# Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Functionally Graded Porous Material

_{c}, $\rho $

_{m}and E

_{c}, E

_{m}are the material densities and the Euler–Young moduli of ceramic and metal, respectively.

- -
- even distribution

- -
- uneven distribution

- -
- even distribution

- -
- uneven distribution

_{y’}= bh

^{3}/12 is the second geometric moment of area of the cross-section about the y’ axis about the geometric center and A = bh is the cross-sectional area.

_{c}= 322.3 GPa) while, on the contrary, for $k$ → ∞, the material properties tend to pure metal (E = E

_{m}= 207.8 GPa).

## 3. Governing Equation

## 4. Stress-Driven Nonlocal Integral Model

#### 4.1. Equation of Motion (SDM)

#### Nonlinear Transverse Free Vibrations (SDM)

## 5. Convergence and Comparison Study

^{+}, 0.01, 0.03, 0.05, 0.1}.

_{NL}) and the local linear one (ω

_{L}), for a perfect simply-supported homogeneous nano-beams, varying the dimensionless amplitude of the nonlinear oscillator ${\mathcal{A}}_{w\text{}}/\mathrm{L}$, and neglecting the elastic foundation coefficient. The results obtained with the present approach are compared with the results of Singh et al., [63], based on the Ritz–Galerkin method (RGM), and with those of Mahmoudpou et al., [59], based on the first-order homotopy analysis method (HAM).

## 6. Results and Discussion

_{NL}, and the linear frequency of a pure ceramic nano-beam, ω

_{cL}, are presented.

#### 6.1. Gradient Index and Porosity Volume Fraction

#### 6.2. Nonlinear Oscillator Amplitude

_{NL}/ω

_{cL}increases as the absolute value of ${\mathcal{A}}_{w\text{}}$increases. Moreover, the curves exhibit a symmetric behavior when a perfect FG nano-beam is considered, while a different trend can be observed for an imperfect FG nano-beam (${\mathcal{A}}_{0}$ = 0.5). It is also found that the nonlinear frequency ratio increases as the porosity volume fraction $\zeta $ increases and that, for a given value of $\zeta $, the ratio ω

_{NL}/ω

_{cL}also increases as the coefficient ${\tilde{K}}_{w}\text{}$increases.

#### 6.3. Winkler Elastic Foundation Coefficient

## 7. Conclusions

- (1)
- the increase in the gradient index, as well as in the porosity volume fraction, cause a decrease in the values of the axial and bending stiffness of porous FG nano-beams;
- (2)
- an increase in the porosity volume fraction of perfect FG nano-beams causes an increase in the nonlinear frequency ratio when $k<1\text{};\text{}$an opposite trend was observed when $k>1$;
- (3)
- the nonlinear frequencies of imperfect porous FG nano-beams are always greater than those obtained for perfect nano-beams;
- (4)
- the nonlinear frequency ratio always increases as the porosity volume fraction increases in the case of an uneven distribution of porosity;
- (5)
- an increase in the elastic foundation coefficient and in the initial imperfection amplitude causes an increase in the nonlinear frequency ratio.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Solution Procedure

_{L}, of a perfect FG nano-beam.

**Table A1.**Expressions of$\mathrm{the}\text{}\mathrm{four}\text{}\mathrm{coefficients}\text{}{\delta}_{1},{\delta}_{2},{\delta}_{3}\text{}\mathrm{and}\text{}{\delta}_{4}\text{}\mathrm{of}\text{}$ Equation (A12).

