•A new I-integral is proposed for inhomogeneous piezoelectrics with initial strains.•The I-integral is domain-independent for complex initial strain and interfaces.•The line integral along the interface is derived for discontinuous initial strains.•The SIFs jump markedly for piezoelectric bi-materials with various substrates. Initial strain in piezoelectric composites is usually formed during the manufacturing process, especially in piezoelectric layer/film and elastic substrate structures. In this work, a novel interaction integral (I-integral) is developed to determine the stress intensity factors (SIFs) and the electric displacement (EDIF) of the interfacial crack between inhomogeneous piezoelectric bi-materials considering the effect of initial strain, and the theoretical derivation rigorously demonstrates that the proposed I-integral does not need to concern the derivatives of any material parameter. Further, complex material interface distribution near the crack tips in piezoelectric composites is considered in the integration domain , and the domain-independence of the established I-integral is theoretically demonstrated. This study derives the effect terms of inhomogeneous and discontinuous initial strain distributions on the I-integral in the interface cracking model of piezoelectric bi-materials. Numerical results show that the graded degree of the inhomogeneous piezoelectric bi-material has a significant effect on both the SIFs and the EDIF of an interfacial crack. As the graded parameter increases from zero to five, the normalized mode-I SIF increases significantly and the relative increment can reach 26.9% for inhomogeneous piezoelectric bi-materials. For the quadratic initial strain, the normalized mode-I SIF can be reduced by 37.5% compared with that for a homogeneous initial strain. The presented I-integral shows excellent domain-independence for continuously inhomogeneous and discontinuous material properties as well as for complex initial strain distribution (relative derivation <1.0%). For piezoelectric/elastic bi-materials, the IFs of the interface crack can be modulated by designing various distributions of material property and initial strain of the elastic substrate, which can provide guidance to reduce interface cracking in engineering. Display omitted