# An Axially Compressed Moving Nanobeam Based on the Nonlocal Couple Stress Theory and the Thermoelastic DPL Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Problem Statement

## 4. Analytical Solution

## 5. Solution Procedure

## 6. Numerical Outcomes and Discussion

## 7. Conclusions

- The salient influence of the nonlocal coefficient on the behavior of all thermo-physical domains;
- Changing the movement speed leads to significant changes in the different transverse oscillatory profiles, while the temperature change is slight.
- Performing mechanical and thermal vibration analysis on systems and nanostructures is crucial to ensuring system integrity and stability. It can thus be used to improve the mechanical properties of core components in MEMS and NEMS.
- It was found that the thermal waves move at limited speeds in the generalized thermal model compared to the traditional Fourier model.
- The length scale of a moving nanobeam can change the patterns of its thermal and mechanical vibrations and the patterns of the physical fields that make it up.
- By considering the combined effects of nanostructure motion, size dependence, and sinusoidal thermal pulsation, the results generated in this work are more comprehensive than those of previous corresponding studies.
- Design engineers working on micro- and nano-electromechanical systems may find the offered numerical solutions helpful.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lyshevski, S.E. MEMS and NEMS: Systems, Devices, and Structures; CRC Press: New York, NY, USA; London, UK, 2002. [Google Scholar]
- Sourki, R.; Hosseini, S.A. Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam. Eur. Phys. J. Plus.
**2017**, 132, 184. [Google Scholar] [CrossRef] - Thai, H.-T.; Vo, T.P.; Nguyen, T.-K.; Kim, S.-E. A review of continuum mechanics models for size-dependent analysis of beams and plates. Comp. Struct.
**2017**, 177, 196. [Google Scholar] [CrossRef] - Dinachandra, M.; Alankar, A. Static and dynamic modeling of functionally graded Euler-Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis. Compos. Struct.
**2022**, 280, 114923. [Google Scholar] [CrossRef] - Aifantis, E.C. Strain gradient interpretation of size effect. Int. J. Fract.
**1999**, 95, 299. [Google Scholar] [CrossRef] - Altan, B.S.; Aifantis, E.C. On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mat.
**1997**, 8, 231. [Google Scholar] [CrossRef] - Eringen, A.C. Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci.
**1972**, 10, 425–435. [Google Scholar] [CrossRef] - Eringen, A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys.
**1983**, 54, 4703. [Google Scholar] [CrossRef] - Eringen, A.C. Nonlocal Continuum Field Theories; Springer: New York, NY, USA, 2002. [Google Scholar]
- Yang, F.; Chong, A.C.M.; Lam, D.C.C.; Tong, P. Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct.
**2002**, 39, 2731. [Google Scholar] [CrossRef] - Mindlin, R.D.; Tiersten, H.F. Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal.
**1962**, 11, 415. [Google Scholar] [CrossRef] - Toupin, R. Elastic materials with couple-stresses. Arch. Ration. Mech. Anal.
**1962**, 11, 385. [Google Scholar] [CrossRef] - Mindlin, R.D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal
**1964**, 16, 51. [Google Scholar] [CrossRef] - Mindlin, R.D. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct.
**1965**, 1, 417–438. [Google Scholar] [CrossRef] - Attar, F.; Khordad, R.; Zarifi, A.; Modabberasl, A. Application of nonlocal modified couple stress to study of functionally graded piezoelectric plates. Phys. B Condens. Matter
**2021**, 600, 412623. [Google Scholar] [CrossRef] - Zhang, R.; Bai, H.; Chen, X. The Consistent Couple Stress Theory-Based Vibration and Post-Buckling Analysis of Bi-directional Functionally Graded Microbeam. Symmetry
**2022**, 14, 602. [Google Scholar] [CrossRef] - Hadjesfandiari, A.R.; Dargush, G.F. Couple stress theory for solids. Int. J. Solids Struct.
**2011**, 48, 2496–2510. [Google Scholar] [CrossRef] - Hadjesfandiari, A.R.; Dargush, G.F. Fundamental solutions for isotropic size-dependent couple stress elasticity. Int. J. Solids Struct.
