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Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory

Year 2020, Volume: 12 Issue: 4, 163 - 172, 29.12.2020

Mustafa Arda

https://doi.org/10.24107/ijeas.842499
Cited By: 2

Abstract

Buckling of axially loaded cantilever nanobeams with intermediate support have been studied in the current study. Higher order size dependent strain gradient theory has been utilized to capture the scale effect in nano dimension. Minimum total potential energy formulation has been used in modeling of nanobeam. Approximate Ritz method has been applied to the energy formulation for obtaining critical buckling loads. Position of the intermediate support has been varied and its effect on the critical buckling load has been investigated in the analysis. Mode shapes in critical buckling loads have been shown for various intermediate support positions. Present results could be useful in design of carbon nanotube resonators.

Keywords

Nanobeam Strain Gradient Intermediate Support Ritz Method

References

  • Ajiki H., Ando T., Energy Bands of Carbon Nanotubes in Magnetic Fields, Journal of the Physical Society of Japan, 65, 505–14, 1996. doi:10.1143/JPSJ.65.505
  • Craighead H.G., Nanoelectromechanical Systems, Science, 290, 1532–5, 2000. doi:10.1126/science.290.5496.1532
  • Huang X.M.H., Zorman C.A., Mehregany M., Roukes M.L., Nanoelectromechanical systems: Nanodevice motion at microwave frequencies, Nature, 421, 496–496, 2003. doi:10.1038/421496a
  • Ghorbanpour Arani A., Shokravi M., Vibration response of visco-elastically coupled double-layered visco-elastic graphene sheet systems subjected to magnetic field via strain gradient theory considering surface stress effects, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 229, 180–90, 2015. doi:10.1177/1740349914529102
  • Arda M., Aydogdu M., Torsional statics and dynamics of nanotubes embedded in an elastic medium, Composite Structures, 114, 80–91, 2014. doi:10.1016/j.compstruct.2014.03.053
  • Li C., Li S., Zhu Z., Prediction of mechanical properties of microstructures through a nonlocal stress field theory, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 229, 50–4, 2015. doi:10.1177/1740349913519437
  • Kumar M., Reddy G.J., Kumar N.N., Bég O.A., Computational study of unsteady couple stress magnetic nanofluid flow from a stretching sheet with Ohmic dissipation, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 233, 49–63, 2019. doi:10.1177/2397791419843730
  • Gul U., Aydogdu M., Gaygusuzoglu G., Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics, Composite Structures, 160, 1268–78, 2017. doi:10.1016/j.compstruct.2016.11.023
  • Oterkus E., Diyaroglu C., Zhu N., Oterkus S., Madenci E., Utilization of Peridynamic Theory for Modeling at the Nano-Scale, 2015, p. 1–16. doi:10.1007/978-3-319-21194-7_1
  • Cauchy A.-L., Mémoire sur les systèmes isotropes de points matériels. Oeuvres complètes, Cambridge University Press, 1882. doi:10.1017/CBO9780511702280.023
  • Voigt W., Theoretische Studien über die Elasticitätsverhältnisse der Krystalle, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 34, 3–52, 1887
  • Cosserat E., Cosserat F., Theorie des corps dédormables. A. Hermann et fils, 1909
  • Kunin I.A., Elastic Media with Microstructure I. vol. 26. Springer Berlin Heidelberg, 1982. doi:10.1007/978-3-642-81748-9
  • Toupin R.A., Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17, 85–112, 1964. doi:10.1007/BF00253050
  • Mindlin R.D., Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 51–78, 1964. doi:10.1007/BF00248490
