Skip to main content
Springer Nature Link
Log in

Magneto-thermoelastic behaviour of a finite viscoelastic rotating rod by incorporating Eringen’s theory and heat equation including Caputo–Fabrizio fractional derivative

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This paper addresses a modified constitutive equation by incorporating the size effect of nanostructured materials and a new formulation of Fourier's law including Caputo–Fabrizio fractional heat conduction equation with a non-singular kernel. The Kelvin–Voigt model is used to characterize the viscoelastic behavior of the material. In the absence of mechanical relaxation and nonlocal effects, the results of different generalized theories of thermoelasticity can be achieved as specific cases. The presented model is then applied to analyze the magneto-thermoelastic interactions in a viscoelastic rotating rod subject to a moving heat source. The analytical solutions are obtained through Laplace transform method and its reversal followed by residue calculations. Several illustrations are presented to evaluate the effects of system parameters, fractional order, nonlocal and vicsocity parameters, magnetic field and the velocity of heat source on the features of physical variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

source c

Fig. 11

source c

Fig. 12

source c

Fig. 13

source c

Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Said SM (2020) Novel model of thermo-magneto-viscoelastic medium with variable thermal conductivity under effect of gravity. Appl Math Mech-Engl Ed 41:819–832

    MathSciNet  MATH  Google Scholar 

  2. Abouelregal AE, Ahmad H, Yao S-W, Abu-Zinadah H (2021) Thermo-viscoelastic orthotropic constraint cylindrical cavity with variable thermal properties heated by laser pulse via the MGT thermoelasticity model. Open Phys 19:504–518

    Google Scholar 

  3. Abouelregal AE (2021) Thermoelastic fractional derivative model for exciting viscoelastic microbeam resting on Winkler foundation. J Vib Control 27(17–18):2123–2135

    MathSciNet  Google Scholar 

  4. Vu AT, Vu AN, Grunwald T, Bergs T (2020) Modeling of thermo-viscoelastic material behavior of glass over a wide temperature range in glass compression molding. J Am Ceram Soc 103(4):2791–2807

    Google Scholar 

  5. Hendy MH, El-Attar SI, Ezzat MA (2021) Thermoelectric viscoelastic spherical cavity with memory-dependent derivative. Mater Phys Mech 47:170–185

    Google Scholar 

  6. Fahmy MA (2019) DRBEM sensitivity analysis and shape optimization of rotating magneto-thermo-viscoelastic FGA structures using DRBEM-GSS and DRBEM-NGGP algorithms. Adv Math Comput Sci 1:83–104

    Google Scholar 

  7. Fahmy MA (2019) Design optimization for a simulation of rotating anisotropic viscoelastic porous structures using time-domain OQBEM. Math Comput Simul 166:193–205

    MathSciNet  MATH  Google Scholar 

  8. Povstenko Y (2020) Fractional nonlocal elasticity and solutions for straight screw and edge dislocations. Phys Mesomech 23:547–555

    Google Scholar 

  9. Samko SG, Kilbas AA, Marichev OI (2002) Fractional integrals and derivatives: theory and applications. Taylor & Francis, London

    MATH  Google Scholar 

  10. Saïd A, Mouffak B, Johnny H (2021) Random Caputo–Fabrizio fractional differential inclusions. Math Model Control 1(2):102–111

    Google Scholar 

  11. Kaabar MK, Kalvandi V, Eghbali N, Samei ME, Siri Z, Martínez F (2021) A generalized ML-Hyers-Ulam stability of quadratic fractional integral equation. Nonlinear Eng 10(1):414–427

    Google Scholar 

  12. Abro KA, Atangana A, Khoso AR (2021) Dynamical behavior of fractionalized simply supported beam: an application of fractional operators to Bernoulli–Euler theory. Nonlinear Eng 10(1):231–239

    Google Scholar 

  13. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):73–85

    Google Scholar 

  14. Caputo M, Fabrizio M (2017) On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica 52:3043–3052

    MathSciNet  MATH  Google Scholar 

  15. Losada J, Nieto J (2015) Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):87–92

    Google Scholar 

  16. Al-Refai M, Abdeljawad T (2017) Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel. Adv Difference Equ 2017:315

    MathSciNet  MATH  Google Scholar 

  17. Atangana A (2016) On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl Math Comput 273:948–956

    MathSciNet  MATH  Google Scholar 

  18. Kaczorek T, Borawski K (2016) Fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. Int J Appl Math Comput Sci 26(3):533–541

