Heat conduction in a functionally graded, infinitely-long hollow cylinder is studied based on the three-phase-lag model. Material properties except the phase-lags vary according to a power-law within the cylinder. The phase-lag heat conduction equation is written in a form in which various models of heat conduction theories can be generated. The governing differential equations in the Laplace domain are solved exactly and a numerical Laplace inversion technique is employed for restoring results in the time domain. The effects of different heat conduction theories, phase-lags, geometries, and non-homogeneity indices are studied on the spatial distribution and time-history of temperature.

This article presents the transient thermoelastic analysis in a long solid cylinder with a circumferential crack using the C–V heat conduction theory. The outer surface of the cylinder is subjected to a sudden temperature change. The Laplace transform technique is adopted to solve the one-dimensional hyperbolic heat conduction equation, and the axial thermal stress is obtained for the un-cracked cylinder in the Laplace domain. Then this axial thermal stress with a minus sign is applied to the crack surface to form a mixed boundary value problem in the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier and Hankel transforms to solve the mode I crack problem. The transient thermal stress intensity factors are obtained by solving the singular integral equation numerically. The influences of thermal relaxation time, crack geometry, and Biot's number upon transient temperature distributions, axial stress fields, and stress intensity factors are analyzed.