(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 3×3 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?
Best Answer
A rank-2 tensor is a linear combination of dyadic products, simply because the space of all such tensors is spanned by the dyadic products of the basis vectors of the underlying vector space. Each dyadic product is also known as a rank-1 operator, where rank here refers to the matrix rank rather than the order of the tensor. On inner product vector spaces they are usually denoted as $$\theta_{x,y}(z):=(y,z)x$$ but when the product is between a vector and a covector one can replace the inner product with the natural pairing between the vector space and its dual.