Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects
A variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler an...
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MDPI AG
2025-06-01
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| author | Sarp Adali |
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| description | A variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler and Pasternak interlayers. The equations governing the vibrations of the coupled system were expressed as a system of four partial differential equations based on nonlocal elastic theory. After deriving the variational principle for the double BNNT system, Hamilton’s principle was expressed in terms of potential and kinetic energies. Next, the differential equations for the free vibration case were presented and the variational form for this case was derived. The Rayleigh quotient was formulated for the vibration frequency, which indicated that piezoelectric and surface effects led to higher vibration frequencies. Next, the variationally consistent boundary conditions were formulated in terms of moment and shear force expressions. It was observed that the presence of the Pasternak interlayer between the nanotubes led to coupled boundary conditions when a shear force and/or a moment was specified at the boundaries. |
| format | Article |
| id | doaj-art-f7173ddc895a4e1ca8a462047cb079b1 |
| institution | Kabale University |
| issn | 2673-8716 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
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| series | Dynamics |
| spelling | doaj-art-f7173ddc895a4e1ca8a462047cb079b12025-08-20T03:27:21ZengMDPI AGDynamics2673-87162025-06-01522110.3390/dynamics5020021Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface EffectsSarp Adali0Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban 4091, South AfricaA variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler and Pasternak interlayers. The equations governing the vibrations of the coupled system were expressed as a system of four partial differential equations based on nonlocal elastic theory. After deriving the variational principle for the double BNNT system, Hamilton’s principle was expressed in terms of potential and kinetic energies. Next, the differential equations for the free vibration case were presented and the variational form for this case was derived. The Rayleigh quotient was formulated for the vibration frequency, which indicated that piezoelectric and surface effects led to higher vibration frequencies. Next, the variationally consistent boundary conditions were formulated in terms of moment and shear force expressions. It was observed that the presence of the Pasternak interlayer between the nanotubes led to coupled boundary conditions when a shear force and/or a moment was specified at the boundaries.https://www.mdpi.com/2673-8716/5/2/21boron nitride nanotubesvariational principlepiezoelastic effectsurface effectvibrationsvariationally consistent boundary conditions |
| spellingShingle | Sarp Adali Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects Dynamics boron nitride nanotubes variational principle piezoelastic effect surface effect vibrations variationally consistent boundary conditions |
| title | Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects |
| title_full | Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects |
| title_fullStr | Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects |
| title_full_unstemmed | Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects |
| title_short | Variational Principles for Coupled Boron Nitride Nanotubes Undergoing Vibrations, Including Piezoelastic and Surface Effects |
| title_sort | variational principles for coupled boron nitride nanotubes undergoing vibrations including piezoelastic and surface effects |
| topic | boron nitride nanotubes variational principle piezoelastic effect surface effect vibrations variationally consistent boundary conditions |
| url | https://www.mdpi.com/2673-8716/5/2/21 |
| work_keys_str_mv | AT sarpadali variationalprinciplesforcoupledboronnitridenanotubesundergoingvibrationsincludingpiezoelasticandsurfaceeffects |