Skip to main content
SpringerLink
Log in

Nonlocal mixed variational formula for orthotropic nanoplates resting on elastic foundations

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This paper deals with the examination of orthotropic nanoplates utilizing nonlocal mixed variational formula. For this model, the displacement and stress fields are considered to be arbitrary and it includes the impact of transverse normal stress which does not exist in numerous theories. The orthotropic nanoplates are thought to be refreshed on elastic foundations. The mixed theory is utilized to get the governing differential equations as per Eringen’s nonlocal elasticity model. The governing equations incorporate the small-scale effect as well as foundations and mechanical impacts. Analytical solutions of bending response for simply supported orthotropic nanoplates are derived. The present outcomes are compared well with those being in the literature. The effects played by elastic foundation parameters, small-scale parameter, aspect and thickness nanoplate ratios are examined. The present study can be used as benchmarks for future comparisons with other investigators applying models containing the transverse shear and normal strains.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  2. M. Arefi, A.M. Zenkour, A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezomagnetic sandwich nanobeams inmagneto-thermo-electric environment. J. Sand Struct. Mater. 18, 624–651 (2016)

    Google Scholar 

  3. M.R. Nami, M. Janghorban, Static analysis of rectangular nanoplates using trigonometric shear deformation theory based on nonlocal elasticity theory. Beilstein J. Nanotechnol. 4, 968–973 (2013)

    Google Scholar 

  4. M. Arefi, A.M. Zenkour, Employing sinusoidal shear deformation plate theory for transient analysis of three layers sandwich nanoplate integrated with piezomagnetic face-sheets. Smart Mater. Struct. 25, 115040 (2016)

    ADS  Google Scholar 

  5. M. Sobhy, A.F. Radwan, A new quasi 3-D nonlocal hyperbolic plate theory for vibration and buckling of FGM nanoplates. Int. J. Appl. Mech. 9, 1750008 (2017)

    Google Scholar 

  6. H.M. Numanoǧlu, B. Akgöz, Ö. Civalek, On dynamic analysis of nanorods. Int. J. Eng. Sci. 130, 33–50 (2018)

    Google Scholar 

  7. B. Akgöz, Ö. Civalek, Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut. 119, 1–12 (2016)

    ADS  Google Scholar 

  8. M. Gürses, B. Akgöz, Ö. Civalek, Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl. Math. Comput. 219, 3226–3240 (2012)

    MathSciNet  MATH  Google Scholar 

  9. E. Allahyari, M. Asgari, F. Pellicano, Nonlinear strain gradient analysis of nanoplates embedded in an elastic medium incorporating surface stress effects. Eur. Phys. J. Plus 134, 191 (2019). https://doi.org/10.1140/epjp/i2019-12575-4

    Article  Google Scholar 

  10. O. Rahmani, M. Shokrnia, H. Golmohammadi, S.A.H. Hosseini, Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory. Eur. Phys. J. Plus 133, 42 (2018). https://doi.org/10.1140/epjp/i2018-11868-4

    Article  Google Scholar 

  11. Ç. Demir, Ö. Civalek, Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Model. 37, 9355–9367 (2013)

    Google Scholar 

  12. Ö. Civalek, C. Demir, A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl. Math. Comput. 289, 335–352 (2016)

    MathSciNet  MATH  Google Scholar 

  13. B. Akgöz, Ö. Civalek, Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech. 224, 2185–2201 (2013)

    MathSciNet  MATH  Google Scholar 

  14. B. Akgöz, Ö. Civalek, A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech. 226, 2277–2294 (2015)

    MathSciNet  MATH  Google Scholar 

  15. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screwd is location and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    ADS  Google Scholar 

  16. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002)

    MATH  Google Scholar 

  17. F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    MATH  Google Scholar 

  18. E.C. Aifantis, Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)

    Google Scholar 

  19. A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972)

    MATH  Google Scholar 

  20. A.C. Eringen, Linear theory of nonlocal microelasticity and dispersion of plane waves. Lett. Appl. Eng. Sci. 1, 129–146 (1973)

    Google Scholar 

  21. A.C. Eringen, Theory of nonlocal electromagnetic elastic solids. J. Math. Phys. 14, 733–740 (1973)

    ADS  MATH  Google Scholar 

  22. A.M. Zenkour, A.F. Radwan, Bending response of FG plates resting on elastic foundations in hygrothermal environment with porosities. Compos. Struct. 213, 133–143 (2019)

    Google Scholar 

  23. E.O. Alzahrani, A.M. Zenkour, M. Sobhy, Small scale effect on hygro-thermo-mechanical bending of nanoplates embedded in an elastic medium. Compos. Struct. 105, 163–172 (2013)

    Google Scholar 

  24. I.S. Radebe, S. Adali, Buckling and sensitivity analysis of nonlocal orthotropic nanoplates with uncertain material properties. Compos. B 56, 840–846 (2014)

    Google Scholar 

  25. N. Radić, D. Jeremić, S. Trifković, M. Milutinović, Buckling analysis of double orthotropic nanoplates embedded in Pasternak elastic medium using nonlocal elasticity theory. Compos. B 61, 162–71 (2014)

    Google Scholar 

  26. S. Pouresmaeeli, S.A. Fazelzadeh, E. Ghavanloo, Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium. Compos. B 43, 3384–3390 (2012)

    Google Scholar 

  27. S. Narendar, S. Gopalakrishnan, Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory. Acta Mech. 223, 395–413 (2012)

    MathSciNet  MATH  Google Scholar 

  28. L.L. Ke, Y.S. Wang, J. Yang, S. Kitipornchai, Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mech. Sin. 30, 516–525 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  29. M. Sobhy, Hygrothermal deformation of orthotropic nanoplates based on the state-space concept. Compos. B 79, 224–235 (2015)

