Recent developments in invariant integrals
Y-H Chen, TJ Lu
Recent developments in invariant integrals (path-independent integrals) are reviewed in this article, with
focus on applications in functional materials as well as multiple interacting crack problems. Although the
topic of invariant integrals is relatively old, and has been extremely attractive for Fracture Mechanics
following the finding of the J-integral (Rice, 1968a,b), novel applications in two distinct research
areas have been identified during the past decade. Firstly, the invariant integrals have been extended to
treat crack-like defects in functional materials. Because the electric, magnetic, thermal, and mechanical
quantities are all coupled in the near-tip field of a crack (or other types of defect), the invariant
integrals exhibit some new features that are quite different from those in purely mechanical problems.
Secondly, the invariant integrals have been shown to be useful for Damage Mechanics, especially in treating
multiple-interacting cracks. For example, it has been established that the invariant integrals can be used to
measure the extent of damage in a brittle solid with microcracks. These results indicate that some new
concepts concerning the invariant integrals should be establised, which seem far apart from the traditional
understandings in Fracture Mechanics. The aim of this review article is therefore to summarise the major
developments in the above two new directions. The topics covered include: (1) invariant integrals in
functional materials with defects; (2) energy momentum tensor (Eshelby 1970, 1975) in functional materials;
(3)Bueckner integral in functional materials and the associated pseudo-orthogonal property (Chen
1985); (4) inherent relations between the Bueckner (1973) work conjugate integrals and the J-integral
(or M-integral); (5) roles of the M- and L-integrals in microcrack damage problems; (6)
applications of invariant integrals in studying damage and fracture of functional materials; and (7)
applications of invariant integrals in nano-structures. This review article includes 171 references.
CUED technical report, CUED/C-MICROMECH/TR.43, January 2001.