Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory

Iijima, 1991, Helical microtubules of graphitic carbon, Nature, 354, 56, 10.1038/354056a0

Iijima, 1993, Single-shell carbon nanotubes of 1-nm diameter, Nature, 363, 603, 10.1038/363603a0

Ru, 2000, Effective bending stiffness of carbon nanotubes, Phys. Rev. B, 62, 9973, 10.1103/PhysRevB.62.9973

Thostenson, 2001, Advances in the science and technology of carbon nanotubes and their composites: a review, Compos. Sci. Technol., 61, 1899, 10.1016/S0266-3538(01)00094-X

Zhang, 2006, Temperature-dependent elastic properties of single-walled carbon nanotubes: prediction from molecular dynamics simulation, Appl. Phys. Lett., 89, 081904, 10.1063/1.2336622

Zhang, 2008, Predicting the elastic properties of double-walled carbon nanotubes by molecular dynamics simulation, J. Phys. D: Appl. Phys., 41, 055404, 10.1088/0022-3727/41/5/055404

Poole, 1996, Micro-hardness of annealed and work- hardened copper polycrystals, Scripta Mater., 34, 559, 10.1016/1359-6462(95)00524-2

Lam, 2003, Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, 51, 1477, 10.1016/S0022-5096(03)00053-X

McFarland, 2005, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, J. Micromech. Microeng., 15, 1060, 10.1088/0960-1317/15/5/024

Mindlin, 1962, Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 415, 10.1007/BF00253946

Koiter, 1964, Couple stresses in the theory of elasticity: I and II, Proc. K. Ned. Akad. Wet. ((B)B, 67, 17

Toupin, 1964, Theory of elasticity with couple stresses, Arch. Ration. Mech. Anal., 17, 85, 10.1007/BF00253050

Eringen, 1967, Theory of micropolar plates, Z. Angew. Math. Phys., 18, 12, 10.1007/BF01593891

Eringen, 1972, Nonlocal polar elastic continua, Int. J. Eng. Sci., 10, 1, 10.1016/0020-7225(72)90070-5

Eringen, 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 4703, 10.1063/1.332803

Vardoulakis, 1995

Aifantis, 1999, Gradient deformation models at nano, micro and macro scales, J. Eng. Mater. Technol., 121, 189, 10.1115/1.2812366

Fleck, 1993, A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids, 41, 1825, 10.1016/0022-5096(93)90072-N

Fleck, 2001, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49, 2245, 10.1016/S0022-5096(01)00049-7

Peddieson, 2003, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci., 41, 305, 10.1016/S0020-7225(02)00210-0

Wang, 2007, Application of nonlocal continuum mechanics to static analysis of micro and nano-structures, Phys. Lett. A, 363, 236, 10.1016/j.physleta.2006.10.093

Reddy, 2007, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 288, 10.1016/j.ijengsci.2007.04.004

Thai, 2012, A nonlocal beam theory for bending, buckling and vibration of nanobeams, Int. J. Eng. Sci., 52, 56, 10.1016/j.ijengsci.2011.11.011

Thai, 2012, A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams, Int. J. Eng. Sci., 54, 58, 10.1016/j.ijengsci.2012.01.009

Aydogdu, 2009, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E, 41, 1651, 10.1016/j.physe.2009.05.014

Kong, 2009, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, Int. J. Eng. Sci., 47, 487, 10.1016/j.ijengsci.2008.08.008

Wang, 2010, A micro scale Timoshenko beam model based on strain gradient elasticity theory, Eur. J. Mech. A/Solids, 29, 591, 10.1016/j.euromechsol.2009.12.005

Akgöz, 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, Int. J. Eng. Sci., 49, 1268, 10.1016/j.ijengsci.2010.12.009

Akgöz, 2012, Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Arch. Appl. Mech., 82, 423, 10.1007/s00419-011-0565-5

Kahrobaiyan, 2012, A strain gradient functionally graded Euler–Bernoulli beam formulation, Int. J. Eng. Sci., 52, 65, 10.1016/j.ijengsci.2011.11.010

Akgöz, 2013, Buckling analysis of functionally graded microbeams based on the strain gradient theory, Acta Mech., 224, 2185, 10.1007/s00707-013-0883-5

Ansari, 2011, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Compos. Struct., 94, 221, 10.1016/j.compstruct.2011.06.024

Yang, 2002, Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct., 39, 2731, 10.1016/S0020-7683(02)00152-X

Park, 2006, Bernoulli–Euler beam model based on a modified couple stress theory, J. Micromech. Microeng., 16, 2355, 10.1088/0960-1317/16/11/015

Kong, 2008, The size-dependent natural frequency of Bernoulli–Euler micro-beams, Int. J. Eng. Sci., 46, 427, 10.1016/j.ijengsci.2007.10.002

Ma, 2008, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, J. Mech. Phys. Solids, 56, 3379, 10.1016/j.jmps.2008.09.007

Şimşek, 2010, Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, Int. J. Eng. Sci., 48, 1721, 10.1016/j.ijengsci.2010.09.027

Reddy, 2011, Microstructure-dependent couples stress theories of functionally graded beams, J. Mech. Phys. Solids, 59, 2382, 10.1016/j.jmps.2011.06.008

Şimşek, 2013, Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory, Compos. Struct., 95, 740, 10.1016/j.compstruct.2012.08.036

Akgöz, 2013, Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM), Compos. Part B Eng., 55, 263, 10.1016/j.compositesb.2013.06.035

Akgöz, 2013, Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory, Compos. Struct., 98, 314, 10.1016/j.compstruct.2012.11.020

Levinson, 1981, A new rectangular beam theory, J. Sound Vib., 74, 81, 10.1016/0022-460X(81)90493-4

