Material tailoring for reducing stress concentration factor at a circular hole in a functionally graded material (FGM) panel

By assuming that Young's modulus and Poisson's ratio of a linearly elastic and isotropic material vary along the radial direction in a panel with a circular hole and deformed by a far field uniaxial tensile traction, we first analytically find the stress concentration factor, K, at the hole. The problem is solved by superposing solutions of two problems - one of uniform biaxial tension and the other of pure shear. The solutions of the first and the second problem are, respectively, in terms of hypergeometric functions and Frobenius series. Subsequently, we analytically study the material tailoring problem for uniform biaxial tension, and give explicit variation of Young's modulus to achieve a prespecified K. For the panel loaded by a far field uniaxial tensile traction, we show that the K can be reduced by a factor of about 8 by appropriately grading Young's modulus and Poisson's ratio in the radial direction. By plotting K versus the two inhomogeneity parameters, we solve the material tailoring problem for a panel loaded with a far field uniaxial traction. The analytical results should serve as benchmarks for verifying the accuracy of approximate/numerical solutions for an inhomogeneous panel.