1. Stress matrix

It is the representation (matrix form) of the stress tensor in a coordinate system. The stress matrix changes from one coordinate system to another but the stress tensor itself remains the same.

A stress matrix is a mathematical representation of the internal forces (stresses) acting within a material or structure. When we change the coordinate system used to analyze these stresses, the representation of the stress in matrix form adjusts accordingly. However, the actual physical stress within the material remains constant, regardless of how we choose to analyze or represent it. This means the stress matrix is dependent on the coordinate system, but the underlying stress tensor (the physical reality) does not change.

Imagine you're looking at a box from different angles. The way you describe what you see (the matrix representation) might change depending on your viewpoint (coordinate system), but the box itself (the stress tensor) remains unchanged.

If we want to get the stress matrix of a stress tensor, then we first choose the coordinate system (let’s say e₁, e₂, e₃). We then find traction on the three planes whose normals are e₁, e₂, and e₃ and form columns out of those tractions.

To create a stress matrix, the first step is selecting a coordinate system. After that, we identify forces acting on specific plane sections of the material. For each section (which corresponds to a normal vector in our chosen coordinate system), we calculate the stress (traction) acting on that section. These tractions become the columns of the stress matrix. So, if we have three planes corresponding to our three coordinate directions, we ultimately create a three-column matrix representing these tractions.

Think about how you might build a report card for a student. You first decide on the subjects (coordinate system), then you collect grades for each subject (tractions on the planes), and finally you arrange these grades into a table (the stress matrix) that summarizes their performance.

In case we have a different coordinate system, we simply have to find traction on planes whose normals are the basis vectors of the new coordinate system and then form columns from these tractions.

If we switch to a different coordinate system, the process of finding stress remains similar: we just need to compute the stresses (or tractions) on planes defined by the new set of basis vectors. By recalculating the stresses for these new planes, we can once again construct the stress matrix for the new coordinate system. This effectively shows how the stress matrix varies while the actual stress tensor remains constant.

Returning to the box analogy, if you change the angle from which you're looking at the box, you will describe the dimensions and angles differently. But the box itself is unchanged. In the same way, how stresses are represented in a new frame changes, while the actual stresses active in the material do not.

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