**2013**, 50, 1253. [Google Scholar] [CrossRef] - Babaei, A.; Arabghahestani, M. Free Vibration Analysis of Rotating Beams Based on the Modified Couple Stress Theory and Coupled Displacement Field. Appl. Mech.
**2021**, 2, 226–238. [Google Scholar] [CrossRef] - Pham, Q.H.; Nguyen, P.C.; Tran, V.K.; Lieu, Q.X.; Tran, T.T. Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates. Eng. Comput.
**2023**, 39, 993. [Google Scholar] [CrossRef] - Rahmani, A.; Faroughi, S.; Friswell, M.I.; Babaei, A. Eringen’s nonlocal and modified couple stress theories applied to vibrating rotating nanobeams with temperature effects. Mech. Adv. Mater. Struct.
**2022**, 29, 4813. [Google Scholar] [CrossRef] - Park, S.K.; Gao, X.-L. Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng.
**2006**, 16, 2355. [Google Scholar] [CrossRef] - Qi, Z.; Peng, W.; He, T. Investigation on the thermoelastic response of a nanobeam in modified couple stress theory considering size-dependent and memory-dependent effects. J. Therm. Stress.
**2022**, 45, 773. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Sedighi, H.M. Thermoelastic characteristics of moving viscoelastic nanobeams based on the nonlocal couple stress theory and dual-phase lag model. Phys. Scr.
**2022**, 97, 114003. [Google Scholar] [CrossRef] - Atta, D.; Abouelregal, A.E.; Alsharari, F. Thermoelastic analysis of functionally graded nanobeams via fractional heat transfer model with nonlocal kernels. Mathematics
**2022**, 10, 4718. [Google Scholar] [CrossRef] - Kaur, I.; Singh, K.; Craciun, E.M. Recent advances in the theory of thermoelasticity and the modified models for the nanobeams: A review. Discov. Mech. Eng.
**2023**, 2, 2. [Google Scholar] [CrossRef] - Hosseini, M.; Mofidi, M.R.; Jamalpoor, A.; Safi Jahanshahi, M. Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory. Microsys. Techn.
**2017**, 24, 2295. [Google Scholar] [CrossRef] - Chen, J.Q.; Hao, Y.X.; Zhang, W.; Liu, L.T. Vibration analysis of the trapezoidal bi-stable composite laminate plate with four free edges. Int. J. Dyn. Control.
**2022**, 10, 1415–1423. [Google Scholar] [CrossRef] - Xu, H.; He, T.; Zhong, N.; Zhao, B.; Liu, Z. Transient thermomechanical analysis of micro cylindrical asperity sliding contact of SnSbCu alloy. Tribol. Int.
**2022**, 167, 107362. [Google Scholar] [CrossRef] - Xiao, X.; Zhang, Q.; Zheng, J.; Li, Z. Analytical model for the nonlinear buckling responses of the confined polyhedral FGP-GPLs lining subjected to crown point loading. Eng. Struct.
**2023**, 282, 115780. [Google Scholar] [CrossRef] - Ye, R.; Liu, P.; Shi, K.; Yan, B. State Damping Control: A Novel Simple Method of Rotor UAV With High Performance. IEEE Access
**2020**, 8, 214346. [Google Scholar] [CrossRef] - Biot, M.A. Thermoelasticity and irreversible thermodynamics. J. Appl. Phys.
**1956**, 27, 240–253. [Google Scholar] [CrossRef] - Cattaneo, C. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comptes Rendus
**1958**, 247, 431. [Google Scholar] - Vernotte, P. Les paradoxes de la theorie continue de l’equation de la chaleur. Comptes Rendus
**1958**, 246, 3154. [Google Scholar] - Tzou, D.Y. Thermal shock phenomena under high rate response in solids. Annu. Rev. Heat Tran.
**1992**, 4, 111. [Google Scholar] [CrossRef] - Tzou, D.Y. A unified field approach for heat conduction from macro- to micro-scales. J. Heat Transf.
**1995**, 117, 8. [Google Scholar] [CrossRef] - Tzou, D.Y. Experimental support for the lagging behavior in heat propagation. J. Thermo. Heat Transf.
**1995**, 9, 686. [Google Scholar] [CrossRef] - Green, A.E.; Lindsay, K.A. Thermoelasticity. J. Elast.
**1972**, 2, 1–7. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. On undamped heat waves in an elastic solid. J. Therm. Stress.
**1992**, 15, 253. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. Thermoelasticity without energy dissipation. J. Elast.
**1993**, 31, 189. [Google Scholar] [CrossRef] - Lord, H.W.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299. [Google Scholar] [CrossRef] - Tzou, D.Y. Macro-to Microscale Heat Transfer: The Lagging Behavior; John Wiley & Sons: West Sussex, UK, 2014. [Google Scholar]
- Quintanilla, R.; Racke, R. Qualitative aspects in dual-phase-lag heat conduction. Proc. R. Soc. Lond. Ser. A
