Surface tractions, or stresses acting on an internal datum plane, are typically decomposed into three mutually orthogonal components. One component is normal to the surface and represents direct stress. The other two components are tangential to the surface and represent shear stresses.
What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses? Direct stresses tend to change the volume of the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus (which depends on the Young's modulus and Poisson ratio). Shear stresses tend to deform the material without changing its volume, and are resisted by the body's shear modulus.
Defining a set of internal datum planes aligned with a Cartesian coordinate system allows the stress state at an internal point P to be described relative to x, y, and z coordinate directions.
For example, the stress state at point P can be represented by an infinitesimal cube with three stress components on each of its six sides (one direct and two shear components).
Since each point in the body is under static equilibrium (no net force in the absense of any body forces), only nine stress components from three planes are needed to describe the stress state at a point P.
These nine components can be organized into the matrix:
where shear stresses across the diagonal are identical (i.e. sxy = syx, syz = szy, and szx = sxz) as a result of static equilibrium (no net moment). This grouping of the nine stress components is known as the stress tensor (or stress matrix).
The subscript notation used for the nine stress components have the following meaning: