A two-dimensional Eshelby problem for two bonded piezoelectric half-planes (Q2748035)

This work deals with a piezoelectric cylindrical inclusion of arbitrary cross-section imbedded in the lower part of two dissimilar piezoelectric half-planes bonded along the \(x\)-axis. All constitutive equations are linear. The inclusion is assumed to have the same material constants as those of the lower half-plane, and is subject to uniform eigenstrains and eigenelectric fields which induce linear displacement components and an electric potential within the inclusion. The plane problem is considered, and the solutions of the governing equations are sought under the conditions of the continuity of displacements and potential on the boundary of the inclusion. Similar half-planes and only the lower half-plane with various boundary conditions on \(x\)-axis are also taken into account. The solution is constructed by employing a represention provided by Stroh functions. These are four analytical functions depending each on a single complex variable in the form \(z = x + p y\), where complex numbers \(p\) are four distinct roots with positive imaginary parts of an eigenvalue problem involving \(4\times 4\) matrices generated by material constants. The explicit solution is then found in terms of some auxiliary functions which can be determined through certain conformal mappings associated with the shape of the inclusion.