Fig. 1.
Dimensions of a GPLRC beam (a), different types of distributions (b)
Tab. 1.
Material constituents and properties
| Materials | Piezoelectric | GPLs |
|---|---|---|
| E(Gpa) | 1.4 | 1010 |
| v | 0.29 | 0.186 |
| ρ(g/cm3) | 1.92 | 1.06 |
| α(10–6K–1) | 60 | 5 |
| A31(10–3C/m2) | 50.535 | 50.535 e0 |
| A33(10–3C/m2) | 13.212 | 13.212 e0 |
| A15(10–3C/m2) | -15.93 | -15.93 e0 |
| s11(10–9C/Vm) | 0.5385 | 0.5385 e0 |
| s33(10–9C/Vm) | 0.59571 | 0.59571 e0 |
Tab. 2.
Convergence study of DQFEM related to linear free vibration nanocomposite beam armed with GPLs
| Ne | N | UD | FG-X | FG-O | FG-A |
|---|---|---|---|---|---|
| 1 | 4 | 0.3061 | 0.3637 | 0.2351 | 0.3062 |
| 6 | 0.2742 | 0.3258 | 0.2105 | 0.2674 | |
| 8 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 10 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 2 | 4 | 0.2752 | 0.3270 | 0.2113 | 0.2686 |
| 6 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 8 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 10 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 3 | 4 | 0.2743 | 0.3259 | 0.2106 | 0.2676 |
| 6 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 8 | 0.2741 | 0.3257 | 0.2104 | 0.2673 | |
| 10 | 0.2741 | 0.3257 | 0.2104 | 0.2673 |
Fig. 2.
Convergent of the vibration frequency of a piezoelectric beam armed with GPLs as a function of the amount of grid points
Fig. 3.
The convergence of the natural frequency of a piezoelectric beam reinforced with graphene platelets as a variable dependent on the quantity of components
Tab. 3.
Comparative examination of the natural frequencies of various boundary conditions with varying L/h ratios
| BC | L/h=10 | L/h=30 | L/h=100 | |
|---|---|---|---|---|
| S-S | Şimşek [37] | 2.695 | 2.737 | 2.742 |
| Present | 2.739 | 2.775 | 2.779 | |
| C-F | Şimşek [37] | 0.969 | 0.976 | 0.977 |
| Present | 0.976 | 0.982 | 0.983 | |
| C-C | Şimşek [37] | 5.811 | 6.167 | 6.212 |
| Present | 5.947 | 6.242 | 6.279 |
Tab. 4.
Comparative of non-dimensional frequency with Wu et al. 7 for various GPL distributions at ∆T = 0 K, L/H = 10, and WGPL=0.3%
| Pure epoxy | UD | FG-X | FG-O | FG-A | |
|---|---|---|---|---|---|
| Wu et al.[6] | 0.5998 | 0.8475 | 0.9293 | 0.7508 | 0.8164 |
| Present | 0.5977 | 0.8445 | 0.9300 | 0.7401 | 0.8158 |
Tab. 5.
Comparative of the non-dimensional fundamental frequency ω1 for Ps/Pcr=0 between the present results and those of Wu et al. under different temperature conditions
| ∆T | Present | Wu et al.[6] |
|---|---|---|
| 0 K | 0.9666 | 0.9289 |
| 50 K | 0.9275 | 0.8883 |
| 100 K | 0.8865 | 0.8501 |
Fig. 4.
The no-dimensional frequency in relation to the quantity of layers (NL) with respect to various forms
Tab. 6.
Dynamic results of FG-GPLRC piezoelectric beam diverse types of distribution and different values of the length-to-thikness pro-portion L/h
| WGPL | Patterns | UD | FG-X | FG-O | FG-A |
|---|---|---|---|---|---|
| 0.1% | L/h=5 | 0.1581 | 0.1765 | 0.1373 | 0.1569 |
| L/h =10 | 0.0448 | 0.0448 | 0.0348 | 0.0398 | |
| L/h 15 | 0.0179 | 0.0200 | 0.0155 | 0.0177 | |
| L/h =20 | 0.0101 | 0.0112 | 0.0087 | 0.0100 | |
| 0.3% | L/h=5 | 0.2237 | 0.2618 | 0.1776 | 0.2194 |
| L/h =10 | 0.0567 | 0.0665 | 0.0450 | 0.0557 | |
| L/h 15 | 0.0253 | 0.0296 | 0.0201 | 0.0248 | |
| L/h =20 | 0.0142 | 0.0167 | 0.0113 | 0.0140 | |
| 0.5% | L/h=5 | 0.2741 | 0.3257 | 0.2104 | 0.2673 |
| L/h =10 | 0.0695 | 0.0827 | 0.0533 | 0.0678 | |
| L/h 15 | 0.0310 | 0.0368 | 0.0238 | 0.0302 | |
| L/h =20 | 0.0174 | 0.0207 | 0.0134 | 0.0170 |
Tab. 7.
Dynamic Change in the non-dimensional frequency of the S-S beams for different temperatures changes, different patterns and various values for weight fraction
| ∆T | wGPL | 0.1% | 0.3% | 0.5% |
|---|---|---|---|---|
| 0 | UD | 0.1581 | 0.2237 | 0.2741 |
| FG-X | 0.1765 | 0.2618 | 0.3257 | |
| FG-O | 0.1373 | 0.1776 | 0.2104 | |
| FG-A | 0.1569 | 0.2194 | 0.2673 | |
| 100 | UD | 0.1413 | 0.2000 | 0.2451 |
| FG-X | 0.1616 | 0.2419 | 0.3017 | |
| FG-O | 0.1175 | 0.1467 | 0.1711 | |
| FG-A | 0.1415 | 0.1983 | 0.2417 | |
| 200 | UD | 0.1221 | 0.1730 | 0.2122 |
| FG-X | 0.1452 | 0.2201 | 0.2757 | |
| FG-O | 0.0936 | 0.1071 | 0.1194 | |
| FG-A | 0.1240 | 0.1741 | 0.2122 |
Fig. 5.
The no-dimensional The impact of the beam's span-to-thickness portion on the natural frequency of GPLRC beams (a) ∆T = 0K and (b) ∆T = 100K
Fig. 6.
The impact of the L/h portion on the no dimensionless natural frequency related to GPLRC beams (a) ∆T = 0K and (b) ∆T = 100K
Fig. 7.
The impact of GPL dimensions and geometry on the vibratory frequency of FG-X type GPRLC beam (a) ∆T = 0K and (b) ∆T = 100
Fig. 8.
The impact of the weight portion and temperature increase on the no dimensional vibration frequency of GPLRC beams
Fig. 9.
The influence of the external electric voltage and piezoelectric component on the vibration frequency of FG-GPLRC beams
Fig. 10.
Natural Frequencies as Three-Dimensional Bar Chart Under different values of External Electric Voltages and Temperature Differences values
Fig. 11.
Impact of boundary conditions on the natural frequency of GPLRC beams