Closed-form solution for Eshelby's elliptic inclusion in antiplane elasticity using complex variable

This paper provides a closed-form solution for the Eshelby's elliptic inclusion in antiplane elasticity. In the formulation, the prescribed eigenstarins are not only for the uniform distribution, but also for the linear form. After using the complex variable and the conformal mapping, the continuation condition for the traction and displacement along the interface in the physical plane can be reduced to a condition along the unit circle. The relevant complex potentials defined in the inclusion and the matrix can be separated from the continuation conditions of the traction and displacement along the interface. The expressions of the real strains and stresses in the inclusion from the assumed eigenstrains are presented. Results for the case of linear distribution of eigenstrain are first obtained in the paper.