Using exterior differential forms, basic equations of continuum mechanics are presented in direct notation. To this end, Elie Cartan’s vector-valued Cauchy stress 2-form is introduced. Its Lie derivative along the world line becomes the Truesdell stress rate. In the presentation, the notation adopted by Theodore Frankel (The Geometry of Physics, Cambridge, New York, 1997) is utilized. With the use of exterior differential forms, complicated computations in tensor analyses in curvilinear coordinates are dramatically simplified. As specific examples, the following subjects are presented: (i) Piola transformations of the Cauchy stress 2-form and (ii) simple shear deformation using the Lie derivative of the Cauchy stress 2-form, i.e., the Truesdell stress rate. It is known that under monotonic shear loading, if inappropriate stress-rates are used, shear stress oscillates. With the use of geometrically correct stress-rate, the shear stress monotonically increases. Thereby, the search for an appropriate stress rate reduces to the correct definition of the stress 2-form and the computation of its Lie derivative with respect to velocity.
Theory of vector-valued differential forms