Nonlinear vibration analysis of fractional viscoelastic Euler–Bernoulli nanobeams based on the surface stress theory

Machado, 2003, A probabilistic interpretation of the fractional-order differentiation, Fract. Calc. Appl. Anal., 6, 73

Heymans, 2008, Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state, J. Vib. Control, 14, 1587, 10.1177/1077546307087428

De Espíndola, 2008, Design of optimum systems of viscoelastic vibration absorbers for a given material based on the fractional calculus model, J. Vib. Control, 14, 1607, 10.1177/1077546308087400

Magin, 2008, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control, 14, 1431, 10.1177/1077546307087439

Machado, 2011, Fractional dynamics in DNA, Commun. Nonlinear Sci. Numer. Simul., 16, 2963, 10.1016/j.cnsns.2010.11.007

Lazarević, 2006, Finite time stability analysis of PD α fractional control of robotic time-delay systems, Mech. Res. Commun., 33, 269, 10.1016/j.mechrescom.2005.08.010

Cervera, 2008, Automatic loop shaping in QFT using CRONE structures, J. Vib. Control, 14, 1513, 10.1177/1077546307087433

Vinagre, 2000, Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal., 3, 231

Frederico, 2008, Fractional conservation laws in optimal control theory, Nonlinear Dyn., 53, 215, 10.1007/s11071-007-9309-z

Calderón, 2006, Fractional order control strategies for power electronic buck converters, Signal Process., 86, 2803, 10.1016/j.sigpro.2006.02.022

Panda, 2006, Fractional generalized splines and signal processing, Signal Process., 86, 2340, 10.1016/j.sigpro.2005.10.017

Vinagre, 2002, Modeling and control of dynamic system using fractional calculus: application to electrochemical processes and flexible structures, 1, 214

Wang, 2010, Photonic crystal slot nanobeam slow light waveguides for refractive index sensing, Appl. Phys. Lett., 97

On, 2010, Stochastic surface effects in nanobeam sensors, Probab. Eng. Mech., 25, 228, 10.1016/j.probengmech.2009.12.001

Duan, 2013, Solution of the model of beam-type micro-and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems, Int. J. Non-Linear Mech., 49, 159, 10.1016/j.ijnonlinmec.2012.10.003

Deotare, 2013, Photonic crystal nanobeam cavities for tunable filter and router applications, IEEE J. Sel. Top. Quantum Electron., 19, 10.1109/JSTQE.2012.2225828

Fegadolli, 2015, Thermally controllable silicon photonic crystal nanobeam cavity without surface cladding for sensing applications, ACS Photonics, 2, 470, 10.1021/ph5004863

Bauer, 2011, Size-effects in TiO2 nanotubes: diameter dependent anatase/rutile stabilization, Electrochem. Commun., 13, 538, 10.1016/j.elecom.2011.03.003

Xiao, 2006, Studies of size effects on carbon nanotubes' mechanical properties by using different potential functions, Fuller. Nanotub. Carbon Nonstruct., 14, 9, 10.1080/15363830500538425

Chowdhury, 2010, A molecular mechanics approach for the vibration of single-walled carbon nanotubes, Comput. Mater. Sci., 48, 730, 10.1016/j.commatsci.2010.03.020

Sun, 2003, Size-dependent elastic moduli of platelike nanomaterials, J. Appl. Phys., 93, 1212, 10.1063/1.1530365

Zhang, 2004, Nanoplate model for platelike nanomaterials, AIAA J., 42, 2002, 10.2514/1.5282

Ansari, 2016, Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory, Compos. Part B Eng., 89, 316, 10.1016/j.compositesb.2015.12.029

Ansari, 2016, Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory, Phys. E Low-dimens. Syst. Nanostruct., 75, 266, 10.1016/j.physe.2015.09.022

Zhang, 2015, Free vibration analysis of four-unknown shear deformable functionally graded cylindrical microshells based on the strain gradient elasticity theory, Compos. Struct., 119, 578, 10.1016/j.compstruct.2014.09.032

