THE GIBBS-EINSTEIN TENSOR ANALYSIS WITH APPLICATION TO CONTINUUM MECHANICS AND CANONICAL FORMS OF GENERAL SECOND-ORDER TENSORS

A new tensor analysis, called the Gibbs-Einstein tensor analysis, is developed based on the concept that directions are algebraic quantities subject to the rule of forming scalar products, tensor products, and linear combinations. The new tensor analysis is explained in this paper by way of reformulating continuum mechanics and the Hamilton-Cayley theorem in matrix theory. The latter reformulation yields an explanation of the deformation dyads introduced in the former reformulation. A scalar product of two deformation dyads yields the strain tensor, which is a thermodynamic state variable for thermodynamically reversible deformations. Mathematics dealing with directions in a flat space becomes much simpler and more understandable when the Gibbs-Einstein tensor expression is used.