# Thermal Buckling of Nanocomposite Stiffened Cylindrical Shells Reinforced by Functionally Graded Wavy Carbon Nanotubes with Temperature-Dependent Properties

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Properties in FG-CNTRC Cylinders

_{R}) and waviness (w). The effective mechanical properties for the CNTRC cylindrical shell, obtained according to the micro mechanical model [20], take the following form:

## 3. Fundamental Equations

## 4. Numerical Study of the Buckling Behaviour

- (1)
- The buckling analysis is carried out for material properties at ambient temperature of $300\text{\hspace{0.17em}}\mathrm{K}$ (i.e., $\Delta T=0$), and the critical temperature $\Delta {T}_{cr}^{1}$ is determined at the first step;
- (2)
- The material properties are updated at the increased temperature of $300\text{\hspace{0.17em}}\mathrm{K}+\Delta {T}_{cr}^{1}$. The buckling analysis is carried out with these new properties, and a new value of critical temperature $\Delta {T}_{cr}^{2}$ is determined;
- (3)
- Step 2 is repeated systematically until $|\Delta T-\Delta {T}^{k}|\text{\hspace{0.17em}}\le 0.0001$ and the solution converges to the critical temperature with TD material properties.

## 5. Numerical Results

^{−6}, as well as a pure ceramic (i.e., an Alumina) with Young’s modulus of 380 GPa, Poisson’s ratio 0.3, and coefficient of thermal expansion 7.4 × 10

^{−6}. As visible from Table 1, the results based on the proposed formulation resemble quite well those predicted in [70]. This demonstrates the accuracy of the GDQ-based solutions of the problem. More in detail, predictions by Mirzavand and Eslami [71] are always more conservative compared to our results, whereas a clear increase of the critical thermal load can be observed for increasing $h/R$ ratios.

_{2}). The material properties in terms of Young’s modulus and thermal expansion coefficient, are expressed as a nonlinear function of temperature, as suggested in [72]. Table 2 reports the thermal buckling loads ${T}_{cr}^{}$ (expressed in $\mathrm{K}$) for the simply supported FGM cylindrical shells with different values of volume fraction index $P$, and subjected to a uniform temperature rise.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Sigmoidal distributions of the ${V}_{CNT}$ through the thickness for different exponents s. CNT: Carbon NanoTubes.

**Figure 3.**Effect of different CNT profiles on the critical thermal buckling load considering $L/R=2,$ ${V}_{CNT}^{*}=0.17,$ ${A}_{R}=1000,$ $w=0,$ ${N}_{r}={N}_{s}=0$. TI: temperature-independent; TD: temperature-dependent.

**Figure 4.**Effect of different CNT profiles on the critical thermal buckling load considering $L/R=2,$ ${V}_{CNT}^{*}=0.17,$ ${A}_{R}=1000,$ $w=0.425,$ ${N}_{r}={N}_{s}=0$.

**Figure 5.**$\Delta {T}_{cr}$ vs. $R/h$ for different aspect ratios ${A}_{R}$ considering $L/R=2,$ ${V}_{CNT}^{*}=0.28,$ $w=0,$ ${N}_{r}={N}_{s}=0$.

**Figure 6.**$\Delta {T}_{cr}$ vs. $R/h$ for different volume fractions ${V}_{CNT}^{\ast}$ considering $L/R=2,$ ${A}_{R}=500,$ $w=0.425,$ ${N}_{r}={N}_{s}=0$.

**Figure 7.**$\Delta {T}_{cr}$ vs. $s$ for different combinations of aspect ratios ${A}_{R}$ and waviness indexes $w$ considering $L/R=2,$ $R/h=200,$ ${V}_{CNT}^{*}=0.28,$ ${N}_{r}={N}_{s}=0$.

**Table 1.**Thermal buckling $\Delta {T}_{cr}$ as predicted by the present work and [71] for a uniform temperature rise and $L/R=1$.

