  Irving and Kirkwood formulism (IK formulism) provides a way to compute
continuum mechanics quantities at certain location in terms of molecular
variables. To make the approach more practical in computer simulation, Hardy
proposed to use a spacial kernel function that couples continuum quantities
with atomistic information. To reduce irrational fluctuations, Murdoch proposed
to use a temporal kernel function to smooth the physical quantities obtained in
Hardy's approach. In this paper, we generalize the original IK formulism to
systematically incorporate both spacial and temporal average. The Cauchy stress
tensor is derived in this generalized IK formulism (g-IK formulism). Analysis
is given to illuminate the connection and difference between g-IK formulism and
traditional temporal post-process approach. The relationship between Cauchy
stress and first Piola-Kirchhoff stress is restudied in the framework of g-IK
formulism. Numerical experiments using molecular dynamics are conducted to
examine the analysis results.
