definition of stress

In introducing the very concept of the stress tensor to the beginning student, text-books always present only indirect relations involving the concept. Thus, you have the relations like "traction = (stress-transposed)(unit normal)" (i.e. Cauchy's formula, for uniform stress), or the relations for the coordinate transformations of the stress tensor, or the divergence theorem (for non-uniform stress). These are immediately followed or interspersed with alternative notations, and the rules for using them.

But what you never ever get to see, in text-books or references, is this: a *direct* definition of the stress tensor, i.e. an equation in which there is only the stress tensor on the left hand-side, and some expression involving some *other* quantities on right hand-side. Why? What possibly could be the conceptual and pedagogical advantages of giving a direct definition of this kind, and its physical meaning? I would like to ponder on these matters here, giving my answers to these and similar questions in the process.