$\begin{array}{c}{\delta}_{1}={\int}_{0}^{1}(-{\tilde{K}}_{w}{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}\left(\tilde{x}\right)+{\tilde{r}}^{2}({\int}_{0}^{1}({\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right)-2{\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(2\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(2\right)}\left(\tilde{x}\right)-{\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{1}{}^{\left(3\right)}\left(\tilde{x}\right)-\\ {\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(3\right)}\left(\tilde{x}\right))dx){\mathcal{A}}_{0}{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{1}{}^{\left(2\right)}\left(\tilde{x}\right)-{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(4\right)}\left(\tilde{x}\right)+{\lambda}^{2}{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(6\right)}\left(\tilde{x}\right))d\tilde{x},\end{array}$ |

$\begin{array}{cc}\hfill {\delta}_{2}={{\displaystyle \int}}_{0}^{1}\left({r}^{2}\right({\int}_{0}^{1}(\frac{1}{2}& {\Psi}_{i}{}^{\left(1\right)}{\left(\tilde{x}\right)}^{2}-{\lambda}^{2}{\Psi}_{i}{}^{\left(2\right)}{\left(\tilde{x}\right)}^{2}-{\lambda}^{2}{\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(3\right)}\left(\tilde{x}\right)\left)d\tilde{x}\right){\mathcal{A}}_{0}{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{1}{}^{\left(2\right)}\left(\tilde{x}\right)\hfill \\ & +{\tilde{r}}^{2}\left({\int}_{0}^{1}\right({\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right)-2{\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(2\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(2\right)}\left(\tilde{x}\right)-{\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{1}{}^{\left(3\right)}\left(\tilde{x}\right)\hfill \\ & -{\lambda}^{2}{\mathcal{A}}_{0}{\Psi}_{1}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(3\right)}\left(\tilde{x}\right)\left)d\tilde{x}\right){\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(2\right)}\left(\tilde{x}\right))d\tilde{x},\hfill \end{array}$ |

${\delta}_{3}={{\displaystyle \int}}_{0}^{1}{\tilde{r}}^{2}\left({\int}_{0}^{1}\left(\frac{1}{2}{\Psi}_{i}{}^{\left(1\right)}{\left(\tilde{x}\right)}^{2}-{\lambda}^{2}{\Psi}_{i}{}^{\left(2\right)}{\left(\tilde{x}\right)}^{2}-{\lambda}^{2}{\Psi}_{i}{}^{\left(1\right)}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(3\right)}\left(\tilde{x}\right)\right)d\tilde{x}\right){\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}{}^{\left(2\right)}\left(\tilde{x}\right)d\tilde{x},$ |

${\delta}_{4}={\int}_{0}^{1}-{\tilde{m}}_{0}{\Psi}_{1}\left(\tilde{x}\right){\Psi}_{i}\left(\tilde{x}\right)d\tilde{x}$. |

