The essential assumption of Cauchy Stress
The essential assumption of Cauchy Stress
is the existence of the limit (\Delta f)/ (\Delta A) as \Delta A-->0.
This means that a infinitesimal small area can only sustain infinitesimal
small surface force. Therefore in the framework of conventional continuum
mechanics, we can only talk about the "force density (stress)" at a point.
The expression int_A (f.n)dA has no physical meaning. Only int_A (stress.n)dA
can give the total force exerting on a closed surface A.