Recent developments and applications of invariant integrals

Although invariant integrals (path independent integrals) have been used extensively in the 20th century, mainly in the calculation of dominant parameters that govern the initiation and propagation of both linear and nonlinear cracks, new applications are increasingly being identified. This article presents developments and applications of the invariant integrals in recent years, focusing on four major application areas: $i)$ fracture mechanics of functional materials (eg, piezoelectric ceramics and ferromagnets), which exhibit features different from those found in purely mechanical problems due to the coupling of electric, magnetic, thermal, and mechanical quantities; $ii)$ damage mechanics of multiple interacting cracks, and new damage measures; $iii)$ domain integrals, two-state integrals, and their applications in determining the dominant parameters of 3D cracks and in clarifying the role of higher order singular terms in the Williams eigenfunction expansions; and $iv)$ nano-structures (eg, stress driven surface evolution in a heteroepitaxial thin film). In writing this review article, we have been able to draw upon a large number of published works on invariant integrals over the last three decades, and yet it is impossible to cover the whole subject in the limited space available. Consequently, the main aim of the article is to summarize the major developments and applications in the four important areas mentioned above. Still, 261 references are reviewed in the article.