Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction.

*(English)*Zbl 07186570Summary: This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.

##### MSC:

74F05 | Thermal effects in solid mechanics |

74R10 | Brittle fracture |

74E05 | Inhomogeneity in solid mechanics |

26A33 | Fractional derivatives and integrals |

80A19 | Diffusive and convective heat and mass transfer, heat flow |

##### Keywords:

time-fractional Cattaneo heat transport; Laplace transform; thermal shock; heat flux phase lag
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\textit{X.-Y. Zhang} et al., Appl. Math. Modelling 70, 328--349 (2019; Zbl 07186570)

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