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Fractional single-phase lag heat conduction and transient thermal fracture in cracked viscoelastic materials

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Abstract

In the present article, a thermo-viscoelastic model is developed to investigate fractional single-phase lag heat conduction and the associated transient thermal mechanical behavior of a cracked viscoelastic material under a thermal shock. To avoid the negative temperature distribution around cracks, which violates the second law of thermodynamics, the time-fractional single-phase lag heat conduction is introduced to analyze the transient temperature field around the cracks. The Fourier and Laplace transforms, coupled with the singular integral equations, are employed to solve the governing partial differential equations numerically. Both the results of temperature field and stress intensity factors (SIFs) show that the fractional single-phase lag heat conduction model is more accurate and reasonable compared to the conventional hyperbolic heat conduction. A significant difference in transient fracture behavior exists between viscoelastic and elastic materials. A sharp pulse of the SIFs at the early stage is observed and should be consider carefully to meet the requirement of increased application of viscoelastic composites under thermal loading.

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Acknowledgements

Funding was provided by Natural Sciences and Engineering Research Council of Canada (2017–2022), China Scholarship Council (2016–2020).

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  1. Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada

    Wenzhi Yang & Zengtao Chen

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  1. Wenzhi Yang
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Appendix

Appendix

$$\begin{aligned} C_1 (\xi ,p)= & {} [m^{2}-\xi ^{2}]^{-2}D(\xi ,p)p^{3}f_1 (p)f_3 (p) \\ C_{21} (\xi ,p)= & {} [m^{2}-\xi ^{2}]^{-2}\frac{-D(\xi ,p)}{1+\exp (-2mh)}p^{3}f_1 (p)f_3 (p) \\ C_{22} (\xi ,p)= & {} [m^{2}-\xi ^{2}]^{-2}\frac{D(\xi ,p)\exp (-2mh)}{1+\exp (-2mh)}p^{3}f_1 (p)f_3 (p) \\ k_{11} (x,\tau )= & {} \int \limits _0^\infty {[1-4\xi f_{11} (\xi )]} \sin [(x-\tau )\xi ]{\mathrm{d}}\xi \\ k_{22} (x,\tau )= & {} \int \limits _0^\infty {[1-4\xi ^{2}f_{22} (\xi )]} \sin [(x-\tau )\xi ]{\mathrm{d}}\xi \\ k_{12} (x,\tau )= & {} \int \limits _0^\infty {-4\xi f_{12} (\xi )} \cos [(x-\tau )\xi ]{\mathrm{d}}\xi \\ k_{21} (x,\tau )= & {} \int \limits _0^\infty {-4\xi ^{2}f_{21} (\xi )} \cos [(x-\tau )\xi ]{\mathrm{d}}\xi \\ W_1 ^{*}(x,p)= & {} 2\int \limits _0^\infty {\xi w_1 ^{*}(\xi ,p)} \sin (x\xi ){\mathrm{d}}\xi \\ W_2 ^{*}(x,p)= & {} -2\int \limits _0^\infty {\xi ^{2}w_2 ^{*}(\xi ,p)} \cos (x\xi ){\mathrm{d}}\xi \\ w_1 ^{*}(\xi ,p)= & {} -\frac{2g_2 h_{11} +2\left| \xi \right| h_{12} (s_2 g_1 -g_2 )}{8\left| \xi \right| \xi ^{2}}-g_3 \\ w_2 ^{*}(\xi ,p)= & {} -\frac{2g_2 h_{21} +2\left| \xi \right| h_{22} (s_2 g_1 -g_2 )}{8\left| \xi \right| \xi ^{2}}-g_4 \\ h_{11} (\xi )= & {} \left| \xi \right| +\exp (-2\left| \xi \right| h)(-\left| \xi \right| +2h\xi ^{2}) \\ h_{12} (\xi )= & {} 1-\exp (-2\left| \xi \right| h)(1-2h\left| \xi \right| +2h^{2}\xi ^{2}) \\ h_{21} (\xi )= & {} 1-\exp (-2\left| \xi \right| h)(1+2h\left| \xi \right| ) \\ h_{22} (\xi )= & {} 2\left| \xi \right| h^{2}\exp (-2\left| \xi \right| h) \\ f_{11} (\xi )= & {} h_{12} (4\left| \xi \right| )^{-1} \\ f_{12} (\xi )= & {} (-2\xi h_{11} +2\xi \left| \xi \right| h_{12} )(-2\left| \xi \right| )^{-3} \\ f_{21} (\xi )= & {} h_{22} (4\left| \xi \right| )^{-1} \\ f_{22} (\xi )= & {} [-2\xi h_{21} +2\xi \left| \xi \right| )h_{22} ](-2\left| \xi \right| )^{-3} \\ g_1 (\xi )= & {} -\xi ^{2}f_3 ^{{\prime }}-2\left| \xi \right| f_4 ^{{\prime }}+f_5 ^{{\prime }} \\ g_2 (\xi )= & {} -2\left| \xi \right| \xi ^{2}f_3 ^{{\prime }}-3\xi ^{2}f_4 ^{{\prime }}-f_6 ^{{\prime }} \\ g_3 (\xi )= & {} \exp (-\left| \xi \right| h)[(1-h\left| \xi \right| )f_2 ^{{\prime }}-h\xi ^{2}f_1 ^{{\prime }}]-mI_{21} +mI_{22} \\ g_4 (\xi )= & {} \exp (-\left| \xi \right| h)[(1+h\left| \xi \right| )f_1 ^{{\prime }}+hf_2 ^{{\prime }}]-I_{21} -I_{22} \\ I_1 (\xi ,p)= & {} \frac{1}{p^{2}f_1 (p)f_3 (p)}C_1 (\xi ,p) \\ I_{21} (\xi ,p)= & {} \frac{1}{p^{2}f_1 (p)f_3 (p)}C_{21} (\xi ,p) \\ I_{22} (\xi ,p)= & {} \frac{1}{p^{2}f_1 (p)f_3 (p)}C_{22} (\xi ,p) \\ f_1 ^{{\prime }}(\xi )= & {} I_{21} \exp (-mh)+I_{22} \exp (mh) \\ f_2 ^{{\prime }}(\xi )= & {} mI_{21} \exp (-mh)-mI_{22} \exp (mh) \\ f_3 ^{{\prime }}(\xi )= & {} I_1 -I_{21} -I_{22} \\ f_4 ^{{\prime }}(\xi )= & {} -mI_{21} +m(I_{22} -I_1 ) \\ f_5 ^{{\prime }}(\xi )= & {} m^{2}I_{21} +m^{2}(I_{22} -I_1 )+\frac{2}{1+e^{-2mh}}D(\xi ) \\ f_6 ^{{\prime }}(\xi )= & {} m^{3}I_{21} -m^{3}(I_{22} -I_1 ) \end{aligned}$$

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Yang, W., Chen, Z. Fractional single-phase lag heat conduction and transient thermal fracture in cracked viscoelastic materials. Acta Mech 230, 3723–3740 (2019). https://doi.org/10.1007/s00707-019-02474-z

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