Consistent continuous defect theory

Abstract

By investigating the benefits and shortcomings of the existing form of continuous defect theory (CDT) and using recent advancements in size-dependent continuum mechanics, we develop a fully coherent theoretical framework, denoted as Consistent Continuous Defect Theory (C-CDT). Among several important potential applications, C-CDT may provide a proper foundation to study the continuum theory of crystal plasticity. The development presented here includes an examination of the character of the bend-twist tensor, Weingarten's theorem, Burgers and Frank vectors, continuous dislocation and disclination density tensors, and the dualism between geometry and statics of CDT based on couple stress theory (CST). Then, by using Consistent Couple Stress Theory (C-CST), the new C-CDT is derived in a totally systematic manner. In this development, the geometry of C-CDT is dual to the statics formulation in C-CST. Previously, the fundamental step in the creation of C-CST was recognizing the skew-symmetric character of the couple-stress tensor, which requires the skew-symmetrical part of the bend-twist tensor to be the additional measure of deformation in size-dependent continuum mechanics. Via Weingarten's theorem and arguments from C-CST, we establish that in defect theory the dislocation density tensor must be skew-symmetric and thus can be represented by an equivalent dislocation density vector. In addition, we investigate the character of a classical version of C-CDT with unexpected consequences. For full consistency, there can be no continuous dislocation density tensor within classical continuum mechanics, and the continuous disclination density tensor becomes symmetric. This clearly is analogous to the absence of couple-stresses and the symmetry of force-stresses in classical continuum mechanics.