Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory

  • 37 Accesses

Abstract

This research investigates transient vibrational characteristics of a porous functionally graded cylindrical nanoshell under different impulsive loadings with the use of nonlocal strain gradient theory (NSGT). Based on NSGT, two size parameters accounting for stiffness softening and hardening effects are incorporated in modeling of the nanoshell. Impulse forces have three forms of triangular, rectangular and sinusoidal. Two sorts of porosity distributions called even and uneven have been taken into account. Governing equations obtained for porous nanoshell have been solved through inverse Laplace transforms technique to derive dynamical deflections. It is shown that transient responses of a nanoshell are affected by the form and position of impulse loading, amount of porosities, porosities dispensation, nonlocal and strain gradient parameters.

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

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

References

  1. 1.

    L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos. Struct. 94(6), 2038–2047 (2012)

  2. 2.

    Boutaleb et al., Dynamic Analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT. Adv. Nano Res. 7(3), 189–206 (2019)

  3. 3.

    M.A. Eltaher, S.A. Emam, F.F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218(14), 7406–7420 (2012)

  4. 4.

    Draoui et al., Static and dynamic behavior of nanotubes-reinforced sandwich plates using (FSDT). J. Nano Res. 57, 117–135 (2019)

  5. 5.

    Bellifa et al., A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams. Struct. Eng. Mech. 62(6), 695–702 (2017)

  6. 6.

    Cherif et al., Vibration analysis of nano beam using differential transform method including thermal effect. J. Nano Res. 54, 1–14 (2018)

  7. 7.

    Semmah et al., Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT. Adv. Nano Res. 7(2), 89–98 (2019)

  8. 8.

    Karami et al., Wave propagation of functionally graded anisotropic nanoplates resting on Winkler–Pasternak foundation. Struct. Eng. Mech. 7(1), 55–66 (2019)

  9. 9.

    Bounouara et al., A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel Compos. Struct. 20(2), 227–249 (2016)

  10. 10.

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

  11. 11.

    L. Li, Y. Hu, Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)

  12. 12.

    F. Ebrahimi, M.R. Barati, Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Compos. Struct. 159, 433–444 (2017)

  13. 13.

    X. Li, L. Li, Y. Hu, Z. Ding, W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos. Struct. 165, 250–265 (2017)

  14. 14.

    F. Ebrahimi, M.R. Barati, A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Compos. Struct. 159, 174–182 (2017)

  15. 15.

    Karami et al., Effects of triaxial magnetic field on the anisotropic nanoplates. Steel Compos. Struct. 25(3), 361–374 (2017)

  16. 16.

    M.R. Barati, N.M. Faleh, A.M. Zenkour, Dynamic response of nanobeams subjected to moving nanoparticles and hygro-thermal environments based on nonlocal strain gradient theory. Mech. Adv. Mater. Struct. 2, 1–9 (2018)

  17. 17.

    M.S.A. Houari, A. Bessaim, F. Bernard, A. Tounsi, S.R. Mahmoud, Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter. Steel. Compos. Struct. 28(1), 13–24 (2018)

  18. 18.

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

  19. 19.

    M.H. Ghayesh, A. Farajpour, Nonlinear coupled mechanics of nanotubes incorporating both nonlocal and strain gradient effects. Mech. Adv. Mater. Struct. 20, 1–10 (2018)

  20. 20.

    Y. Tang, T. Yang, Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material. Compos. Struct. 185, 393–400 (2018)

  21. 21.

    Y. Tang, X. Lv, T. Yang, Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration. Compos. B Eng. 156, 319–331 (2019)

  22. 22.

    G.L. She, F.G. Yuan, Y.R. Ren, H.B. Liu, W.S. Xiao, Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos. Struct. 203, 614–623 (2018)

  23. 23.

    F. Ebrahimi, M. Mokhtari, Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method. J. Braz. Soc. Mech. Sci. Eng. 37(4), 1435–1444 (2015)

  24. 24.

    N. Shafiei, A. Mousavi, M. Ghadiri, On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. Int. J. Eng. Sci. 106, 42–56 (2016)

  25. 25.

    H.A. Atmane, A. Tounsi, F. Bernard, S.R. Mahmoud, A computational shear displacement model for vibrational analysis of functionally graded beams with porosities. Steel Compos. Struct. 19(2), 369–384 (2015)

  26. 26.