    Google Scholar 

  30. P. Lu, P.Q. Zhang, H.P. Lee, C.M. Wang, J.N. Reddy, Non-local elastic plate theories. Math. Phys. Eng. Sci. 463, 3225–3240 (2007)

    MathSciNet  MATH  Google Scholar 

  31. N. Satish, S. Narendar, S. Gopalakrishnan, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics. Physica E 44, 1950–1962 (2012)

    ADS  Google Scholar 

  32. P. Malekzadeh, A.R. Setoodeh, A.A. Beni, Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates. Compos. Struct. 93, 1631–1639 (2011)

    Google Scholar 

  33. R. Aghababaei, J.N. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289 (2009)

    ADS  Google Scholar 

  34. Y.Z. Wang, F.M. Li, K. Kishimoto, Thermal effects on vibration properties of double-layered nanoplates at small scales. Compos. B 42, 1311–1317 (2011)

    Google Scholar 

  35. R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  36. A.L. Dobyns, Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA J. 19, 642–650 (1981)

    ADS  MATH  Google Scholar 

  37. H. Irschik, Membrane-type eigenmotions of Mindlin plates. Acta Mech. 55, 1–20 (1985)

    MathSciNet  MATH  Google Scholar 

  38. R. Heuer, H. Irschik, A boundary element method for eigenvalue problems of polygonal membranes and plates. Acta Mech. 66, 9–20 (1987)

    MATH  Google Scholar 

  39. E. Reissner, On the theory of bending of elastic plates. J. Math. Phys. 23, 184–191 (1944)

    MathSciNet  MATH  Google Scholar 

  40. E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech. 12, 69–77 (1945)

    MathSciNet  MATH  Google Scholar 

  41. E. Reissner, On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids Struct. 11, 569–573 (1975)

    MATH  Google Scholar 

  42. M. Karama, K.S. Afaq, S. Mistou, Mechanical behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003)

    MATH  Google Scholar 

  43. J.N. Reddy, A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–52 (1984)

    ADS  MATH  Google Scholar 

  44. M. Touratier, An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991)

    MATH  Google Scholar 

  45. A.M. Zenkour, M.N.M. Allam, M.O. Shaker, A.F. Radwan, On the simple and mixed first-order theories for plates resting on elastic foundations. Acta Mech. 220, 33–46 (2011)

    MATH  Google Scholar 

  46. A.M. Zenkour, A.F. Radwan, On the simple and mixed first-order theories for functionally graded plates resting on elastic foundations. Meccanica 48, 1501–1516 (2013)

    MathSciNet  MATH  Google Scholar 

  47. A.M. Zenkour, Maupertuis–Lagrange mixed variational formula for laminated composite structures with a refined higher-order beam theory. Int. J. Non-Linear Mech. 32, 989–1001 (1997)

    ADS  MATH  Google Scholar 

  48. A.M. Zenkour, Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mech. 146, 183–197 (2001)

    MATH  Google Scholar 

  49. A.M. Zenkour, A state of stress and displacement of elastic plates using simple and mixed shear deformation theories. J. Eng. Math. 44, 1–20 (2002)

    MathSciNet  MATH  Google Scholar 

  50. A.M. Zenkour, Exact mixed-classical solutions for the bending analysis of shear deformable rectangular plates. Appl. Math. Model. 27, 515–534 (2003)

    MATH  Google Scholar 

  51. A.M. Zenkour, Bending of orthotropic plates resting on Pasternak’s foundations by mixed shear deformation theory. Acta Mech. Sin. 27, 956–962 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  52. M. Kashtalyan, Three-dimensional elasticity solution for bending of functionally graded rectangular plates. Eur. J. Mech. A Solids 23, 853–864 (2004)

    ADS  MATH  Google Scholar 

  53. J.N. Reddy, Z.Q. Cheng, Three-dimensional thermomechanical deformations of functionally graded rectangular plates. Eur. J. Mech. A/Solids 20, 841–855 (2001)

    ADS  MATH  Google Scholar 

  54. D.S. Mashat, A.M. Zenkour, A.F. Radwan, Aquasi 3-D higher-order plate theory for bending of FG plates resting on elastic foundations under hygro-thermo-mechanical loads with porosity. Eur. J. Mech. A/Solids 82, 103985 (2020)

  55. A.M. Zenkour, A novel mixed nonlocal theory for thermoelastic vibration of nanoplates. Compos. Struct. 185, 821–833 (2018)

    Google Scholar 

  56. J.N. Reddy, A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)

    ADS  MATH  Google Scholar 

  57. K.Y. Lam, C.M. Wang, X.Q. He, Canonical exact solution for Levyplates on two parameter foundation using Green’s functions. Eng. Struct. 22, 364–378 (2000)

    Google Scholar 

  58. S. Srinivas, A.K. Rao, Bending, vibration and buckling of simply-supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct. 6, 1463–1481 (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    A. M. Zenkour

  2. Department of Mathematics, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

    A. M. Zenkour

  3. Department of Mathematics and Statistics, Higher Institute of Management and Information Technology, Nile for Science and Technology, Kafrelsheikh, 33514, Egypt

    A. F. Radwan

Authors
  1. A. M. Zenkour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. F. Radwan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Zenkour.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zenkour, A.M., Radwan, A.F. Nonlocal mixed variational formula for orthotropic nanoplates resting on elastic foundations. Eur. Phys. J. Plus 135, 493 (2020). https://doi.org/10.1140/epjp/s13360-020-00504-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00504-7

Search

Navigation