# 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).

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

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

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

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

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**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