On variational symmetry of defect potentials and multiscale configurational force

This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In this work, we study invariant properties of defect potentials that are capable of describing defect motions in a continuum. By formulating two canonical defect theories, a generalized Nye theory and the Kro¨ner–de Wit theory, we have found three defect potentials that are variational, i.e. their associated Euler–Lagrange equations are differential compatibility conditions of the continuum and defects. Consequently, symmetry properties of these variational functionals render several classes of new conservation laws and invariant integrals that are related with continuum compatibility conditions, which are independent of the constitutive relations of the continuum. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by such an integral is specifically defect-free. The invariant integral is applied to study macroscopically brittle fracture, and a multiscale Griffith criterion is proposed, which leads to a rigorous justification of the well-known Griffith–Irwin theory. 1. Introduction It is well established today that the mathematical structure of the configurational force is the conservation law of continuum mechanics, which is based on Noether's invariant theory, e.g. [1,2], and the physical origin of the configurational force is from the balance law of continuum thermodynamics, e.g. [3], which is a manifestation of the symmetry properties of the free-energy density. The configurational force that we refer to is a material force acting on defects in the sense of Eshelby [4,5], which has been eloquently elaborated in several monographs [6–9]. However, classical elasticity, both linear and finite deformation theories, does not have an intrinsic length-scale. Therefore, in the realm of classical elasticity, the configurational mechanics do not have a multiscale character, which is in contrast or in conflict with physical reality where defects and the effects of defect evolutions are multiscale in nature. To bring …