Analytical solution for the Eshelby problem of an inhomogeneous inclusion with the same shear module as the matrix and arbitrary shape in an infinite plane
Abstract：The elastic fields of the matrix in two-dimensional space with an inhomogeneous inclusion undergoing a uniform eigenstrain and/or due to a uniform remote load are studied, where the inclusion shaped by the Laurent polynomial has distinct properties to the matrix but shares the same shear modulus. The equivalent method is used to convert the problem of the perturbance field due to the remote uniform loadings into that of an equivalent uniform eigenstrain, and the interface continuity conditions are expressed by the K-M potentials. Then, by virtue of the Riemann mapping theorem, the exterior of the inclusion is mapped on to the exterior of the unit disk by the Laurent polynomial and making use of the Cauchy integral formula and the Faber polynomial, the explicit analytical solutions of the K-M potentials are carried out in the inclusion and the matrix, where the relative rigid-body displacement of the inclusion to the matrix is considered. The obtained results are compared with those of previous literatures, to show that the method and results of this paper are effective and correct.
2021, Vol. 42 