    MathSciNet  MATH  Google Scholar 

  19. Alkahtani BST, Atangana A (2016) Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order. Chaos Solitons Fract 89:539–546

    MathSciNet  MATH  Google Scholar 

  20. Al-Salti N, Karimov E, Sadarangani K (2016) On a differential equation with Caputo–Fabrizio fractional derivative of order 1<β<2 and application to mass-spring-damper system. Progr Fract Differ Appl 2(4):257–263

    Google Scholar 

  21. Caputo M, Fabrizio M (2016) Applications of new time and spatial fractional derivatives with exponential kernels. Progress in Fractional Differentiation and Applications 2:1–11

    Google Scholar 

  22. Gomez-Aguilar JF, Torres L, Yepez-Martınez H, Baleanu D, Reyes JM, Sosa IO (2016) Fractional Li ́enard type model of a pipeline within the fractional derivative without singular kernel. Adv Difference Equ 2016:173

    MATH  Google Scholar 

  23. Goufo EFD (2016) Application of the Caputo–Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Math Model Anal 21(2):188–198

    MathSciNet  MATH  Google Scholar 

  24. Abouelregal AE, Ahmad H, Elagan SK, Alshehri NA (2021) Modified Moore–Gibson–Thompson photo-thermoelastic model for a rotating semiconductor half-space subjected to a magnetic field. Int J Mod Phys C. https://doi.org/10.1142/S0129183121501631

    Article  MathSciNet  Google Scholar 

  25. Hobiny AD, Abbas IA (2020) A dual-phase-lag model of photothermoelastic waves in a two-dimensional semiconducting medium. Phys Mesomech 23:167–175

    Google Scholar 

  26. Kaur I, Lata P, Singh K (2020) Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures. Int J Mech Mater Eng 15:10

    Google Scholar 

  27. Biswas S (2021) Eigenvalue approach to a magneto-thermoelastic problem in transversely isotropic hollow cylinder: comparison of three theories. Waves Random Complex Media 31(3):403–419

    MathSciNet  Google Scholar 

  28. Mondal S (2020) Memory response in a magneto-thermoelastic rod with moving heat source based on Eringen’s nonlocal theory under dual-phase lag heat conduction. Int J Comput Methods 17(09):1950072

    MathSciNet  MATH  Google Scholar 

  29. Allam MN, Elsibai KA, Abouelregal AE (2010) Magnetothermoelasticity for an infinite body with a spherical cavity and variable material properties without energy dissipation. Int J Solids Struct 47(20):2631–2638

    MATH  Google Scholar 

  30. Abouelregal AE, Abo-Dahab SM (2012) Dual phase lag model on magneto-thermoelasticity infinite non-homogeneous solid having a spherical cavity. J Therm Stresses 35(9):820–841

    Google Scholar 

  31. Abouelregal AE, Abo-Dahab SM (2014) Dual-phase-lag diffusion model for Thomson’s phenomenon on electromagneto-thermoelastic an infinitely long solid cylinder. J Comput Theor Nanosci 11(4):1031–1039

    Google Scholar 

  32. Abouelregal AE (2020) Modified fractional photo-thermoelastic model for a rotating semiconductor half-space subjected to a magnetic field. SILICON 12:2837–2850

    Google Scholar 

  33. Abouelregal AE, Ahmad H, Yao S-W (2020) Functionally graded piezoelectric medium exposed to a movable heat flow based on a heat equation with a memory-dependent derivative. Materials 13(18):3953

    Google Scholar 

  34. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16

    MathSciNet  MATH  Google Scholar 

  35. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248

    MathSciNet  MATH  Google Scholar 

  36. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710

    Google Scholar 

  37. Pinnola FP, Vaccaro MS, Barretta R (2022) Finite element method for stress-driven nonlocal beams. Eng Anal Boundary Elem 134:22–34

    MathSciNet  MATH  Google Scholar 

  38. Vaccaro MS, Marotti de Sciarra F, Barretta R (2021) On the regularity of curvature fields in stress-driven nonlocal elastic beams. Acta Mech 232(7):2595–2603

    MathSciNet  MATH  Google Scholar 

  39. Vaccaro, MS, Pinnola, FP, Marotti de Sciarra, F, Canadija, M, Barretta, R (2021) Stress-driven two-phase integral elasticity for Timoshenko curved beams. Proc Inst Mech Eng N J Nanomater Nanoeng Nanosyst. https://doi.org/10.1177/2397791421990514