    S.S. Mirjavadi, B.M. Afshari, N. Shafiei, A.M.S. Hamouda, M. Kazemi, Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams. Steel Compos. Struct. 25(4), 415–426 (2017)

  27. 27.

    N.M. Faleh, R.A. Ahmed, R.M. Fenjan, On vibrations of porous FG nanoshells. Int. J. Eng. Sci. 133, 1–14 (2018)

  28. 28.

    N.M. Faleh, R.M. Fenjan, R.A. Ahmed, Dynamic analysis of graded small-scale shells with porosity distributions under transverse dynamic loads. Eur. Phys. J. Plus 133(9), 348 (2018)

  29. 29.

    L. Li, H. Tang, Y. Hu, Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature. Compos. Struct. 184, 1177–1188 (2018)

  30. 30.

    F. Mehralian, Y.T. Beni, M.K. Zeverdejani, Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes. Phys. B 514, 61–69 (2017)

  31. 31.

    K. Mohammadi, M. Mahinzare, K. Ghorbani, M. Ghadiri, Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory. Microsyst. Technol. 24(2), 1133–1146 (2018)

  32. 32.

    M.R. Barati, Vibration analysis of porous FG nanoshells with even and uneven porosity distributions using nonlocal strain gradient elasticity. Acta Mech. 229(3), 1183–1196 (2018)

  33. 33.

    B. Karami, M. Janghorban, A. Tounsi, Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin-Walled Struct. 129, 251–264 (2018)

  34. 34.

    Y. Qu, S. Wu, H. Li, G. Meng, Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds: beams, plates and solids. Compos. B Eng. 73, 96–110 (2015)

  35. 35.

    Medani et al., Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate. Steel Compos. Struct. 32(5), 595–610 (2019)

  36. 36.

    Ait Atmane et al., Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations. Int. J. Mech. Mater. Des. 13(1), 71–84 (2017)

  37. 37.

    Zine et al., A novel higher-order shear deformation theory for bending and free vibration analysis of isotropic and multilayered plates and shells. Steel Compos. Struct. 26(2), 125–137 (2018)

  38. 38.

    Zarga et al., Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory. Steel Compos. Struct. 32(3), 389–410 (2019)

  39. 39.

    Chaabane et al., Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation. Struct. Eng. Mech. 71(2), 185–196 (2019)

  40. 40.

    Boukhlif et al., A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation. Steel Compos. Struct. 31(5), 503–516 (2019)

  41. 41.

    Bourada et al., Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory. Wind Struct. 28(1), 19–30 (2019)

  42. 42.

    Boulefrakh et al., The effect of parameters of visco-Pasternak foundation on the bending and vibration properties of a thick FG plate. Geomech. Eng. 18(2), 161–178 (2019)

  43. 43.

    Meksi et al., An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J. Sandw. Struct. Mater. 21(2), 727–757 (2019)

  44. 44.

    Bakhadda et al., Dynamic and bending analysis of carbon nanotube-reinforced composite plates with elastic foundation. Wind Struct. 27(5), 311–324 (2018)

  45. 45.

    Younsi et al., Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates. Geomech. Eng. 14(6), 519–532 (2018)

  46. 46.

    Abdelaziz et al., An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions. Steel Compos. Struct. 25(6), 693–704 (2017)

  47. 47.

    Y.S. Lee, K.D. Lee, On the dynamic response of laminated circular cylindrical shells under impulse loads. Comput. Struct. 63(1), 149–157 (1997)

Download references

Author information

Correspondence to Seyed Sajad Mirjavadi.

