# Thermoelastic Stress and Deformation Analyses of Functionally Graded Doubly Curved Shells

^{*}

*Journal of Composites Science*in 2019)

## Abstract

**:**

## 1. Introduction

## 2. Effective Material Properties

_{l}layers with a small thickness for each individual layer, as compared with the in-surface dimensions. In order to derive the formulation, a global doubly curved coordinate system (i.e., $\xi ,\text{\hspace{0.17em}}\eta $ and $\zeta $ coordinates) is located on the mid-surface of the shell, and a set of local thickness coordinates ${z}_{m}\text{\hspace{0.17em}}(m=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,...,\text{\hspace{0.17em}}{N}_{l})$ is located at the mid-surface of each individual layer, as shown in Figure 1b, in which N

_{l}is taken to be five. The in-surface dimensions of the shell in the $\xi \text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}\eta $ directions are defined as ${L}_{\xi}\text{}\mathrm{and}\text{}{L}_{\eta}$, respectively, and the curvature radii of the shell are ${R}_{\xi}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{R}_{\eta}$. The thicknesses of each individual layer and the shell are ${h}_{m}\text{\hspace{0.17em}}(m=1,\text{\hspace{0.17em}}2,...,\text{\hspace{0.17em}}{N}_{l})$ and h, respectively, while $h={\displaystyle \sum _{m=1}^{{N}_{l}}\text{\hspace{0.17em}}{h}_{m}}$. The relationship between the global and local thickness coordinates in the mth-layer is $\zeta ={\overline{\zeta}}_{m}+{z}_{m}$, in which ${\overline{\zeta}}_{m}=({\zeta}_{m}+{\zeta}_{m-1})/2$, as well as ${\zeta}_{m}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{\zeta}_{m-1}$ denote the global thickness coordinates measured from the mid-surface of the shell to the top and bottom surfaces of the mth-layer, respectively.

_{p}is taken to be κ

_{p}= 0.2, 0.5, 1, 2, and 5.

## 3. Heat Conduction Analysis

## 4. Coupled Thermoelastic Analysis

#### 4.1. Kinematic and Kinetic Assumptions

#### 4.2. An RMVT-Based Weak-form Formulation

_{l}finite layers and take the elastic displacement components and transverse shear and normal stress components as the primary variables subject to variation. A weak-form formulation can be derived by expressing the energy functional in terms of the primary variables, and then letting the first-order variation of the Reissner energy functional be zero, which yields

#### 4.3. System Equations and Boundary Conditions

## 5. Illustrative Examples

#### 5.1. Sandwiched Composite Spherical Shells

#### 5.2. FG Cylindrical Shells

_{2}) and the matrix-phase material being nickel-based alloy (Monel, 70Ni-30Cu). The volume fractions of the ceramic and metal materials (i.e., ${V}_{c}(\zeta )$ and ${V}_{m}(\zeta )$) are taken to be ${V}_{c}(\zeta )={\left[(\zeta +0.5h)/h\right]}^{{\kappa}_{p}}$ and ${V}_{m}(\zeta )=1-{V}_{c}(\zeta )$, in which the subscripts m and c denote the metal and ceramic materials, respectively, while the effective material properties are estimated using the Mori–Tanaka micromechanics scheme [33]. The material properties of these two phase materials [23,32] are given as ${B}_{m}=227.24\text{\hspace{0.17em}}\mathrm{GPa}$, ${G}_{m}=65.55\text{\hspace{0.17em}}\mathrm{GPa}$, ${\alpha}_{m}=15\times {10}^{-6}\text{\hspace{0.17em}}(1/\mathrm{K})$, ${\lambda}_{m}=25\text{\hspace{0.17em}}\mathrm{W}/\mathrm{mK}$, and ${B}_{c}=125.83\text{\hspace{0.17em}}\mathrm{GPa}$, ${G}_{c}=58.08\text{\hspace{0.17em}}\mathrm{GPa}$, ${\alpha}_{c}=10\times {10}^{-6}\text{\hspace{0.17em}}(1/\mathrm{K})$, ${\lambda}_{c}=2.09\text{\hspace{0.17em}}\mathrm{W}/\mathrm{mK}$. The material properties in this example are assumed to be temperature-independent. The geometric parameters of the shell are ${L}_{\xi}=1\text{\hspace{0.17em}}\mathrm{m}$, ${L}_{\eta}=(10\pi /3)\text{\hspace{0.17em}}\mathrm{m}$, ${R}_{\xi}\to \infty \text{\hspace{0.17em}}\mathrm{m}$, ${R}_{\eta}=10\text{\hspace{0.17em}}\mathrm{m}$, and ${R}_{\eta}/h=50\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}1000$.

