Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations

<table class="table-group" id="tab6"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td align="left">FGM</td><td align="center"><svg height="11.175" id="M172" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 10.8625 11.175" width="10.8625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5
l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g>
</svg> (N/m<sup>2</sup>)</td><td align="center">Poisson ratio (<svg height="7.9499998" id="M173" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.1625004 7.9499998" width="9.1625004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M509 373q0 -52 -42.5 -118.5t-124.5 -159.5l-94 -107h-27q-22 209 -78 342q-21 48 -41 48q-18 0 -63 -62l-22 21q77 111 125 111q30 0 52 -21.5t38 -75.5q37 -130 52 -285q48 62 74.5 99.5t53.5 87t27 76.5q0 43 -17 60q-5 5 -5 15q0 17 12.5 30.5t28.5 13.5
q21 0 36 -22.5t15 -52.5z" id="x1D708"></path></g>
</svg>)</td><td align="center">Density <svg height="12.175" id="M174" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 9.2875004 12.175" width="9.2875004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M516 277q0 -126 -140 -231q-36 -27 -75.5 -42.5t-63.5 -15.5q-38 0 -73 43q-10 -44 -18 -106q-18 -134 -32 -151q-12 -16 -40 -25t-46 -10l-5 14q22 38 84 406q16 92 44.5 146.5t88.5 95.5q68 47 136 47q63 0 101.5 -47.5t38.5 -123.5zM432 259q0 61 -27 101t-79 40
t-82 -49t-49 -148l-24 -127q42 -36 88 -36q79 0 126 61.5t47 157.5z" id="x1D70C"></path></g>
</svg> (Kg/m<sup>3</sup>)</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td align="left">Nickel</td><td align="center"><span style="width: 97.550003px;"><svg height="11.425" id="M175" style="vertical-align:-0.20474pt" version="1.1" viewbox="0 0 97.550003 11.425" width="97.550003" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g><g transform="matrix(.017,-0,0,-.017,67.412,11.112)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,81.164,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,89.323,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center">0.310</td><td align="center">8900</td></tr><tr><td align="left">Zirconia</td><td align="center"><span style="width: 97.550003px;"><svg height="11.375" id="M176" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 97.550003 11.375" width="97.550003" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g><g transform="matrix(.017,-0,0,-.017,67.412,11.112)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,81.164,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,89.323,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center">0.298</td><td align="center">5700</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Material properties of Nickel and Zirconia.

Advances in Acoustics and Vibration

tab6

Table 6

Table 6: Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations 