A complete stress-function representation of stresses for a continuum of second grade

Abstract

At present, Cauchy's equations of linear momentum and of moment of momentum seem to demand a modification, advancing forward Cauchy's classical considerations in the sense of incorporating the concept of couple-stresses in those equations. This concept is well-known in the theory of Cosserat-continua, less so in the theory of point-continua. However, the theory of point-continua is widely known and has been applied with great success e.g. to such materials as steel, water or air in engineering mechanics. Incorporating the concept of couple-stresses in the theory of point-continua, however, involves many serious problems. One such problem is the type of structure of the couple-stress tensor. An-other problem is the question of the completeness of a stress-function representation of stresses. In the following paper the author presents a complete stress-function representation of stresses for his «refined theory» of the point-continuum. The main features of incorporation of the concept of couple-stresses in that theory—e.g. the reason why the couple-stress tensor must be of skew-symmetric structure—have already been presented in [1].