## References

- Saleh, B.; Jiang, J.; Fathi, R.; Al-Hababi, T.; Xu, Q.; Wang, L.; Song, D.; Ma, A. 30 Years of functionally graded materials: An overview of manufacturing methods, Applications and Future Challenges. Compos. Part B Eng.
**2020**, 201, 108376. [Google Scholar] [CrossRef] - Mahmoud, N.; Zahra, I.; Mohaddeseh, S.; Monireh, A. Chapter 2-Types of Nanostructures. Interface Sci. Technol.
**2019**, 28, 29–80. [Google Scholar] - El-Galy, I.M.; Saleh, B.I.; Ahmed, M.H. Functionally graded materials classifications and development trends from industrial point of view. SN Appl. Sci.
**2019**, 1, 1378. [Google Scholar] [CrossRef][Green Version] - Kumar, R.; Singh, R.; Hui, D.; Feo, L.; Fraternali, F. Graphene as biomedical sensing element: State of art review and potential engineering applications. Compos. Part B Eng.
**2018**, 134, 193–206. [Google Scholar] [CrossRef] - Kar, V.R.; Panda, S.K. Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel. Steel Compos. Struct.
**2015**, 18, 693–709. [Google Scholar] [CrossRef] - Avcar, M. Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam. Struct. Eng. Mech.
**2015**, 55, 871–884. [Google Scholar] [CrossRef] - Barati, M.R.; Shahverdi, H. A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions. Struct. Eng. Mech.
**2016**, 60, 707–727. [Google Scholar] [CrossRef] - Houari, M.S.A.; Tounsi, A.; Bessaim, A.; Mahmoud, S. A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates. Steel Compos. Struct.
**2016**, 22, 257–276. [Google Scholar] [CrossRef] - Sekkal, M.; Fahsi, B.; Tounsi, A.; Mahmoud, S.R. A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate. Steel Compos. Struct.
**2017**, 25, 389–401. [Google Scholar] - Avcar, M.; Mohammed, W.K.M. Free vibration of functionally graded beams resting on Winkler-Pasternak foundation. Arab. J. Geosci.
**2018**, 11, 232. [Google Scholar] [CrossRef] - Tlidji, Y.; Zidour, M.; Draiche, K.; Safa, A.; Bourada, M.; Tounsi, A.; Bousahla, A.A.; Mahmoud, S.R. Vibration analysis of different material distributions of functionally graded microbeam. Struct. Eng. Mech. Int. J.
**2019**, 69, 637–649. [Google Scholar] - Karami, B.; Shahsavari, D.; Janghorban, M.; Tounsi, A. Resonance behavior of functionally graded polymer composite nanoplates reinforced with grapheme nanoplatelets. Int. J. Mech. Sci.
**2019**, 156, 94–105. [Google Scholar] [CrossRef] - Meksi, R.; Benyoucef, S.; Mahmoudi, A.; Tounsi, A.; Bedia, E.A.A.; Mahmoud, S. An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J. Sandw. Struct. Mater.
**2017**, 21, 727–757. [Google Scholar] [CrossRef] - Cortes, H.J.; Pfeiffer, C.D.; Richter, B.E.; Stevens, T.S. Porous ceramic bed supports for fused silica packed capillary columns used in liquid chromatography. J. High. Resolut. Chromatogr.
**1987**, 10, 446–448. [Google Scholar] [CrossRef] - Kresge, C.T.; Leonowicz, M.E.; Roth, W.J.; Vartuli, J.C.; Beck, J.S. Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism. Nat. Cell Biol.
**1992**, 359, 710–712. [Google Scholar] [CrossRef] - Beck, J.S.; Vartuli, J.C.; Roth, W.J.; Leonowicz, M.E.; Kresge, C.T.; Schmitt, K.D.; Chu, C.T.W.; Olson, D.H.; Sheppard, E.W.; McCullen, S.B.; et al. A new family of mesoporous molecular sieves prepared with liquid crystal templates. J. Am. Chem. Soc.
**1992**, 114, 10834–10843. [Google Scholar] [CrossRef] - Velev, O.D.; Jede, T.A.; Lobo, R.F.; Lenhoff, A.M. Porous silica via colloidal crystallization. Nat. Cell Biol.
**1997**, 389, 447–448. [Google Scholar] [CrossRef] - Wattanasakulpong, N.; Prusty, B.G.; Kelly, D.W.; Hoffman, M. Free vibration analysis of layered functionally graded beams with experimental validation. Mater. Des.
**2012**, 36, 182–190. [Google Scholar] [CrossRef] - Lefebvre, L.-P.; Banhart, J.; Dunand, D.C. Porous Metals and Metallic Foams: Current Status and Recent Developments. Adv. Eng. Mater.
