An arbitrarily shaped Eshelby inclusion interacting with a circular piezoelectric inhomogeneity penetrated by a semi-infinite crack

摘要

We study the interaction between an Eshelby inclusion of arbitrary shape and a circular piezoelectric inhomogeneity penetrated by a semi-infinite crack under antiplane mechanical and in-plane electrical loading in a linear piezoelectric solid. The Eshelby inclusion undergoes uniform antiplane eigenstrains and in-plane eigenelectric fields. Through the use of a conformal mapping, the cracked piezoelectric plane is first mapped onto the lower half of the image plane. The corresponding boundary value problem is then studied in this image plane. The interaction problem is solved through the construction of an auxiliary function and the application of analytic continuation across straight and circular boundaries. We obtain concise expressions for the resultant stress and electric displacement intensity factors at the crack tip.