  40. Vaccaro MS, Pinnola FP, Marotti de Sciarra F, Barretta R (2021) Elastostatics of Bernoulli-Euler beams resting on displacement-driven nonlocal foundation. Nanomaterials 11(3):573

    Google Scholar 

  41. Zare J, Shateri A, Beni YT, Ahmadi A (2020) Vibration analysis of shell-like curved carbon nanotubes using nonlocal strain gradient theory. Math Methods Appl Sci. https://doi.org/10.1002/mma.6599

    Article  Google Scholar 

  42. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos Struct 186:68–78

    Google Scholar 

  43. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Compos Struct 198:51–62

    Google Scholar 

  44. Sahmani S, Aghdam MM, Rabczuk T (2018) A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Mater Res Express 5(4):045048

    Google Scholar 

  45. Rabczuk T, Ren H, Zhuang X (2019) A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Comput Mater Contin 59:31–55

    Google Scholar 

  46. Koochi A, Goharimanesh M (2021) Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: the energy balance method. Rep Mech Eng 2(1):41–50

    Google Scholar 

  47. Sedighi HM (2014) Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut 95:111–123

    Google Scholar 

  48. Abouelregal AE, Mohammad-Sedighi H, Faghidian SA, Shirazi AH (2021) Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load. Facta Univer Ser Mech Eng 19(4):633–656. https://doi.org/10.22190/FUME201222024A

    Article  Google Scholar 

  49. Abouelregal AE, Sedighi HM, Malikan M, Eremeyev VA (2021) Nonlocalized thermal behavior of rotating micromachined beams under dynamic and thermodynamic loads. ZAMM J Appl Math Mech/Z Angew Math Mech. https://doi.org/10.1002/zamm.202100310

    Article  Google Scholar 

  50. Avlović IR et al (2020) Dynamic behavior of two elastically connected nanobeams under a white noise process. Facta Univer Ser Mech Eng 18(2):219–227

    Google Scholar 

  51. Lam D, Yang F, Chong A, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508

    MATH  Google Scholar 

  52. Ebrahimi F, Barati MR, Dabbagh A (2016) A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 107:169–182

    Google Scholar 

  53. Sari S, Al-Kouz G, Atieh M (2020) Transverse vibration of functionally graded tapered double nanobeams resting on elastic foundation. Appl Sci 10(2):493

    Google Scholar 

  54. Mindlin R (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438

    Google Scholar 

  55. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    MATH  Google Scholar 

  56. Inan E, Eringen AC (1991) Nonlocal theory of wave propagation in thermoelastic plates. Int J Eng Sci 29:831–843

    MATH  Google Scholar 

  57. Wang J, Dhaliwal RS (1993) Uniqueness in generalized nonlocal thermoelasticity. J Therm Stresses 16:71–77

    MathSciNet  Google Scholar 

  58. Abouelregal AE, Mohammed WW (2020) Effects of nonlocal thermoelasticity on nanoscale beams based on couple stress theory. Math Methods Appl Sci. https://doi.org/10.1002/mma.6764

    Article  Google Scholar 

  59. Abouelregal AE, Marin M (2020) The response of nanobeams with temperature-dependent properties using state-space method via modified couple stress theory. Symmetry 12:1276

    Google Scholar 

  60. Abouelregal AE (2020) A novel model of nonlocal thermoelasticity with time derivatives of higher order. Math Methods Appl Sci. https://doi.org/10.1002/mma.6416

    Article  MathSciNet  MATH  Google Scholar 

  61. Sae-Long W et al (2021) Fourth-order strain gradient bar-substrate model with nonlocal and surface effects for the analysis of nanowires embedded in substrate media. Facta Univer Ser Mech Eng 19(4):657–680

    Google Scholar 

  62. Az-Z’obi EA, Jhangeer A, Rezazadeh H, Ali MN, Kaabar MK (2021) New soliton solutions for the higher-dimensional non-local Ito equation. Nonlinear Eng 10(1):374–384

    Google Scholar 

  63. Ren YM, Qing H (2021) Bending and buckling analysis of functionally graded Euler-Bernoulli beam using stress-driven nonlocal integral model with Bi-Helmholtz kernel. Int J Appl Mech 13(4):2150041