  • Kröner E., Elasticity theory of materials with long range cohesive forces, International Journal of Solids and Structures, 3, 731–42, 1967. doi:10.1016/0020-7683(67)90049-2
  • Green A.E., Rivlin R.S., Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17, 113–47, 1964. doi:10.1007/BF00253051
  • Eringen A.C., Edelen D.G.B., On nonlocal elasticity, International Journal of Engineering Science, 10, 233–48, 1972. doi:10.1016/0020-7225(72)90039-0
  • Ru C.Q., Aifantis E.C., A simple approach to solve boundary-value problems in gradient elasticity, Acta Mechanica, 101, 59–68, 1993. doi:10.1007/BF01175597
  • Altan B.S., Aifantis E.C., On Some Aspects in the Special Theory of Gradient Elasticity, Journal of the Mechanical Behavior of Materials, 8, 1997. doi:10.1515/JMBM.1997.8.3.231
  • Aifantis E.C., Strain gradient interpretation of size effects, International Journal of Fracture, 95, 299–314, 1999
  • Aifantis E.C., Higher Order Gradients and Self-Organization at Nano, Micro, and Macro Scales, Materials Science Forum, 123–125, 553–66, 1993. doi:10.4028/www.scientific.net/msf.123-125.553
  • Akgöz B., Civalek Ö., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, 49, 1268–80, 2011. doi:10.1016/j.ijengsci.2010.12.009
  • Akgöz B., Civalek Ö., Buckling analysis of functionally graded microbeams based on the strain gradient theory, Acta Mechanica, 224, 2185–201, 2013. doi:10.1007/s00707-013-0883-5
  • Akgöz B., Civalek Ö., A new trigonometric beam model for buckling of strain gradient microbeams, International Journal of Mechanical Sciences, 81, 88–94, 2014. doi:10.1016/j.ijmecsci.2014.02.013
  • Mercan K., Civalek Ö., A Simple Buckling Analysis Of Aorta Artery, International Journal Of Engineering & Applied Sciences, 7, 34–34, 2015. doi:10.24107/ijeas.251256
  • Demir Ç., Mercan K., Civalek O., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel, Composites Part B: Engineering, 94, 1–10, 2016. doi:10.1016/j.compositesb.2016.03.031
  • Mercan K., Civalek Ö., Buckling Analysis of Silicon Carbide Nanotubes (SiCNTs), International Journal Of Engineering & Applied Sciences, 8, 101–101, 2016. doi:10.24107/ijeas.252148
  • Arda M., Aydogdu M., Buckling of Eccentrically Loaded Carbon Nanotubes, Solid State Phenomena, 267, 151–6, 2017. doi:10.4028/www.scientific.net/SSP.267.151
  • Mercan K., Civalek Ö., Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composites Part B: Engineering, 114, 34–45, 2017. doi:10.1016/j.compositesb.2017.01.067
  • Mercan K., Civalek Ö., Comparison of small scale effect theories for buckling analysis of nanobeams, International Journal Of Engineering & Applied Sciences, 9, 87–97, 2017. doi:10.24107/ijeas.340958
  • Mercan K., Numanoglu H.M., Akgöz B., Demir C., Civalek., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix, Archive of Applied Mechanics, 87, 1797–814, 2017. doi:10.1007/s00419-017-1288-z
  • Civalek Ö., Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations, International Journal of Pressure Vessels and Piping, 113, 1–9, 2014. doi:10.1016/j.ijpvp.2013.10.014
  • Arda M., Aydogdu M., Analysis of Free Torsional Vibration in Carbon Nanotubes Embedded in a Viscoelastic Medium, Advances in Science and Technology Research Journal, 9, 28–33, 2015. doi:10.12913/22998624/2361
  • Ebrahimi F., Barati M.R., Civalek Ö., Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures, Engineering with Computers, 36, 953–64, 2020. doi:10.1007/s00366-019-00742-z
  • AlSaid-Alwan H.H.S., Avcar M., AlSaid-Alwan H.H.S., Avcar M., Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study, Computers and Concrete, 26, 285, 2020. doi:10.12989/CAC.2020.26.3.285
