Nonlinear Vibration of a Functionally Graded Nanobeam Based on the Nonlocal Strain Gradient Theory considering Thickness Effect

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Materials</td><td class="align_center">Young’s modulus <i>E</i> (GPa)</td><td class="align_center">Mass density <i>ρ</i> (kg/m<sup>3</sup>)</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">Aluminum (metal)</td><td class="align_center"><span style="width: 49.0159ptpx;"><svg height="11.927pt" id="M116" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 49.0159 11.927" width="49.0159pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.943,3.132)"><path d="M757 650H506L497 618C579 615 596 608 596 570S591 477 571 380L560 327C516 117 435 41 309 41C199 41 108 114 144 303L187 529C202 609 208 612 291 618L296 650H30L24 618C109 612 113 606 99 529L58 299C33 161 62 83 113 41C161 0 222 -17 283 -17C348 -17 417 3 470 48C546 111 584 203 603 311L615 380C631 474 647 531 658 567C667 599 672 616 750 618L757 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.805,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.068,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 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class="align_center"><span style="width: 53.4916ptpx;"><svg height="11.7782pt" id="M117" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 53.4916 11.7782" width="53.4916pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.162,3.132)"><path d="M757 650H506L497 618C579 615 596 608 596 570S591 477 571 380L560 327C516 117 435 41 309 41C199 41 108 114 144 303L187 529C202 609 208 612 291 618L296 650H30L24 618C109 612 113 606 99 529L58 299C33 161 62 83 113 41C161 0 222 -17 283 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<div>The material properties of aluminum and alumina [<a href="/journals/ace/2020/9407673/#B50" target="_blank">50</a>].</div>

Advances in Civil Engineering

tab1

Table 1

Table 1: Nonlinear Vibration of a Functionally Graded Nanobeam Based on the Nonlocal Strain Gradient Theory considering Thickness Effect 