Tahani, 2015, Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory, Int. J. Mech. Sci., 94, 185, 10.1016/j.ijmecsci.2015.03.004

Mohammadi, 2015, An analytical solution for buckling analysis of size-dependent rectangular micro-plates according to the modified strain gradient and couple stress theories, Acta Mech., 226, 3477, 10.1007/s00707-015-1384-5

Ansari, 2015, An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory, Thin-Walled Struct., 93, 169, 10.1016/j.tws.2015.03.013

Gurtin, 1975, A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291, 10.1007/BF00261375

Gurtin, 1978, Surface stress in solids, Int. J. Solids Struct., 14, 431, 10.1016/0020-7683(78)90008-2

Wang, 2010, The effects of surface tension on the elastic properties of nano structures, Int. J. Eng. Sci., 48, 140, 10.1016/j.ijengsci.2009.07.007

Ren, 2004, Influence of surface stress on frequency of microcantilever-based biosensors, Microsyst. Technol., 10, 307, 10.1007/s00542-003-0329-4

He, 2004, A continuum model for size-dependent deformation of elastic films of nano-scale thickness, Int. J. Solids Struct., 41, 847, 10.1016/j.ijsolstr.2003.10.001

Wang, 2007, Effects of surface elasticity and residual surface tension on the natural frequency of microbeams, Appl. Phys. Lett., 90, 23, 10.1063/1.2746950

Ansari, 2011, Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories, Int. J. Eng. Sci., 49, 1244, 10.1016/j.ijengsci.2011.01.007

Wang, 2009, Self-instability and bending behaviors of nano plates, Acta Mech. Solida Sin., 22, 630, 10.1016/S0894-9166(09)60393-1

Sahmani, 2015, On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects, Compos. Struct., 121, 377, 10.1016/j.compstruct.2014.11.033

Pourkiaee, 2015, Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions, J. Vib. Control

Ansari, 2014, Surface stress effect on the postbuckling and free vibrations of axisymmetric circular Mindlin nanoplates subject to various edge supports, Compos. Struct., 112, 358, 10.1016/j.compstruct.2014.02.028

Gheshlaghi, 2011, Surface effects on nonlinear free vibration of nanobeams, Compos. Part B Eng., 42, 934, 10.1016/j.compositesb.2010.12.026

Rao, 2007

Ke, 2010, Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Compos. Struct., 92, 676, 10.1016/j.compstruct.2009.09.024

Schiessel, 1995, Generalized viscoelastic models: their fractional equations and solutions, J. Phys. A Math. Gen., 28, 6567, 10.1088/0305-4470/28/23/012

Caji´c, 2015, Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle, Theor. Appl. Mech., 42, 167, 10.2298/TAM1503167C

Rossikhin, 2010, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids, Appl. Mech. Rev., 63, 10.1115/1.4000246

Grzesikiewicz, 2013, Non-linear problems of fractional calculus in modeling of mechanical systems, Int. J. Mech. Sci., 70, 90, 10.1016/j.ijmecsci.2013.02.007

Ansari, 2015, Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticitytheory, Phys. E Low-dimens. Syst. Nanostruct., 74, 318, 10.1016/j.physe.2015.07.013

Ansari, 2014, On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory, Compos. Part B Eng., 60, 158, 10.1016/j.compositesb.2013.12.066

Lu, 2006, Thin plate theory including surface effects, Int. J. Solids Struct., 43, 4631, 10.1016/j.ijsolstr.2005.07.036

C.F. Lu, C.W. Lim, W.Q. Chen, Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. 46 (2009) 1176–1185.

Sun, 2010, Bifurcations and chaos in fractional-order simplified Lorenz system, Int. J. Bifurc. Chaos, 20, 1209, 10.1142/S0218127410026411

Diethelm, 1997, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1

Diethelm, 2002, Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229, 10.1006/jmaa.2000.7194

Diethelm, 2002, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3, 10.1023/A:1016592219341