  • Zhang J.S.Y.M.O., Analysis of orthotropic plates by the two-dimensional generalized FIT method, Computers and Concrete, 26, 421–7, 2020. doi:10.12989/CAC.2020.26.5.421
  • Hadji L., Avcar M., Free Vibration Analysis of FG Porous Sandwich Plates under Various Boundary Conditions, J Appl Comput Mech, 0, 1–15, 2020. doi:10.22055/JACM.2020.35328.2628
  • Arda M., Aydogdu M., Bending of CNTs Under The Partial Uniform Load, International Journal Of Engineering & Applied Sciences, 8, 21–21, 2016. doi:10.24107/ijeas.252142
  • Arda M., Aydogdu M., Longitudinal Vibration of CNTs Viscously Damped in Span, International Journal Of Engineering & Applied Sciences, 9, 22–22, 2017. doi:10.24107/ijeas.305348
  • Mercan K., Civalek Ö., What is The Correct Mechanical Model of Aorta Artery, International Journal Of Engineering & Applied Sciences, 9, 138–138, 2017. doi:10.24107/ijeas.322526
  • Akgöz B., Civalek Ö., A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation, Composite Structures, 176, 1028–38, 2017. doi:10.1016/j.compstruct.2017.06.039
  • Arda M., Vibration Analysis of an Axially Loaded Viscoelastic Nanobeam, International Journal Of Engineering & Applied Sciences, 10, 252–63, 2018. doi:10.24107/ijeas.468769
  • Arda M., Aydogdu M., Dynamic stability of harmonically excited nanobeams including axial inertia, JVC/Journal of Vibration and Control, 25, 820–33, 2019. doi:10.1177/1077546318802430
  • Arda M., Aydogdu M., Torsional dynamics of coaxial nanotubes with different lengths in viscoelastic medium, Microsystem Technologies, 25, 3943–57, 2019. doi:10.1007/s00542-019-04446-8
  • Jalaei M.H., Civalek., On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam, International Journal of Engineering Science, 143, 14–32, 2019. doi:10.1016/j.ijengsci.2019.06.013
  • Aydogdu M., Arda M., Filiz S., Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter, Advances in Nano Research, 6, 257–78, 2018. doi:10.12989/anr.2018.6.3.257
  • Arda M., Aydogdu M., Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass, Mechanics Based Design of Structures and Machines, 0, 1–17, 2020. doi:10.1080/15397734.2020.1728548
  • Arda M., Axial dynamics of functionally graded Rayleigh-Bishop nanorods, Microsystem Technologies, 2, 2020. doi:10.1007/s00542-020-04950-2
  • Civalek O., Yavas A., Large Deflection Static Analysis of Rectangular Plates On Two Parameter Elastic Foundations, International Journal of Science and Technology, 1, 43–50, 2006
  • Civalek Ö., Kiracioglu O., Free vibration analysis of Timoshenko beams by DSC method, International Journal for Numerical Methods in Biomedical Engineering, 26, 1890–8, 2010. doi:10.1002/cnm.1279
  • Mercan K., Demir Ç., Civalek Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique, Curved and Layered Structures, 3, 82–90, 2016. doi:10.1515/cls-2016-0007
  • Civalek Ö., Avcar M., Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method, Engineering with Computers, 1–33, 2020. doi:10.1007/s00366-020-01168-8
  • Civalek Ö., Uzun B., Yaylı M.Ö., Akgöz B., Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method, European Physical Journal Plus, 135, 381, 2020. doi:10.1140/epjp/s13360-020-00385-w
  • Wright E.M., Kantorovich L. V., Krylov V.I., Benster C.D., Approximate Methods of Higher Analysis, The Mathematical Gazette, 44, 145, 1960. doi:10.2307/3612589
  • Arda M., Evaluation of optimum length scale parameters in longitudinal wave propagation on nonlocal strain gradient carbon nanotubes by lattice dynamics, Mechanics Based Design of Structures and Machines, 1–24, 2020. doi:10.1080/15397734.2020.1835488

