New perspective on Couple Stress theory resolving the indeterminacy issue

Dear fellow Mechanicians,We have done this research on couple stress theory which we believe has resolved the problem of indeterminacy in the spherical part of the couple-stress tensor and the appearance of the body couple in the constitutive relation for the force-stress tensor  (Mindlin and Tiersten, 1962).I humbly request you to read the paper and give me your feedback. I believe you would agree this is an important issue in Continuum Mechanics which can open up new frontiers.Sincerely,

Ali Reza Hadjesfandiari 

 Encl.: please find the attached pdf file

Couple stress theory for polar solids

Abstract

 The existing couple stress theory for polar media suffers from an indeterminacy of the spherical part of the couple-stress tensor, which limits significantly its applicability in the study of micro and nanoscale mechanics.  Here we rely on concepts from virtual work, along with some kinematical considerations, to establish a consistent polar theory for solids that resolves all of the indeterminacies by recognizing the character of the couple stress tensor.  We then develop the corresponding theory of small deformations in elastic bodies, including the energy and constitutive relations, displacement formulations, the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory.  Next, we consider the more restrictive case of isotropic materials and present general solutions for two-dimensional problems based on stress functions and for problems of anti-plane deformation.  Finally, we examine several additional elementary boundary value problems within this consistent theory of polar elasticity.