Circular loadings on the surface of an anisotropic and magnetoelectroelastic half-space

In this paper, we derive the analytical solutions for two types of surface loadings over an anisotropic magnetoelectroelastic half-space: a uniform and an indentation-type load. Our solutions in terms of the simple line integral over [0,π] contain various decoupled materials and materials with high symmetry as special cases. Furthermore, the surface results of the solutions of the half-space are also provided. It is shown that on the surface: (1) for the uniform loading, the integrands for the extended displacements are continuous whilst those for the extended stresses have two weak singularities of order 1/r ^1/2 along the integral interval [0,π], which are integrable in the sense of Cauchy principal value and (2) for the indentation-type loading, the order of the singularity in the extended stress is exactly Cauchy type of 1/r, which can also be integrated via the Cauchy principal value. For a general anisotropic magnetoelectroelastic half-space, numerical examples in the vertical plane are presented for a uniform horizontal/vertical load and a vertical indentation-type load. The physical quantities presented are the magnitude of the elastic displacement vector, the electric and magnetic potentials, the hydrostatic and effective stresses, and the magnitudes of the electric and magnetic field vectors. These numerical results not only show some distinguished features associated with the loadings, but also can serve as benchmarks for future numerical endeavors in this field.