Year 2020, Volume: 12 Issue: 4, 163 - 172, 29.12.2020

Mustafa Arda

https://doi.org/10.24107/ijeas.842499
Cited By: 2

Abstract

References

  • Ajiki H., Ando T., Energy Bands of Carbon Nanotubes in Magnetic Fields, Journal of the Physical Society of Japan, 65, 505–14, 1996. doi:10.1143/JPSJ.65.505
  • Craighead H.G., Nanoelectromechanical Systems, Science, 290, 1532–5, 2000. doi:10.1126/science.290.5496.1532
  • Huang X.M.H., Zorman C.A., Mehregany M., Roukes M.L., Nanoelectromechanical systems: Nanodevice motion at microwave frequencies, Nature, 421, 496–496, 2003. doi:10.1038/421496a
  • Ghorbanpour Arani A., Shokravi M., Vibration response of visco-elastically coupled double-layered visco-elastic graphene sheet systems subjected to magnetic field via strain gradient theory considering surface stress effects, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 229, 180–90, 2015. doi:10.1177/1740349914529102
  • Arda M., Aydogdu M., Torsional statics and dynamics of nanotubes embedded in an elastic medium, Composite Structures, 114, 80–91, 2014. doi:10.1016/j.compstruct.2014.03.053
  • Li C., Li S., Zhu Z., Prediction of mechanical properties of microstructures through a nonlocal stress field theory, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 229, 50–4, 2015. doi:10.1177/1740349913519437
  • Kumar M., Reddy G.J., Kumar N.N., Bég O.A., Computational study of unsteady couple stress magnetic nanofluid flow from a stretching sheet with Ohmic dissipation, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 233, 49–63, 2019. doi:10.1177/2397791419843730
  • Gul U., Aydogdu M., Gaygusuzoglu G., Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics, Composite Structures, 160, 1268–78, 2017. doi:10.1016/j.compstruct.2016.11.023
  • Oterkus E., Diyaroglu C., Zhu N., Oterkus S., Madenci E., Utilization of Peridynamic Theory for Modeling at the Nano-Scale, 2015, p. 1–16. doi:10.1007/978-3-319-21194-7_1
  • Cauchy A.-L., Mémoire sur les systèmes isotropes de points matériels. Oeuvres complètes, Cambridge University Press, 1882. doi:10.1017/CBO9780511702280.023
  • Voigt W., Theoretische Studien über die Elasticitätsverhältnisse der Krystalle, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 34, 3–52, 1887
  • Cosserat E., Cosserat F., Theorie des corps dédormables. A. Hermann et fils, 1909
  • Kunin I.A., Elastic Media with Microstructure I. vol. 26. Springer Berlin Heidelberg, 1982. doi:10.1007/978-3-642-81748-9
  • Toupin R.A., Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17, 85–112, 1964. doi:10.1007/BF00253050
  • Mindlin R.D., Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 51–78, 1964. doi:10.1007/BF00248490
  • Kröner E., Elasticity theory of materials with long range cohesive forces, International Journal of Solids and Structures, 3, 731–42, 1967. doi:10.1016/0020-7683(67)90049-2
  • Green A.E., Rivlin R.S., Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17, 113–47, 1964. doi:10.1007/BF00253051
  • Eringen A.C., Edelen D.G.B., On nonlocal elasticity, International Journal of Engineering Science, 10, 233–48, 1972. doi:10.1016/0020-7225(72)90039-0
  • Ru C.Q., Aifantis E.C., A simple approach to solve boundary-value problems in gradient elasticity, Acta Mechanica, 101, 59–68, 1993. doi:10.1007/BF01175597
  • Altan B.S., Aifantis E.C., On Some Aspects in the Special Theory of Gradient Elasticity, Journal of the Mechanical Behavior of Materials, 8, 1997. doi:10.1515/JMBM.1997.8.3.231
  • Aifantis E.C., Strain gradient interpretation of size effects, International Journal of Fracture, 95, 299–314, 1999
  • Aifantis E.C., Higher Order Gradients and Self-Organization at Nano, Micro, and Macro Scales, Materials Science Forum, 123–125, 553–66, 1993. doi:10.4028/www.scientific.net/msf.123-125.553
  • Akgöz B., Civalek Ö., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, 49, 1268–80, 2011. doi:10.1016/j.ijengsci.2010.12.009
  • Akgöz B., Civalek Ö., Buckling analysis of functionally graded microbeams based on the strain gradient theory, Acta Mechanica, 224, 2185–201, 2013. doi:10.1007/s00707-013-0883-5
  • Akgöz B., Civalek Ö., A new trigonometric beam model for buckling of strain gradient microbeams, International Journal of Mechanical Sciences, 81, 88–94, 2014. doi:10.1016/j.ijmecsci.2014.02.013
  • Mercan K., Civalek Ö., A Simple Buckling Analysis Of Aorta Artery, International Journal Of Engineering & Applied Sciences, 7, 34–34, 2015. doi:10.24107/ijeas.251256
  • Demir Ç., Mercan K., Civalek O., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel, Composites Part B: Engineering, 94, 1–10, 2016. doi:10.1016/j.compositesb.2016.03.031
  • Mercan K., Civalek Ö., Buckling Analysis of Silicon Carbide Nanotubes (SiCNTs), International Journal Of Engineering & Applied Sciences, 8, 101–101, 2016. doi:10.24107/ijeas.252148
  • Arda M., Aydogdu M., Buckling of Eccentrically Loaded Carbon Nanotubes, Solid State Phenomena, 267, 151–6, 2017. doi:10.4028/www.scientific.net/SSP.267.151