**2008**, 10, 775–787. [Google Scholar] [CrossRef][Green Version] - Smith, B.; Szyniszewski, S.; Hajjar, J.; Schafer, B.; Arwade, S. Steel foam for structures: A review of applications, manufacturing and material properties. J. Constr. Steel Res.
**2012**, 71, 1–10. [Google Scholar] [CrossRef][Green Version] - Zhao, C. Review on thermal transport in high porosity cellular metal foams with open cells. Int. J. Heat Mass Transf.
**2012**, 55, 3618–3632. [Google Scholar] [CrossRef] - Betts, C. Benefits of metal foams and developments in modelling techniques to assess their materials bahaviour: A review. Mater. Sci. Technol.
**2012**, 28, 129–143. [Google Scholar] [CrossRef] - Jena, S.K.; Chakraverty, S.; Tornabene, F. Dynamical behavior of nanobeam embedded in constant, linear, parabolic, and sinusoidal types of Winkler elastic foundation using first-Order nonlocal strain gradient model. Mater. Res. Express
**2019**, 6, 0850f2. [Google Scholar] [CrossRef] - Togun, N.; Bağdatlı, S.M. Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory. Math. Comput. Appl.
**2016**, 21, 3. [Google Scholar] [CrossRef][Green Version] - Zenkour, A.; Ebrahimi, F.; Barati, M.R. Buckling analysis of a size-dependent functionally graded nanobeam resting on Pasternak’s foundations. Int. J. Nano Dimens.
**2019**, 10, 141–153. [Google Scholar] - Aydogdu, M.; Gul, U. Buckling analysis of double nanofibers embeded in an elastic medium using doublet mechanics theory. Compos. Struct.
**2018**, 202, 355–363. [Google Scholar] [CrossRef] - Kara, H.F.; Aydogdu, M. Dynamic response of a functionally graded tube embedded in an elastic medium due to SH-Waves. Compos. Struct.
**2018**, 206, 22–32. [Google Scholar] [CrossRef] - Lam, D.C.; Yang, F.; Chong, A.; Wang, J.; Tong, P. Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids
**2003**, 51, 1477–1508. [Google Scholar] [CrossRef] - Acierno, S.; Barretta, R.; Luciano, R.; De Sciarra, F.M.; Russo, P. Experimental evaluations and modeling of the tensile behavior of polypropylene/single-walled carbon nanotubes fibers. Compos. Struct.
**2017**, 174, 12–18. [Google Scholar] [CrossRef] - Chiu, H.-Y.; Hung, P.; Postma, H.W.C.; Bockrath, M. Atomic-Scale Mass Sensing Using Carbon Nanotube Resonators. Nano Lett.
**2008**, 8, 4342–4346. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Chou, T.-W. Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mech. Mater.
**2004**, 36, 1047–1055. [Google Scholar] [CrossRef] - Maranganti, R.; Sharma, P. A novel atomistic approach to determine strain gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and their relevance for nanotechnologies. J. Mech. Phys. Solids
**2007**, 55, 1823–1852. [Google Scholar] [CrossRef] - Aifantis, E. Internal Length Gradient (ILG) Material Mechanics Across Scales and Disciplines. Adv. Appl. Mech.
**2016**, 49, 1–110. [Google Scholar] [CrossRef] - Mindlin, R. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct.
**1965**, 1, 417–438. [Google Scholar] [CrossRef] - Eringen, A. 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, 4703–4710. [Google Scholar] [CrossRef] - Eringen, A.C. Theory of Nonlocal Elasticity and Some Applications. Princet. Univ. NJ Dept. Civ. Eng.
**1984**, 21, 313–342. [Google Scholar] [CrossRef] - Peddieson, J.; Buchanan, G.R.; McNitt, R.P. Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci.
**2003**, 41, 305–312. [Google Scholar] [CrossRef] - Romano, G.; Barretta, R.; Diaco, M.; Marotti de Sciarra, F. Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int. J. Mech. Sci.
**2017**, 121, 151–156. [Google Scholar] [CrossRef] - Lim, C.; Zhang, G.; Reddy, J. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids
**2015**, 78, 298–313. [Google Scholar] [CrossRef] - Gurtin, M.E.; Murdoch, A.I. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal.
**1975**, 57, 291–323. [Google Scholar] [CrossRef] - Zhu, X.; Li, L. Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos. Struct.
**2017**, 178, 87–96. [Google Scholar] [CrossRef] - Li, L.; Li, X.; Hu, Y. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci.