$\mathit{h}/\mathit{R}$ | Method | Pure Metal (Aluminum) | Pure Ceramic (Alumina) |
---|---|---|---|

0.003 | Present | 55.1114 | 171.2921 |

[71] | 52.1867 | 181.375 | |

0.005 | Present | 91.7741 | 285.2439 |

[71] | 90.8012 | 289.537 | |

0.007 | Present | 128.4551 | 399.2525 |

[71] | 124.447 | 402.683 | |

0.01 | Present | 182.6926 | 567.8284 |

[71] | 187.33 | 597.253 | |

0.012 | Present | 219.6345 | 682.6477 |

[71] | 216.007 | 688.039 | |

0.015 | Present | 273.1372 | 848.9400 |

[71] | 268.953 | 862.72 |

**Table 2.**Buckling temperatures ${T}_{cr}$ (in K) as computed by the Generalized Differential Quadrature (GDQ) approach, compared to predictions by [73,74] for a Functionally Graded Material (FGM) cylindrical shell, with temperature-dependent (TD) or temperature-independent (TI) material properties considering $H=1/400$ and ${L}^{2}/H=300$.

p | Method | TD | TI |
---|---|---|---|

0.2 | Present | 354.5263 | 363.8471 |

[73] | 365.0509 | 378.5620 | |

[74] | 353.7 | - | |

0.5 | Present | 360.8525 | 371.0475 |

[73] | 372.6016 | 387.4075 | |

[74] | 360.119 | - | |

1 | Present | 369.8812 | 380.9365 |

[73] | 383.4589 | 399.5912 | |

[74] | 368.785 | - | |

2 | Present | 384.4789 | 395.7673 |

[73] | 401.2534 | 417.9189 | |

[74] | 383.244 | - | |

3 | Present | 395.9562 | 406.1519 |

[73] | 415.4941 | 430.7708 | |

[74] | 394.319 | - | |

5 | Present | 413.3029 | 419.3226 |

[73] | 437.9041 | 447.0620 | |

[74] | 411.669 | - |

**Table 3.**Critical thermal buckling load for different distributions of the CNTs within the structure made of TI material properties considering $L/R=1,R/h=200,{N}_{r}=0,{N}_{s}=0$.

Profiles | ${\mathit{A}}_{\mathit{R}}$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{\ast}=0.12$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{\ast}=0.28$ | ||
---|---|---|---|---|---|

