Solution of multiple confocally elliptical layers with dissimilar properties in antiplane elasticity with eigenstrains and remote loading

• This paper provides a general solution for the problem of multiple confocally elliptical layers in antiplane elasticity.
• The elastic medium is composed of an inclusion and many confocally elliptical layers with different elastic properties.
• The eigenstrain is placed in the inclusion and the remote loading is applied.
• The complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients.
• The continuity conditions for the traction and the displacement along the interfaces are formulated exactly.

This paper provides a general solution for the problem of multiple confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of many confocally elliptical layers with different elastic properties, and it is assumed to be isotropic. The eigenstrain is placed in the inclusion. The complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the traction and the displacement along the interfaces are formulated exactly. From those conditions, the relation between two sets for two undetermined coefficients in two adjacent layers can be evaluated. By using the above-mentioned relation, a definite relation among (1) the remote loading, (2) the stress in inclusion and (3) the eigenstrian in inclusion can be evaluated. Within the three components, two of them are independent. The proposed condition of the uniform remote stress and the eigenstrain admits an internal uniform stress field. Many computed results are provided, which give a relation among the remote loading, the eigenstrain and stress in inclusion.