General theory of continuous media with conserved dislocations

Abstract

A comprehensive rigorous continuous media model with conserved dislocations is developed in the present paper on the basis of the variational kinematic principle. This model generalizes those previously derived by Cosserat and Aero–Kuvshinskii, and it refines some assertions of the models by Mindlin and Toupin from the point of view of the calculation of adhesive interactions. Our model is justified by the application of the kinematic variational principle. It is based on the consistent formal description of the media kinematics and the construction of the relevant potential strain energy with the use of the Lagrangian multipliers. The set of constitutive relations is derived and the corresponding consistent boundary-value problems are formulated. On the basis of our analysis a novel classification of dislocations is proposed. The important feature of the newly developed classification is a new kinematic interpretation of dislocations, which reflects the connection of dislocations with distortion, change in volume (porosity), and twisting. The proposed classification allows prediction of certain particular cases of dislocations when only one or two types of dislocations are dominating in the medium. In the present work principal attention is focused on analysis of the physical side of the developed continuous media model. The general model of continuous media proposed in the present paper allows analysis of various particular scale effects, e.g., cohesive interactions, surface tensions, etc., both in volume and on the surface within the unified continuum description.