**2016**, 102, 77–92. [Google Scholar] [CrossRef] - Zhu, X.; Li, L. A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect. Appl. Math. Mech.
**2019**, 40, 1561–1588. [Google Scholar] [CrossRef] - Romano, G.; Barretta, R. Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos. Part B Eng.
**2017**, 114, 184–188. [Google Scholar] [CrossRef] - Barretta, R.; Fazelzadeh, S.; Feo, L.; Ghavanloo, E.; Luciano, R. Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type. Compos. Struct.
**2018**, 200, 239–245. [Google Scholar] [CrossRef] - Barretta, R.; Luciano, R.; De Sciarra, F.M.; Ruta, G.C. Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur. J. Mech. A Solids
**2018**, 72, 275–286. [Google Scholar] [CrossRef] - Barretta, R.; Diaco, M.; Feo, L.; Luciano, R.; De Sciarra, F.M.; Penna, R. Stress-driven integral elastic theory for torsion of nano-beams. Mech. Res. Commun.
**2018**, 87, 35–41. [Google Scholar] [CrossRef] - Barretta, R.; Čanađija, M.; Luciano, R.; De Sciarra, F.M. Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. Int. J. Eng. Sci.
**2018**, 126, 53–67. [Google Scholar] [CrossRef][Green Version] - Barretta, R.; Fabbrocino, F.; Luciano, R.; De Sciarra, F.M. Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams. Phys. E Low-Dimens. Syst. Nanostruct.
**2018**, 97, 13–30. [Google Scholar] [CrossRef] - Barretta, R.; Faghidian, S.A.; Luciano, R.; Medaglia, C.M.; Penna, R. Stress-driven two-phase integral elasticity for torsion of nano-beams. Compos. Part B Eng.
**2018**, 145, 62–69. [Google Scholar] [CrossRef] - Barretta, R.; Faghidian, S.A.; Luciano, R. Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech. Adv. Mater. Struct.
**2018**, 26, 1307–1315. [Google Scholar] [CrossRef] - Darban, H.; Fabbrocino, F.; Feo, L.; Luciano, R. Size-dependent buckling analysis of nanobeams resting on two-parameter elastic foundation through stress-driven nonlocal elasticity model. Mech. Adv. Mater. Struct.
**2020**, 1–9. [Google Scholar] [CrossRef] - Apuzzo, A.; Barretta, R.; Luciano, R.; De Sciarra, F.M.; Penna, R. Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model. Compos. Part B Eng.
**2017**, 123, 105–111. [Google Scholar] [CrossRef] - Barretta, R.; Čanađija, M.; Feo, L.; Luciano, R.; De Sciarra, F.M.; Penna, R. Exact solutions of inflected functionally graded nano-beams in integral elasticity. Compos. Part B Eng.
**2018**, 142, 273–286. [Google Scholar] [CrossRef] - Penna, R.; Feo, L.; Fortunato, A.; Luciano, R. Nonlinear free vibrations analysis of geometrically imperfect FG nano-beams based on stress-driven nonlocal elasticity with initial pretension force. Compos. Struct.
**2020**, 255, 112856. [Google Scholar] [CrossRef] - He, J.-H. Hamiltonian approach to nonlinear oscillators. Phys. Lett. A
**2010**, 374, 2312–2314. [Google Scholar] [CrossRef] - Dvorak, G. Micromechanics of Composite Materials; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Mahmoudpour, E.; Hosseini-Hashemi, S.H.; Faghidian, S.A. Non linear vibration of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl. Math. Model.
**2018**, 57, 302–315. [Google Scholar] [CrossRef] - Aifantis, E. Update on a class of gradient theories. Mech. Mater.
**2003**, 35, 259–280. [Google Scholar] [CrossRef] - Liu, H.; Lv, Z.; Wu, H. Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory. Compos. Struct.
**2019**, 214, 47–61. [Google Scholar] [CrossRef] - Dehrouyeh-Semnani, A.M.; Mostafaei, H.; Nikkhah-Bahrami, M. Free flexural vibration of geometrically imperfect functionally graded microbeams. Int. J. Eng. Sci.
**2016**, 105, 56–79. [Google Scholar] [CrossRef] - Singh, G.; Sharma, A.; Rao, G.V. Large-amplitude free vibrations of beams—A discussion on various formulations and assumptions. J. Sound Vib.
**1990**, 142, 77–85. [Google Scholar] [CrossRef] - Apuzzo, A.; Barretta, R.; Faghidian, S.A.; Luciano, R.; De Sciarra, F.M. Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int. J. Eng. Sci.
**2018**, 133, 99–108. [Google Scholar] [CrossRef]

