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
- -
- even distribution
- -
- uneven distribution
- -
- even distribution
- -
- uneven distribution
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
6. Results and Discussion
6.1. Gradient Index and Porosity Volume Fraction
6.2. Nonlinear Oscillator Amplitude
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 an opposite trend was observed when ;
- (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
. |
References
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Material | Young’s Modulus [GPa] | Poisson’s Ratio | Mass Density [kg/m3] |
---|---|---|---|
SuS3O4 (stainless steel) | 207.8 | 0.3178 | 8166 |
Si3N4 (silicon nitride) | 322.3 | 0.24 | 2370 |
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 |
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 |
Present Approach | RGM Ref. [63] | HAM Ref. [59] | |||
---|---|---|---|---|---|
ωNL | ωL | ωNL/ωL | ωNL/ωL | ωNL/ωL | |
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 |
Present Approach | Ref. [59] | Δ1 [%] | Present Approach | Ref. [59] | Δ2 [%] | |
---|---|---|---|---|---|---|
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 |
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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
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 StylePenna, 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
APA StylePenna, R., & Feo, L. (2020). Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation. Technologies, 8(4), 56. https://doi.org/10.3390/technologies8040056