Thermal phase lag in a solid containing periodic planar cracks

The thermal response of a solid containing a periodic planar array of cracks is considered. Special emphasis is placed on the extra phase lag of the thermal wave due to each crack when the solid is heated by a periodic point source. An exact solution for this phase lag is obtained in the form of an integral that requires numerical evaluation. Additionally, simpler asymptotic solutions are obtained for both low and high heating frequencies and their validity ranges are determined through comparisons with the exact numerical results. Furthermore, comparisons are made with the prediction of an equivalent solid model wherein the effective thermal conductivity perpendicular to the cracks is obtained through a simple resistance summation procedure. It is demonstrated that the extra phase lag due to each crack reaches a constant value (independent of location) at distances from the heat source that are greater than about five times the crack spacing. In most cases, the variation in the phase lag with frequency and the thermal properties of both the solid and the cracks is accurately described by the asymptotic solutions.