3D elasticity solutions for stress field around square hole in functionally graded plate

Since the square holes are widely used in many structures and machines, the purpose of this work is to obtain analytical solutions for the three-dimensional (3D) elastic stress field of transversely isotropic functionally graded (FG) infinite plates containing a single square hole with small round corners subjected to far-field loads or on the edges of the holes. Based on the generalized England-Spencer plate theory, material parameters can vary continuously and arbitrarily along the thickness direction of the plate, the solutions are obtained by using the conformal mapping technology combined with the Cauchy integral method. Concentration factors of internal forces Kij around the square hole are explicitly expressed. The obtained explicit expressions of solutions can be validated against the classical two-dimensional (2D) solutions when the material degenerates into an isotropic and homogeneous material. The detailed numerical examples are given to further investigate the influence of several parameters such as the material gradient factor and load conditions on the stress field around the hole. The present 3D analytical solutions can not only provide an in-depth understanding of the 3D stress field of FG plates with holes, but also be utilized as a benchmark to verify the numerical solutions of the same problems.