Figure 1.
Coordinate system and configuration of a porous FG Bernoulli–Euler nano-beam.
Figure 1.
Coordinate system and configuration of a porous FG Bernoulli–Euler nano-beam.
Figure 2.
Influence of the temperature rise on the non-dimensional gyration radius.
Figure 2.
Influence of the temperature rise on the non-dimensional gyration radius.
Figure 3.
Effects of the gradient index (k) on the frequency ratio of FG cantilever nanobeam (C-F) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying nonlocal parameter, λc, in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0}.
Figure 3.
Effects of the gradient index (k) on the frequency ratio of FG cantilever nanobeam (C-F) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying nonlocal parameter, λc, in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0}.
Figure 4.
Effects of the gradient index (k) on the frequency ratio of FG fully clamped nanobeam (C-C) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying nonlocal parameter, λc, in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0}.
Figure 4.
Effects of the gradient index (k) on the frequency ratio of FG fully clamped nanobeam (C-C) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying nonlocal parameter, λc, in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0}.
Figure 5.
Effects of the gradient index (k) on the frequency ratio of FG cantilever nanobeam (C-F) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying the gradient length parameter, λl, in the set {0.1, 0.3, 0.5}.
Figure 5.
Effects of the gradient index (k) on the frequency ratio of FG cantilever nanobeam (C-F) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying the gradient length parameter, λl, in the set {0.1, 0.3, 0.5}.
Figure 6.
Effects of the gradient index (k) on the frequency ratio of FG fully clamped nanobeam (C-C) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying the gradient length parameter, λl, in the set {0.1, 0.3, 0.5}.
Figure 6.
Effects of the gradient index (k) on the frequency ratio of FG fully clamped nanobeam (C-C) for two different values of porosity volume fraction (ζ = 0.0, 0.15) and for two different values of mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b), varying the gradient length parameter, λl, in the set {0.1, 0.3, 0.5}.
Figure 7.
Cantilever porous FG nano-beam. 3D-plot of the frequency ratio in terms of the gradient index (k) and the porosity volume fraction (ζ) carrying the nonlocal parameter λc in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0} and for two different values of the mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b).
Figure 7.
Cantilever porous FG nano-beam. 3D-plot of the frequency ratio in terms of the gradient index (k) and the porosity volume fraction (ζ) carrying the nonlocal parameter λc in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0} and for two different values of the mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b).
Figure 8.
Fully clamped porous FG nano-beam. 3D-plot of the frequency ratio in terms of the gradient index (k) and the porosity volume fraction (ζ) carrying the nonlocal parameter λc in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0} and for two different values of the mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b).
Figure 8.
Fully clamped porous FG nano-beam. 3D-plot of the frequency ratio in terms of the gradient index (k) and the porosity volume fraction (ζ) carrying the nonlocal parameter λc in the set {0+, 0.2, 0.4, 0.6, 0.8, 1.0} and for two different values of the mixture parameter: ξ1 = 0.0 (a) and ξ1 = 0.5 (b).
Table 1.
Thermo-elastic properties of metal (SuS3O4) and ceramic (Si3N4).
Table 1.
Thermo-elastic properties of metal (SuS3O4) and ceramic (Si3N4).
Material | Properties | Unit | P0 |
---|
Ceramic (Si3N4) | Ec | (GPa) | 348.40 |
ρc | (kg/m3) | 2325 |
αc | (K−1) | 0.00000587 |
βc | (wt% H2O)−1 | 0.0 |
Metal (SuS3O4) | Em | (GPa) | 201.04 |
ρm | (kg/m3) | 8011 |
αm | (K−1) | 0.00001233 |
βm | (wt% H2O)−1 | 0.0005 |
Table 2.
Coefficients of material phases for metal (SuS3O4) and ceramic (Si3N4).
Table 2.
Coefficients of material phases for metal (SuS3O4) and ceramic (Si3N4).
| | Ceramic (Si3N4) | Metal (SuS3O4) |
---|
Coefficients | Unit | Ec | ρc | αc | βc | Em | ρm | αm | βm |
---|
X−1 | (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
X1 | (K−1) | −0.0003070 | 0 | 0.0009095 | 0 | 0.0003079 | 0 | 0.0008086 | 0 |
X2 | (K−2) | 2.160 × 10−7 | 0 | 0 | 0 | −6.534 × 10−7 | 0 | 0 | 0 |
X3 | (K−3) | −8.946 × 10−11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table 3.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) for .
