Green’s functions for the bending of thin plates under various boundary conditions and applications: a review

Green’s functions of a point dislocation for the bending problem of an infinite thin plate containing an arbitrarily shaped hole under displacement, stress, and mixed boundary conditions and for a bimaterial problem are stated and the review of the related works is given. The point dislocation is defined by the discontinuity of the plate deflection angle. The original formulation for these problems consists of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. The Green’s functions were obtained in closed forms by using the complex stress function method in junction with the rational mapping function. The applications of the Green’s functions are demonstrated in studying the interaction of a hole or inclusion with a line crack in an infinite plate subjected to bending. The Green’s functions can also be used as the kernels for boundary integral representations in boundary element method analysis, where they can notably simplify the procedure of the standard boundary integral equations.