  • Mercan K., Civalek Ö., Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composites Part B: Engineering, 114, 34–45, 2017. doi:10.1016/j.compositesb.2017.01.067
  • Mercan K., Civalek Ö., Comparison of small scale effect theories for buckling analysis of nanobeams, International Journal Of Engineering & Applied Sciences, 9, 87–97, 2017. doi:10.24107/ijeas.340958
  • Mercan K., Numanoglu H.M., Akgöz B., Demir C., Civalek., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix, Archive of Applied Mechanics, 87, 1797–814, 2017. doi:10.1007/s00419-017-1288-z
  • Civalek Ö., Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations, International Journal of Pressure Vessels and Piping, 113, 1–9, 2014. doi:10.1016/j.ijpvp.2013.10.014
  • Arda M., Aydogdu M., Analysis of Free Torsional Vibration in Carbon Nanotubes Embedded in a Viscoelastic Medium, Advances in Science and Technology Research Journal, 9, 28–33, 2015. doi:10.12913/22998624/2361
  • Ebrahimi F., Barati M.R., Civalek Ö., Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures, Engineering with Computers, 36, 953–64, 2020. doi:10.1007/s00366-019-00742-z
  • AlSaid-Alwan H.H.S., Avcar M., AlSaid-Alwan H.H.S., Avcar M., Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study, Computers and Concrete, 26, 285, 2020. doi:10.12989/CAC.2020.26.3.285
  • Zhang J.S.Y.M.O., Analysis of orthotropic plates by the two-dimensional generalized FIT method, Computers and Concrete, 26, 421–7, 2020. doi:10.12989/CAC.2020.26.5.421
  • Hadji L., Avcar M., Free Vibration Analysis of FG Porous Sandwich Plates under Various Boundary Conditions, J Appl Comput Mech, 0, 1–15, 2020. doi:10.22055/JACM.2020.35328.2628
  • Arda M., Aydogdu M., Bending of CNTs Under The Partial Uniform Load, International Journal Of Engineering & Applied Sciences, 8, 21–21, 2016. doi:10.24107/ijeas.252142
  • Arda M., Aydogdu M., Longitudinal Vibration of CNTs Viscously Damped in Span, International Journal Of Engineering & Applied Sciences, 9, 22–22, 2017. doi:10.24107/ijeas.305348
  • Mercan K., Civalek Ö., What is The Correct Mechanical Model of Aorta Artery, International Journal Of Engineering & Applied Sciences, 9, 138–138, 2017. doi:10.24107/ijeas.322526
  • Akgöz B., Civalek Ö., A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation, Composite Structures, 176, 1028–38, 2017. doi:10.1016/j.compstruct.2017.06.039
  • Arda M., Vibration Analysis of an Axially Loaded Viscoelastic Nanobeam, International Journal Of Engineering & Applied Sciences, 10, 252–63, 2018. doi:10.24107/ijeas.468769
  • Arda M., Aydogdu M., Dynamic stability of harmonically excited nanobeams including axial inertia, JVC/Journal of Vibration and Control, 25, 820–33, 2019. doi:10.1177/1077546318802430
  • Arda M., Aydogdu M., Torsional dynamics of coaxial nanotubes with different lengths in viscoelastic medium, Microsystem Technologies, 25, 3943–57, 2019. doi:10.1007/s00542-019-04446-8
  • Jalaei M.H., Civalek., On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam, International Journal of Engineering Science, 143, 14–32, 2019. doi:10.1016/j.ijengsci.2019.06.013
  • Aydogdu M., Arda M., Filiz S., Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter, Advances in Nano Research, 6, 257–78, 2018. doi:10.12989/anr.2018.6.3.257
  • Arda M., Aydogdu M., Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass, Mechanics Based Design of Structures and Machines, 0, 1–17, 2020. doi:10.1080/15397734.2020.1728548
  • Arda M., Axial dynamics of functionally graded Rayleigh-Bishop nanorods, Microsystem Technologies, 2, 2020. doi:10.1007/s00542-020-04950-2
  • Civalek O., Yavas A., Large Deflection Static Analysis of Rectangular Plates On Two Parameter Elastic Foundations, International Journal of Science and Technology, 1, 43–50, 2006
  • Civalek Ö., Kiracioglu O., Free vibration analysis of Timoshenko beams by DSC method, International Journal for Numerical Methods in Biomedical Engineering, 26, 1890–8, 2010. doi:10.1002/cnm.1279
  • Mercan K., Demir Ç., Civalek Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique, Curved and Layered Structures, 3, 82–90, 2016. doi:10.1515/cls-2016-0007
  • Civalek Ö., Avcar M., Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method, Engineering with Computers, 1–33, 2020. doi:10.1007/s00366-020-01168-8
  • Civalek Ö., Uzun B., Yaylı M.Ö., Akgöz B., Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method, European Physical Journal Plus, 135, 381, 2020. doi:10.1140/epjp/s13360-020-00385-w
  • Wright E.M., Kantorovich L. V., Krylov V.I., Benster C.D., Approximate Methods of Higher Analysis, The Mathematical Gazette, 44, 145, 1960. doi:10.2307/3612589
  • Arda M., Evaluation of optimum length scale parameters in longitudinal wave propagation on nonlocal strain gradient carbon nanotubes by lattice dynamics, Mechanics Based Design of Structures and Machines, 1–24, 2020. doi:10.1080/15397734.2020.1835488
There are 56 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Mustafa Arda 0000-0002-0314-3950

