Cauchy Stress Tensor Components from Forces $(x,y,z)$ Background: I'm using numerical modeling software to investigate fluid-structure interaction. One of the outputs I get from the model are the forces imposed by the fluid on the solid object (given as $\vec{F}_x,\vec{F}_y,\vec{F}_z$), and these forces are available for a given elemental area. Here's an example of the elemental area (bounded by the black box) with forces acting on the red point:

My Question: Presuming a knowledge of the dimensions & orientation of the elemental area and a knowledge of the forces acting on that area ($\vec{F}_x,\vec{F}_y,\vec{F}_z$), is it possible to calculate each of the 9 components in the Cauchy stress tensor? Or is there missing information that is required to resolve the stress tensor components? If you're able to provide an answer, perhaps you could do so in terms of the following simple case:
Let's say that $\vec{F}_x=4 N$, $\vec{F}_y=2 N$, and $\vec{F}_z=-12 N$, and the elemental area ($A=1cm^2$) is oriented parallel to the x-axis and perpendicular to the z-axis, as depicted below. What are the steps to producing the 9 tensorial components of $\sigma_{ij}$? It seems like dividing each of the force components by the area would yield normal stresses ($\sigma_{xx},\sigma_{yy},\sigma_{zz}$), but what about the shear components?

 A: You can calculate some of the stress tensor components, but not all of them.
If the stress tensor is $\sigma$, the force $\vec F$ acting on a surface with normal $\vec n$ and area $dA$ is given by $\vec F = \sigma \cdot \vec n\: dA$.
Converting that equation into matrix form, for your example the normal direction is $\begin{bmatrix}0 & 0 & 1\end{bmatrix}^T$ and the equation becomes $$
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}
\begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}dA =
\begin{bmatrix}F_x \\ F_y \\ F_z \end{bmatrix} 
$$
or 
$$\begin{align}
\sigma_{xz}\:dA &= F_x \\
\sigma_{yz}\:dA &= F_y \\
\sigma_{zz}\:dA &= F_y \end{align}
$$
Since the stress tensor is symmetric, you also have $\sigma_{zx} = \sigma_{xz}$
and $\sigma_{zy} = \sigma_{yz}$, but that still leaves three unknown components of $\sigma$.
You need to impose the components of $\vec F$ as forces on the boundary of the solid, and do a stress analysis of the solid component. That will give you all the components of $\sigma$ on the boundary (and inside the solid as well!) that are consistent with all the loads and constraints on it.
It should be fairly obvious physically why your idea can't work: for example if you imposed any uniform compressive or tensile stress in the solid, in the $x$ direction (i.e. an arbitrary value of $\sigma_{xx}$), that would produce no force on your plane element.
