Inherent relations between the Bueckner integral and the J k -integral or the M-integral in piezoelectric materials containing multiple defects

Abstract

This paper deals with the inherent relations between the Bueckner work conjugate integral and the Jk-integral (k = 1, 2) or the M-integral in piezoelectric materials with a number of arbitrarily oriented and distributed defects such as cracks, voids and inclusions. The explicit expression of the Bueckner integral is derived in piezoelectric materials by using its path-independent property and asymptotic feature for each pair of the complex potential functions in the series expansion forms. It is concluded that the Jk-integral or the M-integral are only three different special cases of the Bueckner integral when three different complementary fields are introduced, respectively, and when the closed integral contour encloses all the defects. In other words, there are universal relations between the Bueckner integral and the Jk-integral or M-integral whatever the detailed configurations of the multiple defects are. It is also concluded that both components of the Jk-integral vanish when the contour selected to calculate the invariant integral encloses all defects, providing that no resultant force acting on each defect exists. This leads to the independence of the M-integral from the global coordinate shifts, and its value is significantly influenced by the mechanical–electrical feature of the damaged piezoelectric materials, e.g., the material properties, the remote loading conditions, the defect configurations, etc.