Reddy, 1984, A simple higher-order theory for laminated composite plates, J. Appl. Mech., 51, 745, 10.1115/1.3167719

Touratier, 1991, An efficient standard plate theory, Int. J. Eng. Sci., 29, 901, 10.1016/0020-7225(91)90165-Y

Soldatos, 1992, A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mech., 94, 195, 10.1007/BF01176650

Karama, 2003, Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, Int. J. Solids Struct., 40, 1525, 10.1016/S0020-7683(02)00647-9

Aydogdu, 2009, A new shear deformation theory for laminated composite plates, Compos. Struct., 89, 94, 10.1016/j.compstruct.2008.07.008

Salamat-talab, 2012, Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory, Int. J. Mech. Sci., 57, 63, 10.1016/j.ijmecsci.2012.02.004

Nateghi, 2012, Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory, Appl. Math. Modell., 36, 4971, 10.1016/j.apm.2011.12.035

Şimşek, 2013, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, Int. J. Eng. Sci., 64, 37, 10.1016/j.ijengsci.2012.12.002

Şimşek, 2013, A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory, Compos. Struct., 101, 47, 10.1016/j.compstruct.2013.01.017

Akgöz, 2013, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, Int. J. Eng. Sci., 70, 1, 10.1016/j.ijengsci.2013.04.004

Akgöz, 2014, A new trigonometric beam model for buckling of strain gradient microbeams, Int. J. Mech. Sci., 81, 88, 10.1016/j.ijmecsci.2014.02.013

Akgöz, 2014, Shear deformation beam models for functionally graded microbeams with new shear correction factors, Compos. Struct., 112, 214, 10.1016/j.compstruct.2014.02.022

Akgöz, 2014, Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium, Int. J. Eng. Sci., 85, 90, 10.1016/j.ijengsci.2014.08.011

Lei, 2013, Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory, Int. J. Eng. Sci., 72, 36, 10.1016/j.ijengsci.2013.06.012

Sudak, 2003, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys., 94, 7281, 10.1063/1.1625437

Zhang, 2006, Investigation of buckling of double-walled carbon nanotubes embedded in an elastic medium using the energy method, Int. J. Mech. Sci., 48, 53, 10.1016/j.ijmecsci.2005.09.010

Murmu, 2009, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E, 41, 1232, 10.1016/j.physe.2009.02.004

Murmu, 2009, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Comput. Mater. Sci., 46, 854, 10.1016/j.commatsci.2009.04.019

Lim, 2010, New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes, J. Comput. Theor. Nanosci., 7, 988, 10.1166/jctn.2010.1443

Zhang, 2007, Buckling and postbuckling of single-walled carbon nanotubes under combined axial compression and torsion in thermal environments, Phys. Rev. B, 75, 045408, 10.1103/PhysRevB.75.045408

Zhang, 2007, Thermal buckling of initially compressed single-walled carbon nanotubes by molecular dynamics simulation, Carbon, 45, 2614, 10.1016/j.carbon.2007.08.007

Yuan, 2009, Effects of vacancy defect reconstruction on the elastic properties of carbon nanotubes, Carbon, 47, 1526, 10.1016/j.carbon.2009.01.048

Lim, 2007, Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams, J. Appl. Phys., 101, 054312, 10.1063/1.2435878

Li, 2004, Vibrational behaviors of multi-walled carbon nanotube- based nanomechanical resonators, Appl. Phys. Lett., 84, 121, 10.1063/1.1638623

Li, 2008, Structures of water molecular nanotube induced by axial tensile strains, Phys. Lett. A, 372, 6288, 10.1016/j.physleta.2008.08.038

Liew, 2008, Flexural wave propagation in single-walled carbon nanotubes, J. Comput. Theor. Nanosci., 5, 581, 10.1166/jctn.2008.019

Sun, 2008, The buckling of single-walled carbon nanotubes upon bending: The higher order gradient continuum and mesh-free method, Comput, Meth. Appl. Mech. Eng, 197, 3001, 10.1016/j.cma.2008.02.003

Lim, 2010, On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection, Appl. Math. Mech, 31, 37, 10.1007/s10483-010-0105-7

Lim, 2010, Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales, J. Comput. Theor. Nanosci., 7, 2104, 10.1166/jctn.2010.1591

Sedighi, 2013, Vibrations of microbeams actuated by an electric field via parameter expansion method, Acta Astronaut., 85, 19, 10.1016/j.actaastro.2012.11.014

Sedighi, 2014, Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronaut., 95, 111, 10.1016/j.actaastro.2013.10.020

Ebrahimi, 2015, Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment, Acta Astronaut., 113, 29, 10.1016/j.actaastro.2015.03.031

Reddy, 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, J. Appl. Phys., 103, 023511, 10.1063/1.2833431

Arda, 2014, Torsional statics and dynamics of nanotubes embedded in an elastic medium, Compos. Struct., 114, 80, 10.1016/j.compstruct.2014.03.053

Ansari, 2012, Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Compos. Part B: Eng., 43, 2985, 10.1016/j.compositesb.2012.05.049

Brischetto, 2014, A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes, Compos. Part B: Eng., 61, 222, 10.1016/j.compositesb.2014.01.046

Lu, 2012, Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling, Compos. Part B: Eng., 43, 1902, 10.1016/j.compositesb.2012.02.002

Lei, 2012, Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model, Compos. Part B: Eng., 43, 64, 10.1016/j.compositesb.2011.04.032

Kiani, 2015, Vibration analysis of two orthogonal slender single-walled carbon nanotubes with a new insight into continuum-based modeling of van der Waals forces, Part B: Eng., 73, 72, 10.1016/j.compositesb.2014.12.025