243–251 (2015)
Tarasov, V.E., Aifantis, E.C.: Toward fractional gradient elasticity. J. Mech. Behav. Mater. 23(1–2), 41–46 (2014)
Tarasov, V.E., Aifantis, E.C.: Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 197–227 (2015)
Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277(5334), 1971–1975 (1997)
Yang, X.J.: Advanced Local Fractional Calculus and its Applications. World Science Publisher, New York (2012)
Zhu, R., Pan, E., Chung, P.W., Cai, X., Liew, K.M., Buldum, A.: Atomistic calculation of elastic moduli in strained silicon. Semicond. Sci. Technol. 21(7), 906 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rahimi, Z., Rezazadeh, G. & Sumelka, W. A non-local fractional stress–strain gradient theory. Int J Mech Mater Des 16, 265–278 (2020). https://doi.org/10.1007/s10999-019-09469-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10999-019-09469-7