(As I understand it ... qualifies every sentence in what follows).. a stress tensor is a rank 2 tensor that maps a unit vector normal to a surface to the stress (or traction) vector corresponding to that surface. A rank 2 tensor can be represented by a 3x3 matrix, and that matrix maps the components of the unit vector to the components of a stress (or traction) vector.
A rank 2 tensor can be written as a dyad, that is, the vector dyadic product of two vectors. Is there a geometric interpretation of the two vectors making up the dyad corresponding to the stress tensor?
Related question - the product of a dyad $UV$ and a vector $D$, say $UV$ dot $D$, corresponds to the matrix product of the matrix representing the dyad and the vector $D$, and is always a vector that equals $sU$ where $s$ is a scalar, so the stress (or traction) for any surface at a point always points in the same direction. T or F?