**2007**, 463, 659. [Google Scholar] [CrossRef] - Reddy, J.N. Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci.
**2007**, 45, 288–307. [Google Scholar] [CrossRef] - Chen, L.Q. Analysis and control of transverse vibrations of axially moving strings. ASME Appl. Mech. Rev.
**2005**, 58, 91–116. [Google Scholar] [CrossRef] - Lee, U.; Jang, I. On the boundary conditions for axially moving beams. J. Sound Vib.
**2007**, 306, 675. [Google Scholar] [CrossRef] - Arda, M.; Aydogdu, M. Dynamic stability of harmonically excited nanobeams including axial inertia. J. Vib. Control
**2019**, 25, 820. [Google Scholar] [CrossRef] - Kulkarni, R.G. Solving sextic equations, Atlantic Electronic. J. Math.
**2008**, 3, 1. [Google Scholar] - Honig, G.; Hirdes, U. A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math.
**1984**, 10, 113. [Google Scholar] [CrossRef] - Attia, M.A.; Mahmoud, F.F. Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int. J. Mech. Sci.
**2016**, 105, 126. [Google Scholar] [CrossRef] - Wang, Q. Axi-symmetric wave propagation of carbon nanotubes with non-local elastic shell model. Int. J. Struct. Stab. Dyn.
**2006**, 06, 285. [Google Scholar] [CrossRef] - Lim, C.W. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: Equilibrium, governing equation and static deflection. Appl. Math. Mech.
**2010**, 31, 37. [Google Scholar] [CrossRef] - Li, C.; Yao, L.; Chen, W.; Li, S. Comments on nonlocal effects in nano-cantilever beams. Int. J. Eng. Sci.
**2015**, 87, 47. [Google Scholar] [CrossRef] - Scutaru, M.L.; Vlase, S.; Marin, M.; Modrea, A. New analytical method based on dynamic response of planar mechanical elastic systems. Bound. Value Probl.
**2020**, 2020, 104. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Mohammad-Sedighi, H.; Faghidian, S.A.; Shirazi, A.H. Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load. Facta Univ. Ser. Mech. Eng.
**2021**, 19, 633. [Google Scholar] [CrossRef] - Abo-Dahab, S.M.; Abouelregal, A.E.; Marin, M. Generalized thermoelastic functionally graded on a thin slim strip non-Gaussian laser beam. Symmetry
**2020**, 12, 1094. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Sedighi, H.M.; Malikan, M.; Eremeyev, V.A. Nonlocalized thermal behavior of rotating micromachined beams under dynamic and thermodynamic loads. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech.
**2022**, 102, e202100310. [Google Scholar] [CrossRef] - Ma, H.; Gao, X.; Reddy, J. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids
**2008**, 56, 3379–3391. [Google Scholar] [CrossRef] - Moaaz, O.; Abouelregal, A.E.; Alsharari, F. Lateral vibration of an axially moving thermoelastic nanobeam subjected to an external transverse excitation. AIMS Math.
**2023**, 8, 2272. [Google Scholar] [CrossRef] - Shariati, A.; Jung, D.; Mohammad-Sedighi, H.; Żur, K.K.; Habibi, M.; Safa, M. On the Vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials
**2020**, 13, 1707. [Google Scholar] [CrossRef] - Wang, J.; Shen, H. Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory. J. Phys. Condens. Matter
**2019**, 31, 485403. [Google Scholar] [CrossRef] - Tilmans, H.A.C.; Legtenberg, R. Electrostatically driven vacuum-encapsulated polysilicon resonators. Sens. Actuators A Phys.
**1994**, 45, 67. [Google Scholar] [CrossRef] - Marin, M.; Ellahi, R.; Vlase, S.; Bhatti, M.M. On the decay of exponential type for the solutions in a dipolar elastic body. J. Taibah Univ. Sci.
**2020**, 14, 534. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abouelregal, A.E.; Askar, S.S.; Marin, M.
An Axially Compressed Moving Nanobeam Based on the Nonlocal Couple Stress Theory and the Thermoelastic DPL Model. *Mathematics* **2023**, *11*, 2155.
https://doi.org/10.3390/math11092155

**AMA Style**

Abouelregal AE, Askar SS, Marin M.
An Axially Compressed Moving Nanobeam Based on the Nonlocal Couple Stress Theory and the Thermoelastic DPL Model. *Mathematics*. 2023; 11(9):2155.
https://doi.org/10.3390/math11092155

**Chicago/Turabian Style**

Abouelregal, Ahmed E., S. S. Askar, and Marin Marin.
2023. "An Axially Compressed Moving Nanobeam Based on the Nonlocal Couple Stress Theory and the Thermoelastic DPL Model" *Mathematics* 11, no. 9: 2155.
https://doi.org/10.3390/math11092155