#### 5.3. FGDC Shells

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The configuration of a FGDC shell; (

**b**) the global and local coordinates of the DC shell.

**Figure 2.**Through-thickness distributions of the volume fraction of the FGDC shell for different values of material–property gradient indices ${\kappa}_{p}$ of the power–law model.

**Figure 3.**Through-thickness distributions of thermal and elastic field variables induced in a [0°/core/0°] laminated spherical shell with different values of (${L}_{\xi}/h$). (

**a**) In-surface displacements; (

**b**) Out-of-surface displacement; (

**c**) In-surface normal stress; (

**d**) In-surface shear stress; (

**e**) Transverse shear stress; (

**f**) Transverse normal stress.

**Figure 4.**Through-thickness distributions of thermal and elastic field variables induced in an FG cylindrical shell with different values of ${\kappa}_{p}$. (

**a**) Temperature; (

**b**) Out-of-surface displacement; (

**c**) In-surface normal stress; (

**d**) In-surface shear stress; (

**e**) Transverse shear stress; (

**f**) Transverse normal stress.

**Figure 5.**Through-thickness distributions of thermal and elastic field variables induced in an FGDC shell with different values of ${\kappa}_{p}$. (

**a**) Temperature; (

**b**) Out-of-surface displacement; (

**c**) In-surface normal stress; (

**d**) In-surface shear stress; (

**e**) Transverse shear stress; (

**f**) Transverse normal stress.

**Figure 6.**Through-thickness distributions of thermal and elastic field variables induced in an FGDC shell with different values of (${L}_{\xi}/h$). (

**a**) Temperature; (

**b**) Out-of-surface displacement; (

**c**) In-surface normal stress; (

**d**) In-surface shear stress; (

**e**) Transverse shear stress; (

**f**) Transverse normal stress.

**Table 1.**Results of convergence and accuracy studies for the dimensionless deflections induced in thick sandwiched spherical shells ([0°/core/0°]) under a thermal load.

Theories $({\mathit{N}}_{\mathit{l}}=2{\mathit{N}}_{\mathit{f}}+{\mathit{N}}_{\mathit{c}})$ | ${\mathit{R}}_{\mathit{\xi}}/{\mathit{L}}_{\mathit{\xi}}=5$ | ${\mathit{R}}_{\mathit{\xi}}/{\mathit{L}}_{\mathit{\xi}}=10$ | ${\mathit{R}}_{\mathit{\xi}}/{\mathit{L}}_{\mathit{\xi}}=20$ | ${\mathit{R}}_{\mathit{\xi}}/{\mathit{L}}_{\mathit{\xi}}=\mathit{\infty}$ (Plates) |
---|---|---|---|---|

