Singularities at the vertex of connected angular inhomogeneities under thermal and elastic loading

Abstract: This paper investigates the singularities at the vertex of multiply connected angular inhomogeneities for heat conduction and elastic deformation. With the aid of Eshelby's equivalent inclusion method (EIM), each inhomogeneity is simulated as an equivalent inclusion, exhibiting the same material properties as the matrix but containing a continuously distributed eigen-field with potential singularities at the vertices and edge lines. Specifically, the eigen-temperature-gradient (ETG) and eigenstrain are utilized to simulate material mismatch of thermal conductivity and stiffness, respectively. Using the separation of variables, the eigen-fields can be formulated in terms of distance to vertices and opening angles, and disturbed thermal/elastic fields are evaluated by domain integrals of Green's function multiplied by eigen-fields, which form Fredholm's integral equation of the second kind. The boundary value problem is reduced to solve for eigenvalues, which are used to determine the order of singularity. The present solution is versatile - by placing two identical inhomogeneities together, it recovers the classic solutions for a single wedge in a bimaterial media or infinite domain. The general and analytical formulae take full consideration of interactions of multiple inhomogeneities and reveal the effects of opening angles and material properties on the thermal and elastic singularities.