## 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 to check access.

## 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) - 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) - 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) - 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) - 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) - 6.
H.M. Numanoǧlu, B. Akgöz, Ö. Civalek, On dynamic analysis of nanorods. Int. J. Eng. Sci.

**130**, 33–50 (2018) - 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) - 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) - 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 - 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 - 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) - 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) - 13.
B. Akgöz, Ö. Civalek, Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech.

**224**, 2185–2201 (2013) - 14.
B. Akgöz, Ö. Civalek, A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech.

**226**, 2277–2294 (2015) - 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) - 16.
A.C. Eringen,

*Nonlocal Continuum Field Theories*(Springer, New York, 2002) - 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) - 18.
E.C. Aifantis, Strain gradient interpretation of size effects. Int. J. Fract.

**95**, 299–314 (1999) - 19.
A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci.

**10**, 425–435 (1972) - 20.
A.C. Eringen, Linear theory of nonlocal microelasticity and dispersion of plane waves. Lett. Appl. Eng. Sci.

**1**, 129–146 (1973) - 21.
A.C. Eringen, Theory of nonlocal electromagnetic elastic solids. J. Math. Phys.

**14**, 733–740 (1973) - 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) - 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) - 24.
I.S. Radebe, S. Adali, Buckling and sensitivity analysis of nonlocal orthotropic nanoplates with uncertain material properties. Compos. B

**56**, 840–846 (2014) - 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) - 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) - 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) - 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) - 29.
M. Sobhy, Hygrothermal deformation of orthotropic nanoplates based on the state-space concept. Compos. B

**79**, 224–235 (2015) - 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) - 31.
N. Satish, S. Narendar, S. Gopalakrishnan, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics. Physica E

**44**, 1950–1962 (2012) - 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) - 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) - 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) - 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) - 36.
A.L. Dobyns, Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA J.

**19**, 642–650 (1981) - 37.
H. Irschik, Membrane-type eigenmotions of Mindlin plates. Acta Mech.

**55**, 1–20 (1985) - 38.
R. Heuer, H. Irschik, A boundary element method for eigenvalue problems of polygonal membranes and plates. Acta Mech.

**66**, 9–20 (1987) - 39.
E. Reissner, On the theory of bending of elastic plates. J. Math. Phys.

**23**, 184–191 (1944) - 40.
E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech.

**12**, 69–77 (1945) - 41.
E. Reissner, On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids Struct.

**11**, 569–573 (1975) - 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) - 43.
J.N. Reddy, A simple higher-order theory for laminated composite plates. J. Appl. Mech.

**51**, 745–52 (1984) - 44.
M. Touratier, An efficient standard plate theory. Int. J. Eng. Sci.

**29**, 901–916 (1991) - 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) - 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) - 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) - 48.
A.M. Zenkour, Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mech.

**146**, 183–197 (2001) - 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) - 50.
A.M. Zenkour, Exact mixed-classical solutions for the bending analysis of shear deformable rectangular plates. Appl. Math. Model.

**27**, 515–534 (2003) - 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) - 52.
M. Kashtalyan, Three-dimensional elasticity solution for bending of functionally graded rectangular plates. Eur. J. Mech. A Solids

**23**, 853–864 (2004) - 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) - 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) - 56.
J.N. Reddy, A simple higher-order theory for laminated composite plates. J. Appl. Mech.

**51**, 745–752 (1984) - 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) - 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)

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### 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

Received:

Accepted:

Published: