A new twist on torsion: A connection between vector calculus and mechanics of materials

In engineering education, the topic of torsion is taught in both calculus and mechanics of materials, but using different definitions. Apparently, a clear connection between these two concepts is not currently found in the literature. In this paper, for the case of a homogeneous linear-elastic cylindrical shaft subjected to external torques, mathematical expressions are derived for relating torsional stress, strain and angle of twist to the torsion of a curve as taught in calculus. A physical connection is also presented, in that the curve torsion resulting from the deformation of a longitudinal line close to the shaft axis is shown to be approximately equal to the mechanical twist per length in the shaft. Although some researchers have hinted at this connection, to our knowledge the full precise connection has not been previously established.