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The goal of this effort is to determine the interaction among the heating and cooling processes in order to understand how solids behave when subjected to temperature changes. In this instance, the temperature, displacement, and stress relations are determined analytically and numerically while a thin annular disc is subjected to both the heating and cooling processes. The ability of a material to withstand stress is essential for the design of diverse mechanical structures that aim to enhance performance, durability, characteristics, and strength. This ability is demonstrated in many physical processes where the material structure crosses over into heating and cooling processes. Furthermore, memory derivatives used in the modelling of heat transfer equations more accurately depict the memory behaviour of an imagined disc and explain its physical significance.