Table 1 Convergence study for bending, buckling, and free vibration solutions of the square nanoplates with b = 10 nm, \(\mu = 1\;{\text{nm}}^{2}\), and \(\overline{k}_{p} =\) 20 under different BCs.
Mechanical quantity | Number of series terms | BC | |||||
|---|---|---|---|---|---|---|---|
CCCC | CCCS | CCSS | SCSC | SCSS | SSSS | ||
\({{10^{5} Dw\left( {{a \mathord{\left/ {\vphantom {a {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right. \kern-\nulldelimiterspace} {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right)} \mathord{\left/ {\vphantom {{10^{5} Dw\left( {{a \mathord{\left/ {\vphantom {a {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right. \kern-\nulldelimiterspace} {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right)} {\left( {q_{u} b^{4} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {q_{u} b^{4} } \right)}}\) | 10 | 70.575 | 85.081 | 106.25 | 100.52 | 129.81 | 163.76 |
20 | 70.532 | 85.028 | 106.22 | 100.51 | 129.81 | 163.76 | |
40 | 70.527 | 85.022 | 106.22 | 100.51 | 129.81 | 163.76 | |
80 | 70.526 | 85.021 | 106.22 | 100.51 | 129.81 | 163.76 | |
120 | 70.526 | 85.021 | 106.216 | 100.510 | 129.810 | 163.760 | |
\({{10^{5} Dw\left( {{a \mathord{\left/ {\vphantom {a {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right. \kern-\nulldelimiterspace} {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right)} \mathord{\left/ {\vphantom {{10^{5} Dw\left( {{a \mathord{\left/ {\vphantom {a {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right. \kern-\nulldelimiterspace} {2,{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right)} {\left( {q_{c} b^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {q_{c} b^{2} } \right)}}\) | 40 | 330.37 | 351.37 | 380.94 | 373.81 | 413.82 | 460.12 |
80 | 330.52 | 351.53 | 381.10 | 373.97 | 413.98 | 460.28 | |
160 | 330.55 | 351.56 | 381.14 | 374.01 | 414.02 | 460.32 | |
240 | 330.56 | 351.57 | 381.14 | 374.02 | 414.02 | 460.33 | |
320 | 330.57 | 351.57 | 381.15 | 374.02 | 414.03 | 460.33 | |
400 | 330.57 | 351.57 | 381.15 | 374.02 | 414.03 | 460.33 | |
\({{ - N_{x}^{{{\text{critical}}}} b^{2} } \mathord{\left/ {\vphantom {{ - N_{x}^{{{\text{critical}}}} b^{2} } D}} \right. \kern-\nulldelimiterspace} D}\) | 3 | 34.352 | 29.844 | 24.271 | – | – | – |
5 | 34.359 | 29.848 | 24.272 | – | – | – | |
10 | 34.359 | 29.848 | 24.273 | – | – | – | |
20 | 34.359 | 29.848 | 24.273 | – | – | – | |
\(\omega b^{2} \sqrt {{{\rho h} \mathord{\left/ {\vphantom {{\rho h} D}} \right. \kern-\nulldelimiterspace} D}}\) | 3 | 32.240 | 28.620 | 24.418 | – | – | – |
5 | 32.251 | 28.626 | 24.421 | – | – | – | |
10 | 32.252 | 28.626 | 24.421 | – | – | – | |
20 | 32.252 | 28.626 | 24.421 | – | – | – | |
