Thermoelastic stresses alleviation for two-dimensional functionally graded cylinders under asymmetric loading

Figures & data

Figure 1. Contour plot for the distribution of $E$ using Eq. (11)(11) $$\beta \left(r\right)={\beta }_{\mathrm{i}}+\left({\beta }_{\mathrm{o}}-{\beta }_{\mathrm{i}}\right){\left(\frac{r-r_{\mathrm{i}}}{r_{\mathrm{o}}-r_{\mathrm{i}}}\right)}^{\eta }$$(11) .

Figure 2. Dimensionless stresses distribution from the current study and Ref. [37]. ${\overline{\sigma }}_r$ and ${\overline{\sigma }}_{\theta }$ are plotted at $\theta =0°,$ and ${\overline{\tau }}_{r\theta }$ is plotted at $\theta =+45°.$

Figure 3. Contour plots for $E$ using Eq. (14)(14) $$\beta \left(r,\theta \right)={\beta }_{\mathrm{i}}+\left(\frac{\check{\beta }\left(r,\theta \right)-\min \hspace{0.17em}\left(\check{\beta }\right)}{\max \hspace{0.17em}\left(\check{\beta }\right)-\min \hspace{0.17em}\left(\check{\beta }\right)}\right)\left({\beta }_{\mathrm{o}}-{\beta }_{\mathrm{i}}\right)$$(14) .

Figure 4. Dimensionless stress contours based on FEM solution for 1D-FGC. (a) Dimensionless radial stress, (b) dimensionless tangential stress, and (c) dimensionless shear stress.

Figure 5. Dimensionless stress contours for $f\left(\theta \right)=\cos \hspace{0.17em}(2\theta +0°).$ (a) Dimensionless radial stress, (b) dimensionless tangential stress, and (c) dimensionless shear stress.

Figure 6. Dimensionless stress contours for $f\left(\theta \right)=\cos \hspace{0.17em}(2\theta -30°).$ (a) Dimensionless radial stress, (b) dimensionless tangential stress, and (c) dimensionless shear stress.

Figure 7. Dimensionless stress contours for $f\left(\theta \right)=\cos \hspace{0.17em}(2\theta -90°).$ (a) Dimensionless radial stress, (b) dimensionless tangential stress, and (c) dimensionless shear stress.

Figure 8. Dimensionless stress contours for $f\left(\theta \right)=\cos \hspace{0.17em}(2\theta +180°).$ (a) Dimensionless radial stress, (b) dimensionless tangential stress, and (c) dimensionless shear stress.

Figure 9. Percentages of stresses reduction at different values of $s$ for $f\left(\theta \right)=\cos (n_{\theta }\theta +s):$ (a) $n_{\theta }=2,$ and (b) $n_{\theta }=1.$

Figure 10. Resulting contours of the thermoelastic problem of 1D-FGC ($f\left(\theta \right)=1$). (a) Temperature, (b) dimensionless radial stress, (c) dimensionless tangential stress, (d) dimensionless axial stress, (e) dimensionless shear stress, and (f) dimensionless von Mises stress.

Figure 11. Resulting contours of the thermoelastic problem of 2D-FGC with $f\left(\theta \right)=\cos \hspace{0.17em}\left(2\theta +90°\right).$ (a) Temperature, (b) dimensionless radial stress, (c) dimensionless tangential stress, (d) dimensionless axial stress, (e) dimensionless shear stress, and (f) dimensionless von Mises stress.

Figure 12. Resulting contours of the thermoelastic problem of 2D-FGC with $f\left(\theta \right)=\cos \hspace{0.17em}\left(2\theta -30°\right).$ (a) Temperature, (b) dimensionless radial stress, (c) dimensionless tangential stress, (d) dimensionless axial stress, (e) dimensionless shear stress, and (f) dimensionless von Mises stress.

Figure 13. Resulting contours of the thermoelastic problem of 2D-FGC with $f\left(\theta \right)=\cos \hspace{0.17em}\left(\theta +180°\right).$ (a) Temperature, (b) dimensionless radial stress, (c) dimensionless tangential stress, (d) dimensionless axial stress, (e) dimensionless shear stress, and (f) dimensionless von Mises stress.

Figure 14. Resulting contours of the thermoelastic problem of 2D-FGC with $f\left(\theta \right)=\cos \hspace{0.17em}\left(\theta -120°\right).$ (a) Temperature, (b) dimensionless radial stress, (c) dimensionless tangential stress, (d) dimensionless axial stress, (e) dimensionless shear stress, and (f) dimensionless von Mises stress.

Figure 15. Variation’s percentage for each stress component for the thermoelastic problem of 2D-FGC with $f\left(\theta \right)=\cos \hspace{0.17em}\left(n_{\theta }\theta +s\right).$ (a) $n_{\theta }=2,$ and (b) $n_{\theta }=1.$

Figure 16. Maximum value of the dimensionless von Mises stress (${\overline{\sigma }}_{\mathrm{V}{\mathrm{M}}_{\max }}$) reduction’s percentage.

Table 1. The pertinent numerical values of some parameters used while investigating different values of ${\mathit{n}}_{\mathit{\theta }}$ of 2D-FGC under thermomechanical loading conditions.

	Case (1)(1) $${\left(r{\sigma }_r\right)}_{,r}+{\tau }_{r\theta ,\theta }-{\sigma }_{\theta }=0$$(1)	Case (2)(2) $${\left(r{\tau }_{r\theta }\right)}_{,r}+{\sigma }_{\theta ,\theta }+{\tau }_{r\theta }=0$$(2)	Case (3)(3) $$\mathbf{\sigma }=\mathbf{C}[\mathbf{\epsilon }-\mathbf{\alpha }\mathrm{T}]$$(3)	Case (4)(4) $$\mathbf{\sigma }=\left\{\begin{array}{c}{\sigma }_r\\ {\sigma }_{\theta }\\ {\sigma }_z\\ {\tau }_{r\theta }\end{array}\right\}, \mathbf{\epsilon }=\left\{\begin{array}{c}{\epsilon }_r=u_{,r}\\ {\epsilon }_{\theta }=\left(u+{\vartheta }_{,\theta }\right)/r\\ {\epsilon }_z=w_{,z}\\ {2\epsilon }_{r\theta }={\vartheta }_{,r}+\left(u_{,\theta }-\vartheta \right)/r\end{array}\right\}$$(4)
${\mathit{n}}_{\mathit{p}}$	$2$	$1$	$2$	$2$
${\mathit{n}}_{\mathit{T}}$	$2$	$2$	$1$	$1$
${\mathit{p}}_0$ ($\mathrm{MPa}$)	$100$	$100$	$100$	$2$
${\mathit{T}}_0$ ($\mathrm{°C}$)	$100$	$100$	$100$	$1000$

Figure 17. Variation of ${\overline{\sigma }}_{\mathrm{V}{\mathrm{M}}_{\max }}$ with $n_{\theta }$ using different values of $n_p,$ $n_T,$ $p_0,$ and $T_0.$

H. Li and Y. Liu, “Functionally graded hollow cylinders with arbitrary varying material properties under nonaxisymmetric loads,” Mech. Res. Commun., vol. 55, pp. 1–9, 2014. DOI: 10.1016/j.mechrescom.2013.10.011.

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Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.