    Google Scholar 

  64. Abouelregal AE, Mohamed BO (2018) Fractional order thermoelasticity for a functionally graded thermoelastic nanobeam induced by a sinusoidal pulse heating. J Comput Theor Nanosci 15:1233–1242

    Google Scholar 

  65. Ouakad HM, Sedighi HM, Al-Qahtani HM (2020) Forward and backward whirling of a spinning nanotube nano-rotor assuming gyroscopic effects. Adv Nano Res 8:245–254

    Google Scholar 

  66. Adnan IK, Ahmed AD, Eltaher MA (2021) Novel four-unknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates. Adv Nano Res 11:621–640

    Google Scholar 

  67. Ghandourah EE, Ahmed HM, Eltaher MA, Attia MA, Abdraboh AM (2021) Free vibration of porous FG nonlocal modified couple nanobeams via a modified porosity model. Advances in Nano Research 11(4):405–422

    Google Scholar 

  68. Abouelregal AE (2019) Rotating magneto-thermoelastic rod with finite length due to moving heat sources via Eringen’s nonlocal model. J Comput Appl Mech 50(1):118–126

    Google Scholar 

  69. Barretta R, Faghidian SA, Luciano R (2019) Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech Adv Mater Struct 26(15):1307–1315

    Google Scholar 

  70. Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solid 15:299–309

    MATH  Google Scholar 

  71. Lei Y, Adhikari S, Friswell MI (2013) Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. Int J Eng Sci 66–67:1–13

    MathSciNet  MATH  Google Scholar 

  72. Atangana A, Gómez-Aguilar JF (2018) Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu. Numer Methods Partial Differ Equ 34:1502–1523

    MathSciNet  MATH  Google Scholar 

  73. Furati KM, Kassim MD, Tatar NT (2012) Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput Math Appl 64:1616–1626

    MathSciNet  MATH  Google Scholar 

  74. Shaikh A, Tassaddiq A, Nisar KS et al (2019) Analysis of differential equations involving Caputo-Fabrizio fractional operator and its applications to reaction–diffusion equations. Adv Differ Equ 2019:173

    MathSciNet  MATH  Google Scholar 

  75. Wang H, Dong K, Men F, Yan YJ, Wang X (2010) Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix. Appl Math Model 34:878–889

    MathSciNet  MATH  Google Scholar 

  76. Schoenberg M, Censor D (1973) Elastic waves in rotating media. Q Appl Math 31:115–125

    MATH  Google Scholar 

  77. Abouelregal AE, Abo-Dahab SM (2018) A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model. Sådhanå 43:13

    MathSciNet  MATH  Google Scholar 

  78. Roychoudhuri SK, Mukhopadhyay S (2000) Effect of rotation and relaxation times on plane waves in generalized thermo-viscoelasticity. Int J Math Math Sci 23:497–505

    MathSciNet  MATH  Google Scholar 

  79. Honig G, Hirdes U (1984) A method for the numerical inversion of Laplace transforms. J Comput Appl Math 10(1):113–132

    MathSciNet  MATH  Google Scholar 

  80. He T, Cao L (2009) A problem of generalized magnetothermoelastic thin slim strip subjected to a moving heat source. Math Comput Model 49(7–8):1710–1720

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to sincerely thank Mr. Ali H. Shirazi for his cooperation in producing the quality image for the considered problem. H.M. Sedighi is grateful to the Research Council of the Shahid Chamran University of Ahvaz for its financial support (Grant No. SCU.EM1400.98).

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayat, Saudi Arabia

    Ahmed E. Abouelregal

  2. Basic Sciences Research Unit, Jouf University, Sakaka, Saudi Arabia

    Ahmed E. Abouelregal

  3. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Ahmed E. Abouelregal

  4. Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Hamid M. Sedighi

  5. ‏Drilling Center of Excellence and Research Center, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Hamid M. Sedighi

Authors
  1. Ahmed E. Abouelregal
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Hamid M. Sedighi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Hamid M. Sedighi.

Ethics declarations

Conflict of interest

The authors declared no potential conflicts of interest concerning the research, authorship, and publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abouelregal, A.E., Sedighi, H.M. Magneto-thermoelastic behaviour of a finite viscoelastic rotating rod by incorporating Eringen’s theory and heat equation including Caputo–Fabrizio fractional derivative. Engineering with Computers 39, 655–668 (2023). https://doi.org/10.1007/s00366-022-01645-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-022-01645-2

Keywords