Table 3.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) for .
| |
---|
| |
---|
| Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] |
---|
0+ | 0.99145 | 0.99145 | 0.92475 | 0.92475 | 0.82685 | 0.82685 | 0.99096 | 0.99096 | 0.92435 | 0.92435 | 0.82656 | 0.82656 |
0.2 | 1.21717 | 1.21717 | 1.14936 | 1.14936 | 1.04530 | 1.04530 | 1.08962 | 1.08962 | 1.04005 | 1.04005 | 0.96049 | 0.96049 |
0.4 | 1.44445 | 1.44445 | 1.38385 | 1.38385 | 1.28526 | 1.28526 | 1.16113 | 1.16113 | 1.12886 | 1.12886 | 1.07277 | 1.07277 |
0.6 | 1.64847 | 1.64847 | 1.59405 | 1.59405 | 1.50187 | 1.50187 | 1.20831 | 1.20831 | 1.18632 | 1.18632 | 1.12796 | 1.12796 |
0.8 | 1.83235 | 1.83235 | 1.78272 | 1.78272 | 1.69628 | 1.69628 | 1.24099 | 1.24099 | 1.22520 | 1.22520 | 1.16337 | 1.16337 |
1.0 | 2.00041 | 2.00041 | 1.95455 | 1.95455 | 1.87306 | 1.87306 | 1.26483 | 1.26483 | 1.25298 | 1.25298 | 1.18922 | 1.18922 |
Table 4.
Normalized fundamental flexural frequency of fully clamped (C-C) nano-beam for .
Table 4.
Normalized fundamental flexural frequency of fully clamped (C-C) nano-beam for .
| |
---|
| |
---|
| Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] | | Ref. [45] |
---|
0+ | 0.89165 | 0.89165 | 0.52522 | 0.52522 | 0.34619 | 0.34619 | 0.88416 | 0.88416 | 0.52314 | 0.52314 | 0.34529 | 0.34529 |
0.2 | 1.58127 | 1.58127 | 0.89822 | 0.89822 | 0.58545 | 0.58545 | 1.14531 | 1.14531 | 0.77938 | 0.77938 | 0.54126 | 0.54126 |
0.4 | 2.57577 | 2.57577 | 1.38724 | 1.38724 | 0.93713 | 0.93713 | 1.28946 | 1.28946 | 1.02374 | 1.02374 | 0.77625 | 0.77625 |
0.6 | 3.61940 | 3.61940 | 2.01640 | 2.01640 | 1.30727 | 1.30727 | 1.34633 | 1.34633 | 1.16750 | 1.16750 | 0.95453 | 0.95453 |
0.8 | 4.67784 | 4.67784 | 2.59796 | 2.59796 | 1.68291 | 1.68291 | 1.37237 | 1.37237 | 1.24944 | 1.24944 | 1.07846 | 1.07846 |
1.0 | 5.74258 | 5.74258 | 3.18308 | 3.18308 | 2.06089 | 2.06089 | 1.38608 | 1.38608 | 1.29819 | 1.29819 | 1.16320 | 1.16320 |
Table 5.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) assuming .
Table 5.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) assuming .
−−. |
---|
| | | | |
---|
| | | | | | | | | | | |
---|
0+ | 0.98544 | 0.91112 | 0.81096 | 0.98543 | 0.91061 | 0.81076 | 0.95774 | 0.89743 | 0.78232 | 0.89187 | 0.85324 | 0.71161 |
0.1 | 1.13066 | 1.06061 | 0.95190 | 1.07907 | 1.00592 | 0.89094 | 1.02226 | 0.94452 | 0.81943 | 0.95918 | 0.87469 | 0.73300 |
0.2 | 1.36641 | 1.30477 | 1.20421 | 1.32124 | 1.25788 | 1.15415 | 1.27247 | 1.20677 | 1.09847 | 1.21960 | 1.15076 | 1.03596 |
0.3 | 1.57748 | 1.52261 | 1.42958 | 1.53723 | 1.48128 | 1.38631 | 2.49424 | 1.43689 | 1.33931 | 1.44822 | 1.38908 | 1.28804 |
0.4 | 1.76687 | 1.71711 | 1.63040 | 1.73033 | 1.67981 | 1.59176 | 1.69154 | 1.64008 | 1.55031 | 1.65033 | 1.59771 | 1.50575 |
0.5 | 1.93929 | 1.89349 | 1.81207 | 1.90563 | 1.85925 | 1.77684 | 1.87005 | 1.82299 | 1.73933 | 1.83242 | 1.78454 | 1.69935 |
Table 6.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) assuming .
Table 6.
