On variational symmetry of defect potentials and multiscale configurational force

Abstract

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 Kröner–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.

Keywords:

• compatibility
• conservation laws
• configurational force
• dislocation
• fracture
• multiscale analysis
• path-independent integral

Acknowledgements

This work is supported by a grant from NSF (Grant No. CMS-0239130), which is greatly appreciated.

Notes

Note

1. See the comments made in 22, pp. 245–246.