  We establish the interior and exterior Gauss-Green formulas for
divergence-measure fields in $L^p$ over general open sets, motivated by the
rigorous mathematical formulation of the physical principle of balance law via
the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing
discontinuities and singularities. The method, based on a distance function,
allows to give a representation of the interior (resp. exterior) normal trace
of the field on the boundary of any given open set as the limit of classical
normal traces over the boundaries of interior (resp. exterior) smooth
approximations of the open set. In the particular case of open sets with
continuous boundary, the approximating smooth sets can explicitly be
characterized by using a regularized distance. We also show that any open set
with Lipschitz boundary has a regular Lipschitz deformable boundary from the
interior. In addition, some new product rules for divergence-measure fields and
suitable scalar functions are presented, and the connection between these
product rules and the representation of the normal trace of the field as a
Radon measure is explored. With these formulas at hand, we introduce the notion
of Cauchy fluxes as functionals defined on the boundaries of general bounded
open sets for the rigorous mathematical formulation of the physical principle
of balance law, and show that the Cauchy fluxes can be represented by
corresponding divergence-measure fields.