Normalized fundamental flexural frequency of cantilever nano-beam (C-F) assuming .
| −−. |
---|
| | | | |
---|
| | | | | | | | | | | |
---|
0+ | 0.96700 | 0.89741 | 0.79637 | 0.91465 | 0.85428 | 0.76812 | 0.85375 | 0.79756 | 0.71106 | 0.75051 | 0.71109 | 0.59731 |
0.1 | 1.00082 | 0.94827 | 0.86294 | 0.94694 | 0.89096 | 0.79867 | 0.88681 | 0.82572 | 0.72204 | 0.81887 | 0.75007 | 0.62700 |
0.2 | 1.07717 | 1.04295 | 0.98332 | 1.02690 | 0.99065 | 0.92713 | 1.07134 | 0.93237 | 0.86316 | 0.90968 | 0.86669 | 0.78904 |
0.3 | 1.12808 | 1.10478 | 1.06236 | 1.08041 | 1.05580 | 1.01090 | 1.02814 | 1.00180 | 0.95349 | 0.97042 | 0.94176 | 0.88701 |
0.4 | 1.16349 | 1.14676 | 1.11549 | 1.11764 | 1.10001 | 1.06701 | 1.06757 | 1.04878 | 1.01348 | 1.01256 | 0.99224 | 0.95388 |
0.5 | 1.18936 | 1.17682 | 1.15296 | 1.14484 | 1.13164 | 1.10651 | 1.09635 | 1.08231 | 1.05554 | 1.04325 | 1.02813 | 0.99921 |
Table 7.
Normalized fundamental flexural frequency of fully clamped nano-beam (C-C) for .
Table 7.
Normalized fundamental flexural frequency of fully clamped nano-beam (C-C) for .
| −−. |
---|
| | | | |
---|
| | | | | | | | | | | |
---|
0+ | 0.87793 | 0.51401 | 0.34059 | 0.87788 | 0.51401 | 0.34059 | 0.87784 | 0.51358 | 0.34056 | 0.87775 | 0.51313 | 0.34050 |
0.1 | 1.56938 | 0.88682 | 0.57162 | 1.56718 | 0.88284 | 0.56538 | 1.56490 | 0.87869 | 0.55883 | 1.56254 | 0.87438 | 0.55194 |
0.2 | 2.55988 | 1.35548 | 0.92572 | 2.55854 | 1.35840 | 0.92207 | 2.55716 | 1.36141 | 0.91826 | 2.55572 | 1.29619 | 0.91430 |
0.3 | 2.59834 | 2.00256 | 1.29562 | 3.59739 | 2.00090 | 1.29307 | 3.59639 | 1.99917 | 1.29041 | 3.59536 | 1.99737 | 1.28765 |
0.4 | 4.65122 | 2.58140 | 1.67011 | 4.65046 | 2.58011 | 1.66815 | 4.64967 | 2.57877 | 1.66611 | 4.64886 | 2.57738 | 1.66399 |
0.5 | 5.71022 | 3.16352 | 2.04652 | 5.70959 | 3.16247 | 2.04492 | 5.70892 | 3.16137 | 2.04326 | 5.70824 | 3.16023 | 2.04154 |
Table 8.
Normalized fundamental flexural frequency of fully clamped nano-beam (C-C) for .
Table 8.
Normalized fundamental flexural frequency of fully clamped nano-beam (C-C) for .
| −−. |
---|
| | | | |
---|
| | | | | | | | | | | |
---|
0+ | 0.86807 | 0.50998 | 0.34009 | 0.84925 | 0.49179 | 0.31288 | 0.84524 | 0.49166 | 0.30845 | 0.80905 | 0.48723 | 0.30840 |
0.1 | 1.13373 | 0.76745 | 0.52694 | 1.13089 | 0.76307 | 0.52041 | 1.12794 | 0.75850 | 0.51354 | 1.12487 | 0.75374 | 0.50631 |
0.2 | 1.27747 | 1.01222 | 0.76431 | 1.27501 | 1.00906 | 0.76006 | 1.27244 | 1.00576 | 0.75562 | 1.26987 | 1.00233 | 0.75100 |
0.3 | 1.33415 | 1.15579 | 0.94297 | 1.33180 | 1.15304 | 0.93960 | 1.32936 | 1.15020 | 0.93608 | 1.32682 | 1.14726 | 0.93243 |
0.4 | 1.36010 | 1.23753 | 1.06684 | 1.35780 | 1.23499 | 1.06389 | 1.35541 | 1.23235 | 1.06081 | 1.35293 | 1.22962 | 1.05762 |
0.5 | 1.37377 | 1.28614 | 1.15145 | 1.37149 | 1.28371 | 1.14873 | 1.36912 | 1.28118 | 1.14589 | 1.36667 | 1.27855 | 1.14295 |