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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.

 Phyically speaking, in the framework of  conventional continuum
mechanics, every "point" at macro-scale is a RVE at micro-scale. The possibility of "seperation of scale" is a prerequisite of the application of the conventional continuum mechanics theory.  I think that Cauchy's consideration is a mixture of the thoughts from both Mechanical (considering the interaction between "particles") and Mathematical point of view (taking limit "mathematically").  Therefore, if you believethe theory of continuum mechanics, then the basic variable must be the "force density (stress)". This is the "first principle" of conventional continuum mechanics theory.  Then you can use the tool of calculus to establish the PDEs and solving the corresponding BVPs. In this solution process, the operations such as taking limit or differentiation are only Mathematical in nature and has nothing to do with the physics of the problem.  Hope this may be helpful to answer Mr.Falk's questions. 


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