Cauchy's stress formula relates the traction vector on an arbitrarily oriented plane through a point to the six independent components of the Cartesian stress tensor via the plane's direction cosines. A principal plane is defined as the orientation on which the traction vector is purely normal (zero shear); the associated principal stresses are the three real roots of the characteristic cubic equation obtained by requiring a non-trivial solution for the direction cosines, and the coefficients of that cubic — the stress invariants — are combinations of the normal and shear stress components that remain unchanged under any rotation of the reference axes. This belongs to the theory of stress analysis within mechanics of solids (strength of materials), underpinning subsequent transformation-of-stress and failure-theory topics.
Cauchy's stress formula relates the traction vector on an arbitrarily oriented plane through a point to the six independent components of the Cartesian stress tensor via the plane's direction cosines…