**Figure 1.**Coordinate system and configuration of a porous FG Bernoulli–Euler nano-beam resting on an elastic foundation: even (

**A**) and uneven (

**B**) porosity distributions across the thickness of the nano-beam.

**Figure 2.**Effect of the gradient index of the FG material ($k$) on the dimensionless position of the elastic center C ($\text{}{\tilde{z}}_{c}={z}_{c}^{\prime}/h)\text{}$ of the rectangular cross section of the FG nano-beam, varying the porosity volume fraction ($\zeta $): even distribution (continuous lines) and uneven distribution (dashed lines).

**Figure 3.**Variations of the dimensionless axial stiffness, $\tilde{EA}=\text{}\frac{EA}{{E}_{c}A}$ (

**A**), and of the bending stiffness, $\tilde{EI}=\text{}\frac{EI}{{E}_{c}I}$ (

**B**). The variations are shown for the porous FG nano-beam in terms of the gradient index of the FG material ($k$), varying the porosity volume fraction ($\zeta $): even distribution (continuous lines) and uneven distribution (dashed lines).

**Figure 4.**Variations of the dimensionless cross-sectional mass (

**A**) and of the rotatory inertia (

**B**) of a porous FG nano-beam. The variations are shown in terms of the gradient index of the FG material ($k$), varying the porosity volume fraction ($\zeta $): even distribution (continuous lines) and uneven distribution (dashed lines).

**Figure 5.**Variations of the Euler–Young modulus (E) in terms of dimensionless thickness $\left(\tilde{z}=z/h\right)\text{}$with k ranging in the set {0, 0.3, 0.5, 1, 3, 5, → ∞} and $\zeta $ = 0.2: (

**A**) FG material (continuous lines); (

**B**) even distribution (dashed lines); and (

**C**) uneven distribution (dashed-points lines).

**Figure 6.**Simply-supported porous FG nano-beams: 3D-plot of the frequency ratio in terms of porosity volume fraction ($\zeta $) and gradient index ($k$) with$\text{}{\mathcal{A}}_{w\text{}}=$ 0.2, ${\mathcal{A}}_{0\text{}}=$ 0.0 and ${\tilde{K}}_{w}=$ 0: (

**A**) even distribution of porosity, and (

**B**) uneven distribution of porosity.

**Figure 7.**Simply-supported porous FG nano-beams: frequency ratio in terms of porosity volume fraction ($\zeta $) and gradient index ($k$) with$\text{}{\mathcal{A}}_{w\text{}}=$ 0.2, ${\mathcal{A}}_{0\text{}}=$ 0.0 and ${\tilde{K}}_{w}=$ 0: (

**A**) even distribution of porosity and $k$ < 1, and (

**B**) even distribution of porosity and $k$ > 1.

**Figure 8.**Simply-supported porous FG nano-beams: frequency ratio in terms of porosity volume fraction ($\zeta $) and gradient index ($k$) with$\text{}{\mathcal{A}}_{w\text{}}=$ 0.2, ${\mathcal{A}}_{0\text{}}=$ 0.0 and ${\tilde{K}}_{w}=$ 0: (

