We present a comparative study on several aspects of atomistic-stress evaluation. Firstly, we derived an Irving-Kirkwood formulation for Cauchy stress evaluation in Eulerian coordinates. Secondly, different from that in the Lagrangian coordinates case, where the time averaging procedure can be performed in a post-processing manner, the stress evaluation in Eulerian system must be evaluated spatially and temporally at the same time. Thirdly, the relation between Cauchy and first Piola-Kirchhoff stress was investigated. At zero temperature, traditional linear relation between these two is satisfied both in Virial and Hardy formulation. While at finite temperature, temporal averaging is required to guarantee this relation for Virial formulation. For Hardy formulation, Cauchy stress differs from first Piola-Kirchhoff stress, which increases linearly with temperature. Numerical examples were provided to illustrate our investigation.