Publication Date December 29, 2020
Acceptance Date December 29, 2020
Published in Issue Year 2020 Volume: 12 Issue: 4

Cite

APA Arda, M. (2020). Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory. International Journal of Engineering and Applied Sciences, 12(4), 163-172. https://doi.org/10.24107/ijeas.842499
AMA Arda M. Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory. IJEAS. December 2020;12(4):163-172. doi:10.24107/ijeas.842499
Chicago Arda, Mustafa. “Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory”. International Journal of Engineering and Applied Sciences 12, no. 4 (December 2020): 163-72. https://doi.org/10.24107/ijeas.842499.
EndNote Arda M (December 1, 2020) Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory. International Journal of Engineering and Applied Sciences 12 4 163–172.
IEEE M. Arda, “Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory”, IJEAS, vol. 12, no. 4, pp. 163–172, 2020, doi: 10.24107/ijeas.842499.
ISNAD Arda, Mustafa. “Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory”. International Journal of Engineering and Applied Sciences 12/4 (December 2020), 163-172. https://doi.org/10.24107/ijeas.842499.
JAMA Arda M. Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory. IJEAS. 2020;12:163–172.
MLA Arda, Mustafa. “Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory”. International Journal of Engineering and Applied Sciences, vol. 12, no. 4, 2020, pp. 163-72, doi:10.24107/ijeas.842499.
Vancouver Arda M. Buckling Analysis of Intermediately Supported Nanobeams via Strain Gradient Elasticity Theory. IJEAS. 2020;12(4):163-72.

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