$\mathit{w}=0$ | $\mathit{w}=0.425$ | $\mathit{w}=0$ | $\mathit{w}=0.425$ | ||

$\wedge $-profile | ${A}_{R}$ = 50 | 355.6195 | 409.7686 | 346.9656 | 446.356 |

${A}_{R}$ = 100 | 342.6400 | 407.8564 | 338.9678 | 447.5867 | |

${A}_{R}$ = 500 | 336.1531 | 404.8622 | 334.6581 | 448.6354 | |

${A}_{R}$ = 1000 | 335.5539 | 404.3232 | 333.9651 | 448.7409 | |

$X$-profile | ${A}_{R}$ = 50 | 375.9555 | 413.1924 | 366.8814 | 450.1936 |

${A}_{R}$ = 100 | 356.2644 | 409.7816 | 351.3718 | 449.1031 | |

${A}_{R}$ = 500 | 344.5911 | 404.1034 | 343.0042 | 446.3097 | |

${A}_{R}$ = 1000 | 343.5005 | 403.1505 | 342.1739 | 445.7818 | |

$O$-profile | ${A}_{R}$ = 50 | 356.0734 | 406.7710 | 343.8665 | 431.8363 |

${A}_{R}$ = 100 | 341.6468 | 404.8307 | 335.4600 | 432.9287 | |

${A}_{R}$ = 500 | 334.4003 | 402.2474 | 331.3018 | 432.8269 | |

${A}_{R}$ = 1000 | 333.7368 | 401.8458 | 330.8940 | 432.7693 | |

$U$-profile | ${A}_{R}$ = 50 | 367.6717 | 410.8309 | 358.5940 | 437.8143 |

${A}_{R}$ = 100 | 352.3555 | 408.3589 | 344.8158 | 437.9515 | |

${A}_{R}$ = 500 | 341.3101 | 405.0617 | 336.4925 | 437.7304 | |

${A}_{R}$ = 1000 | 340.1902 | 404.5425 | 335.6926 | 437.6768 | |

Sigmoidal-profile | ${A}_{R}$ = 50 | 384.5763 | 401.4333 | 375.2713 | 412.1084 |

${A}_{R}$ = 100 | 366.8624 | 397.5881 | 357.3245 | 411.5575 | |

${A}_{R}$ = 500 | 354.0331 | 391.1778 | 347.7359 | 408.1837 | |

${A}_{R}$ = 1000 | 352.8324 | 390.1325 | 346.8347 | 407.5554 |

**Table 4.**Critical thermal buckling load for different distributions of the CNTs within the structure made of TD properties considering $L/R=1,\text{\hspace{0.17em}}R/h=200,\text{\hspace{0.17em}}{N}_{r}=0,\text{\hspace{0.17em}}{N}_{s}=0$.

Profiles | ${\mathit{A}}_{\mathit{R}}$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{\ast}=0.12$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{\ast}=0.28$ | ||
---|---|---|---|---|---|

$\mathit{w}=0$ | $\mathit{w}=0.425$ | $\mathit{w}=0$ | $\mathit{w}=0.425$ | ||

$\wedge $-profile | ${A}_{R}$ = 50 | 350.5315 | 401.0361 | 342.4973 | 433.3668 |

${A}_{R}$ = 100 | 338.9949 | 398.8109 | 335.5588 | 433.8537 | |

${A}_{R}$ = 500 | 333.2777 | 395.5772 | 331.5408 | 434.1682 | |

${A}_{R}$ = 1000 | 332.7525 | 395.0126 | 330.9463 | 434.1723 | |

$X$-profile | ${A}_{R}$ = 50 | 367.6339 | 403.9401 | 358.4562 | 436.5301 |

${A}_{R}$ = 100 | 350.0884 | 400.3097 | 345.5374 | 434.9102 | |

${A}_{R}$ = 500 | 340.1044 | 394.5435 | 338.5318 | 431.6631 | |

${A}_{R}$ = 1000 | 339.1756 | 393.5870 | 337.8357 | 431.0726 | |

$O$-profile | ${A}_{R}$ = 50 | 350.8708 | 398.4375 | 339.8652 | 420.952 |

${A}_{R}$ = 100 | 338.0843 | 396.1992 | 332.5311 | 421.3494 | |

${A}_{R}$ = 500 | 331.6981 | 393.3625 | 328.9072 | 420.6853 | |

${A}_{R}$ = 1000 | 331.1157 | 392.9302 | 328.5520 | 420.5504 | |

$U$-profile | ${A}_{R}$ = 50 | 360.7792 | 401.5390 | 352.3864 | 425.8502 |

${A}_{R}$ = 100 | 347.2884 | 398.8054 | 340.2270 | 425.4899 | |

${A}_{R}$ = 500 | 337.3306 | 395.3611 | 333.1242 | 424.8383 | |

${A}_{R}$ = 1000 | 336.3705 | 394.8309 | 332.4403 | 424.7290 | |

Sigmoidal-profile | ${A}_{R}$ = 50 | 376.4948 | 394.1399 | 366.8533 | 403.9861 |

${A}_{R}$ = 100 | 359.9199 | 390.0668 | 351.1715 | 402.9769 | |

${A}_{R}$ = 500 | 348.5575 | 383.5699 | 342.9259 | 399.2144 | |

${A}_{R}$ = 1000 | 347.5143 | 382.5242 | 342.1573 | 398.5467 |

**Table 5.**Effect of the stiffeners on the critical thermal buckling load of the cylindrical shell reinforced by wavy graded CNTs for different profiles considering L/R = 2, R/h = 200, w = 0.425, s = 1, A

_{R}= 1000, ${V}_{CNT}^{*}=0.17$.

