A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.