Heat conduction phenomena and induced thermal stresses are simulated from a microscopic viewpoint by using a molecular dynamics method. Three dimensional rectangular parallelepiped model composed of 2000 atoms surrounded by periodic boundary is imposed for the simulations. Thermo-physical properties such as melting temperature, specific heat, heat conductivity and latent heat are evaluated at the first step to obtain the fundamental data for the following heat conduction simulation. The central part of the model is heated for two cases with and without melting, and the variation of temperature, potential energy and thermal stresses are simulated. These simulations are carried out with two different potential functions, Lennard-Jones and Morse type, in order to clarify the effects of the interatomic potential, which follows to qualitatively demonstrate similar tendency in spite of remarkable quantitative differences. Then the variation of temperature is compared with the numerical solution of macroscopic heat conduction equation. It is clarified that the results show good agreement with each other if they are plotted against non-dimensional time, or Fourier number. Stresses calculated by molecular dynamics method are also compared with macroscopic thermal stresses.