Ring and Stringer Number | TD | TI | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\wedge $-Profile | $\mathit{X}$-Profile | $\mathit{O}$-Profile | $\mathit{U}$-Profile | Sigmoidal-Profile | $\wedge $-Profile | $\mathit{X}$-Profile | $\mathit{O}$-Profile | $\mathit{U}$-Profile | Sigmoidal-Profile | |

Unstiffened | 411.2273 | 408.5345 | 405.5307 | 408.2149 | 392.371 | 421.7055 | 419.1132 | 415.1666 | 418.5061 | 400.4704 |

${N}_{r}=5$ | 459.2519 | 454.296 | 463.9345 | 460.9213 | 439.7066 | 472.9684 | 468.3331 | 476.1383 | 474.1699 | 450.2883 |

${N}_{s}=5$ | 432.5377 | 430.5091 | 424.9031 | 429.7281 | 409.3133 | 443.834 | 442.0279 | 435.547 | 440.66 | 418.4353 |

${N}_{r}={N}_{s}=5$ | 508.4548 | 504.2132 | 515.1755 | 513.4648 | 480.9117 | 515.2078 | 512.2288 | 519.7585 | 519.4056 | 487.7804 |

${N}_{r}=7$ | 468.2045 | 463.572 | 477.4953 | 471.558 | 451.0799 | 482.7139 | 478.9919 | 490.3426 | 487.1171 | 462.3776 |

${N}_{s}=7$ | 441.8178 | 439.6434 | 433.3342 | 438.8732 | 416.2962 | 453.5255 | 451.6761 | 444.2205 | 450.2502 | 425.8656 |

${N}_{r}={N}_{s}=7$ | 555.4144 | 553.2414 | 569.8029 | 568.5566 | 520.3784 | 550.762 | 548.6321 | 557.7685 | 557.7418 | 519.8421 |

${N}_{r}=10$ | 480.1307 | 475.8978 | 491.597 | 484.6137 | 464.2851 | 494.8228 | 491.5553 | 505.6045 | 500.4459 | 476.2802 |

${N}_{s}=10$ | 456.9784 | 454.8882 | 446.7671 | 453.9572 | 427.7454 | 468.3527 | 466.4062 | 457.7995 | 465.3265 | 437.6988 |

${N}_{r}={N}_{s}=10$ | 595.4563 | 586.3257 | 605.6545 | 604.3793 | 563.6043 | 606.5443 | 606.4783 | 618.9873 | 617.3305 | 571.0981 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nejati, M.; Dimitri, R.; Tornabene, F.; Hossein Yas, M. Thermal Buckling of Nanocomposite Stiffened Cylindrical Shells Reinforced by Functionally Graded Wavy Carbon Nanotubes with Temperature-Dependent Properties. *Appl. Sci.* **2017**, *7*, 1223.
https://doi.org/10.3390/app7121223

**AMA Style**

Nejati M, Dimitri R, Tornabene F, Hossein Yas M. Thermal Buckling of Nanocomposite Stiffened Cylindrical Shells Reinforced by Functionally Graded Wavy Carbon Nanotubes with Temperature-Dependent Properties. *Applied Sciences*. 2017; 7(12):1223.
https://doi.org/10.3390/app7121223

**Chicago/Turabian Style**

Nejati, Mohammad, Rossana Dimitri, Francesco Tornabene, and Mohammad Hossein Yas. 2017. "Thermal Buckling of Nanocomposite Stiffened Cylindrical Shells Reinforced by Functionally Graded Wavy Carbon Nanotubes with Temperature-Dependent Properties" *Applied Sciences* 7, no. 12: 1223.
https://doi.org/10.3390/app7121223