This book summarizes two significant tendencies for application of conservation laws and energy release rates. The first is to establish a bridge between some famous invariant integrals and microcrack damage descriptions. The second is the direct extension from the understandings established in Fracture Mechanics for conventional materials to those for functional materials. In the first point it discusses the vanishing nature for both components of the Jk-integral vector when the closed contour encloses all discontinuities completely. Both mathematical manipulations and numerical examinations are given. Thus the M-integral and the L-integral are independent of coordinate shifts and, more significantly, the M-integral presents a new description for the damage level of a microcracking brittle solid. In the second point it discusses the direct extension from the basic understandings established in Linear Elastic Fracture Mechanics to those for functional materials, e.g., piezoelectric ceramics. Owing to the mechanical and electric coupling, some new insights of energy release rates are discussed in detail.