**A**) uneven distribution of porosity and $k$ < 1, and (

**B**) uneven distribution of porosity and $k$ > 1.

**Figure 9.**Simply-supported porous FG nano-beams. Frequency ratio versus porosity volume fraction with k ranging in the set {0, 0.3, 0.5, 1, 3, 5} and ${\tilde{K}}_{w}$ varying in the set {0, 20}: (

**A**) even distribution of porosity and (

**B**) uneven distribution of porosity.

**Figure 10.**Effect of porosity volume fraction ($\zeta $) and Winkler elastic foundation coefficient (${K}_{w}$) on the frequency ratio of simply-supported (SS) porous perfect $({\mathcal{A}}_{0\text{}}$ = 0.0) FG nano-beams: (

**A**) even distribution of porosity (blue lines), (

**B**) uneven distribution of porosity (red lines). Effect of the porosity volume fraction ($\zeta $) and the Winkler elastic foundation coefficient (${K}_{w}$) on the frequency ratio of simply-supported (SS) porous imperfect $({\mathcal{A}}_{0\text{}}$ = 0.5) FG nano-beams: (

**C**) even distribution of porosity (blue lines), and (

**D**) uneven distribution of porosity (red lines).

**Figure 11.**Simply-supported (SS) porous FG nano-beams. Frequency ratio versus Winkler elastic foundation coefficient, ${\mathcal{A}}_{w\text{}}=$ 0.2 and k = 1, and different initial imperfection amplitudes: (

**A**) even distribution of porosity, and (

**B**) uneven distribution of porosity.

Material | Young’s Modulus [GPa] | Poisson’s Ratio | Mass Density [kg/m^{3}] |
---|---|---|---|

SuS3O4 (stainless steel) | 207.8 | 0.3178 | 8166 |

Si3N4 (silicon nitride) | 322.3 | 0.24 | 2370 |

**Table 2.**Linear dimensionless natural frequencies of homogeneous simply-supported (S-S) and clamped-clamped (C-C) nano-beams (${\mathcal{A}}_{w}=\text{}{\mathcal{A}}_{0}=0;\text{}{\tilde{K}}_{w}=0).$

λ | S-S | C-C | ||
---|---|---|---|---|

Present Approach | Ref. [55] | Present Approach | Ref. [55] | |

0.00+ | 9.8696 | 9.8696 | 22.3733 | 22.3733 |

0.01 | 9.8744 | 9.8744 | 22.8518 | 22.8518 |

0.03 | 9.9107 | 9.9107 | 23.9976 | 23.9976 |

0.05 | 9.9786 | 9.9786 | 25.3918 | 25.3918 |

0.1 | 10.2534 | 10.2534 | 29.8303 | 29.8303 |

**Table 3.**Linear dimensionless natural frequencies of homogeneous simply-supported (S-S) and clamped-clamped (C-C) nano-beams (${\mathcal{A}}_{w}=\text{}{\mathcal{A}}_{0}=0;\text{}{\tilde{K}}_{w}=50).$

λ | S-S | C-C | ||
---|---|---|---|---|

Present Approach | Ref. [59] | Present Approach | Ref. [59] | |

0.00+ | 12.1412 | 12.1412 | 23.4641 | 23.4641 |

0.01 | 12.1451 | 12.1451 | 23.9208 | 23.9208 |

0.03 | 12.1747 | 12.1747 | 25.0177 | 25.0177 |

0.05 | 12.2300 | 12.2300 | 26.3579 | 26.3579 |

0.1 | 12.4552 | 12.4552 | 30.6569 | 30.6569 |

**Table 4.**Local nonlinear frequency ratios of perfect simply-supported (S-S) nano-beams ($\lambda =0;\text{}{\mathcal{A}}_{0}=0;\text{}{\tilde{K}}_{w}=0).$

${\mathcal{A}}_{\mathit{w}}/\mathit{L}$ | Present Approach | RGM Ref. [63] | HAM Ref. [59] | ||
---|---|---|---|---|---|