Appendix

Appendix

$$ k_{1,1} = A_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{A_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(44)
$$\begin{aligned} & k_{2,1} = n\left( {\frac{{A_{12} }}{R} + \frac{{A_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad \;k_{1,2} = - n\left( {\frac{{A_{12} }}{R} + \frac{{A_{66} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(45)
$$ k_{3,1} = + \frac{{A_{12} }}{R}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \;k_{1,3} = - \frac{{A_{12} }}{R}\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right), $$
(46)
$$ \begin{aligned} k_{4,1} = & + B_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{B_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \\ k_{1,4} = & B_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{B_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \\ \end{aligned} $$
(47)
$$ \begin{aligned} & k_{5,1} = n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad \;k_{1,5} = - n\left( {\frac{{B_{66} }}{R} + \frac{{B_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(48)
$$\begin{aligned} & k_{2,2} = A_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{A_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right) \\ & \qquad - \frac{{\tilde{A}_{66} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(49)
$$\begin{aligned} & k_{3,2} = - n\left( {\frac{{A_{11} }}{{R^{2} }} + \frac{{\tilde{A}_{66} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\\ & \qquad \;k_{2,3} = - n\left( {\frac{{\tilde{A}_{66} }}{{R^{2} }} + \frac{{A_{11} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(50)
$$\begin{aligned} & k_{4,2} = - n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\\ & \qquad \;k_{2,4} = + n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(51)
$$\begin{aligned} k_{5,2} &= B_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{B_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right)\\ & \quad + \frac{{\tilde{A}_{66} }}{R}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right),} \right)\end{aligned} $$
(52)
$$\begin{aligned} k_{3,3} &= \tilde{A}_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}}}}\varPsi_{20} } \right)} \right) - n^{2} \frac{{\tilde{A}_{66} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right) \\ & \quad- \frac{{A_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned}$$
(53)
$$\begin{aligned} & k_{4,3} = \left( {\tilde{A}_{66} - \frac{{B_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\\ & \qquad\;k_{3,4} = + \left( {\frac{{B_{12} }}{R} - \tilde{A}_{66} } \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(54)
$$\begin{aligned} & k_{5,3} = n\left( { + \frac{{\tilde{A}_{66} }}{R} - \frac{{B_{11} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\\ & \qquad \;k_{3,5} = - n\left( { + \frac{{B_{11} }}{{R^{2} }} - \frac{{\tilde{A}_{66} }}{R}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(55)
$$\begin{aligned} k_{4,4} & = + D_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{D_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right)\\ &\quad - \tilde{A}_{66} \left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(56)
$$\begin{aligned} & k_{5,4} = + n\left( {\frac{{D_{12} }}{R} + \frac{{D_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad\;k_{4,5} = - n\left( {\frac{{D_{66} }}{R} + \frac{{D_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(57)
$$\begin{aligned} k_{5,5}& = + D_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{D_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right)\\ & \quad - \tilde{A}_{66} \left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(58)
$$ m_{1,1} = + I_{0} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(59)
$$ m_{2,2} = m_{3,3} = m_{5,5} = + I_{0} \left( {\varPsi_{00} - ea^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right), $$
(60)
$$ m_{4,4} = + I_{2} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(61)
$$ m_{4,1} = + I_{1} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\;m_{5,2} = m_{2,5} = + I_{1} \left( {\varPsi_{00} - ea^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right), $$
(62)
$$ Q_{\text{dynamic}} = Q_{n} \varPsi_{00} , $$
(63)

in which

$$ \varPsi_{00} = \int_{0}^{L} {F_{m} } F_{m} {\text{d}}x, $$
(64)
$$ \varPsi_{20} = \int_{0}^{L} {\frac{{{\text{d}}^{2} F_{m}^{{}} }}{{{\text{d}}x^{2} }}} F_{m} {\text{d}}x, $$
(65)
$$ \varPsi_{11} = \int_{0}^{L} {\frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(66)
$$ \varPsi_{31} = \int\limits_{0}^{L} {\frac{{{\text{d}}^{3} F_{m}^{{}} }}{{{\text{d}}x^{3} }}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(67)
$$ \varPsi_{40} = \int_{0}^{L} {\frac{{{\text{d}}^{4} F_{m}^{{}} }}{{{\text{d}}x^{4} }}} F_{m} {\text{d}}x, $$
(68)
$$ \varPsi_{51} = \int_{0}^{L} {\frac{{{\text{d}}^{5} F_{m}^{{}} }}{{{\text{d}}x^{5} }}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(69)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Forsat, M., Badnava, S., Mirjavadi, S.S. et al. Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory. Eur. Phys. J. Plus 135, 81 (2020). https://doi.org/10.1140/epjp/s13360-019-00042-x

Download citation