Linear FDCL method | ||||

(${N}_{f}=1$, ${N}_{c}=1$) | 4.3444 | 4.3683 | 4.3744 | 4.3764 |

(${N}_{f}=1$, ${N}_{c}=2$) | 4.3518 | 4.375 | 4.3809 | 4.3828 |

(${N}_{f}=1$, ${N}_{c}=4$) | 4.3422 | 4.3656 | 4.3715 | 4.3734 |

(${N}_{f}=2$, ${N}_{c}=8$) | 4.3497 | 4.3732 | 4.3791 | 4.3811 |

(${N}_{f}=4$, ${N}_{c}=8$) | 4.3498 | 4.3733 | 4.3792 | 4.3811 |

(${N}_{f}=4$, ${N}_{c}=16$) | 4.3496 | 4.373 | 4.3789 | 4.3809 |

Quadratic FDCL method | ||||

(${N}_{f}=1$, ${N}_{c}=1$) | 4.3493 | 4.3728 | 4.3787 | 4.3806 |

(${N}_{f}=1$, ${N}_{c}=2$) | 4.3494 | 4.3728 | 4.3787 | 4.3807 |

(${N}_{f}=1$, ${N}_{c}=4$) | 4.3495 | 4.373 | 4.3789 | 4.3808 |

(${N}_{f}=2$, ${N}_{c}=8$) | 4.3496 | 4.373 | 4.3789 | 4.3809 |

Cubic FDCL method | ||||

(${N}_{f}=1$, ${N}_{c}=1$) | 4.3497 | 4.3731 | 4.379 | 4.381 |

(${N}_{f}=1$, ${N}_{c}=2$) | 4.3496 | 4.373 | 4.3789 | 4.3809 |

(${N}_{f}=1$, ${N}_{c}=4$) | 4.3496 | 4.373 | 4.3789 | 4.3809 |

(${N}_{f}=2$, ${N}_{c}=8$) | 4.3496 | 4.373 | 4.3789 | 4.3809 |

CLT [20] | 1.8043 | 1.8025 | 1.8021 | 1.8019 |

FSDT [20] | 3.1472 | 3.1632 | 3.1672 | 3.1685 |

FSDT [57] | 3.2618 | 3.2745 | 3.2775 | 3.2784 |

HSDT [57] | 4.2032 | 4.2343 | 3.2422 | 4.2448 |

ED1 [20] | 3.1466 | 3.1631 | 3.1672 | 3.1685 |

ED2 [20] | 3.0306 | 3.0471 | 3.0512 | 3.0525 |

ED3 [20] | 4.1867 | 4.2308 | 4.2419 | 4.2456 |

ED4 [20] | 4.1928 | 4.236 | 4.2469 | 4.2505 |

EDZ1 [20] | 4.3705 | 4.419 | 4.4312 | 4.4352 |

EDZ2 [20] | 4.3228 | 4.372 | 4.3843 | 4.3885 |

EDZ3 [20] | 4.3261 | 4.3754 | 4.3878 | 4.3919 |

LD1 [20] | 4.3417 | 4.3653 | 4.3712 | 4.3732 |

LD2 [20] | 4.342 | 4.3651 | 4.3709 | 4.3729 |

LD3 [20] | 4.3427 | 4.3658 | 4.3716 | 4.3736 |

LD4 [20] | 4.3426 | 4.3657 | 4.3715 | 4.3735 |

**Table 2.**Results of cubic FDCL methods for various field variables induced in a single-layered FGDC shell under a thermal load.

${\mathit{R}}_{\mathit{\eta}}/\mathit{h}$ | Cubic FDCL Methods | ${\widehat{\mathit{u}}}_{\mathit{\xi}}(\mathit{\zeta}=0.5\mathit{h})$ | ${\widehat{\mathit{u}}}_{\mathit{\xi}}(\mathit{\zeta}=-0.5\mathit{h})$ | ${\widehat{\mathit{u}}}_{\mathit{\zeta}}(\mathit{\zeta}=0.5\mathit{h})$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\eta}}(\mathit{\zeta}=0.5\mathit{h})$ | ${\widehat{\mathit{\tau}}}_{\mathit{\xi}\mathit{\zeta}}(\mathit{\zeta}=0)$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\zeta}}(\mathit{\zeta}=0)$ |
---|---|---|---|---|---|---|---|