ω_{NL} | ω_{L} | ω_{NL}/ω_{L} | ω_{NL}/ω_{L} | ω_{NL}/ω_{L} | |

1.0 | 10.7552 | 9.8696 | 1.0897 | 1.0897 | 1.0892 |

2.0 | 13.0563 | 9.8696 | 1.3229 | 1.3229 | 1.3178 |

3.0 | 16.1798 | 9.8696 | 1.6394 | 1.6394 | 1.6259 |

4.0 | 19.7392 | 9.8696 | 2.0000 | - | 1.9766 |

**Table 5.**Nonlocal dimensionless natural frequencies of a homogeneous simply-supported (S-S) nano-beam resting on an elastic foundation for various dimensionless amplitude ratios ($\lambda =0.1;\text{}{\mathcal{A}}_{0}=0;\text{}{\tilde{K}}_{w}=50,\text{}100).$

${\mathcal{A}}_{\mathit{w}}/\mathit{L}$ | ${\tilde{\mathit{K}}}_{\mathit{w}}=50$ | ${\tilde{\mathit{K}}}_{\mathit{w}}=100$ | ||||
---|---|---|---|---|---|---|

Present Approach | Ref. [59] | Δ_{1} [%] | Present Approach | Ref. [59] | Δ_{2} [%] | |

0.0 | 12.4552 | 12.4552 | 0.00 | 14.3225 | 14.3225 | 0.00 |

0.1 | 12.4694 | - | - | 14.3348 | - | - |

0.2 | 12.5120 | - | - | 14.3719 | - | - |

0.3 | 12.5827 | 12.5639 | 0.15 | 14.4334 | 14.4171 | 0.11 |

0.4 | 12.6810 | - | - | 14.5192 | - | - |

0.5 | 12.8062 | - | - | 14.6287 | - | - |

0.6 | 12.9576 | 12.8835 | 0.57 | 14.7614 | 14.6967 | 0.44 |

**Table 6.**Nonlocal dimensionless natural frequencies of a homogeneous clamped-clamped (C-C) nano-beam resting on an elastic foundation for various dimensionless amplitude ratios ($\lambda =0.1;\text{}{\mathcal{A}}_{0}=0;\text{}{\tilde{K}}_{w}=50,\text{}100).$

${\mathcal{A}}_{\mathit{w}}/\mathit{L}$ | ${\tilde{\mathit{K}}}_{\mathit{w}}=50$ | ${\tilde{\mathit{K}}}_{\mathit{w}}=100$ | ||||
---|---|---|---|---|---|---|

Present Approach | Ref. [59] | Δ_{1} [%] | Present Approach | Ref. [59] | Δ_{2} [%] | |

0.0 | 30.6591 | 30.6575 | 0.01 | 31.4640 | 31.4624 | 0.01 |

0.1 | 30.6606 | - | - | 31.4654 | - | - |

0.2 | 30.6651 | - | - | 31.4698 | - | - |

0.3 | 30.6725 | 30.7440 | 0.23 | 31.4770 | 31.5467 | 0.22 |

0.4 | 30.6830 | - | - | 31.4872 | - | - |

0.5 | 30.6964 | - | - | 31.5003 | - | - |

0.6 | 30.7128 | 31.0017 | 0.93 | 31.5163 | 31.7980 | 0.89 |

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## Share and Cite

**MDPI and ACS Style**

Penna, R.; Feo, L.
Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation. *Technologies* **2020**, *8*, 56.
https://doi.org/10.3390/technologies8040056

**AMA Style**

Penna R, Feo L.
Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation. *Technologies*. 2020; 8(4):56.
https://doi.org/10.3390/technologies8040056

**Chicago/Turabian Style**

Penna, Rosa, and Luciano Feo.
2020. "Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation" *Technologies* 8, no. 4: 56.
https://doi.org/10.3390/technologies8040056