50 | ${N}_{l}=10$ | −3.6818 | 0.5008 | 7.4042 | −1444.4 | 27.552 | 5.2791 |

${N}_{l}=20$ | −3.5948 | 0.4855 | 7.2241 | −1461.3 | 26.87 | 5.1512 | |

${N}_{l}=40$ | −3.5685 | 0.4807 | 7.1693 | −1466.5 | 26.643 | 5.0905 | |

${N}_{l}=80$ | −3.5614 | 0.4794 | 7.1546 | −1467.8 | 26.5804 | 5.0744 | |

${N}_{l}=400$ | −3.5591 | 0.4789 | 7.1498 | −1468.3 | 26.56 | 5.0692 | |

LD2 [25] | −4.162 | 0.9074 | 8.8684 | −1409.8 | −7.3846 | 319.18 | |

LD8 [25] | −3.5545 | 0.488 | 7.1548 | −1470.8 | 26.664 | 7.5271 | |

LD14 [25] | −3.5477 | 0.4837 | 7.1361 | −1470.4 | 26.459 | 5.1982 | |

Quasi-3D [25] | −3.5466 | 0.4833 | 7.1337 | −1481.4 | 26.448 | 5.0753 | |

1000 | ${N}_{l}=10$ | −1.8435 | −0.4301 | 45.001 | −1127.2 | −5.4085 | 0.2023 |

${N}_{l}=20$ | −1.8033 | −0.4214 | 44 | −1150 | −5.2768 | 0.245 | |

${N}_{l}=40$ | −1.7913 | −0.4188 | 43.7 | −1156.9 | −5.2373 | 0.24 | |

${N}_{l}=80$ | −1.7881 | −0.4181 | 43.62 | −1158.7 | −5.2268 | 0.2393 | |

${N}_{l}=400$ | −1.787 | −0.4179 | 43.593 | −1159.3 | −5.2233 | 0.2391 | |

LD2 [25] | −1.8872 | −0.3785 | 48.034 | −1098.6 | −6.6837 | 259.6 | |

LD8 [25] | −1.7886 | −0.4176 | 43.653 | −1159.3 | −5.2415 | −1.7681 | |

LD14 [25] | −1.7871 | −0.4178 | 43.6 | −1159.2 | −5.2262 | 0.3165 | |

Quasi-3D [25] | −1.7868 | −0.4178 | 43.59 | −1170.2 | −5.2242 | 0.2428 |

**Table 3.**Temperature-dependent material properties of metal and ceramic materials (Ti-6Al-4V and zirconia), in which $P={P}_{0}\lfloor \left({P}_{-1}/\widehat{T}\right)+1+\left({P}_{1}\text{\hspace{0.17em}}\widehat{T}\right)+\left({P}_{2}\text{\hspace{0.17em}}{\widehat{T}}^{2}\right)+\left({P}_{3}\text{\hspace{0.17em}}{\widehat{T}}^{3}\right)\rfloor $, and $\widehat{T}$ denotes the current temperature and its unit is K.

Materials | Properties P(T) | P_{0} | P_{−1} | P_{1} | P_{2} | P_{3} |
---|---|---|---|---|---|---|

Zirconia [34] | E (GPa) | 244.27 | 0 | −1.371 × 10^{−3} | 1.214 × 10^{−6} | −3.681 × 10^{−10} |

ν | 0.2882 | 0 | 1.133 × 10^{−4} | 0 | 0 | |

α (1/K) | 12.766 × 10^{−6} | 0 | −1.491 × 10^{−3} | 1.00 × 10^{−5} | −6.778 × 10^{−11} | |

λ (W/m K) | 1.7000 | 0 | 1.276 × 10^{−4} | 6.648 × 10^{−8} | 0 | |

Ti-6Al-4V [34] | E (GPa) | 122.56 | 0 | −4.586 × 10^{−4} | 0 | 0 |

ν | 0.2884 | 0 | 1.121 × 10^{−4} | 0 | 0 | |

α (1/K) | 7.5788 × 10^{−6} | 0 | 6.638 × 10^{−4} | −3.147 × 10^{−6} | 0 | |

λ (W/m K) | 1.0000 | 0 | 1.704 × 10^{−2} | 0 | 0 |

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**MDPI and ACS Style**

Wu, C.-P.; He, Y.-W. Thermoelastic Stress and Deformation Analyses of Functionally Graded Doubly Curved Shells. *J. Compos. Sci.* **2019**, *3*, 94.
https://doi.org/10.3390/jcs3040094

**AMA Style**

Wu C-P, He Y-W. Thermoelastic Stress and Deformation Analyses of Functionally Graded Doubly Curved Shells. *Journal of Composites Science*. 2019; 3(4):94.
https://doi.org/10.3390/jcs3040094

**Chicago/Turabian Style**

Wu, Chih-Ping, and Yu-Wen He. 2019. "Thermoelastic Stress and Deformation Analyses of Functionally Graded Doubly Curved Shells" *Journal of Composites Science* 3, no. 4: 94.
https://doi.org/10.3390/jcs3040094